1 use crate::{Category, ExpInt, IEK_INF, IEK_NAN, IEK_ZERO};
2 use crate::{Float, FloatConvert, ParseError, Round, Status, StatusAnd};
4 use smallvec::{SmallVec, smallvec};
5 use std::cmp::{self, Ordering};
6 use std::convert::TryFrom;
7 use std::fmt::{self, Write};
8 use std::marker::PhantomData;
13 pub struct IeeeFloat<S> {
14 /// Absolute significand value (including the integer bit).
17 /// The signed unbiased exponent of the value.
20 /// What kind of floating point number this is.
23 /// Sign bit of the number.
26 marker: PhantomData<S>,
29 /// Fundamental unit of big integer arithmetic, but also
30 /// large to store the largest significands by itself.
32 const LIMB_BITS: usize = 128;
33 fn limbs_for_bits(bits: usize) -> usize {
34 (bits + LIMB_BITS - 1) / LIMB_BITS
37 /// Enum that represents what fraction of the LSB truncated bits of an fp number
40 /// This essentially combines the roles of guard and sticky bits.
42 #[derive(Copy, Clone, PartialEq, Eq, Debug)]
44 // Example of truncated bits:
45 ExactlyZero, // 000000
46 LessThanHalf, // 0xxxxx x's not all zero
47 ExactlyHalf, // 100000
48 MoreThanHalf, // 1xxxxx x's not all zero
51 /// Represents floating point arithmetic semantics.
52 pub trait Semantics: Sized {
53 /// Total number of bits in the in-memory format.
56 /// Number of bits in the significand. This includes the integer bit.
57 const PRECISION: usize;
59 /// The largest E such that 2<sup>E</sup> is representable; this matches the
60 /// definition of IEEE 754.
61 const MAX_EXP: ExpInt;
63 /// The smallest E such that 2<sup>E</sup> is a normalized number; this
64 /// matches the definition of IEEE 754.
65 const MIN_EXP: ExpInt = -Self::MAX_EXP + 1;
67 /// The significand bit that marks NaN as quiet.
68 const QNAN_BIT: usize = Self::PRECISION - 2;
70 /// The significand bitpattern to mark a NaN as quiet.
71 /// NOTE: for X87DoubleExtended we need to set two bits instead of 2.
72 const QNAN_SIGNIFICAND: Limb = 1 << Self::QNAN_BIT;
74 fn from_bits(bits: u128) -> IeeeFloat<Self> {
75 assert!(Self::BITS > Self::PRECISION);
77 let sign = bits & (1 << (Self::BITS - 1));
78 let exponent = (bits & !sign) >> (Self::PRECISION - 1);
79 let mut r = IeeeFloat {
80 sig: [bits & ((1 << (Self::PRECISION - 1)) - 1)],
81 // Convert the exponent from its bias representation to a signed integer.
82 exp: (exponent as ExpInt) - Self::MAX_EXP,
83 category: Category::Zero,
88 if r.exp == Self::MIN_EXP - 1 && r.sig == [0] {
89 // Exponent, significand meaningless.
90 r.category = Category::Zero;
91 } else if r.exp == Self::MAX_EXP + 1 && r.sig == [0] {
92 // Exponent, significand meaningless.
93 r.category = Category::Infinity;
94 } else if r.exp == Self::MAX_EXP + 1 && r.sig != [0] {
95 // Sign, exponent, significand meaningless.
96 r.category = Category::NaN;
98 r.category = Category::Normal;
99 if r.exp == Self::MIN_EXP - 1 {
101 r.exp = Self::MIN_EXP;
104 sig::set_bit(&mut r.sig, Self::PRECISION - 1);
111 fn to_bits(x: IeeeFloat<Self>) -> u128 {
112 assert!(Self::BITS > Self::PRECISION);
114 // Split integer bit from significand.
115 let integer_bit = sig::get_bit(&x.sig, Self::PRECISION - 1);
116 let mut significand = x.sig[0] & ((1 << (Self::PRECISION - 1)) - 1);
117 let exponent = match x.category {
118 Category::Normal => {
119 if x.exp == Self::MIN_EXP && !integer_bit {
127 // FIXME(eddyb) Maybe we should guarantee an invariant instead?
131 Category::Infinity => {
132 // FIXME(eddyb) Maybe we should guarantee an invariant instead?
136 Category::NaN => Self::MAX_EXP + 1,
139 // Convert the exponent from a signed integer to its bias representation.
140 let exponent = (exponent + Self::MAX_EXP) as u128;
142 ((x.sign as u128) << (Self::BITS - 1)) | (exponent << (Self::PRECISION - 1)) | significand
146 impl<S> Copy for IeeeFloat<S> {}
147 impl<S> Clone for IeeeFloat<S> {
148 fn clone(&self) -> Self {
153 macro_rules! ieee_semantics {
154 ($($name:ident = $sem:ident($bits:tt : $exp_bits:tt)),*) => {
156 $(pub type $name = IeeeFloat<$sem>;)*
157 $(impl Semantics for $sem {
158 const BITS: usize = $bits;
159 const PRECISION: usize = ($bits - 1 - $exp_bits) + 1;
160 const MAX_EXP: ExpInt = (1 << ($exp_bits - 1)) - 1;
167 Single = SingleS(32:8),
168 Double = DoubleS(64:11),
172 pub struct X87DoubleExtendedS;
173 pub type X87DoubleExtended = IeeeFloat<X87DoubleExtendedS>;
174 impl Semantics for X87DoubleExtendedS {
175 const BITS: usize = 80;
176 const PRECISION: usize = 64;
177 const MAX_EXP: ExpInt = (1 << (15 - 1)) - 1;
179 /// For x87 extended precision, we want to make a NaN, not a
180 /// pseudo-NaN. Maybe we should expose the ability to make
182 const QNAN_SIGNIFICAND: Limb = 0b11 << Self::QNAN_BIT;
184 /// Integer bit is explicit in this format. Intel hardware (387 and later)
185 /// does not support these bit patterns:
186 /// exponent = all 1's, integer bit 0, significand 0 ("pseudoinfinity")
187 /// exponent = all 1's, integer bit 0, significand nonzero ("pseudoNaN")
188 /// exponent = 0, integer bit 1 ("pseudodenormal")
189 /// exponent != 0 nor all 1's, integer bit 0 ("unnormal")
190 /// At the moment, the first two are treated as NaNs, the second two as Normal.
191 fn from_bits(bits: u128) -> IeeeFloat<Self> {
192 let sign = bits & (1 << (Self::BITS - 1));
193 let exponent = (bits & !sign) >> Self::PRECISION;
194 let mut r = IeeeFloat {
195 sig: [bits & ((1 << (Self::PRECISION - 1)) - 1)],
196 // Convert the exponent from its bias representation to a signed integer.
197 exp: (exponent as ExpInt) - Self::MAX_EXP,
198 category: Category::Zero,
203 if r.exp == Self::MIN_EXP - 1 && r.sig == [0] {
204 // Exponent, significand meaningless.
205 r.category = Category::Zero;
206 } else if r.exp == Self::MAX_EXP + 1 && r.sig == [1 << (Self::PRECISION - 1)] {
207 // Exponent, significand meaningless.
208 r.category = Category::Infinity;
209 } else if r.exp == Self::MAX_EXP + 1 && r.sig != [1 << (Self::PRECISION - 1)] {
210 // Sign, exponent, significand meaningless.
211 r.category = Category::NaN;
213 r.category = Category::Normal;
214 if r.exp == Self::MIN_EXP - 1 {
216 r.exp = Self::MIN_EXP;
223 fn to_bits(x: IeeeFloat<Self>) -> u128 {
224 // Get integer bit from significand.
225 let integer_bit = sig::get_bit(&x.sig, Self::PRECISION - 1);
226 let mut significand = x.sig[0] & ((1 << Self::PRECISION) - 1);
227 let exponent = match x.category {
228 Category::Normal => {
229 if x.exp == Self::MIN_EXP && !integer_bit {
237 // FIXME(eddyb) Maybe we should guarantee an invariant instead?
241 Category::Infinity => {
242 // FIXME(eddyb) Maybe we should guarantee an invariant instead?
243 significand = 1 << (Self::PRECISION - 1);
246 Category::NaN => Self::MAX_EXP + 1,
249 // Convert the exponent from a signed integer to its bias representation.
250 let exponent = (exponent + Self::MAX_EXP) as u128;
252 ((x.sign as u128) << (Self::BITS - 1)) | (exponent << Self::PRECISION) | significand
256 float_common_impls!(IeeeFloat<S>);
258 impl<S: Semantics> PartialEq for IeeeFloat<S> {
259 fn eq(&self, rhs: &Self) -> bool {
260 self.partial_cmp(rhs) == Some(Ordering::Equal)
264 impl<S: Semantics> PartialOrd for IeeeFloat<S> {
265 fn partial_cmp(&self, rhs: &Self) -> Option<Ordering> {
266 match (self.category, rhs.category) {
268 (_, Category::NaN) => None,
270 (Category::Infinity, Category::Infinity) => Some((!self.sign).cmp(&(!rhs.sign))),
272 (Category::Zero, Category::Zero) => Some(Ordering::Equal),
274 (Category::Infinity, _) |
275 (Category::Normal, Category::Zero) => Some((!self.sign).cmp(&self.sign)),
277 (_, Category::Infinity) |
278 (Category::Zero, Category::Normal) => Some(rhs.sign.cmp(&(!rhs.sign))),
280 (Category::Normal, Category::Normal) => {
281 // Two normal numbers. Do they have the same sign?
282 Some((!self.sign).cmp(&(!rhs.sign)).then_with(|| {
283 // Compare absolute values; invert result if negative.
284 let result = self.cmp_abs_normal(*rhs);
286 if self.sign { result.reverse() } else { result }
293 impl<S> Neg for IeeeFloat<S> {
295 fn neg(mut self) -> Self {
296 self.sign = !self.sign;
301 /// Prints this value as a decimal string.
303 /// \param precision The maximum number of digits of
304 /// precision to output. If there are fewer digits available,
305 /// zero padding will not be used unless the value is
306 /// integral and small enough to be expressed in
307 /// precision digits. 0 means to use the natural
308 /// precision of the number.
309 /// \param width The maximum number of zeros to
310 /// consider inserting before falling back to scientific
311 /// notation. 0 means to always use scientific notation.
313 /// \param alternate Indicate whether to remove the trailing zero in
314 /// fraction part or not. Also setting this parameter to true forces
315 /// producing of output more similar to default printf behavior.
316 /// Specifically the lower e is used as exponent delimiter and exponent
317 /// always contains no less than two digits.
319 /// Number precision width Result
320 /// ------ --------- ----- ------
321 /// 1.01E+4 5 2 10100
322 /// 1.01E+4 4 2 1.01E+4
323 /// 1.01E+4 5 1 1.01E+4
324 /// 1.01E-2 5 2 0.0101
325 /// 1.01E-2 4 2 0.0101
326 /// 1.01E-2 4 1 1.01E-2
327 impl<S: Semantics> fmt::Display for IeeeFloat<S> {
328 fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
329 let width = f.width().unwrap_or(3);
330 let alternate = f.alternate();
332 match self.category {
333 Category::Infinity => {
335 return f.write_str("-Inf");
337 return f.write_str("+Inf");
341 Category::NaN => return f.write_str("NaN"),
351 if let Some(n) = f.precision() {
356 f.write_str("e+00")?;
358 f.write_str("0.0E+0")?;
366 Category::Normal => {}
373 // We use enough digits so the number can be round-tripped back to an
374 // APFloat. The formula comes from "How to Print Floating-Point Numbers
375 // Accurately" by Steele and White.
376 // FIXME: Using a formula based purely on the precision is conservative;
377 // we can print fewer digits depending on the actual value being printed.
379 // precision = 2 + floor(S::PRECISION / lg_2(10))
380 let precision = f.precision().unwrap_or(2 + S::PRECISION * 59 / 196);
382 // Decompose the number into an APInt and an exponent.
383 let mut exp = self.exp - (S::PRECISION as ExpInt - 1);
384 let mut sig = vec![self.sig[0]];
386 // Ignore trailing binary zeros.
387 let trailing_zeros = sig[0].trailing_zeros();
388 let _: Loss = sig::shift_right(&mut sig, &mut exp, trailing_zeros as usize);
390 // Change the exponent from 2^e to 10^e.
395 let shift = exp as usize;
396 sig.resize(limbs_for_bits(S::PRECISION + shift), 0);
397 sig::shift_left(&mut sig, &mut exp, shift);
400 let mut texp = -exp as usize;
402 // We transform this using the identity:
403 // (N)(2^-e) == (N)(5^e)(10^-e)
405 // Multiply significand by 5^e.
406 // N * 5^0101 == N * 5^(1*1) * 5^(0*2) * 5^(1*4) * 5^(0*8)
407 let mut sig_scratch = vec![];
409 let mut p5_scratch = vec![];
414 p5_scratch.resize(p5.len() * 2, 0);
416 sig::mul(&mut p5_scratch, &mut 0, &p5, &p5, p5.len() * 2 * LIMB_BITS);
417 while p5_scratch.last() == Some(&0) {
420 mem::swap(&mut p5, &mut p5_scratch);
423 sig_scratch.resize(sig.len() + p5.len(), 0);
424 let _: Loss = sig::mul(
429 (sig.len() + p5.len()) * LIMB_BITS,
431 while sig_scratch.last() == Some(&0) {
434 mem::swap(&mut sig, &mut sig_scratch);
441 let mut buffer = vec![];
443 // Ignore digits from the significand until it is no more
444 // precise than is required for the desired precision.
445 // 196/59 is a very slight overestimate of lg_2(10).
446 let required = (precision * 196 + 58) / 59;
447 let mut discard_digits = sig::omsb(&sig).saturating_sub(required) * 59 / 196;
448 let mut in_trail = true;
449 while !sig.is_empty() {
450 // Perform short division by 10 to extract the rightmost digit.
455 // Use 64-bit division and remainder, with 32-bit chunks from sig.
456 sig::each_chunk(&mut sig, 32, |chunk| {
457 let chunk = chunk as u32;
458 let combined = ((rem as u64) << 32) | (chunk as u64);
459 rem = (combined % 10) as u8;
460 (combined / 10) as u32 as Limb
463 // Reduce the sigificand to avoid wasting time dividing 0's.
464 while sig.last() == Some(&0) {
470 // Ignore digits we don't need.
471 if discard_digits > 0 {
477 // Drop trailing zeros.
478 if in_trail && digit == 0 {
482 buffer.push(b'0' + digit);
486 assert!(!buffer.is_empty(), "no characters in buffer!");
488 // Drop down to precision.
489 // FIXME: don't do more precise calculations above than are required.
490 if buffer.len() > precision {
491 // The most significant figures are the last ones in the buffer.
492 let mut first_sig = buffer.len() - precision;
495 // FIXME: this probably shouldn't use 'round half up'.
497 // Rounding down is just a truncation, except we also want to drop
498 // trailing zeros from the new result.
499 if buffer[first_sig - 1] < b'5' {
500 while first_sig < buffer.len() && buffer[first_sig] == b'0' {
504 // Rounding up requires a decimal add-with-carry. If we continue
505 // the carry, the newly-introduced zeros will just be truncated.
506 for x in &mut buffer[first_sig..] {
516 exp += first_sig as ExpInt;
517 buffer.drain(..first_sig);
519 // If we carried through, we have exactly one digit of precision.
520 if buffer.is_empty() {
525 let digits = buffer.len();
527 // Check whether we should use scientific notation.
528 let scientific = if width == 0 {
533 // But we shouldn't make the number look more precise than it is.
534 exp as usize > width || digits + exp as usize > precision
536 // Power of the most significant digit.
537 let msd = exp + (digits - 1) as ExpInt;
544 -msd as usize > width
548 // Scientific formatting is pretty straightforward.
550 exp += digits as ExpInt - 1;
552 f.write_char(buffer[digits - 1] as char)?;
554 let truncate_zero = !alternate;
555 if digits == 1 && truncate_zero {
558 for &d in buffer[..digits - 1].iter().rev() {
559 f.write_char(d as char)?;
562 // Fill with zeros up to precision.
563 if !truncate_zero && precision > digits - 1 {
564 for _ in 0..=precision - digits {
568 // For alternate we use lower 'e'.
569 f.write_char(if alternate { 'e' } else { 'E' })?;
571 // Exponent always at least two digits if we do not truncate zeros.
573 write!(f, "{:+}", exp)?;
575 write!(f, "{:+03}", exp)?;
581 // Non-scientific, positive exponents.
583 for &d in buffer.iter().rev() {
584 f.write_char(d as char)?;
592 // Non-scientific, negative exponents.
593 let unit_place = -exp as usize;
594 if unit_place < digits {
595 for &d in buffer[unit_place..].iter().rev() {
596 f.write_char(d as char)?;
599 for &d in buffer[..unit_place].iter().rev() {
600 f.write_char(d as char)?;
604 for _ in digits..unit_place {
607 for &d in buffer.iter().rev() {
608 f.write_char(d as char)?;
616 impl<S: Semantics> fmt::Debug for IeeeFloat<S> {
617 fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
618 write!(f, "{}({:?} | {}{:?} * 2^{})",
620 if self.sign { "-" } else { "+" },
626 impl<S: Semantics> Float for IeeeFloat<S> {
627 const BITS: usize = S::BITS;
628 const PRECISION: usize = S::PRECISION;
629 const MAX_EXP: ExpInt = S::MAX_EXP;
630 const MIN_EXP: ExpInt = S::MIN_EXP;
632 const ZERO: Self = IeeeFloat {
635 category: Category::Zero,
640 const INFINITY: Self = IeeeFloat {
643 category: Category::Infinity,
648 // FIXME(eddyb) remove when qnan becomes const fn.
649 const NAN: Self = IeeeFloat {
650 sig: [S::QNAN_SIGNIFICAND],
652 category: Category::NaN,
657 fn qnan(payload: Option<u128>) -> Self {
660 S::QNAN_SIGNIFICAND |
661 payload.map_or(0, |payload| {
662 // Zero out the excess bits of the significand.
663 payload & ((1 << S::QNAN_BIT) - 1)
667 category: Category::NaN,
673 fn snan(payload: Option<u128>) -> Self {
674 let mut snan = Self::qnan(payload);
676 // We always have to clear the QNaN bit to make it an SNaN.
677 sig::clear_bit(&mut snan.sig, S::QNAN_BIT);
679 // If there are no bits set in the payload, we have to set
680 // *something* to make it a NaN instead of an infinity;
681 // conventionally, this is the next bit down from the QNaN bit.
682 if snan.sig[0] & !S::QNAN_SIGNIFICAND == 0 {
683 sig::set_bit(&mut snan.sig, S::QNAN_BIT - 1);
689 fn largest() -> Self {
690 // We want (in interchange format):
692 // significand = 1..1
694 sig: [(1 << S::PRECISION) - 1],
696 category: Category::Normal,
702 // We want (in interchange format):
704 // significand = 0..01
705 const SMALLEST: Self = IeeeFloat {
708 category: Category::Normal,
713 fn smallest_normalized() -> Self {
714 // We want (in interchange format):
716 // significand = 10..0
718 sig: [1 << (S::PRECISION - 1)],
720 category: Category::Normal,
726 fn add_r(mut self, rhs: Self, round: Round) -> StatusAnd<Self> {
727 let status = match (self.category, rhs.category) {
728 (Category::Infinity, Category::Infinity) => {
729 // Differently signed infinities can only be validly
731 if self.sign != rhs.sign {
739 // Sign may depend on rounding mode; handled below.
740 (_, Category::Zero) |
742 (Category::Infinity, Category::Normal) => Status::OK,
744 (Category::Zero, _) |
746 (_, Category::Infinity) => {
751 // This return code means it was not a simple case.
752 (Category::Normal, Category::Normal) => {
753 let loss = sig::add_or_sub(
762 self = unpack!(status=, self.normalize(round, loss));
764 // Can only be zero if we lost no fraction.
765 assert!(self.category != Category::Zero || loss == Loss::ExactlyZero);
771 // If two numbers add (exactly) to zero, IEEE 754 decrees it is a
772 // positive zero unless rounding to minus infinity, except that
773 // adding two like-signed zeroes gives that zero.
774 if self.category == Category::Zero &&
775 (rhs.category != Category::Zero || self.sign != rhs.sign)
777 self.sign = round == Round::TowardNegative;
783 fn mul_r(mut self, rhs: Self, round: Round) -> StatusAnd<Self> {
784 self.sign ^= rhs.sign;
786 match (self.category, rhs.category) {
787 (Category::NaN, _) => {
792 (_, Category::NaN) => {
794 self.category = Category::NaN;
799 (Category::Zero, Category::Infinity) |
800 (Category::Infinity, Category::Zero) => Status::INVALID_OP.and(Self::NAN),
802 (_, Category::Infinity) |
803 (Category::Infinity, _) => {
804 self.category = Category::Infinity;
808 (Category::Zero, _) |
809 (_, Category::Zero) => {
810 self.category = Category::Zero;
814 (Category::Normal, Category::Normal) => {
816 let mut wide_sig = [0; 2];
824 self.sig = [wide_sig[0]];
826 self = unpack!(status=, self.normalize(round, loss));
827 if loss != Loss::ExactlyZero {
828 status |= Status::INEXACT;
835 fn mul_add_r(mut self, multiplicand: Self, addend: Self, round: Round) -> StatusAnd<Self> {
836 // If and only if all arguments are normal do we need to do an
837 // extended-precision calculation.
838 if !self.is_finite_non_zero() || !multiplicand.is_finite_non_zero() || !addend.is_finite() {
840 self = unpack!(status=, self.mul_r(multiplicand, round));
842 // FS can only be Status::OK or Status::INVALID_OP. There is no more work
843 // to do in the latter case. The IEEE-754R standard says it is
844 // implementation-defined in this case whether, if ADDEND is a
845 // quiet NaN, we raise invalid op; this implementation does so.
847 // If we need to do the addition we can do so with normal
849 if status == Status::OK {
850 self = unpack!(status=, self.add_r(addend, round));
852 return status.and(self);
855 // Post-multiplication sign, before addition.
856 self.sign ^= multiplicand.sign;
858 // Allocate space for twice as many bits as the original significand, plus one
859 // extra bit for the addition to overflow into.
860 assert!(limbs_for_bits(S::PRECISION * 2 + 1) <= 2);
861 let mut wide_sig = sig::widening_mul(self.sig[0], multiplicand.sig[0]);
863 let mut loss = Loss::ExactlyZero;
864 let mut omsb = sig::omsb(&wide_sig);
865 self.exp += multiplicand.exp;
867 // Assume the operands involved in the multiplication are single-precision
868 // FP, and the two multiplicants are:
869 // lhs = a23 . a22 ... a0 * 2^e1
870 // rhs = b23 . b22 ... b0 * 2^e2
871 // the result of multiplication is:
872 // lhs = c48 c47 c46 . c45 ... c0 * 2^(e1+e2)
873 // Note that there are three significant bits at the left-hand side of the
874 // radix point: two for the multiplication, and an overflow bit for the
875 // addition (that will always be zero at this point). Move the radix point
876 // toward left by two bits, and adjust exponent accordingly.
879 if addend.is_non_zero() {
880 // Normalize our MSB to one below the top bit to allow for overflow.
881 let ext_precision = 2 * S::PRECISION + 1;
882 if omsb != ext_precision - 1 {
883 assert!(ext_precision > omsb);
884 sig::shift_left(&mut wide_sig, &mut self.exp, (ext_precision - 1) - omsb);
887 // The intermediate result of the multiplication has "2 * S::PRECISION"
888 // significant bit; adjust the addend to be consistent with mul result.
889 let mut ext_addend_sig = [addend.sig[0], 0];
891 // Extend the addend significand to ext_precision - 1. This guarantees
892 // that the high bit of the significand is zero (same as wide_sig),
893 // so the addition will overflow (if it does overflow at all) into the top bit.
897 ext_precision - 1 - S::PRECISION,
899 loss = sig::add_or_sub(
908 omsb = sig::omsb(&wide_sig);
911 // Convert the result having "2 * S::PRECISION" significant-bits back to the one
912 // having "S::PRECISION" significant-bits. First, move the radix point from
913 // position "2*S::PRECISION - 1" to "S::PRECISION - 1". The exponent need to be
914 // adjusted by "2*S::PRECISION - 1" - "S::PRECISION - 1" = "S::PRECISION".
915 self.exp -= S::PRECISION as ExpInt + 1;
917 // In case MSB resides at the left-hand side of radix point, shift the
918 // mantissa right by some amount to make sure the MSB reside right before
919 // the radix point (i.e., "MSB . rest-significant-bits").
920 if omsb > S::PRECISION {
921 let bits = omsb - S::PRECISION;
922 loss = sig::shift_right(&mut wide_sig, &mut self.exp, bits).combine(loss);
925 self.sig[0] = wide_sig[0];
928 self = unpack!(status=, self.normalize(round, loss));
929 if loss != Loss::ExactlyZero {
930 status |= Status::INEXACT;
933 // If two numbers add (exactly) to zero, IEEE 754 decrees it is a
934 // positive zero unless rounding to minus infinity, except that
935 // adding two like-signed zeroes gives that zero.
936 if self.category == Category::Zero && !status.intersects(Status::UNDERFLOW) &&
937 self.sign != addend.sign
939 self.sign = round == Round::TowardNegative;
945 fn div_r(mut self, rhs: Self, round: Round) -> StatusAnd<Self> {
946 self.sign ^= rhs.sign;
948 match (self.category, rhs.category) {
949 (Category::NaN, _) => {
954 (_, Category::NaN) => {
955 self.category = Category::NaN;
961 (Category::Infinity, Category::Infinity) |
962 (Category::Zero, Category::Zero) => Status::INVALID_OP.and(Self::NAN),
964 (Category::Infinity, _) |
965 (Category::Zero, _) => Status::OK.and(self),
967 (Category::Normal, Category::Infinity) => {
968 self.category = Category::Zero;
972 (Category::Normal, Category::Zero) => {
973 self.category = Category::Infinity;
974 Status::DIV_BY_ZERO.and(self)
977 (Category::Normal, Category::Normal) => {
979 let dividend = self.sig[0];
988 self = unpack!(status=, self.normalize(round, loss));
989 if loss != Loss::ExactlyZero {
990 status |= Status::INEXACT;
997 fn c_fmod(mut self, rhs: Self) -> StatusAnd<Self> {
998 match (self.category, rhs.category) {
1000 (Category::Zero, Category::Infinity) |
1001 (Category::Zero, Category::Normal) |
1002 (Category::Normal, Category::Infinity) => Status::OK.and(self),
1004 (_, Category::NaN) => {
1006 self.category = Category::NaN;
1008 Status::OK.and(self)
1011 (Category::Infinity, _) |
1012 (_, Category::Zero) => Status::INVALID_OP.and(Self::NAN),
1014 (Category::Normal, Category::Normal) => {
1015 while self.is_finite_non_zero() && rhs.is_finite_non_zero() &&
1016 self.cmp_abs_normal(rhs) != Ordering::Less
1018 let mut v = rhs.scalbn(self.ilogb() - rhs.ilogb());
1019 if self.cmp_abs_normal(v) == Ordering::Less {
1025 self = unpack!(status=, self - v);
1026 assert_eq!(status, Status::OK);
1028 Status::OK.and(self)
1033 fn round_to_integral(self, round: Round) -> StatusAnd<Self> {
1034 // If the exponent is large enough, we know that this value is already
1035 // integral, and the arithmetic below would potentially cause it to saturate
1036 // to +/-Inf. Bail out early instead.
1037 if self.is_finite_non_zero() && self.exp + 1 >= S::PRECISION as ExpInt {
1038 return Status::OK.and(self);
1041 // The algorithm here is quite simple: we add 2^(p-1), where p is the
1042 // precision of our format, and then subtract it back off again. The choice
1043 // of rounding modes for the addition/subtraction determines the rounding mode
1044 // for our integral rounding as well.
1045 // NOTE: When the input value is negative, we do subtraction followed by
1046 // addition instead.
1047 assert!(S::PRECISION <= 128);
1049 let magic_const = unpack!(status=, Self::from_u128(1 << (S::PRECISION - 1)));
1050 let magic_const = magic_const.copy_sign(self);
1052 if status != Status::OK {
1053 return status.and(self);
1057 r = unpack!(status=, r.add_r(magic_const, round));
1058 if status != Status::OK && status != Status::INEXACT {
1059 return status.and(self);
1062 // Restore the input sign to handle 0.0/-0.0 cases correctly.
1063 r.sub_r(magic_const, round).map(|r| r.copy_sign(self))
1066 fn next_up(mut self) -> StatusAnd<Self> {
1067 // Compute nextUp(x), handling each float category separately.
1068 match self.category {
1069 Category::Infinity => {
1071 // nextUp(-inf) = -largest
1072 Status::OK.and(-Self::largest())
1074 // nextUp(+inf) = +inf
1075 Status::OK.and(self)
1079 // IEEE-754R 2008 6.2 Par 2: nextUp(sNaN) = qNaN. Set Invalid flag.
1080 // IEEE-754R 2008 6.2: nextUp(qNaN) = qNaN. Must be identity so we do not
1081 // change the payload.
1082 if self.is_signaling() {
1083 // For consistency, propagate the sign of the sNaN to the qNaN.
1084 Status::INVALID_OP.and(Self::NAN.copy_sign(self))
1086 Status::OK.and(self)
1090 // nextUp(pm 0) = +smallest
1091 Status::OK.and(Self::SMALLEST)
1093 Category::Normal => {
1094 // nextUp(-smallest) = -0
1095 if self.is_smallest() && self.sign {
1096 return Status::OK.and(-Self::ZERO);
1099 // nextUp(largest) == INFINITY
1100 if self.is_largest() && !self.sign {
1101 return Status::OK.and(Self::INFINITY);
1104 // Excluding the integral bit. This allows us to test for binade boundaries.
1105 let sig_mask = (1 << (S::PRECISION - 1)) - 1;
1107 // nextUp(normal) == normal + inc.
1109 // If we are negative, we need to decrement the significand.
1111 // We only cross a binade boundary that requires adjusting the exponent
1113 // 1. exponent != S::MIN_EXP. This implies we are not in the
1114 // smallest binade or are dealing with denormals.
1115 // 2. Our significand excluding the integral bit is all zeros.
1116 let crossing_binade_boundary = self.exp != S::MIN_EXP &&
1117 self.sig[0] & sig_mask == 0;
1119 // Decrement the significand.
1121 // We always do this since:
1122 // 1. If we are dealing with a non-binade decrement, by definition we
1123 // just decrement the significand.
1124 // 2. If we are dealing with a normal -> normal binade decrement, since
1125 // we have an explicit integral bit the fact that all bits but the
1126 // integral bit are zero implies that subtracting one will yield a
1127 // significand with 0 integral bit and 1 in all other spots. Thus we
1128 // must just adjust the exponent and set the integral bit to 1.
1129 // 3. If we are dealing with a normal -> denormal binade decrement,
1130 // since we set the integral bit to 0 when we represent denormals, we
1131 // just decrement the significand.
1132 sig::decrement(&mut self.sig);
1134 if crossing_binade_boundary {
1135 // Our result is a normal number. Do the following:
1136 // 1. Set the integral bit to 1.
1137 // 2. Decrement the exponent.
1138 sig::set_bit(&mut self.sig, S::PRECISION - 1);
1142 // If we are positive, we need to increment the significand.
1144 // We only cross a binade boundary that requires adjusting the exponent if
1145 // the input is not a denormal and all of said input's significand bits
1146 // are set. If all of said conditions are true: clear the significand, set
1147 // the integral bit to 1, and increment the exponent. If we have a
1148 // denormal always increment since moving denormals and the numbers in the
1149 // smallest normal binade have the same exponent in our representation.
1150 let crossing_binade_boundary = !self.is_denormal() &&
1151 self.sig[0] & sig_mask == sig_mask;
1153 if crossing_binade_boundary {
1155 sig::set_bit(&mut self.sig, S::PRECISION - 1);
1159 "We can not increment an exponent beyond the MAX_EXP \
1160 allowed by the given floating point semantics."
1164 sig::increment(&mut self.sig);
1167 Status::OK.and(self)
1172 fn from_bits(input: u128) -> Self {
1173 // Dispatch to semantics.
1177 fn from_u128_r(input: u128, round: Round) -> StatusAnd<Self> {
1180 exp: S::PRECISION as ExpInt - 1,
1181 category: Category::Normal,
1183 marker: PhantomData,
1184 }.normalize(round, Loss::ExactlyZero)
1187 fn from_str_r(mut s: &str, mut round: Round) -> Result<StatusAnd<Self>, ParseError> {
1189 return Err(ParseError("Invalid string length"));
1192 // Handle special cases.
1194 "inf" | "INFINITY" => return Ok(Status::OK.and(Self::INFINITY)),
1195 "-inf" | "-INFINITY" => return Ok(Status::OK.and(-Self::INFINITY)),
1196 "nan" | "NaN" => return Ok(Status::OK.and(Self::NAN)),
1197 "-nan" | "-NaN" => return Ok(Status::OK.and(-Self::NAN)),
1201 // Handle a leading minus sign.
1202 let minus = s.starts_with("-");
1203 if minus || s.starts_with("+") {
1206 return Err(ParseError("String has no digits"));
1210 // Adjust the rounding mode for the absolute value below.
1215 let r = if s.starts_with("0x") || s.starts_with("0X") {
1218 return Err(ParseError("Invalid string"));
1220 Self::from_hexadecimal_string(s, round)?
1222 Self::from_decimal_string(s, round)?
1225 Ok(r.map(|r| if minus { -r } else { r }))
1228 fn to_bits(self) -> u128 {
1229 // Dispatch to semantics.
1233 fn to_u128_r(self, width: usize, round: Round, is_exact: &mut bool) -> StatusAnd<u128> {
1234 // The result of trying to convert a number too large.
1235 let overflow = if self.sign {
1236 // Negative numbers cannot be represented as unsigned.
1239 // Largest unsigned integer of the given width.
1245 match self.category {
1246 Category::NaN => Status::INVALID_OP.and(0),
1248 Category::Infinity => Status::INVALID_OP.and(overflow),
1251 // Negative zero can't be represented as an int.
1252 *is_exact = !self.sign;
1256 Category::Normal => {
1259 // Step 1: place our absolute value, with any fraction truncated, in
1261 let truncated_bits = if self.exp < 0 {
1262 // Our absolute value is less than one; truncate everything.
1263 // For exponent -1 the integer bit represents .5, look at that.
1264 // For smaller exponents leftmost truncated bit is 0.
1265 S::PRECISION - 1 + (-self.exp) as usize
1267 // We want the most significant (exponent + 1) bits; the rest are
1269 let bits = self.exp as usize + 1;
1271 // Hopelessly large in magnitude?
1273 return Status::INVALID_OP.and(overflow);
1276 if bits < S::PRECISION {
1277 // We truncate (S::PRECISION - bits) bits.
1278 r = self.sig[0] >> (S::PRECISION - bits);
1281 // We want at least as many bits as are available.
1282 r = self.sig[0] << (bits - S::PRECISION);
1287 // Step 2: work out any lost fraction, and increment the absolute
1288 // value if we would round away from zero.
1289 let mut loss = Loss::ExactlyZero;
1290 if truncated_bits > 0 {
1291 loss = Loss::through_truncation(&self.sig, truncated_bits);
1292 if loss != Loss::ExactlyZero &&
1293 self.round_away_from_zero(round, loss, truncated_bits)
1295 r = r.wrapping_add(1);
1297 return Status::INVALID_OP.and(overflow); // Overflow.
1302 // Step 3: check if we fit in the destination.
1304 return Status::INVALID_OP.and(overflow);
1307 if loss == Loss::ExactlyZero {
1311 Status::INEXACT.and(r)
1317 fn cmp_abs_normal(self, rhs: Self) -> Ordering {
1318 assert!(self.is_finite_non_zero());
1319 assert!(rhs.is_finite_non_zero());
1321 // If exponents are equal, do an unsigned comparison of the significands.
1322 self.exp.cmp(&rhs.exp).then_with(
1323 || sig::cmp(&self.sig, &rhs.sig),
1327 fn bitwise_eq(self, rhs: Self) -> bool {
1328 if self.category != rhs.category || self.sign != rhs.sign {
1332 if self.category == Category::Zero || self.category == Category::Infinity {
1336 if self.is_finite_non_zero() && self.exp != rhs.exp {
1343 fn is_negative(self) -> bool {
1347 fn is_denormal(self) -> bool {
1348 self.is_finite_non_zero() && self.exp == S::MIN_EXP &&
1349 !sig::get_bit(&self.sig, S::PRECISION - 1)
1352 fn is_signaling(self) -> bool {
1353 // IEEE-754R 2008 6.2.1: A signaling NaN bit string should be encoded with the
1354 // first bit of the trailing significand being 0.
1355 self.is_nan() && !sig::get_bit(&self.sig, S::QNAN_BIT)
1358 fn category(self) -> Category {
1362 fn get_exact_inverse(self) -> Option<Self> {
1363 // Special floats and denormals have no exact inverse.
1364 if !self.is_finite_non_zero() {
1368 // Check that the number is a power of two by making sure that only the
1369 // integer bit is set in the significand.
1370 if self.sig != [1 << (S::PRECISION - 1)] {
1375 let mut reciprocal = Self::from_u128(1).value;
1377 reciprocal = unpack!(status=, reciprocal / self);
1378 if status != Status::OK {
1382 // Avoid multiplication with a denormal, it is not safe on all platforms and
1383 // may be slower than a normal division.
1384 if reciprocal.is_denormal() {
1388 assert!(reciprocal.is_finite_non_zero());
1389 assert_eq!(reciprocal.sig, [1 << (S::PRECISION - 1)]);
1394 fn ilogb(mut self) -> ExpInt {
1401 if self.is_infinite() {
1404 if !self.is_denormal() {
1408 let sig_bits = (S::PRECISION - 1) as ExpInt;
1409 self.exp += sig_bits;
1410 self = self.normalize(Round::NearestTiesToEven, Loss::ExactlyZero)
1415 fn scalbn_r(mut self, exp: ExpInt, round: Round) -> Self {
1416 // If exp is wildly out-of-scale, simply adding it to self.exp will
1417 // overflow; clamp it to a safe range before adding, but ensure that the range
1418 // is large enough that the clamp does not change the result. The range we
1419 // need to support is the difference between the largest possible exponent and
1420 // the normalized exponent of half the smallest denormal.
1422 let sig_bits = (S::PRECISION - 1) as i32;
1423 let max_change = S::MAX_EXP as i32 - (S::MIN_EXP as i32 - sig_bits) + 1;
1425 // Clamp to one past the range ends to let normalize handle overflow.
1426 let exp_change = cmp::min(cmp::max(exp as i32, -max_change - 1), max_change);
1427 self.exp = self.exp.saturating_add(exp_change as ExpInt);
1428 self = self.normalize(round, Loss::ExactlyZero).value;
1430 sig::set_bit(&mut self.sig, S::QNAN_BIT);
1435 fn frexp_r(mut self, exp: &mut ExpInt, round: Round) -> Self {
1436 *exp = self.ilogb();
1438 // Quiet signalling nans.
1439 if *exp == IEK_NAN {
1440 sig::set_bit(&mut self.sig, S::QNAN_BIT);
1444 if *exp == IEK_INF {
1448 // 1 is added because frexp is defined to return a normalized fraction in
1449 // +/-[0.5, 1.0), rather than the usual +/-[1.0, 2.0).
1450 if *exp == IEK_ZERO {
1455 self.scalbn_r(-*exp, round)
1459 impl<S: Semantics, T: Semantics> FloatConvert<IeeeFloat<T>> for IeeeFloat<S> {
1460 fn convert_r(self, round: Round, loses_info: &mut bool) -> StatusAnd<IeeeFloat<T>> {
1461 let mut r = IeeeFloat {
1464 category: self.category,
1466 marker: PhantomData,
1469 // x86 has some unusual NaNs which cannot be represented in any other
1470 // format; note them here.
1471 fn is_x87_double_extended<S: Semantics>() -> bool {
1472 S::QNAN_SIGNIFICAND == X87DoubleExtendedS::QNAN_SIGNIFICAND
1474 let x87_special_nan = is_x87_double_extended::<S>() && !is_x87_double_extended::<T>() &&
1475 r.category == Category::NaN &&
1476 (r.sig[0] & S::QNAN_SIGNIFICAND) != S::QNAN_SIGNIFICAND;
1478 // If this is a truncation of a denormal number, and the target semantics
1479 // has larger exponent range than the source semantics (this can happen
1480 // when truncating from PowerPC double-double to double format), the
1481 // right shift could lose result mantissa bits. Adjust exponent instead
1482 // of performing excessive shift.
1483 let mut shift = T::PRECISION as ExpInt - S::PRECISION as ExpInt;
1484 if shift < 0 && r.is_finite_non_zero() {
1485 let mut exp_change = sig::omsb(&r.sig) as ExpInt - S::PRECISION as ExpInt;
1486 if r.exp + exp_change < T::MIN_EXP {
1487 exp_change = T::MIN_EXP - r.exp;
1489 if exp_change < shift {
1493 shift -= exp_change;
1494 r.exp += exp_change;
1498 // If this is a truncation, perform the shift.
1499 let loss = if shift < 0 && (r.is_finite_non_zero() || r.category == Category::NaN) {
1500 sig::shift_right(&mut r.sig, &mut 0, -shift as usize)
1505 // If this is an extension, perform the shift.
1506 if shift > 0 && (r.is_finite_non_zero() || r.category == Category::NaN) {
1507 sig::shift_left(&mut r.sig, &mut 0, shift as usize);
1511 if r.is_finite_non_zero() {
1512 r = unpack!(status=, r.normalize(round, loss));
1513 *loses_info = status != Status::OK;
1514 } else if r.category == Category::NaN {
1515 *loses_info = loss != Loss::ExactlyZero || x87_special_nan;
1517 // For x87 extended precision, we want to make a NaN, not a special NaN if
1518 // the input wasn't special either.
1519 if !x87_special_nan && is_x87_double_extended::<T>() {
1520 sig::set_bit(&mut r.sig, T::PRECISION - 1);
1523 // gcc forces the Quiet bit on, which means (float)(double)(float_sNan)
1524 // does not give you back the same bits. This is dubious, and we
1525 // don't currently do it. You're really supposed to get
1526 // an invalid operation signal at runtime, but nobody does that.
1527 status = Status::OK;
1529 *loses_info = false;
1530 status = Status::OK;
1537 impl<S: Semantics> IeeeFloat<S> {
1538 /// Handle positive overflow. We either return infinity or
1539 /// the largest finite number. For negative overflow,
1540 /// negate the `round` argument before calling.
1541 fn overflow_result(round: Round) -> StatusAnd<Self> {
1544 Round::NearestTiesToEven | Round::NearestTiesToAway | Round::TowardPositive => {
1545 (Status::OVERFLOW | Status::INEXACT).and(Self::INFINITY)
1547 // Otherwise we become the largest finite number.
1548 Round::TowardNegative | Round::TowardZero => Status::INEXACT.and(Self::largest()),
1552 /// Returns `true` if, when truncating the current number, with `bit` the
1553 /// new LSB, with the given lost fraction and rounding mode, the result
1554 /// would need to be rounded away from zero (i.e., by increasing the
1555 /// signficand). This routine must work for `Category::Zero` of both signs, and
1556 /// `Category::Normal` numbers.
1557 fn round_away_from_zero(&self, round: Round, loss: Loss, bit: usize) -> bool {
1558 // NaNs and infinities should not have lost fractions.
1559 assert!(self.is_finite_non_zero() || self.is_zero());
1561 // Current callers never pass this so we don't handle it.
1562 assert_ne!(loss, Loss::ExactlyZero);
1565 Round::NearestTiesToAway => loss == Loss::ExactlyHalf || loss == Loss::MoreThanHalf,
1566 Round::NearestTiesToEven => {
1567 if loss == Loss::MoreThanHalf {
1571 // Our zeros don't have a significand to test.
1572 if loss == Loss::ExactlyHalf && self.category != Category::Zero {
1573 return sig::get_bit(&self.sig, bit);
1578 Round::TowardZero => false,
1579 Round::TowardPositive => !self.sign,
1580 Round::TowardNegative => self.sign,
1584 fn normalize(mut self, round: Round, mut loss: Loss) -> StatusAnd<Self> {
1585 if !self.is_finite_non_zero() {
1586 return Status::OK.and(self);
1589 // Before rounding normalize the exponent of Category::Normal numbers.
1590 let mut omsb = sig::omsb(&self.sig);
1593 // OMSB is numbered from 1. We want to place it in the integer
1594 // bit numbered PRECISION if possible, with a compensating change in
1596 let mut final_exp = self.exp.saturating_add(
1597 omsb as ExpInt - S::PRECISION as ExpInt,
1600 // If the resulting exponent is too high, overflow according to
1601 // the rounding mode.
1602 if final_exp > S::MAX_EXP {
1603 let round = if self.sign { -round } else { round };
1604 return Self::overflow_result(round).map(|r| r.copy_sign(self));
1607 // Subnormal numbers have exponent MIN_EXP, and their MSB
1608 // is forced based on that.
1609 if final_exp < S::MIN_EXP {
1610 final_exp = S::MIN_EXP;
1613 // Shifting left is easy as we don't lose precision.
1614 if final_exp < self.exp {
1615 assert_eq!(loss, Loss::ExactlyZero);
1617 let exp_change = (self.exp - final_exp) as usize;
1618 sig::shift_left(&mut self.sig, &mut self.exp, exp_change);
1620 return Status::OK.and(self);
1623 // Shift right and capture any new lost fraction.
1624 if final_exp > self.exp {
1625 let exp_change = (final_exp - self.exp) as usize;
1626 loss = sig::shift_right(&mut self.sig, &mut self.exp, exp_change).combine(loss);
1628 // Keep OMSB up-to-date.
1629 omsb = omsb.saturating_sub(exp_change);
1633 // Now round the number according to round given the lost
1636 // As specified in IEEE 754, since we do not trap we do not report
1637 // underflow for exact results.
1638 if loss == Loss::ExactlyZero {
1639 // Canonicalize zeros.
1641 self.category = Category::Zero;
1644 return Status::OK.and(self);
1647 // Increment the significand if we're rounding away from zero.
1648 if self.round_away_from_zero(round, loss, 0) {
1650 self.exp = S::MIN_EXP;
1653 // We should never overflow.
1654 assert_eq!(sig::increment(&mut self.sig), 0);
1655 omsb = sig::omsb(&self.sig);
1657 // Did the significand increment overflow?
1658 if omsb == S::PRECISION + 1 {
1659 // Renormalize by incrementing the exponent and shifting our
1660 // significand right one. However if we already have the
1661 // maximum exponent we overflow to infinity.
1662 if self.exp == S::MAX_EXP {
1663 self.category = Category::Infinity;
1665 return (Status::OVERFLOW | Status::INEXACT).and(self);
1668 let _: Loss = sig::shift_right(&mut self.sig, &mut self.exp, 1);
1670 return Status::INEXACT.and(self);
1674 // The normal case - we were and are not denormal, and any
1675 // significand increment above didn't overflow.
1676 if omsb == S::PRECISION {
1677 return Status::INEXACT.and(self);
1680 // We have a non-zero denormal.
1681 assert!(omsb < S::PRECISION);
1683 // Canonicalize zeros.
1685 self.category = Category::Zero;
1688 // The Category::Zero case is a denormal that underflowed to zero.
1689 (Status::UNDERFLOW | Status::INEXACT).and(self)
1692 fn from_hexadecimal_string(s: &str, round: Round) -> Result<StatusAnd<Self>, ParseError> {
1693 let mut r = IeeeFloat {
1696 category: Category::Normal,
1698 marker: PhantomData,
1701 let mut any_digits = false;
1702 let mut has_exp = false;
1703 let mut bit_pos = LIMB_BITS as isize;
1704 let mut loss = None;
1706 // Without leading or trailing zeros, irrespective of the dot.
1707 let mut first_sig_digit = None;
1708 let mut dot = s.len();
1710 for (p, c) in s.char_indices() {
1711 // Skip leading zeros and any (hexa)decimal point.
1714 return Err(ParseError("String contains multiple dots"));
1717 } else if let Some(hex_value) = c.to_digit(16) {
1720 if first_sig_digit.is_none() {
1724 first_sig_digit = Some(p);
1727 // Store the number while we have space.
1730 r.sig[0] |= (hex_value as Limb) << bit_pos;
1731 // If zero or one-half (the hexadecimal digit 8) are followed
1732 // by non-zero, they're a little more than zero or one-half.
1733 } else if let Some(ref mut loss) = loss {
1735 if *loss == Loss::ExactlyZero {
1736 *loss = Loss::LessThanHalf;
1738 if *loss == Loss::ExactlyHalf {
1739 *loss = Loss::MoreThanHalf;
1743 loss = Some(match hex_value {
1744 0 => Loss::ExactlyZero,
1745 1..=7 => Loss::LessThanHalf,
1746 8 => Loss::ExactlyHalf,
1747 9..=15 => Loss::MoreThanHalf,
1748 _ => unreachable!(),
1751 } else if c == 'p' || c == 'P' {
1753 return Err(ParseError("Significand has no digits"));
1760 let mut chars = s[p + 1..].chars().peekable();
1762 // Adjust for the given exponent.
1763 let exp_minus = chars.peek() == Some(&'-');
1764 if exp_minus || chars.peek() == Some(&'+') {
1769 if let Some(value) = c.to_digit(10) {
1771 r.exp = r.exp.saturating_mul(10).saturating_add(value as ExpInt);
1773 return Err(ParseError("Invalid character in exponent"));
1777 return Err(ParseError("Exponent has no digits"));
1786 return Err(ParseError("Invalid character in significand"));
1790 return Err(ParseError("Significand has no digits"));
1793 // Hex floats require an exponent but not a hexadecimal point.
1795 return Err(ParseError("Hex strings require an exponent"));
1798 // Ignore the exponent if we are zero.
1799 let first_sig_digit = match first_sig_digit {
1801 None => return Ok(Status::OK.and(Self::ZERO)),
1804 // Calculate the exponent adjustment implicit in the number of
1805 // significant digits and adjust for writing the significand starting
1806 // at the most significant nibble.
1807 let exp_adjustment = if dot > first_sig_digit {
1808 ExpInt::try_from(dot - first_sig_digit).unwrap()
1810 -ExpInt::try_from(first_sig_digit - dot - 1).unwrap()
1812 let exp_adjustment = exp_adjustment
1815 .saturating_add(S::PRECISION as ExpInt)
1816 .saturating_sub(LIMB_BITS as ExpInt);
1817 r.exp = r.exp.saturating_add(exp_adjustment);
1819 Ok(r.normalize(round, loss.unwrap_or(Loss::ExactlyZero)))
1822 fn from_decimal_string(s: &str, round: Round) -> Result<StatusAnd<Self>, ParseError> {
1823 // Given a normal decimal floating point number of the form
1825 // dddd.dddd[eE][+-]ddd
1827 // where the decimal point and exponent are optional, fill out the
1828 // variables below. Exponent is appropriate if the significand is
1829 // treated as an integer, and normalized_exp if the significand
1830 // is taken to have the decimal point after a single leading
1833 // If the value is zero, first_sig_digit is None.
1835 let mut any_digits = false;
1836 let mut dec_exp = 0i32;
1838 // Without leading or trailing zeros, irrespective of the dot.
1839 let mut first_sig_digit = None;
1840 let mut last_sig_digit = 0;
1841 let mut dot = s.len();
1843 for (p, c) in s.char_indices() {
1846 return Err(ParseError("String contains multiple dots"));
1849 } else if let Some(dec_value) = c.to_digit(10) {
1853 if first_sig_digit.is_none() {
1854 first_sig_digit = Some(p);
1858 } else if c == 'e' || c == 'E' {
1860 return Err(ParseError("Significand has no digits"));
1867 let mut chars = s[p + 1..].chars().peekable();
1869 // Adjust for the given exponent.
1870 let exp_minus = chars.peek() == Some(&'-');
1871 if exp_minus || chars.peek() == Some(&'+') {
1877 if let Some(value) = c.to_digit(10) {
1879 dec_exp = dec_exp.saturating_mul(10).saturating_add(value as i32);
1881 return Err(ParseError("Invalid character in exponent"));
1885 return Err(ParseError("Exponent has no digits"));
1894 return Err(ParseError("Invalid character in significand"));
1898 return Err(ParseError("Significand has no digits"));
1901 // Test if we have a zero number allowing for non-zero exponents.
1902 let first_sig_digit = match first_sig_digit {
1904 None => return Ok(Status::OK.and(Self::ZERO)),
1907 // Adjust the exponents for any decimal point.
1908 if dot > last_sig_digit {
1909 dec_exp = dec_exp.saturating_add((dot - last_sig_digit - 1) as i32);
1911 dec_exp = dec_exp.saturating_sub((last_sig_digit - dot) as i32);
1913 let significand_digits = last_sig_digit - first_sig_digit + 1 -
1914 (dot > first_sig_digit && dot < last_sig_digit) as usize;
1915 let normalized_exp = dec_exp.saturating_add(significand_digits as i32 - 1);
1917 // Handle the cases where exponents are obviously too large or too
1918 // small. Writing L for log 10 / log 2, a number d.ddddd*10^dec_exp
1919 // definitely overflows if
1921 // (dec_exp - 1) * L >= MAX_EXP
1923 // and definitely underflows to zero where
1925 // (dec_exp + 1) * L <= MIN_EXP - PRECISION
1927 // With integer arithmetic the tightest bounds for L are
1929 // 93/28 < L < 196/59 [ numerator <= 256 ]
1930 // 42039/12655 < L < 28738/8651 [ numerator <= 65536 ]
1932 // Check for MAX_EXP.
1933 if normalized_exp.saturating_sub(1).saturating_mul(42039) >= 12655 * S::MAX_EXP as i32 {
1934 // Overflow and round.
1935 return Ok(Self::overflow_result(round));
1938 // Check for MIN_EXP.
1939 if normalized_exp.saturating_add(1).saturating_mul(28738) <=
1940 8651 * (S::MIN_EXP as i32 - S::PRECISION as i32)
1942 // Underflow to zero and round.
1943 let r = if round == Round::TowardPositive {
1948 return Ok((Status::UNDERFLOW | Status::INEXACT).and(r));
1951 // A tight upper bound on number of bits required to hold an
1952 // N-digit decimal integer is N * 196 / 59. Allocate enough space
1953 // to hold the full significand, and an extra limb required by
1955 let max_limbs = limbs_for_bits(1 + 196 * significand_digits / 59);
1956 let mut dec_sig: SmallVec<[Limb; 1]> = SmallVec::with_capacity(max_limbs);
1958 // Convert to binary efficiently - we do almost all multiplication
1959 // in a Limb. When this would overflow do we do a single
1960 // bignum multiplication, and then revert again to multiplication
1962 let mut chars = s[first_sig_digit..=last_sig_digit].chars();
1965 let mut multiplier = 1;
1968 let dec_value = match chars.next() {
1969 Some('.') => continue,
1970 Some(c) => c.to_digit(10).unwrap(),
1975 val = val * 10 + dec_value as Limb;
1977 // The maximum number that can be multiplied by ten with any
1978 // digit added without overflowing a Limb.
1979 if multiplier > (!0 - 9) / 10 {
1984 // If we've consumed no digits, we're done.
1985 if multiplier == 1 {
1989 // Multiply out the current limb.
1990 let mut carry = val;
1991 for x in &mut dec_sig {
1992 let [low, mut high] = sig::widening_mul(*x, multiplier);
1995 let (low, overflow) = low.overflowing_add(carry);
1996 high += overflow as Limb;
2002 // If we had carry, we need another limb (likely but not guaranteed).
2004 dec_sig.push(carry);
2008 // Calculate pow(5, abs(dec_exp)) into `pow5_full`.
2009 // The *_calc Vec's are reused scratch space, as an optimization.
2010 let (pow5_full, mut pow5_calc, mut sig_calc, mut sig_scratch_calc) = {
2011 let mut power = dec_exp.abs() as usize;
2013 const FIRST_EIGHT_POWERS: [Limb; 8] = [1, 5, 25, 125, 625, 3125, 15625, 78125];
2015 let mut p5_scratch = smallvec![];
2016 let mut p5: SmallVec<[Limb; 1]> = smallvec![FIRST_EIGHT_POWERS[4]];
2018 let mut r_scratch = smallvec![];
2019 let mut r: SmallVec<[Limb; 1]> = smallvec![FIRST_EIGHT_POWERS[power & 7]];
2023 // Calculate pow(5,pow(2,n+3)).
2024 p5_scratch.resize(p5.len() * 2, 0);
2025 let _: Loss = sig::mul(&mut p5_scratch, &mut 0, &p5, &p5, p5.len() * 2 * LIMB_BITS);
2026 while p5_scratch.last() == Some(&0) {
2029 mem::swap(&mut p5, &mut p5_scratch);
2032 r_scratch.resize(r.len() + p5.len(), 0);
2033 let _: Loss = sig::mul(
2038 (r.len() + p5.len()) * LIMB_BITS,
2040 while r_scratch.last() == Some(&0) {
2043 mem::swap(&mut r, &mut r_scratch);
2049 (r, r_scratch, p5, p5_scratch)
2052 // Attempt dec_sig * 10^dec_exp with increasing precision.
2053 let mut attempt = 0;
2055 let calc_precision = (LIMB_BITS << attempt) - 1;
2058 let calc_normal_from_limbs = |sig: &mut SmallVec<[Limb; 1]>,
2060 -> StatusAnd<ExpInt> {
2061 sig.resize(limbs_for_bits(calc_precision), 0);
2062 let (mut loss, mut exp) = sig::from_limbs(sig, limbs, calc_precision);
2064 // Before rounding normalize the exponent of Category::Normal numbers.
2065 let mut omsb = sig::omsb(sig);
2067 assert_ne!(omsb, 0);
2069 // OMSB is numbered from 1. We want to place it in the integer
2070 // bit numbered PRECISION if possible, with a compensating change in
2072 let final_exp = exp.saturating_add(omsb as ExpInt - calc_precision as ExpInt);
2074 // Shifting left is easy as we don't lose precision.
2075 if final_exp < exp {
2076 assert_eq!(loss, Loss::ExactlyZero);
2078 let exp_change = (exp - final_exp) as usize;
2079 sig::shift_left(sig, &mut exp, exp_change);
2081 return Status::OK.and(exp);
2084 // Shift right and capture any new lost fraction.
2085 if final_exp > exp {
2086 let exp_change = (final_exp - exp) as usize;
2087 loss = sig::shift_right(sig, &mut exp, exp_change).combine(loss);
2089 // Keep OMSB up-to-date.
2090 omsb = omsb.saturating_sub(exp_change);
2093 assert_eq!(omsb, calc_precision);
2095 // Now round the number according to round given the lost
2098 // As specified in IEEE 754, since we do not trap we do not report
2099 // underflow for exact results.
2100 if loss == Loss::ExactlyZero {
2101 return Status::OK.and(exp);
2104 // Increment the significand if we're rounding away from zero.
2105 if loss == Loss::MoreThanHalf || loss == Loss::ExactlyHalf && sig::get_bit(sig, 0) {
2106 // We should never overflow.
2107 assert_eq!(sig::increment(sig), 0);
2108 omsb = sig::omsb(sig);
2110 // Did the significand increment overflow?
2111 if omsb == calc_precision + 1 {
2112 let _: Loss = sig::shift_right(sig, &mut exp, 1);
2114 return Status::INEXACT.and(exp);
2118 // The normal case - we were and are not denormal, and any
2119 // significand increment above didn't overflow.
2120 Status::INEXACT.and(exp)
2124 let mut exp = unpack!(status=,
2125 calc_normal_from_limbs(&mut sig_calc, &dec_sig));
2127 let pow5_exp = unpack!(pow5_status=,
2128 calc_normal_from_limbs(&mut pow5_calc, &pow5_full));
2130 // Add dec_exp, as 10^n = 5^n * 2^n.
2131 exp += dec_exp as ExpInt;
2133 let mut used_bits = S::PRECISION;
2134 let mut truncated_bits = calc_precision - used_bits;
2136 let half_ulp_err1 = (status != Status::OK) as Limb;
2137 let (calc_loss, half_ulp_err2);
2141 sig_scratch_calc.resize(sig_calc.len() + pow5_calc.len(), 0);
2142 calc_loss = sig::mul(
2143 &mut sig_scratch_calc,
2149 mem::swap(&mut sig_calc, &mut sig_scratch_calc);
2151 half_ulp_err2 = (pow5_status != Status::OK) as Limb;
2155 sig_scratch_calc.resize(sig_calc.len(), 0);
2156 calc_loss = sig::div(
2157 &mut sig_scratch_calc,
2163 mem::swap(&mut sig_calc, &mut sig_scratch_calc);
2165 // Denormal numbers have less precision.
2166 if exp < S::MIN_EXP {
2167 truncated_bits += (S::MIN_EXP - exp) as usize;
2168 used_bits = calc_precision.saturating_sub(truncated_bits);
2170 // Extra half-ulp lost in reciprocal of exponent.
2172 (pow5_status != Status::OK || calc_loss != Loss::ExactlyZero) as Limb;
2175 // Both sig::mul and sig::div return the
2176 // result with the integer bit set.
2177 assert!(sig::get_bit(&sig_calc, calc_precision - 1));
2179 // The error from the true value, in half-ulps, on multiplying two
2180 // floating point numbers, which differ from the value they
2181 // approximate by at most half_ulp_err1 and half_ulp_err2 half-ulps, is strictly less
2182 // than the returned value.
2184 // See "How to Read Floating Point Numbers Accurately" by William D Clinger.
2186 half_ulp_err1 < 2 || half_ulp_err2 < 2 || (half_ulp_err1 + half_ulp_err2 < 8)
2189 let inexact = (calc_loss != Loss::ExactlyZero) as Limb;
2190 let half_ulp_err = if half_ulp_err1 + half_ulp_err2 == 0 {
2191 inexact * 2 // <= inexact half-ulps.
2193 inexact + 2 * (half_ulp_err1 + half_ulp_err2)
2196 let ulps_from_boundary = {
2197 let bits = calc_precision - used_bits - 1;
2199 let i = bits / LIMB_BITS;
2200 let limb = sig_calc[i] & (!0 >> (LIMB_BITS - 1 - bits % LIMB_BITS));
2201 let boundary = match round {
2202 Round::NearestTiesToEven | Round::NearestTiesToAway => 1 << (bits % LIMB_BITS),
2206 let delta = limb.wrapping_sub(boundary);
2207 cmp::min(delta, delta.wrapping_neg())
2208 } else if limb == boundary {
2209 if !sig::is_all_zeros(&sig_calc[1..i]) {
2214 } else if limb == boundary.wrapping_sub(1) {
2215 if sig_calc[1..i].iter().any(|&x| x.wrapping_neg() != 1) {
2218 sig_calc[0].wrapping_neg()
2225 // Are we guaranteed to round correctly if we truncate?
2226 if ulps_from_boundary.saturating_mul(2) >= half_ulp_err {
2227 let mut r = IeeeFloat {
2230 category: Category::Normal,
2232 marker: PhantomData,
2234 sig::extract(&mut r.sig, &sig_calc, used_bits, calc_precision - used_bits);
2235 // If we extracted less bits above we must adjust our exponent
2236 // to compensate for the implicit right shift.
2237 r.exp += (S::PRECISION - used_bits) as ExpInt;
2238 let loss = Loss::through_truncation(&sig_calc, truncated_bits);
2239 return Ok(r.normalize(round, loss));
2246 /// Combine the effect of two lost fractions.
2247 fn combine(self, less_significant: Loss) -> Loss {
2248 let mut more_significant = self;
2249 if less_significant != Loss::ExactlyZero {
2250 if more_significant == Loss::ExactlyZero {
2251 more_significant = Loss::LessThanHalf;
2252 } else if more_significant == Loss::ExactlyHalf {
2253 more_significant = Loss::MoreThanHalf;
2260 /// Returns the fraction lost were a bignum truncated losing the least
2261 /// significant `bits` bits.
2262 fn through_truncation(limbs: &[Limb], bits: usize) -> Loss {
2264 return Loss::ExactlyZero;
2267 let half_bit = bits - 1;
2268 let half_limb = half_bit / LIMB_BITS;
2269 let (half_limb, rest) = if half_limb < limbs.len() {
2270 (limbs[half_limb], &limbs[..half_limb])
2274 let half = 1 << (half_bit % LIMB_BITS);
2275 let has_half = half_limb & half != 0;
2276 let has_rest = half_limb & (half - 1) != 0 || !sig::is_all_zeros(rest);
2278 match (has_half, has_rest) {
2279 (false, false) => Loss::ExactlyZero,
2280 (false, true) => Loss::LessThanHalf,
2281 (true, false) => Loss::ExactlyHalf,
2282 (true, true) => Loss::MoreThanHalf,
2287 /// Implementation details of IeeeFloat significands, such as big integer arithmetic.
2288 /// As a rule of thumb, no functions in this module should dynamically allocate.
2290 use std::cmp::Ordering;
2292 use super::{ExpInt, Limb, LIMB_BITS, limbs_for_bits, Loss};
2294 pub(super) fn is_all_zeros(limbs: &[Limb]) -> bool {
2295 limbs.iter().all(|&l| l == 0)
2298 /// One, not zero, based LSB. That is, returns 0 for a zeroed significand.
2299 pub(super) fn olsb(limbs: &[Limb]) -> usize {
2300 limbs.iter().enumerate().find(|(_, &limb)| limb != 0).map_or(0,
2301 |(i, limb)| i * LIMB_BITS + limb.trailing_zeros() as usize + 1)
2304 /// One, not zero, based MSB. That is, returns 0 for a zeroed significand.
2305 pub(super) fn omsb(limbs: &[Limb]) -> usize {
2306 limbs.iter().enumerate().rfind(|(_, &limb)| limb != 0).map_or(0,
2307 |(i, limb)| (i + 1) * LIMB_BITS - limb.leading_zeros() as usize)
2310 /// Comparison (unsigned) of two significands.
2311 pub(super) fn cmp(a: &[Limb], b: &[Limb]) -> Ordering {
2312 assert_eq!(a.len(), b.len());
2313 for (a, b) in a.iter().zip(b).rev() {
2315 Ordering::Equal => {}
2323 /// Extracts the given bit.
2324 pub(super) fn get_bit(limbs: &[Limb], bit: usize) -> bool {
2325 limbs[bit / LIMB_BITS] & (1 << (bit % LIMB_BITS)) != 0
2328 /// Sets the given bit.
2329 pub(super) fn set_bit(limbs: &mut [Limb], bit: usize) {
2330 limbs[bit / LIMB_BITS] |= 1 << (bit % LIMB_BITS);
2333 /// Clear the given bit.
2334 pub(super) fn clear_bit(limbs: &mut [Limb], bit: usize) {
2335 limbs[bit / LIMB_BITS] &= !(1 << (bit % LIMB_BITS));
2338 /// Shifts `dst` left `bits` bits, subtract `bits` from its exponent.
2339 pub(super) fn shift_left(dst: &mut [Limb], exp: &mut ExpInt, bits: usize) {
2341 // Our exponent should not underflow.
2342 *exp = exp.checked_sub(bits as ExpInt).unwrap();
2344 // Jump is the inter-limb jump; shift is the intra-limb shift.
2345 let jump = bits / LIMB_BITS;
2346 let shift = bits % LIMB_BITS;
2348 for i in (0..dst.len()).rev() {
2354 // dst[i] comes from the two limbs src[i - jump] and, if we have
2355 // an intra-limb shift, src[i - jump - 1].
2356 limb = dst[i - jump];
2360 limb |= dst[i - jump - 1] >> (LIMB_BITS - shift);
2370 /// Shifts `dst` right `bits` bits noting lost fraction.
2371 pub(super) fn shift_right(dst: &mut [Limb], exp: &mut ExpInt, bits: usize) -> Loss {
2372 let loss = Loss::through_truncation(dst, bits);
2375 // Our exponent should not overflow.
2376 *exp = exp.checked_add(bits as ExpInt).unwrap();
2378 // Jump is the inter-limb jump; shift is the intra-limb shift.
2379 let jump = bits / LIMB_BITS;
2380 let shift = bits % LIMB_BITS;
2382 // Perform the shift. This leaves the most significant `bits` bits
2383 // of the result at zero.
2384 for i in 0..dst.len() {
2387 if i + jump >= dst.len() {
2390 limb = dst[i + jump];
2393 if i + jump + 1 < dst.len() {
2394 limb |= dst[i + jump + 1] << (LIMB_BITS - shift);
2406 /// Copies the bit vector of width `src_bits` from `src`, starting at bit SRC_LSB,
2407 /// to `dst`, such that the bit SRC_LSB becomes the least significant bit of `dst`.
2408 /// All high bits above `src_bits` in `dst` are zero-filled.
2409 pub(super) fn extract(dst: &mut [Limb], src: &[Limb], src_bits: usize, src_lsb: usize) {
2414 let dst_limbs = limbs_for_bits(src_bits);
2415 assert!(dst_limbs <= dst.len());
2417 let src = &src[src_lsb / LIMB_BITS..];
2418 dst[..dst_limbs].copy_from_slice(&src[..dst_limbs]);
2420 let shift = src_lsb % LIMB_BITS;
2421 let _: Loss = shift_right(&mut dst[..dst_limbs], &mut 0, shift);
2423 // We now have (dst_limbs * LIMB_BITS - shift) bits from `src`
2424 // in `dst`. If this is less that src_bits, append the rest, else
2425 // clear the high bits.
2426 let n = dst_limbs * LIMB_BITS - shift;
2428 let mask = (1 << (src_bits - n)) - 1;
2429 dst[dst_limbs - 1] |= (src[dst_limbs] & mask) << (n % LIMB_BITS);
2430 } else if n > src_bits && src_bits % LIMB_BITS > 0 {
2431 dst[dst_limbs - 1] &= (1 << (src_bits % LIMB_BITS)) - 1;
2434 // Clear high limbs.
2435 for x in &mut dst[dst_limbs..] {
2440 /// We want the most significant PRECISION bits of `src`. There may not
2441 /// be that many; extract what we can.
2442 pub(super) fn from_limbs(dst: &mut [Limb], src: &[Limb], precision: usize) -> (Loss, ExpInt) {
2443 let omsb = omsb(src);
2445 if precision <= omsb {
2446 extract(dst, src, precision, omsb - precision);
2448 Loss::through_truncation(src, omsb - precision),
2452 extract(dst, src, omsb, 0);
2453 (Loss::ExactlyZero, precision as ExpInt - 1)
2457 /// For every consecutive chunk of `bits` bits from `limbs`,
2458 /// going from most significant to the least significant bits,
2459 /// call `f` to transform those bits and store the result back.
2460 pub(super) fn each_chunk<F: FnMut(Limb) -> Limb>(limbs: &mut [Limb], bits: usize, mut f: F) {
2461 assert_eq!(LIMB_BITS % bits, 0);
2462 for limb in limbs.iter_mut().rev() {
2464 for i in (0..LIMB_BITS / bits).rev() {
2465 r |= f((*limb >> (i * bits)) & ((1 << bits) - 1)) << (i * bits);
2471 /// Increment in-place, return the carry flag.
2472 pub(super) fn increment(dst: &mut [Limb]) -> Limb {
2474 *x = x.wrapping_add(1);
2483 /// Decrement in-place, return the borrow flag.
2484 pub(super) fn decrement(dst: &mut [Limb]) -> Limb {
2486 *x = x.wrapping_sub(1);
2495 /// `a += b + c` where `c` is zero or one. Returns the carry flag.
2496 pub(super) fn add(a: &mut [Limb], b: &[Limb], mut c: Limb) -> Limb {
2499 for (a, &b) in a.iter_mut().zip(b) {
2500 let (r, overflow) = a.overflowing_add(b);
2501 let (r, overflow2) = r.overflowing_add(c);
2503 c = (overflow | overflow2) as Limb;
2509 /// `a -= b + c` where `c` is zero or one. Returns the borrow flag.
2510 pub(super) fn sub(a: &mut [Limb], b: &[Limb], mut c: Limb) -> Limb {
2513 for (a, &b) in a.iter_mut().zip(b) {
2514 let (r, overflow) = a.overflowing_sub(b);
2515 let (r, overflow2) = r.overflowing_sub(c);
2517 c = (overflow | overflow2) as Limb;
2523 /// `a += b` or `a -= b`. Does not preserve `b`.
2524 pub(super) fn add_or_sub(
2532 // Are we bigger exponent-wise than the RHS?
2533 let bits = *a_exp - b_exp;
2535 // Determine if the operation on the absolute values is effectively
2536 // an addition or subtraction.
2537 // Subtraction is more subtle than one might naively expect.
2538 if *a_sign ^ b_sign {
2539 let (reverse, loss);
2542 reverse = cmp(a_sig, b_sig) == Ordering::Less;
2543 loss = Loss::ExactlyZero;
2544 } else if bits > 0 {
2545 loss = shift_right(b_sig, &mut 0, (bits - 1) as usize);
2546 shift_left(a_sig, a_exp, 1);
2549 loss = shift_right(a_sig, a_exp, (-bits - 1) as usize);
2550 shift_left(b_sig, &mut 0, 1);
2554 let borrow = (loss != Loss::ExactlyZero) as Limb;
2556 // The code above is intended to ensure that no borrow is necessary.
2557 assert_eq!(sub(b_sig, a_sig, borrow), 0);
2558 a_sig.copy_from_slice(b_sig);
2561 // The code above is intended to ensure that no borrow is necessary.
2562 assert_eq!(sub(a_sig, b_sig, borrow), 0);
2565 // Invert the lost fraction - it was on the RHS and subtracted.
2567 Loss::LessThanHalf => Loss::MoreThanHalf,
2568 Loss::MoreThanHalf => Loss::LessThanHalf,
2572 let loss = if bits > 0 {
2573 shift_right(b_sig, &mut 0, bits as usize)
2575 shift_right(a_sig, a_exp, -bits as usize)
2577 // We have a guard bit; generating a carry cannot happen.
2578 assert_eq!(add(a_sig, b_sig, 0), 0);
2583 /// `[low, high] = a * b`.
2585 /// This cannot overflow, because
2587 /// `(n - 1) * (n - 1) + 2 * (n - 1) == (n - 1) * (n + 1)`
2589 /// which is less than n<sup>2</sup>.
2590 pub(super) fn widening_mul(a: Limb, b: Limb) -> [Limb; 2] {
2591 let mut wide = [0, 0];
2593 if a == 0 || b == 0 {
2597 const HALF_BITS: usize = LIMB_BITS / 2;
2599 let select = |limb, i| (limb >> (i * HALF_BITS)) & ((1 << HALF_BITS) - 1);
2602 let mut x = [select(a, i) * select(b, j), 0];
2603 shift_left(&mut x, &mut 0, (i + j) * HALF_BITS);
2604 assert_eq!(add(&mut wide, &x, 0), 0);
2611 /// `dst = a * b` (for normal `a` and `b`). Returns the lost fraction.
2612 pub(super) fn mul<'a>(
2619 // Put the narrower number on the `a` for less loops below.
2620 if a.len() > b.len() {
2621 mem::swap(&mut a, &mut b);
2624 for x in &mut dst[..b.len()] {
2628 for i in 0..a.len() {
2630 for j in 0..b.len() {
2631 let [low, mut high] = widening_mul(a[i], b[j]);
2634 let (low, overflow) = low.overflowing_add(carry);
2635 high += overflow as Limb;
2637 // And now `dst[i + j]`, and store the new low part there.
2638 let (low, overflow) = low.overflowing_add(dst[i + j]);
2639 high += overflow as Limb;
2644 dst[i + b.len()] = carry;
2647 // Assume the operands involved in the multiplication are single-precision
2648 // FP, and the two multiplicants are:
2649 // a = a23 . a22 ... a0 * 2^e1
2650 // b = b23 . b22 ... b0 * 2^e2
2651 // the result of multiplication is:
2652 // dst = c48 c47 c46 . c45 ... c0 * 2^(e1+e2)
2653 // Note that there are three significant bits at the left-hand side of the
2654 // radix point: two for the multiplication, and an overflow bit for the
2655 // addition (that will always be zero at this point). Move the radix point
2656 // toward left by two bits, and adjust exponent accordingly.
2659 // Convert the result having "2 * precision" significant-bits back to the one
2660 // having "precision" significant-bits. First, move the radix point from
2661 // poision "2*precision - 1" to "precision - 1". The exponent need to be
2662 // adjusted by "2*precision - 1" - "precision - 1" = "precision".
2663 *exp -= precision as ExpInt + 1;
2665 // In case MSB resides at the left-hand side of radix point, shift the
2666 // mantissa right by some amount to make sure the MSB reside right before
2667 // the radix point (i.e., "MSB . rest-significant-bits").
2669 // Note that the result is not normalized when "omsb < precision". So, the
2670 // caller needs to call IeeeFloat::normalize() if normalized value is
2672 let omsb = omsb(dst);
2673 if omsb <= precision {
2676 shift_right(dst, exp, omsb - precision)
2680 /// `quotient = dividend / divisor`. Returns the lost fraction.
2681 /// Does not preserve `dividend` or `divisor`.
2683 quotient: &mut [Limb],
2685 dividend: &mut [Limb],
2686 divisor: &mut [Limb],
2690 // Normalize the divisor.
2691 let bits = precision - omsb(divisor);
2692 shift_left(divisor, &mut 0, bits);
2693 *exp += bits as ExpInt;
2695 // Normalize the dividend.
2696 let bits = precision - omsb(dividend);
2697 shift_left(dividend, exp, bits);
2700 let olsb_divisor = olsb(divisor);
2701 if olsb_divisor == precision {
2702 quotient.copy_from_slice(dividend);
2703 return Loss::ExactlyZero;
2706 // Ensure the dividend >= divisor initially for the loop below.
2707 // Incidentally, this means that the division loop below is
2708 // guaranteed to set the integer bit to one.
2709 if cmp(dividend, divisor) == Ordering::Less {
2710 shift_left(dividend, exp, 1);
2711 assert_ne!(cmp(dividend, divisor), Ordering::Less)
2714 // Helper for figuring out the lost fraction.
2715 let lost_fraction = |dividend: &[Limb], divisor: &[Limb]| {
2716 match cmp(dividend, divisor) {
2717 Ordering::Greater => Loss::MoreThanHalf,
2718 Ordering::Equal => Loss::ExactlyHalf,
2720 if is_all_zeros(dividend) {
2729 // Try to perform a (much faster) short division for small divisors.
2730 let divisor_bits = precision - (olsb_divisor - 1);
2731 macro_rules! try_short_div {
2732 ($W:ty, $H:ty, $half:expr) => {
2733 if divisor_bits * 2 <= $half {
2734 // Extract the small divisor.
2735 let _: Loss = shift_right(divisor, &mut 0, olsb_divisor - 1);
2736 let divisor = divisor[0] as $H as $W;
2738 // Shift the dividend to produce a quotient with the unit bit set.
2739 let top_limb = *dividend.last().unwrap();
2740 let mut rem = (top_limb >> (LIMB_BITS - (divisor_bits - 1))) as $H;
2741 shift_left(dividend, &mut 0, divisor_bits - 1);
2743 // Apply short division in place on $H (of $half bits) chunks.
2744 each_chunk(dividend, $half, |chunk| {
2745 let chunk = chunk as $H;
2746 let combined = ((rem as $W) << $half) | (chunk as $W);
2747 rem = (combined % divisor) as $H;
2748 (combined / divisor) as $H as Limb
2750 quotient.copy_from_slice(dividend);
2752 return lost_fraction(&[(rem as Limb) << 1], &[divisor as Limb]);
2757 try_short_div!(u32, u16, 16);
2758 try_short_div!(u64, u32, 32);
2759 try_short_div!(u128, u64, 64);
2761 // Zero the quotient before setting bits in it.
2762 for x in &mut quotient[..limbs_for_bits(precision)] {
2767 for bit in (0..precision).rev() {
2768 if cmp(dividend, divisor) != Ordering::Less {
2769 sub(dividend, divisor, 0);
2770 set_bit(quotient, bit);
2772 shift_left(dividend, &mut 0, 1);
2775 lost_fraction(dividend, divisor)