1 // Copyright 2015 The Rust Project Developers. See the COPYRIGHT
2 // file at the top-level directory of this distribution and at
3 // http://rust-lang.org/COPYRIGHT.
5 // Licensed under the Apache License, Version 2.0 <LICENSE-APACHE or
6 // http://www.apache.org/licenses/LICENSE-2.0> or the MIT license
7 // <LICENSE-MIT or http://opensource.org/licenses/MIT>, at your
8 // option. This file may not be copied, modified, or distributed
9 // except according to those terms.
11 //! Bit fiddling on positive IEEE 754 floats. Negative numbers aren't and needn't be handled.
12 //! Normal floating point numbers have a canonical representation as (frac, exp) such that the
13 //! value is 2^exp * (1 + sum(frac[N-i] / 2^i)) where N is the number of bits. Subnormals are
14 //! slightly different and weird, but the same principle applies.
16 //! Here, however, we represent them as (sig, k) with f positive, such that the value is f * 2^e.
17 //! Besides making the "hidden bit" explicit, this changes the exponent by the so-called
20 //! Put another way, normally floats are written as (1) but here they are written as (2):
22 //! 1. `1.101100...11 * 2^m`
23 //! 2. `1101100...11 * 2^n`
25 //! We call (1) the **fractional representation** and (2) the **integral representation**.
27 //! Many functions in this module only handle normal numbers. The dec2flt routines conservatively
28 //! take the universally-correct slow path (Algorithm M) for very small and very large numbers.
29 //! That algorithm needs only next_float() which does handle subnormals and zeros.
32 use cmp::Ordering::{Less, Equal, Greater};
33 use ops::{Mul, Div, Neg};
34 use fmt::{Debug, LowerExp};
36 use num::diy_float::Fp;
37 use num::FpCategory::{Infinite, Zero, Subnormal, Normal, Nan};
39 use num::dec2flt::num::{self, Big};
40 use num::dec2flt::table;
42 #[derive(Copy, Clone, Debug)]
49 pub fn new(sig: u64, k: i16) -> Self {
50 Unpacked { sig: sig, k: k }
54 /// A helper trait to avoid duplicating basically all the conversion code for `f32` and `f64`.
56 /// See the parent module's doc comment for why this is necessary.
58 /// Should **never ever** be implemented for other types or be used outside the dec2flt module.
59 /// Inherits from `Float` because there is some overlap, but all the reused methods are trivial.
60 /// The "methods" (pseudo-constants) with default implementation should not be overriden.
61 pub trait RawFloat : Float + Copy + Debug + LowerExp
62 + Mul<Output=Self> + Div<Output=Self> + Neg<Output=Self>
64 /// Get the raw binary representation of the float.
65 fn transmute(self) -> u64;
67 /// Transmute the raw binary representation into a float.
68 fn from_bits(bits: u64) -> Self;
71 fn unpack(self) -> Unpacked;
73 /// Cast from a small integer that can be represented exactly. Panic if the integer can't be
74 /// represented, the other code in this module makes sure to never let that happen.
75 fn from_int(x: u64) -> Self;
77 /// Get the value 10^e from a pre-computed table. Panics for e >= ceil_log5_of_max_sig().
78 fn short_fast_pow10(e: usize) -> Self;
80 // FIXME Everything that follows should be associated constants, but taking the value of an
81 // associated constant from a type parameter does not work (yet?)
82 // A possible workaround is having a `FloatInfo` struct for all the constants, but so far
83 // the methods aren't painful enough to rewrite.
85 /// What the name says. It's easier to hard code than juggling intrinsics and
86 /// hoping LLVM constant folds it.
87 fn ceil_log5_of_max_sig() -> i16;
89 // A conservative bound on the decimal digits of inputs that can't produce overflow or zero or
90 /// subnormals. Probably the decimal exponent of the maximum normal value, hence the name.
91 fn max_normal_digits() -> usize;
93 /// When the most significant decimal digit has a place value greater than this, the number
94 /// is certainly rounded to infinity.
95 fn inf_cutoff() -> i64;
97 /// When the most significant decimal digit has a place value less than this, the number
98 /// is certainly rounded to zero.
99 fn zero_cutoff() -> i64;
101 /// The number of bits in the exponent.
104 /// The number of bits in the singificand, *including* the hidden bit.
107 /// The number of bits in the singificand, *excluding* the hidden bit.
108 fn explicit_sig_bits() -> u8 {
112 /// The maximum legal exponent in fractional representation.
113 fn max_exp() -> i16 {
114 (1 << (Self::exp_bits() - 1)) - 1
117 /// The minimum legal exponent in fractional representation, excluding subnormals.
118 fn min_exp() -> i16 {
122 /// `MAX_EXP` for integral representation, i.e., with the shift applied.
123 fn max_exp_int() -> i16 {
124 Self::max_exp() - (Self::sig_bits() as i16 - 1)
127 /// `MAX_EXP` encoded (i.e., with offset bias)
128 fn max_encoded_exp() -> i16 {
129 (1 << Self::exp_bits()) - 1
132 /// `MIN_EXP` for integral representation, i.e., with the shift applied.
133 fn min_exp_int() -> i16 {
134 Self::min_exp() - (Self::sig_bits() as i16 - 1)
137 /// The maximum normalized singificand in integral representation.
138 fn max_sig() -> u64 {
139 (1 << Self::sig_bits()) - 1
142 /// The minimal normalized significand in integral representation.
143 fn min_sig() -> u64 {
144 1 << (Self::sig_bits() - 1)
148 impl RawFloat for f32 {
149 fn sig_bits() -> u8 {
153 fn exp_bits() -> u8 {
157 fn ceil_log5_of_max_sig() -> i16 {
161 fn transmute(self) -> u64 {
162 let bits: u32 = unsafe { transmute(self) };
166 fn from_bits(bits: u64) -> f32 {
167 assert!(bits < u32::MAX as u64, "f32::from_bits: too many bits");
168 unsafe { transmute(bits as u32) }
171 fn unpack(self) -> Unpacked {
172 let (sig, exp, _sig) = self.integer_decode();
173 Unpacked::new(sig, exp)
176 fn from_int(x: u64) -> f32 {
177 // rkruppe is uncertain whether `as` rounds correctly on all platforms.
178 debug_assert!(x as f32 == fp_to_float(Fp { f: x, e: 0 }));
182 fn short_fast_pow10(e: usize) -> Self {
183 table::F32_SHORT_POWERS[e]
186 fn max_normal_digits() -> usize {
190 fn inf_cutoff() -> i64 {
194 fn zero_cutoff() -> i64 {
200 impl RawFloat for f64 {
201 fn sig_bits() -> u8 {
205 fn exp_bits() -> u8 {
209 fn ceil_log5_of_max_sig() -> i16 {
213 fn transmute(self) -> u64 {
214 let bits: u64 = unsafe { transmute(self) };
218 fn from_bits(bits: u64) -> f64 {
219 unsafe { transmute(bits) }
222 fn unpack(self) -> Unpacked {
223 let (sig, exp, _sig) = self.integer_decode();
224 Unpacked::new(sig, exp)
227 fn from_int(x: u64) -> f64 {
228 // rkruppe is uncertain whether `as` rounds correctly on all platforms.
229 debug_assert!(x as f64 == fp_to_float(Fp { f: x, e: 0 }));
233 fn short_fast_pow10(e: usize) -> Self {
234 table::F64_SHORT_POWERS[e]
237 fn max_normal_digits() -> usize {
241 fn inf_cutoff() -> i64 {
245 fn zero_cutoff() -> i64 {
251 /// Convert an Fp to the closest f64. Only handles number that fit into a normalized f64.
252 pub fn fp_to_float<T: RawFloat>(x: Fp) -> T {
253 let x = x.normalize();
254 // x.f is 64 bit, so x.e has a mantissa shift of 63
256 if e > T::max_exp() {
257 panic!("fp_to_float: exponent {} too large", e)
258 } else if e > T::min_exp() {
259 encode_normal(round_normal::<T>(x))
261 panic!("fp_to_float: exponent {} too small", e)
265 /// Round the 64-bit significand to 53 bit with half-to-even. Does not handle exponent overflow.
266 pub fn round_normal<T: RawFloat>(x: Fp) -> Unpacked {
267 let excess = 64 - T::sig_bits() as i16;
268 let half: u64 = 1 << (excess - 1);
269 let (q, rem) = (x.f >> excess, x.f & ((1 << excess) - 1));
270 assert_eq!(q << excess | rem, x.f);
271 // Adjust mantissa shift
272 let k = x.e + excess;
275 } else if rem == half && (q % 2) == 0 {
277 } else if q == T::max_sig() {
278 Unpacked::new(T::min_sig(), k + 1)
280 Unpacked::new(q + 1, k)
284 /// Inverse of `RawFloat::unpack()` for normalized numbers.
285 /// Panics if the significand or exponent are not valid for normalized numbers.
286 pub fn encode_normal<T: RawFloat>(x: Unpacked) -> T {
287 debug_assert!(T::min_sig() <= x.sig && x.sig <= T::max_sig(),
288 "encode_normal: significand not normalized");
289 // Remove the hidden bit
290 let sig_enc = x.sig & !(1 << T::explicit_sig_bits());
291 // Adjust the exponent for exponent bias and mantissa shift
292 let k_enc = x.k + T::max_exp() + T::explicit_sig_bits() as i16;
293 debug_assert!(k_enc != 0 && k_enc < T::max_encoded_exp(),
294 "encode_normal: exponent out of range");
295 // Leave sign bit at 0 ("+"), our numbers are all positive
296 let bits = (k_enc as u64) << T::explicit_sig_bits() | sig_enc;
300 /// Construct the subnormal. A mantissa of 0 is allowed and constructs zero.
301 pub fn encode_subnormal<T: RawFloat>(significand: u64) -> T {
302 assert!(significand < T::min_sig(), "encode_subnormal: not actually subnormal");
303 // Encoded exponent is 0, the sign bit is 0, so we just have to reinterpret the bits.
304 T::from_bits(significand)
307 /// Approximate a bignum with an Fp. Rounds within 0.5 ULP with half-to-even.
308 pub fn big_to_fp(f: &Big) -> Fp {
309 let end = f.bit_length();
310 assert!(end != 0, "big_to_fp: unexpectedly, input is zero");
311 let start = end.saturating_sub(64);
312 let leading = num::get_bits(f, start, end);
313 // We cut off all bits prior to the index `start`, i.e., we effectively right-shift by
314 // an amount of `start`, so this is also the exponent we need.
315 let e = start as i16;
316 let rounded_down = Fp { f: leading, e: e }.normalize();
317 // Round (half-to-even) depending on the truncated bits.
318 match num::compare_with_half_ulp(f, start) {
319 Less => rounded_down,
320 Equal if leading % 2 == 0 => rounded_down,
321 Equal | Greater => match leading.checked_add(1) {
322 Some(f) => Fp { f: f, e: e }.normalize(),
323 None => Fp { f: 1 << 63, e: e + 1 },
328 /// Find the largest floating point number strictly smaller than the argument.
329 /// Does not handle subnormals, zero, or exponent underflow.
330 pub fn prev_float<T: RawFloat>(x: T) -> T {
332 Infinite => panic!("prev_float: argument is infinite"),
333 Nan => panic!("prev_float: argument is NaN"),
334 Subnormal => panic!("prev_float: argument is subnormal"),
335 Zero => panic!("prev_float: argument is zero"),
337 let Unpacked { sig, k } = x.unpack();
338 if sig == T::min_sig() {
339 encode_normal(Unpacked::new(T::max_sig(), k - 1))
341 encode_normal(Unpacked::new(sig - 1, k))
347 // Find the smallest floating point number strictly larger than the argument.
348 // This operation is saturating, i.e. next_float(inf) == inf.
349 // Unlike most code in this module, this function does handle zero, subnormals, and infinities.
350 // However, like all other code here, it does not deal with NaN and negative numbers.
351 pub fn next_float<T: RawFloat>(x: T) -> T {
353 Nan => panic!("next_float: argument is NaN"),
354 Infinite => T::infinity(),
355 // This seems too good to be true, but it works.
356 // 0.0 is encoded as the all-zero word. Subnormals are 0x000m...m where m is the mantissa.
357 // In particular, the smallest subnormal is 0x0...01 and the largest is 0x000F...F.
358 // The smallest normal number is 0x0010...0, so this corner case works as well.
359 // If the increment overflows the mantissa, the carry bit increments the exponent as we
360 // want, and the mantissa bits become zero. Because of the hidden bit convention, this
361 // too is exactly what we want!
362 // Finally, f64::MAX + 1 = 7eff...f + 1 = 7ff0...0 = f64::INFINITY.
363 Zero | Subnormal | Normal => {
364 let bits: u64 = x.transmute();
365 T::from_bits(bits + 1)