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<sup>exp</sup> * (1 + sum(frac[N-i] / 2<sup>i</sup>)) where N is the number of bits.
14 //! Subnormals are 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 *
17 //! 2<sup>e</sup>. Besides making the "hidden bit" explicit, this changes the exponent by the
18 //! so-called mantissa shift.
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.
31 use cmp::Ordering::{Less, Equal, Greater};
32 use ops::{Mul, Div, Neg};
33 use fmt::{Debug, LowerExp};
35 use num::diy_float::Fp;
36 use num::FpCategory::{Infinite, Zero, Subnormal, Normal, Nan};
38 use num::dec2flt::num::{self, Big};
39 use num::dec2flt::table;
41 #[derive(Copy, Clone, Debug)]
48 pub fn new(sig: u64, k: i16) -> Self {
49 Unpacked { sig: sig, k: k }
53 /// A helper trait to avoid duplicating basically all the conversion code for `f32` and `f64`.
55 /// See the parent module's doc comment for why this is necessary.
57 /// Should **never ever** be implemented for other types or be used outside the dec2flt module.
58 /// Inherits from `Float` because there is some overlap, but all the reused methods are trivial.
59 pub trait RawFloat : Float + Copy + Debug + LowerExp
60 + Mul<Output=Self> + Div<Output=Self> + Neg<Output=Self>
66 // suffix of "2" because Float::integer_decode is deprecated
68 fn integer_decode2(self) -> (u64, i16, i8) {
69 Float::integer_decode(self)
72 /// Get the raw binary representation of the float.
73 fn transmute(self) -> u64;
75 /// Transmute the raw binary representation into a float.
76 fn from_bits(bits: u64) -> Self;
79 fn unpack(self) -> Unpacked;
81 /// Cast from a small integer that can be represented exactly. Panic if the integer can't be
82 /// represented, the other code in this module makes sure to never let that happen.
83 fn from_int(x: u64) -> Self;
85 /// Get the value 10<sup>e</sup> from a pre-computed table.
86 /// Panics for `e >= CEIL_LOG5_OF_MAX_SIG`.
87 fn short_fast_pow10(e: usize) -> Self;
89 /// What the name says. It's easier to hard code than juggling intrinsics and
90 /// hoping LLVM constant folds it.
91 const CEIL_LOG5_OF_MAX_SIG: i16;
93 // A conservative bound on the decimal digits of inputs that can't produce overflow or zero or
94 /// subnormals. Probably the decimal exponent of the maximum normal value, hence the name.
95 const MAX_NORMAL_DIGITS: usize;
97 /// When the most significant decimal digit has a place value greater than this, the number
98 /// is certainly rounded to infinity.
99 const INF_CUTOFF: i64;
101 /// When the most significant decimal digit has a place value less than this, the number
102 /// is certainly rounded to zero.
103 const ZERO_CUTOFF: i64;
105 /// The number of bits in the exponent.
108 /// The number of bits in the singificand, *including* the hidden bit.
111 /// The number of bits in the singificand, *excluding* the hidden bit.
112 const EXPLICIT_SIG_BITS: u8;
114 /// The maximum legal exponent in fractional representation.
117 /// The minimum legal exponent in fractional representation, excluding subnormals.
120 /// `MAX_EXP` for integral representation, i.e., with the shift applied.
121 const MAX_EXP_INT: i16;
123 /// `MAX_EXP` encoded (i.e., with offset bias)
124 const MAX_ENCODED_EXP: i16;
126 /// `MIN_EXP` for integral representation, i.e., with the shift applied.
127 const MIN_EXP_INT: i16;
129 /// The maximum normalized singificand in integral representation.
132 /// The minimal normalized significand in integral representation.
136 // Mostly a workaround for #34344.
137 macro_rules! other_constants {
139 const EXPLICIT_SIG_BITS: u8 = Self::SIG_BITS - 1;
140 const MAX_EXP: i16 = (1 << (Self::EXP_BITS - 1)) - 1;
141 const MIN_EXP: i16 = -Self::MAX_EXP + 1;
142 const MAX_EXP_INT: i16 = Self::MAX_EXP - (Self::SIG_BITS as i16 - 1);
143 const MAX_ENCODED_EXP: i16 = (1 << Self::EXP_BITS) - 1;
144 const MIN_EXP_INT: i16 = Self::MIN_EXP - (Self::SIG_BITS as i16 - 1);
145 const MAX_SIG: u64 = (1 << Self::SIG_BITS) - 1;
146 const MIN_SIG: u64 = 1 << (Self::SIG_BITS - 1);
148 const INFINITY: Self = $crate::$type::INFINITY;
149 const NAN: Self = $crate::$type::NAN;
150 const ZERO: Self = 0.0;
154 impl RawFloat for f32 {
155 const SIG_BITS: u8 = 24;
156 const EXP_BITS: u8 = 8;
157 const CEIL_LOG5_OF_MAX_SIG: i16 = 11;
158 const MAX_NORMAL_DIGITS: usize = 35;
159 const INF_CUTOFF: i64 = 40;
160 const ZERO_CUTOFF: i64 = -48;
161 other_constants!(f32);
163 fn transmute(self) -> u64 {
164 let bits: u32 = unsafe { transmute(self) };
168 fn from_bits(bits: u64) -> f32 {
169 assert!(bits < u32::MAX as u64, "f32::from_bits: too many bits");
170 unsafe { transmute(bits as u32) }
173 fn unpack(self) -> Unpacked {
174 let (sig, exp, _sig) = self.integer_decode2();
175 Unpacked::new(sig, exp)
178 fn from_int(x: u64) -> f32 {
179 // rkruppe is uncertain whether `as` rounds correctly on all platforms.
180 debug_assert!(x as f32 == fp_to_float(Fp { f: x, e: 0 }));
184 fn short_fast_pow10(e: usize) -> Self {
185 table::F32_SHORT_POWERS[e]
190 impl RawFloat for f64 {
191 const SIG_BITS: u8 = 53;
192 const EXP_BITS: u8 = 11;
193 const CEIL_LOG5_OF_MAX_SIG: i16 = 23;
194 const MAX_NORMAL_DIGITS: usize = 305;
195 const INF_CUTOFF: i64 = 310;
196 const ZERO_CUTOFF: i64 = -326;
197 other_constants!(f64);
199 fn transmute(self) -> u64 {
200 let bits: u64 = unsafe { transmute(self) };
204 fn from_bits(bits: u64) -> f64 {
205 unsafe { transmute(bits) }
208 fn unpack(self) -> Unpacked {
209 let (sig, exp, _sig) = self.integer_decode2();
210 Unpacked::new(sig, exp)
213 fn from_int(x: u64) -> f64 {
214 // rkruppe is uncertain whether `as` rounds correctly on all platforms.
215 debug_assert!(x as f64 == fp_to_float(Fp { f: x, e: 0 }));
219 fn short_fast_pow10(e: usize) -> Self {
220 table::F64_SHORT_POWERS[e]
224 /// Convert an Fp to the closest machine float type.
225 /// Does not handle subnormal results.
226 pub fn fp_to_float<T: RawFloat>(x: Fp) -> T {
227 let x = x.normalize();
228 // x.f is 64 bit, so x.e has a mantissa shift of 63
231 panic!("fp_to_float: exponent {} too large", e)
232 } else if e > T::MIN_EXP {
233 encode_normal(round_normal::<T>(x))
235 panic!("fp_to_float: exponent {} too small", e)
239 /// Round the 64-bit significand to T::SIG_BITS bits with half-to-even.
240 /// Does not handle exponent overflow.
241 pub fn round_normal<T: RawFloat>(x: Fp) -> Unpacked {
242 let excess = 64 - T::SIG_BITS as i16;
243 let half: u64 = 1 << (excess - 1);
244 let (q, rem) = (x.f >> excess, x.f & ((1 << excess) - 1));
245 assert_eq!(q << excess | rem, x.f);
246 // Adjust mantissa shift
247 let k = x.e + excess;
250 } else if rem == half && (q % 2) == 0 {
252 } else if q == T::MAX_SIG {
253 Unpacked::new(T::MIN_SIG, k + 1)
255 Unpacked::new(q + 1, k)
259 /// Inverse of `RawFloat::unpack()` for normalized numbers.
260 /// Panics if the significand or exponent are not valid for normalized numbers.
261 pub fn encode_normal<T: RawFloat>(x: Unpacked) -> T {
262 debug_assert!(T::MIN_SIG <= x.sig && x.sig <= T::MAX_SIG,
263 "encode_normal: significand not normalized");
264 // Remove the hidden bit
265 let sig_enc = x.sig & !(1 << T::EXPLICIT_SIG_BITS);
266 // Adjust the exponent for exponent bias and mantissa shift
267 let k_enc = x.k + T::MAX_EXP + T::EXPLICIT_SIG_BITS as i16;
268 debug_assert!(k_enc != 0 && k_enc < T::MAX_ENCODED_EXP,
269 "encode_normal: exponent out of range");
270 // Leave sign bit at 0 ("+"), our numbers are all positive
271 let bits = (k_enc as u64) << T::EXPLICIT_SIG_BITS | sig_enc;
275 /// Construct a subnormal. A mantissa of 0 is allowed and constructs zero.
276 pub fn encode_subnormal<T: RawFloat>(significand: u64) -> T {
277 assert!(significand < T::MIN_SIG, "encode_subnormal: not actually subnormal");
278 // Encoded exponent is 0, the sign bit is 0, so we just have to reinterpret the bits.
279 T::from_bits(significand)
282 /// Approximate a bignum with an Fp. Rounds within 0.5 ULP with half-to-even.
283 pub fn big_to_fp(f: &Big) -> Fp {
284 let end = f.bit_length();
285 assert!(end != 0, "big_to_fp: unexpectedly, input is zero");
286 let start = end.saturating_sub(64);
287 let leading = num::get_bits(f, start, end);
288 // We cut off all bits prior to the index `start`, i.e., we effectively right-shift by
289 // an amount of `start`, so this is also the exponent we need.
290 let e = start as i16;
291 let rounded_down = Fp { f: leading, e: e }.normalize();
292 // Round (half-to-even) depending on the truncated bits.
293 match num::compare_with_half_ulp(f, start) {
294 Less => rounded_down,
295 Equal if leading % 2 == 0 => rounded_down,
296 Equal | Greater => match leading.checked_add(1) {
297 Some(f) => Fp { f: f, e: e }.normalize(),
298 None => Fp { f: 1 << 63, e: e + 1 },
303 /// Find the largest floating point number strictly smaller than the argument.
304 /// Does not handle subnormals, zero, or exponent underflow.
305 pub fn prev_float<T: RawFloat>(x: T) -> T {
307 Infinite => panic!("prev_float: argument is infinite"),
308 Nan => panic!("prev_float: argument is NaN"),
309 Subnormal => panic!("prev_float: argument is subnormal"),
310 Zero => panic!("prev_float: argument is zero"),
312 let Unpacked { sig, k } = x.unpack();
313 if sig == T::MIN_SIG {
314 encode_normal(Unpacked::new(T::MAX_SIG, k - 1))
316 encode_normal(Unpacked::new(sig - 1, k))
322 // Find the smallest floating point number strictly larger than the argument.
323 // This operation is saturating, i.e. next_float(inf) == inf.
324 // Unlike most code in this module, this function does handle zero, subnormals, and infinities.
325 // However, like all other code here, it does not deal with NaN and negative numbers.
326 pub fn next_float<T: RawFloat>(x: T) -> T {
328 Nan => panic!("next_float: argument is NaN"),
329 Infinite => T::INFINITY,
330 // This seems too good to be true, but it works.
331 // 0.0 is encoded as the all-zero word. Subnormals are 0x000m...m where m is the mantissa.
332 // In particular, the smallest subnormal is 0x0...01 and the largest is 0x000F...F.
333 // The smallest normal number is 0x0010...0, so this corner case works as well.
334 // If the increment overflows the mantissa, the carry bit increments the exponent as we
335 // want, and the mantissa bits become zero. Because of the hidden bit convention, this
336 // too is exactly what we want!
337 // Finally, f64::MAX + 1 = 7eff...f + 1 = 7ff0...0 = f64::INFINITY.
338 Zero | Subnormal | Normal => {
339 let bits: u64 = x.transmute();
340 T::from_bits(bits + 1)