src/libcore/num/dec2flt/rawfp.rs

   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.
   4 //
   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.
  10
  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.
  15 //!
  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
  18 //! mantissa shift.
  19 //!
  20 //! Put another way, normally floats are written as (1) but here they are written as (2):
  21 //!
  22 //! 1. `1.101100...11 * 2^m`
  23 //! 2. `1101100...11 * 2^n`
  24 //!
  25 //! We call (1) the **fractional representation** and (2) the **integral representation**.
  26 //!
  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.
  30 use u32;
  31 use cmp::Ordering::{Less, Equal, Greater};
  32 use ops::{Mul, Div, Neg};
  33 use fmt::{Debug, LowerExp};
  34 use mem::transmute;
  35 use num::diy_float::Fp;
  36 use num::FpCategory::{Infinite, Zero, Subnormal, Normal, Nan};
  37 use num::Float;
  38 use num::dec2flt::num::{self, Big};
  39 use num::dec2flt::table;
  40
  41 #[derive(Copy, Clone, Debug)]
  42 pub struct Unpacked {
  43     pub sig: u64,
  44     pub k: i16,
  45 }
  46
  47 impl Unpacked {
  48     pub fn new(sig: u64, k: i16) -> Self {
  49         Unpacked { sig: sig, k: k }
  50     }
  51 }
  52
  53 /// A helper trait to avoid duplicating basically all the conversion code for `f32` and `f64`.
  54 ///
  55 /// See the parent module's doc comment for why this is necessary.
  56 ///
  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 /// The "methods" (pseudo-constants) with default implementation should not be overriden.
  60 pub trait RawFloat : Float + Copy + Debug + LowerExp
  61                     + Mul<Output=Self> + Div<Output=Self> + Neg<Output=Self>
  62 {
  63     // suffix of "2" because Float::infinity is deprecated
  64     #[allow(deprecated)]
  65     fn infinity2() -> Self {
  66         Float::infinity()
  67     }
  68
  69     // suffix of "2" because Float::nan is deprecated
  70     #[allow(deprecated)]
  71     fn nan2() -> Self {
  72         Float::nan()
  73     }
  74
  75     // suffix of "2" because Float::zero is deprecated
  76     fn zero2() -> Self;
  77
  78     // suffix of "2" because Float::integer_decode is deprecated
  79     #[allow(deprecated)]
  80     fn integer_decode2(self) -> (u64, i16, i8) {
  81         Float::integer_decode(self)
  82     }
  83
  84     /// Get the raw binary representation of the float.
  85     fn transmute(self) -> u64;
  86
  87     /// Transmute the raw binary representation into a float.
  88     fn from_bits(bits: u64) -> Self;
  89
  90     /// Decode the float.
  91     fn unpack(self) -> Unpacked;
  92
  93     /// Cast from a small integer that can be represented exactly.  Panic if the integer can't be
  94     /// represented, the other code in this module makes sure to never let that happen.
  95     fn from_int(x: u64) -> Self;
  96
  97     /// Get the value 10^e from a pre-computed table. Panics for e >= ceil_log5_of_max_sig().
  98     fn short_fast_pow10(e: usize) -> Self;
  99
 100     // FIXME Everything that follows should be associated constants, but taking the value of an
 101     // associated constant from a type parameter does not work (yet?)
 102     // A possible workaround is having a `FloatInfo` struct for all the constants, but so far
 103     // the methods aren't painful enough to rewrite.
 104
 105     /// What the name says. It's easier to hard code than juggling intrinsics and
 106     /// hoping LLVM constant folds it.
 107     fn ceil_log5_of_max_sig() -> i16;
 108
 109     // A conservative bound on the decimal digits of inputs that can't produce overflow or zero or
 110     /// subnormals. Probably the decimal exponent of the maximum normal value, hence the name.
 111     fn max_normal_digits() -> usize;
 112
 113     /// When the most significant decimal digit has a place value greater than this, the number
 114     /// is certainly rounded to infinity.
 115     fn inf_cutoff() -> i64;
 116
 117     /// When the most significant decimal digit has a place value less than this, the number
 118     /// is certainly rounded to zero.
 119     fn zero_cutoff() -> i64;
 120
 121     /// The number of bits in the exponent.
 122     fn exp_bits() -> u8;
 123
 124     /// The number of bits in the singificand, *including* the hidden bit.
 125     fn sig_bits() -> u8;
 126
 127     /// The number of bits in the singificand, *excluding* the hidden bit.
 128     fn explicit_sig_bits() -> u8 {
 129         Self::sig_bits() - 1
 130     }
 131
 132     /// The maximum legal exponent in fractional representation.
 133     fn max_exp() -> i16 {
 134         (1 << (Self::exp_bits() - 1)) - 1
 135     }
 136
 137     /// The minimum legal exponent in fractional representation, excluding subnormals.
 138     fn min_exp() -> i16 {
 139         -Self::max_exp() + 1
 140     }
 141
 142     /// `MAX_EXP` for integral representation, i.e., with the shift applied.
 143     fn max_exp_int() -> i16 {
 144         Self::max_exp() - (Self::sig_bits() as i16 - 1)
 145     }
 146
 147     /// `MAX_EXP` encoded (i.e., with offset bias)
 148     fn max_encoded_exp() -> i16 {
 149         (1 << Self::exp_bits()) - 1
 150     }
 151
 152     /// `MIN_EXP` for integral representation, i.e., with the shift applied.
 153     fn min_exp_int() -> i16 {
 154         Self::min_exp() - (Self::sig_bits() as i16 - 1)
 155     }
 156
 157     /// The maximum normalized singificand in integral representation.
 158     fn max_sig() -> u64 {
 159         (1 << Self::sig_bits()) - 1
 160     }
 161
 162     /// The minimal normalized significand in integral representation.
 163     fn min_sig() -> u64 {
 164         1 << (Self::sig_bits() - 1)
 165     }
 166 }
 167
 168 impl RawFloat for f32 {
 169     fn zero2() -> Self {
 170         0.0
 171     }
 172
 173     fn sig_bits() -> u8 {
 174         24
 175     }
 176
 177     fn exp_bits() -> u8 {
 178         8
 179     }
 180
 181     fn ceil_log5_of_max_sig() -> i16 {
 182         11
 183     }
 184
 185     fn transmute(self) -> u64 {
 186         let bits: u32 = unsafe { transmute(self) };
 187         bits as u64
 188     }
 189
 190     fn from_bits(bits: u64) -> f32 {
 191         assert!(bits < u32::MAX as u64, "f32::from_bits: too many bits");
 192         unsafe { transmute(bits as u32) }
 193     }
 194
 195     fn unpack(self) -> Unpacked {
 196         let (sig, exp, _sig) = self.integer_decode2();
 197         Unpacked::new(sig, exp)
 198     }
 199
 200     fn from_int(x: u64) -> f32 {
 201         // rkruppe is uncertain whether `as` rounds correctly on all platforms.
 202         debug_assert!(x as f32 == fp_to_float(Fp { f: x, e: 0 }));
 203         x as f32
 204     }
 205
 206     fn short_fast_pow10(e: usize) -> Self {
 207         table::F32_SHORT_POWERS[e]
 208     }
 209
 210     fn max_normal_digits() -> usize {
 211         35
 212     }
 213
 214     fn inf_cutoff() -> i64 {
 215         40
 216     }
 217
 218     fn zero_cutoff() -> i64 {
 219         -48
 220     }
 221 }
 222
 223
 224 impl RawFloat for f64 {
 225     fn zero2() -> Self {
 226         0.0
 227     }
 228
 229     fn sig_bits() -> u8 {
 230         53
 231     }
 232
 233     fn exp_bits() -> u8 {
 234         11
 235     }
 236
 237     fn ceil_log5_of_max_sig() -> i16 {
 238         23
 239     }
 240
 241     fn transmute(self) -> u64 {
 242         let bits: u64 = unsafe { transmute(self) };
 243         bits
 244     }
 245
 246     fn from_bits(bits: u64) -> f64 {
 247         unsafe { transmute(bits) }
 248     }
 249
 250     fn unpack(self) -> Unpacked {
 251         let (sig, exp, _sig) = self.integer_decode2();
 252         Unpacked::new(sig, exp)
 253     }
 254
 255     fn from_int(x: u64) -> f64 {
 256         // rkruppe is uncertain whether `as` rounds correctly on all platforms.
 257         debug_assert!(x as f64 == fp_to_float(Fp { f: x, e: 0 }));
 258         x as f64
 259     }
 260
 261     fn short_fast_pow10(e: usize) -> Self {
 262         table::F64_SHORT_POWERS[e]
 263     }
 264
 265     fn max_normal_digits() -> usize {
 266         305
 267     }
 268
 269     fn inf_cutoff() -> i64 {
 270         310
 271     }
 272
 273     fn zero_cutoff() -> i64 {
 274         -326
 275     }
 276
 277 }
 278
 279 /// Convert an Fp to the closest f64. Only handles number that fit into a normalized f64.
 280 pub fn fp_to_float<T: RawFloat>(x: Fp) -> T {
 281     let x = x.normalize();
 282     // x.f is 64 bit, so x.e has a mantissa shift of 63
 283     let e = x.e + 63;
 284     if e > T::max_exp() {
 285         panic!("fp_to_float: exponent {} too large", e)
 286     }  else if e > T::min_exp() {
 287         encode_normal(round_normal::<T>(x))
 288     } else {
 289         panic!("fp_to_float: exponent {} too small", e)
 290     }
 291 }
 292
 293 /// Round the 64-bit significand to 53 bit with half-to-even. Does not handle exponent overflow.
 294 pub fn round_normal<T: RawFloat>(x: Fp) -> Unpacked {
 295     let excess = 64 - T::sig_bits() as i16;
 296     let half: u64 = 1 << (excess - 1);
 297     let (q, rem) = (x.f >> excess, x.f & ((1 << excess) - 1));
 298     assert_eq!(q << excess | rem, x.f);
 299     // Adjust mantissa shift
 300     let k = x.e + excess;
 301     if rem < half {
 302         Unpacked::new(q, k)
 303     } else if rem == half && (q % 2) == 0 {
 304         Unpacked::new(q, k)
 305     } else if q == T::max_sig() {
 306         Unpacked::new(T::min_sig(), k + 1)
 307     } else {
 308         Unpacked::new(q + 1, k)
 309     }
 310 }
 311
 312 /// Inverse of `RawFloat::unpack()` for normalized numbers.
 313 /// Panics if the significand or exponent are not valid for normalized numbers.
 314 pub fn encode_normal<T: RawFloat>(x: Unpacked) -> T {
 315     debug_assert!(T::min_sig() <= x.sig && x.sig <= T::max_sig(),
 316         "encode_normal: significand not normalized");
 317     // Remove the hidden bit
 318     let sig_enc = x.sig & !(1 << T::explicit_sig_bits());
 319     // Adjust the exponent for exponent bias and mantissa shift
 320     let k_enc = x.k + T::max_exp() + T::explicit_sig_bits() as i16;
 321     debug_assert!(k_enc != 0 && k_enc < T::max_encoded_exp(),
 322         "encode_normal: exponent out of range");
 323     // Leave sign bit at 0 ("+"), our numbers are all positive
 324     let bits = (k_enc as u64) << T::explicit_sig_bits() | sig_enc;
 325     T::from_bits(bits)
 326 }
 327
 328 /// Construct the subnormal. A mantissa of 0 is allowed and constructs zero.
 329 pub fn encode_subnormal<T: RawFloat>(significand: u64) -> T {
 330     assert!(significand < T::min_sig(), "encode_subnormal: not actually subnormal");
 331     // Encoded exponent is 0, the sign bit is 0, so we just have to reinterpret the bits.
 332     T::from_bits(significand)
 333 }
 334
 335 /// Approximate a bignum with an Fp. Rounds within 0.5 ULP with half-to-even.
 336 pub fn big_to_fp(f: &Big) -> Fp {
 337     let end = f.bit_length();
 338     assert!(end != 0, "big_to_fp: unexpectedly, input is zero");
 339     let start = end.saturating_sub(64);
 340     let leading = num::get_bits(f, start, end);
 341     // We cut off all bits prior to the index `start`, i.e., we effectively right-shift by
 342     // an amount of `start`, so this is also the exponent we need.
 343     let e = start as i16;
 344     let rounded_down = Fp { f: leading, e: e }.normalize();
 345     // Round (half-to-even) depending on the truncated bits.
 346     match num::compare_with_half_ulp(f, start) {
 347         Less => rounded_down,
 348         Equal if leading % 2 == 0 => rounded_down,
 349         Equal | Greater => match leading.checked_add(1) {
 350             Some(f) => Fp { f: f, e: e }.normalize(),
 351             None => Fp { f: 1 << 63, e: e + 1 },
 352         }
 353     }
 354 }
 355
 356 /// Find the largest floating point number strictly smaller than the argument.
 357 /// Does not handle subnormals, zero, or exponent underflow.
 358 pub fn prev_float<T: RawFloat>(x: T) -> T {
 359     match x.classify() {
 360         Infinite => panic!("prev_float: argument is infinite"),
 361         Nan => panic!("prev_float: argument is NaN"),
 362         Subnormal => panic!("prev_float: argument is subnormal"),
 363         Zero => panic!("prev_float: argument is zero"),
 364         Normal => {
 365             let Unpacked { sig, k } = x.unpack();
 366             if sig == T::min_sig() {
 367                 encode_normal(Unpacked::new(T::max_sig(), k - 1))
 368             } else {
 369                 encode_normal(Unpacked::new(sig - 1, k))
 370             }
 371         }
 372     }
 373 }
 374
 375 // Find the smallest floating point number strictly larger than the argument.
 376 // This operation is saturating, i.e. next_float(inf) == inf.
 377 // Unlike most code in this module, this function does handle zero, subnormals, and infinities.
 378 // However, like all other code here, it does not deal with NaN and negative numbers.
 379 pub fn next_float<T: RawFloat>(x: T) -> T {
 380     match x.classify() {
 381         Nan => panic!("next_float: argument is NaN"),
 382         Infinite => T::infinity2(),
 383         // This seems too good to be true, but it works.
 384         // 0.0 is encoded as the all-zero word. Subnormals are 0x000m...m where m is the mantissa.
 385         // In particular, the smallest subnormal is 0x0...01 and the largest is 0x000F...F.
 386         // The smallest normal number is 0x0010...0, so this corner case works as well.
 387         // If the increment overflows the mantissa, the carry bit increments the exponent as we
 388         // want, and the mantissa bits become zero. Because of the hidden bit convention, this
 389         // too is exactly what we want!
 390         // Finally, f64::MAX + 1 = 7eff...f + 1 = 7ff0...0 = f64::INFINITY.
 391         Zero | Subnormal | Normal => {
 392             let bits: u64 = x.transmute();
 393             T::from_bits(bits + 1)
 394         }
 395     }
 396 }