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+// Copyright 2015 The Rust Project Developers. See the COPYRIGHT
+// file at the top-level directory of this distribution and at
+// http://rust-lang.org/COPYRIGHT.
+//
+// Licensed under the Apache License, Version 2.0 <LICENSE-APACHE or
+// http://www.apache.org/licenses/LICENSE-2.0> or the MIT license
+// <LICENSE-MIT or http://opensource.org/licenses/MIT>, at your
+// option. This file may not be copied, modified, or distributed
+// except according to those terms.
+
+//! Bit fiddling on positive IEEE 754 floats. Negative numbers aren't and needn't be handled.
+//! Normal floating point numbers have a canonical representation as (frac, exp) such that the
+//! value is 2^exp * (1 + sum(frac[N-i] / 2^i)) where N is the number of bits. Subnormals are
+//! slightly different and weird, but the same principle applies.
+//!
+//! Here, however, we represent them as (sig, k) with f positive, such that the value is f * 2^e.
+//! Besides making the "hidden bit" explicit, this changes the exponent by the so-called
+//! mantissa shift.
+//!
+//! Put another way, normally floats are written as (1) but here they are written as (2):
+//!
+//! 1. `1.101100...11 * 2^m`
+//! 2. `1101100...11 * 2^n`
+//!
+//! We call (1) the **fractional representation** and (2) the **integral representation**.
+//!
+//! Many functions in this module only handle normal numbers. The dec2flt routines conservatively
+//! take the universally-correct slow path (Algorithm M) for very small and very large numbers.
+//! That algorithm needs only next_float() which does handle subnormals and zeros.
+use prelude::v1::*;
+use u32;
+use cmp::Ordering::{Less, Equal, Greater};
+use ops::{Mul, Div, Neg};
+use fmt::{Debug, LowerExp};
+use mem::transmute;
+use num::diy_float::Fp;
+use num::FpCategory::{Infinite, Zero, Subnormal, Normal, Nan};
+use num::Float;
+use num::dec2flt::num::{self, Big};
+use num::dec2flt::table;
+
+#[derive(Copy, Clone, Debug)]
+pub struct Unpacked {
+ pub sig: u64,
+ pub k: i16,
+}
+
+impl Unpacked {
+ pub fn new(sig: u64, k: i16) -> Self {
+ Unpacked { sig: sig, k: k }
+ }
+}
+
+/// A helper trait to avoid duplicating basically all the conversion code for `f32` and `f64`.
+///
+/// See the parent module's doc comment for why this is necessary.
+///
+/// Should **never ever** be implemented for other types or be used outside the dec2flt module.
+/// Inherits from `Float` because there is some overlap, but all the reused methods are trivial.
+/// The "methods" (pseudo-constants) with default implementation should not be overriden.
+pub trait RawFloat : Float + Copy + Debug + LowerExp
+ + Mul<Output=Self> + Div<Output=Self> + Neg<Output=Self>
+{
+ /// Get the raw binary representation of the float.
+ fn transmute(self) -> u64;
+
+ /// Transmute the raw binary representation into a float.
+ fn from_bits(bits: u64) -> Self;
+
+ /// Decode the float.
+ fn unpack(self) -> Unpacked;
+
+ /// Cast from a small integer that can be represented exactly. Panic if the integer can't be
+ /// represented, the other code in this module makes sure to never let that happen.
+ fn from_int(x: u64) -> Self;
+
+ /// Get the value 10^e from a pre-computed table. Panics for e >= ceil_log5_of_max_sig().
+ fn short_fast_pow10(e: usize) -> Self;
+
+ // FIXME Everything that follows should be associated constants, but taking the value of an
+ // associated constant from a type parameter does not work (yet?)
+ // A possible workaround is having a `FloatInfo` struct for all the constants, but so far
+ // the methods aren't painful enough to rewrite.
+
+ /// What the name says. It's easier to hard code than juggling intrinsics and
+ /// hoping LLVM constant folds it.
+ fn ceil_log5_of_max_sig() -> i16;
+
+ // A conservative bound on the decimal digits of inputs that can't produce overflow or zero or
+ /// subnormals. Probably the decimal exponent of the maximum normal value, hence the name.
+ fn max_normal_digits() -> usize;
+
+ /// When the most significant decimal digit has a place value greater than this, the number
+ /// is certainly rounded to infinity.
+ fn inf_cutoff() -> i64;
+
+ /// When the most significant decimal digit has a place value less than this, the number
+ /// is certainly rounded to zero.
+ fn zero_cutoff() -> i64;
+
+ /// The number of bits in the exponent.
+ fn exp_bits() -> u8;
+
+ /// The number of bits in the singificand, *including* the hidden bit.
+ fn sig_bits() -> u8;
+
+ /// The number of bits in the singificand, *excluding* the hidden bit.
+ fn explicit_sig_bits() -> u8 {
+ Self::sig_bits() - 1
+ }
+
+ /// The maximum legal exponent in fractional representation.
+ fn max_exp() -> i16 {
+ (1 << (Self::exp_bits() - 1)) - 1
+ }
+
+ /// The minimum legal exponent in fractional representation, excluding subnormals.
+ fn min_exp() -> i16 {
+ -Self::max_exp() + 1
+ }
+
+ /// `MAX_EXP` for integral representation, i.e., with the shift applied.
+ fn max_exp_int() -> i16 {
+ Self::max_exp() - (Self::sig_bits() as i16 - 1)
+ }
+
+ /// `MAX_EXP` encoded (i.e., with offset bias)
+ fn max_encoded_exp() -> i16 {
+ (1 << Self::exp_bits()) - 1
+ }
+
+ /// `MIN_EXP` for integral representation, i.e., with the shift applied.
+ fn min_exp_int() -> i16 {
+ Self::min_exp() - (Self::sig_bits() as i16 - 1)
+ }
+
+ /// The maximum normalized singificand in integral representation.
+ fn max_sig() -> u64 {
+ (1 << Self::sig_bits()) - 1
+ }
+
+ /// The minimal normalized significand in integral representation.
+ fn min_sig() -> u64 {
+ 1 << (Self::sig_bits() - 1)
+ }
+}
+
+impl RawFloat for f32 {
+ fn sig_bits() -> u8 {
+ 24
+ }
+
+ fn exp_bits() -> u8 {
+ 8
+ }
+
+ fn ceil_log5_of_max_sig() -> i16 {
+ 11
+ }
+
+ fn transmute(self) -> u64 {
+ let bits: u32 = unsafe { transmute(self) };
+ bits as u64
+ }
+
+ fn from_bits(bits: u64) -> f32 {
+ assert!(bits < u32::MAX as u64, "f32::from_bits: too many bits");
+ unsafe { transmute(bits as u32) }
+ }
+
+ fn unpack(self) -> Unpacked {
+ let (sig, exp, _sig) = self.integer_decode();
+ Unpacked::new(sig, exp)
+ }
+
+ fn from_int(x: u64) -> f32 {
+ // rkruppe is uncertain whether `as` rounds correctly on all platforms.
+ debug_assert!(x as f32 == fp_to_float(Fp { f: x, e: 0 }));
+ x as f32
+ }
+
+ fn short_fast_pow10(e: usize) -> Self {
+ table::F32_SHORT_POWERS[e]
+ }
+
+ fn max_normal_digits() -> usize {
+ 35
+ }
+
+ fn inf_cutoff() -> i64 {
+ 40
+ }
+
+ fn zero_cutoff() -> i64 {
+ -48
+ }
+}
+
+
+impl RawFloat for f64 {
+ fn sig_bits() -> u8 {
+ 53
+ }
+
+ fn exp_bits() -> u8 {
+ 11
+ }
+
+ fn ceil_log5_of_max_sig() -> i16 {
+ 23
+ }
+
+ fn transmute(self) -> u64 {
+ let bits: u64 = unsafe { transmute(self) };
+ bits
+ }
+
+ fn from_bits(bits: u64) -> f64 {
+ unsafe { transmute(bits) }
+ }
+
+ fn unpack(self) -> Unpacked {
+ let (sig, exp, _sig) = self.integer_decode();
+ Unpacked::new(sig, exp)
+ }
+
+ fn from_int(x: u64) -> f64 {
+ // rkruppe is uncertain whether `as` rounds correctly on all platforms.
+ debug_assert!(x as f64 == fp_to_float(Fp { f: x, e: 0 }));
+ x as f64
+ }
+
+ fn short_fast_pow10(e: usize) -> Self {
+ table::F64_SHORT_POWERS[e]
+ }
+
+ fn max_normal_digits() -> usize {
+ 305
+ }
+
+ fn inf_cutoff() -> i64 {
+ 310
+ }
+
+ fn zero_cutoff() -> i64 {
+ -326
+ }
+
+}
+
+/// Convert an Fp to the closest f64. Only handles number that fit into a normalized f64.
+pub fn fp_to_float<T: RawFloat>(x: Fp) -> T {
+ let x = x.normalize();
+ // x.f is 64 bit, so x.e has a mantissa shift of 63
+ let e = x.e + 63;
+ if e > T::max_exp() {
+ panic!("fp_to_float: exponent {} too large", e)
+ } else if e > T::min_exp() {
+ encode_normal(round_normal::<T>(x))
+ } else {
+ panic!("fp_to_float: exponent {} too small", e)
+ }
+}
+
+/// Round the 64-bit significand to 53 bit with half-to-even. Does not handle exponent overflow.
+pub fn round_normal<T: RawFloat>(x: Fp) -> Unpacked {
+ let excess = 64 - T::sig_bits() as i16;
+ let half: u64 = 1 << (excess - 1);
+ let (q, rem) = (x.f >> excess, x.f & ((1 << excess) - 1));
+ assert_eq!(q << excess | rem, x.f);
+ // Adjust mantissa shift
+ let k = x.e + excess;
+ if rem < half {
+ Unpacked::new(q, k)
+ } else if rem == half && (q % 2) == 0 {
+ Unpacked::new(q, k)
+ } else if q == T::max_sig() {
+ Unpacked::new(T::min_sig(), k + 1)
+ } else {
+ Unpacked::new(q + 1, k)
+ }
+}
+
+/// Inverse of `RawFloat::unpack()` for normalized numbers.
+/// Panics if the significand or exponent are not valid for normalized numbers.
+pub fn encode_normal<T: RawFloat>(x: Unpacked) -> T {
+ debug_assert!(T::min_sig() <= x.sig && x.sig <= T::max_sig(),
+ "encode_normal: significand not normalized");
+ // Remove the hidden bit
+ let sig_enc = x.sig & !(1 << T::explicit_sig_bits());
+ // Adjust the exponent for exponent bias and mantissa shift
+ let k_enc = x.k + T::max_exp() + T::explicit_sig_bits() as i16;
+ debug_assert!(k_enc != 0 && k_enc < T::max_encoded_exp(),
+ "encode_normal: exponent out of range");
+ // Leave sign bit at 0 ("+"), our numbers are all positive
+ let bits = (k_enc as u64) << T::explicit_sig_bits() | sig_enc;
+ T::from_bits(bits)
+}
+
+/// Construct the subnormal. A mantissa of 0 is allowed and constructs zero.
+pub fn encode_subnormal<T: RawFloat>(significand: u64) -> T {
+ assert!(significand < T::min_sig(), "encode_subnormal: not actually subnormal");
+ // Encoded exponent is 0, the sign bit is 0, so we just have to reinterpret the bits.
+ T::from_bits(significand)
+}
+
+/// Approximate a bignum with an Fp. Rounds within 0.5 ULP with half-to-even.
+pub fn big_to_fp(f: &Big) -> Fp {
+ let end = f.bit_length();
+ assert!(end != 0, "big_to_fp: unexpectedly, input is zero");
+ let start = end.saturating_sub(64);
+ let leading = num::get_bits(f, start, end);
+ // We cut off all bits prior to the index `start`, i.e., we effectively right-shift by
+ // an amount of `start`, so this is also the exponent we need.
+ let e = start as i16;
+ let rounded_down = Fp { f: leading, e: e }.normalize();
+ // Round (half-to-even) depending on the truncated bits.
+ match num::compare_with_half_ulp(f, start) {
+ Less => rounded_down,
+ Equal if leading % 2 == 0 => rounded_down,
+ Equal | Greater => match leading.checked_add(1) {
+ Some(f) => Fp { f: f, e: e }.normalize(),
+ None => Fp { f: 1 << 63, e: e + 1 },
+ }
+ }
+}
+
+/// Find the largest floating point number strictly smaller than the argument.
+/// Does not handle subnormals, zero, or exponent underflow.
+pub fn prev_float<T: RawFloat>(x: T) -> T {
+ match x.classify() {
+ Infinite => panic!("prev_float: argument is infinite"),
+ Nan => panic!("prev_float: argument is NaN"),
+ Subnormal => panic!("prev_float: argument is subnormal"),
+ Zero => panic!("prev_float: argument is zero"),
+ Normal => {
+ let Unpacked { sig, k } = x.unpack();
+ if sig == T::min_sig() {
+ encode_normal(Unpacked::new(T::max_sig(), k - 1))
+ } else {
+ encode_normal(Unpacked::new(sig - 1, k))
+ }
+ }
+ }
+}
+
+// Find the smallest floating point number strictly larger than the argument.
+// This operation is saturating, i.e. next_float(inf) == inf.
+// Unlike most code in this module, this function does handle zero, subnormals, and infinities.
+// However, like all other code here, it does not deal with NaN and negative numbers.
+pub fn next_float<T: RawFloat>(x: T) -> T {
+ match x.classify() {
+ Nan => panic!("next_float: argument is NaN"),
+ Infinite => T::infinity(),
+ // This seems too good to be true, but it works.
+ // 0.0 is encoded as the all-zero word. Subnormals are 0x000m...m where m is the mantissa.
+ // In particular, the smallest subnormal is 0x0...01 and the largest is 0x000F...F.
+ // The smallest normal number is 0x0010...0, so this corner case works as well.
+ // If the increment overflows the mantissa, the carry bit increments the exponent as we
+ // want, and the mantissa bits become zero. Because of the hidden bit convention, this
+ // too is exactly what we want!
+ // Finally, f64::MAX + 1 = 7eff...f + 1 = 7ff0...0 = f64::INFINITY.
+ Zero | Subnormal | Normal => {
+ let bits: u64 = x.transmute();
+ T::from_bits(bits + 1)
+ }
+ }
+}