417 lines
17 KiB
Rust
417 lines
17 KiB
Rust
//! The various algorithms from the paper.
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use crate::cmp::min;
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use crate::cmp::Ordering::{Less, Equal, Greater};
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use crate::num::diy_float::Fp;
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use crate::num::dec2flt::table;
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use crate::num::dec2flt::rawfp::{self, Unpacked, RawFloat, fp_to_float, next_float, prev_float};
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use crate::num::dec2flt::num::{self, Big};
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/// Number of significand bits in Fp
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const P: u32 = 64;
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// We simply store the best approximation for *all* exponents, so the variable "h" and the
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// associated conditions can be omitted. This trades performance for a couple kilobytes of space.
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fn power_of_ten(e: i16) -> Fp {
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assert!(e >= table::MIN_E);
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let i = e - table::MIN_E;
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let sig = table::POWERS.0[i as usize];
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let exp = table::POWERS.1[i as usize];
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Fp { f: sig, e: exp }
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}
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// In most architectures, floating point operations have an explicit bit size, therefore the
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// precision of the computation is determined on a per-operation basis.
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#[cfg(any(not(target_arch="x86"), target_feature="sse2"))]
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mod fpu_precision {
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pub fn set_precision<T>() { }
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}
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// On x86, the x87 FPU is used for float operations if the SSE/SSE2 extensions are not available.
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// The x87 FPU operates with 80 bits of precision by default, which means that operations will
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// round to 80 bits causing double rounding to happen when values are eventually represented as
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// 32/64 bit float values. To overcome this, the FPU control word can be set so that the
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// computations are performed in the desired precision.
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#[cfg(all(target_arch="x86", not(target_feature="sse2")))]
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mod fpu_precision {
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use crate::mem::size_of;
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/// A structure used to preserve the original value of the FPU control word, so that it can be
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/// restored when the structure is dropped.
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///
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/// The x87 FPU is a 16-bits register whose fields are as follows:
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///
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/// | 12-15 | 10-11 | 8-9 | 6-7 | 5 | 4 | 3 | 2 | 1 | 0 |
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/// |------:|------:|----:|----:|---:|---:|---:|---:|---:|---:|
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/// | | RC | PC | | PM | UM | OM | ZM | DM | IM |
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///
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/// The documentation for all of the fields is available in the IA-32 Architectures Software
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/// Developer's Manual (Volume 1).
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///
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/// The only field which is relevant for the following code is PC, Precision Control. This
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/// field determines the precision of the operations performed by the FPU. It can be set to:
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/// - 0b00, single precision i.e., 32-bits
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/// - 0b10, double precision i.e., 64-bits
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/// - 0b11, double extended precision i.e., 80-bits (default state)
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/// The 0b01 value is reserved and should not be used.
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pub struct FPUControlWord(u16);
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fn set_cw(cw: u16) {
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// SAFETY: the `fldcw` instruction has been audited to be able to work correctly with
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// any `u16`
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unsafe { asm!("fldcw $0" :: "m" (cw) :: "volatile") }
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}
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/// Sets the precision field of the FPU to `T` and returns a `FPUControlWord`.
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pub fn set_precision<T>() -> FPUControlWord {
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let cw = 0u16;
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// Compute the value for the Precision Control field that is appropriate for `T`.
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let cw_precision = match size_of::<T>() {
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4 => 0x0000, // 32 bits
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8 => 0x0200, // 64 bits
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_ => 0x0300, // default, 80 bits
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};
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// Get the original value of the control word to restore it later, when the
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// `FPUControlWord` structure is dropped
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// SAFETY: the `fnstcw` instruction has been audited to be able to work correctly with
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// any `u16`
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unsafe { asm!("fnstcw $0" : "=*m" (&cw) ::: "volatile") }
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// Set the control word to the desired precision. This is achieved by masking away the old
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// precision (bits 8 and 9, 0x300) and replacing it with the precision flag computed above.
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set_cw((cw & 0xFCFF) | cw_precision);
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FPUControlWord(cw)
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}
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impl Drop for FPUControlWord {
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fn drop(&mut self) {
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set_cw(self.0)
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}
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}
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}
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/// The fast path of Bellerophon using machine-sized integers and floats.
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///
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/// This is extracted into a separate function so that it can be attempted before constructing
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/// a bignum.
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pub fn fast_path<T: RawFloat>(integral: &[u8], fractional: &[u8], e: i64) -> Option<T> {
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let num_digits = integral.len() + fractional.len();
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// log_10(f64::MAX_SIG) ~ 15.95. We compare the exact value to MAX_SIG near the end,
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// this is just a quick, cheap rejection (and also frees the rest of the code from
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// worrying about underflow).
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if num_digits > 16 {
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return None;
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}
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if e.abs() >= T::CEIL_LOG5_OF_MAX_SIG as i64 {
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return None;
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}
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let f = num::from_str_unchecked(integral.iter().chain(fractional.iter()));
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if f > T::MAX_SIG {
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return None;
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}
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// The fast path crucially depends on arithmetic being rounded to the correct number of bits
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// without any intermediate rounding. On x86 (without SSE or SSE2) this requires the precision
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// of the x87 FPU stack to be changed so that it directly rounds to 64/32 bit.
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// The `set_precision` function takes care of setting the precision on architectures which
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// require setting it by changing the global state (like the control word of the x87 FPU).
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let _cw = fpu_precision::set_precision::<T>();
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// The case e < 0 cannot be folded into the other branch. Negative powers result in
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// a repeating fractional part in binary, which are rounded, which causes real
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// (and occasionally quite significant!) errors in the final result.
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if e >= 0 {
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Some(T::from_int(f) * T::short_fast_pow10(e as usize))
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} else {
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Some(T::from_int(f) / T::short_fast_pow10(e.abs() as usize))
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}
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}
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/// Algorithm Bellerophon is trivial code justified by non-trivial numeric analysis.
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///
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/// It rounds ``f`` to a float with 64 bit significand and multiplies it by the best approximation
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/// of `10^e` (in the same floating point format). This is often enough to get the correct result.
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/// However, when the result is close to halfway between two adjacent (ordinary) floats, the
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/// compound rounding error from multiplying two approximation means the result may be off by a
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/// few bits. When this happens, the iterative Algorithm R fixes things up.
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///
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/// The hand-wavy "close to halfway" is made precise by the numeric analysis in the paper.
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/// In the words of Clinger:
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///
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/// > Slop, expressed in units of the least significant bit, is an inclusive bound for the error
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/// > accumulated during the floating point calculation of the approximation to f * 10^e. (Slop is
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/// > not a bound for the true error, but bounds the difference between the approximation z and
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/// > the best possible approximation that uses p bits of significand.)
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pub fn bellerophon<T: RawFloat>(f: &Big, e: i16) -> T {
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let slop = if f <= &Big::from_u64(T::MAX_SIG) {
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// The cases abs(e) < log5(2^N) are in fast_path()
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if e >= 0 { 0 } else { 3 }
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} else {
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if e >= 0 { 1 } else { 4 }
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};
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let z = rawfp::big_to_fp(f).mul(&power_of_ten(e)).normalize();
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let exp_p_n = 1 << (P - T::SIG_BITS as u32);
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let lowbits: i64 = (z.f % exp_p_n) as i64;
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// Is the slop large enough to make a difference when
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// rounding to n bits?
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if (lowbits - exp_p_n as i64 / 2).abs() <= slop {
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algorithm_r(f, e, fp_to_float(z))
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} else {
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fp_to_float(z)
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}
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}
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/// An iterative algorithm that improves a floating point approximation of `f * 10^e`.
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///
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/// Each iteration gets one unit in the last place closer, which of course takes terribly long to
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/// converge if `z0` is even mildly off. Luckily, when used as fallback for Bellerophon, the
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/// starting approximation is off by at most one ULP.
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fn algorithm_r<T: RawFloat>(f: &Big, e: i16, z0: T) -> T {
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let mut z = z0;
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loop {
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let raw = z.unpack();
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let (m, k) = (raw.sig, raw.k);
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let mut x = f.clone();
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let mut y = Big::from_u64(m);
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// Find positive integers `x`, `y` such that `x / y` is exactly `(f * 10^e) / (m * 2^k)`.
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// This not only avoids dealing with the signs of `e` and `k`, we also eliminate the
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// power of two common to `10^e` and `2^k` to make the numbers smaller.
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make_ratio(&mut x, &mut y, e, k);
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let m_digits = [(m & 0xFF_FF_FF_FF) as u32, (m >> 32) as u32];
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// This is written a bit awkwardly because our bignums don't support
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// negative numbers, so we use the absolute value + sign information.
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// The multiplication with m_digits can't overflow. If `x` or `y` are large enough that
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// we need to worry about overflow, then they are also large enough that `make_ratio` has
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// reduced the fraction by a factor of 2^64 or more.
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let (d2, d_negative) = if x >= y {
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// Don't need x any more, save a clone().
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x.sub(&y).mul_pow2(1).mul_digits(&m_digits);
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(x, false)
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} else {
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// Still need y - make a copy.
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let mut y = y.clone();
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y.sub(&x).mul_pow2(1).mul_digits(&m_digits);
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(y, true)
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};
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if d2 < y {
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let mut d2_double = d2;
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d2_double.mul_pow2(1);
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if m == T::MIN_SIG && d_negative && d2_double > y {
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z = prev_float(z);
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} else {
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return z;
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}
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} else if d2 == y {
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if m % 2 == 0 {
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if m == T::MIN_SIG && d_negative {
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z = prev_float(z);
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} else {
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return z;
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}
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} else if d_negative {
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z = prev_float(z);
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} else {
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z = next_float(z);
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}
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} else if d_negative {
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z = prev_float(z);
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} else {
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z = next_float(z);
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}
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}
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}
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/// Given `x = f` and `y = m` where `f` represent input decimal digits as usual and `m` is the
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/// significand of a floating point approximation, make the ratio `x / y` equal to
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/// `(f * 10^e) / (m * 2^k)`, possibly reduced by a power of two both have in common.
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fn make_ratio(x: &mut Big, y: &mut Big, e: i16, k: i16) {
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let (e_abs, k_abs) = (e.abs() as usize, k.abs() as usize);
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if e >= 0 {
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if k >= 0 {
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// x = f * 10^e, y = m * 2^k, except that we reduce the fraction by some power of two.
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let common = min(e_abs, k_abs);
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x.mul_pow5(e_abs).mul_pow2(e_abs - common);
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y.mul_pow2(k_abs - common);
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} else {
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// x = f * 10^e * 2^abs(k), y = m
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// This can't overflow because it requires positive `e` and negative `k`, which can
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// only happen for values extremely close to 1, which means that `e` and `k` will be
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// comparatively tiny.
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x.mul_pow5(e_abs).mul_pow2(e_abs + k_abs);
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}
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} else {
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if k >= 0 {
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// x = f, y = m * 10^abs(e) * 2^k
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// This can't overflow either, see above.
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y.mul_pow5(e_abs).mul_pow2(k_abs + e_abs);
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} else {
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// x = f * 2^abs(k), y = m * 10^abs(e), again reducing by a common power of two.
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let common = min(e_abs, k_abs);
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x.mul_pow2(k_abs - common);
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y.mul_pow5(e_abs).mul_pow2(e_abs - common);
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}
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}
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}
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/// Conceptually, Algorithm M is the simplest way to convert a decimal to a float.
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///
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/// We form a ratio that is equal to `f * 10^e`, then throwing in powers of two until it gives
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/// a valid float significand. The binary exponent `k` is the number of times we multiplied
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/// numerator or denominator by two, i.e., at all times `f * 10^e` equals `(u / v) * 2^k`.
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/// When we have found out significand, we only need to round by inspecting the remainder of the
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/// division, which is done in helper functions further below.
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///
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/// This algorithm is super slow, even with the optimization described in `quick_start()`.
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/// However, it's the simplest of the algorithms to adapt for overflow, underflow, and subnormal
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/// results. This implementation takes over when Bellerophon and Algorithm R are overwhelmed.
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/// Detecting underflow and overflow is easy: The ratio still isn't an in-range significand,
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/// yet the minimum/maximum exponent has been reached. In the case of overflow, we simply return
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/// infinity.
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///
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/// Handling underflow and subnormals is trickier. One big problem is that, with the minimum
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/// exponent, the ratio might still be too large for a significand. See underflow() for details.
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pub fn algorithm_m<T: RawFloat>(f: &Big, e: i16) -> T {
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let mut u;
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let mut v;
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let e_abs = e.abs() as usize;
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let mut k = 0;
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if e < 0 {
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u = f.clone();
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v = Big::from_small(1);
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v.mul_pow5(e_abs).mul_pow2(e_abs);
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} else {
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// FIXME possible optimization: generalize big_to_fp so that we can do the equivalent of
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// fp_to_float(big_to_fp(u)) here, only without the double rounding.
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u = f.clone();
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u.mul_pow5(e_abs).mul_pow2(e_abs);
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v = Big::from_small(1);
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}
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quick_start::<T>(&mut u, &mut v, &mut k);
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let mut rem = Big::from_small(0);
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let mut x = Big::from_small(0);
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let min_sig = Big::from_u64(T::MIN_SIG);
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let max_sig = Big::from_u64(T::MAX_SIG);
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loop {
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u.div_rem(&v, &mut x, &mut rem);
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if k == T::MIN_EXP_INT {
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// We have to stop at the minimum exponent, if we wait until `k < T::MIN_EXP_INT`,
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// then we'd be off by a factor of two. Unfortunately this means we have to special-
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// case normal numbers with the minimum exponent.
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// FIXME find a more elegant formulation, but run the `tiny-pow10` test to make sure
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// that it's actually correct!
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if x >= min_sig && x <= max_sig {
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break;
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}
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return underflow(x, v, rem);
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}
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if k > T::MAX_EXP_INT {
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return T::INFINITY;
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}
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if x < min_sig {
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u.mul_pow2(1);
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k -= 1;
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} else if x > max_sig {
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v.mul_pow2(1);
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k += 1;
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} else {
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break;
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}
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}
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let q = num::to_u64(&x);
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let z = rawfp::encode_normal(Unpacked::new(q, k));
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round_by_remainder(v, rem, q, z)
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}
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/// Skips over most Algorithm M iterations by checking the bit length.
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fn quick_start<T: RawFloat>(u: &mut Big, v: &mut Big, k: &mut i16) {
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// The bit length is an estimate of the base two logarithm, and log(u / v) = log(u) - log(v).
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// The estimate is off by at most 1, but always an under-estimate, so the error on log(u)
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// and log(v) are of the same sign and cancel out (if both are large). Therefore the error
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// for log(u / v) is at most one as well.
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// The target ratio is one where u/v is in an in-range significand. Thus our termination
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// condition is log2(u / v) being the significand bits, plus/minus one.
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// FIXME Looking at the second bit could improve the estimate and avoid some more divisions.
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let target_ratio = T::SIG_BITS as i16;
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let log2_u = u.bit_length() as i16;
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let log2_v = v.bit_length() as i16;
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let mut u_shift: i16 = 0;
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let mut v_shift: i16 = 0;
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assert!(*k == 0);
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loop {
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if *k == T::MIN_EXP_INT {
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// Underflow or subnormal. Leave it to the main function.
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break;
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}
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if *k == T::MAX_EXP_INT {
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// Overflow. Leave it to the main function.
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break;
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}
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let log2_ratio = (log2_u + u_shift) - (log2_v + v_shift);
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if log2_ratio < target_ratio - 1 {
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u_shift += 1;
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*k -= 1;
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} else if log2_ratio > target_ratio + 1 {
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v_shift += 1;
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*k += 1;
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} else {
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break;
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}
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}
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u.mul_pow2(u_shift as usize);
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v.mul_pow2(v_shift as usize);
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}
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fn underflow<T: RawFloat>(x: Big, v: Big, rem: Big) -> T {
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if x < Big::from_u64(T::MIN_SIG) {
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let q = num::to_u64(&x);
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let z = rawfp::encode_subnormal(q);
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return round_by_remainder(v, rem, q, z);
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}
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// Ratio isn't an in-range significand with the minimum exponent, so we need to round off
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// excess bits and adjust the exponent accordingly. The real value now looks like this:
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//
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// x lsb
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// /--------------\/
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// 1010101010101010.10101010101010 * 2^k
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// \-----/\-------/ \------------/
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// q trunc. (represented by rem)
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//
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// Therefore, when the rounded-off bits are != 0.5 ULP, they decide the rounding
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// on their own. When they are equal and the remainder is non-zero, the value still
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// needs to be rounded up. Only when the rounded off bits are 1/2 and the remainder
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// is zero, we have a half-to-even situation.
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let bits = x.bit_length();
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let lsb = bits - T::SIG_BITS as usize;
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let q = num::get_bits(&x, lsb, bits);
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let k = T::MIN_EXP_INT + lsb as i16;
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let z = rawfp::encode_normal(Unpacked::new(q, k));
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let q_even = q % 2 == 0;
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match num::compare_with_half_ulp(&x, lsb) {
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Greater => next_float(z),
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Less => z,
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Equal if rem.is_zero() && q_even => z,
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Equal => next_float(z),
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}
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}
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/// Ordinary round-to-even, obfuscated by having to round based on the remainder of a division.
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fn round_by_remainder<T: RawFloat>(v: Big, r: Big, q: u64, z: T) -> T {
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let mut v_minus_r = v;
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v_minus_r.sub(&r);
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if r < v_minus_r {
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z
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} else if r > v_minus_r {
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next_float(z)
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} else if q % 2 == 0 {
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z
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} else {
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next_float(z)
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}
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}
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