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use crate::{Category, ExpInt, IEK_INF, IEK_NAN, IEK_ZERO};
use crate::{Float, FloatConvert, ParseError, Round, Status, StatusAnd};
use smallvec::{SmallVec, smallvec};
use std::cmp::{self, Ordering};
use std::convert::TryFrom;
use std::fmt::{self, Write};
use std::marker::PhantomData;
use std::mem;
use std::ops::Neg;
#[must_use]
pub struct IeeeFloat<S> {
/// Absolute significand value (including the integer bit).
sig: [Limb; 1],
/// The signed unbiased exponent of the value.
exp: ExpInt,
/// What kind of floating point number this is.
category: Category,
/// Sign bit of the number.
sign: bool,
marker: PhantomData<S>,
}
/// Fundamental unit of big integer arithmetic, but also
/// large to store the largest significands by itself.
type Limb = u128;
const LIMB_BITS: usize = 128;
fn limbs_for_bits(bits: usize) -> usize {
(bits + LIMB_BITS - 1) / LIMB_BITS
}
/// Enum that represents what fraction of the LSB truncated bits of an fp number
/// represent.
///
/// This essentially combines the roles of guard and sticky bits.
#[must_use]
#[derive(Copy, Clone, PartialEq, Eq, Debug)]
enum Loss {
// Example of truncated bits:
ExactlyZero, // 000000
LessThanHalf, // 0xxxxx x's not all zero
ExactlyHalf, // 100000
MoreThanHalf, // 1xxxxx x's not all zero
}
/// Represents floating point arithmetic semantics.
pub trait Semantics: Sized {
/// Total number of bits in the in-memory format.
const BITS: usize;
/// Number of bits in the significand. This includes the integer bit.
const PRECISION: usize;
/// The largest E such that 2<sup>E</sup> is representable; this matches the
/// definition of IEEE 754.
const MAX_EXP: ExpInt;
/// The smallest E such that 2<sup>E</sup> is a normalized number; this
/// matches the definition of IEEE 754.
const MIN_EXP: ExpInt = -Self::MAX_EXP + 1;
/// The significand bit that marks NaN as quiet.
const QNAN_BIT: usize = Self::PRECISION - 2;
/// The significand bitpattern to mark a NaN as quiet.
/// NOTE: for X87DoubleExtended we need to set two bits instead of 2.
const QNAN_SIGNIFICAND: Limb = 1 << Self::QNAN_BIT;
fn from_bits(bits: u128) -> IeeeFloat<Self> {
assert!(Self::BITS > Self::PRECISION);
let sign = bits & (1 << (Self::BITS - 1));
let exponent = (bits & !sign) >> (Self::PRECISION - 1);
let mut r = IeeeFloat {
sig: [bits & ((1 << (Self::PRECISION - 1)) - 1)],
// Convert the exponent from its bias representation to a signed integer.
exp: (exponent as ExpInt) - Self::MAX_EXP,
category: Category::Zero,
sign: sign != 0,
marker: PhantomData,
};
if r.exp == Self::MIN_EXP - 1 && r.sig == [0] {
// Exponent, significand meaningless.
r.category = Category::Zero;
} else if r.exp == Self::MAX_EXP + 1 && r.sig == [0] {
// Exponent, significand meaningless.
r.category = Category::Infinity;
} else if r.exp == Self::MAX_EXP + 1 && r.sig != [0] {
// Sign, exponent, significand meaningless.
r.category = Category::NaN;
} else {
r.category = Category::Normal;
if r.exp == Self::MIN_EXP - 1 {
// Denormal.
r.exp = Self::MIN_EXP;
} else {
// Set integer bit.
sig::set_bit(&mut r.sig, Self::PRECISION - 1);
}
}
r
}
fn to_bits(x: IeeeFloat<Self>) -> u128 {
assert!(Self::BITS > Self::PRECISION);
// Split integer bit from significand.
let integer_bit = sig::get_bit(&x.sig, Self::PRECISION - 1);
let mut significand = x.sig[0] & ((1 << (Self::PRECISION - 1)) - 1);
let exponent = match x.category {
Category::Normal => {
if x.exp == Self::MIN_EXP && !integer_bit {
// Denormal.
Self::MIN_EXP - 1
} else {
x.exp
}
}
Category::Zero => {
// FIXME(eddyb) Maybe we should guarantee an invariant instead?
significand = 0;
Self::MIN_EXP - 1
}
Category::Infinity => {
// FIXME(eddyb) Maybe we should guarantee an invariant instead?
significand = 0;
Self::MAX_EXP + 1
}
Category::NaN => Self::MAX_EXP + 1,
};
// Convert the exponent from a signed integer to its bias representation.
let exponent = (exponent + Self::MAX_EXP) as u128;
((x.sign as u128) << (Self::BITS - 1)) | (exponent << (Self::PRECISION - 1)) | significand
}
}
impl<S> Copy for IeeeFloat<S> {}
impl<S> Clone for IeeeFloat<S> {
fn clone(&self) -> Self {
*self
}
}
macro_rules! ieee_semantics {
($($name:ident = $sem:ident($bits:tt : $exp_bits:tt)),*) => {
$(pub struct $sem;)*
$(pub type $name = IeeeFloat<$sem>;)*
$(impl Semantics for $sem {
const BITS: usize = $bits;
const PRECISION: usize = ($bits - 1 - $exp_bits) + 1;
const MAX_EXP: ExpInt = (1 << ($exp_bits - 1)) - 1;
})*
}
}
ieee_semantics! {
Half = HalfS(16:5),
Single = SingleS(32:8),
Double = DoubleS(64:11),
Quad = QuadS(128:15)
}
pub struct X87DoubleExtendedS;
pub type X87DoubleExtended = IeeeFloat<X87DoubleExtendedS>;
impl Semantics for X87DoubleExtendedS {
const BITS: usize = 80;
const PRECISION: usize = 64;
const MAX_EXP: ExpInt = (1 << (15 - 1)) - 1;
/// For x87 extended precision, we want to make a NaN, not a
/// pseudo-NaN. Maybe we should expose the ability to make
/// pseudo-NaNs?
const QNAN_SIGNIFICAND: Limb = 0b11 << Self::QNAN_BIT;
/// Integer bit is explicit in this format. Intel hardware (387 and later)
/// does not support these bit patterns:
/// exponent = all 1's, integer bit 0, significand 0 ("pseudoinfinity")
/// exponent = all 1's, integer bit 0, significand nonzero ("pseudoNaN")
/// exponent = 0, integer bit 1 ("pseudodenormal")
/// exponent != 0 nor all 1's, integer bit 0 ("unnormal")
/// At the moment, the first two are treated as NaNs, the second two as Normal.
fn from_bits(bits: u128) -> IeeeFloat<Self> {
let sign = bits & (1 << (Self::BITS - 1));
let exponent = (bits & !sign) >> Self::PRECISION;
let mut r = IeeeFloat {
sig: [bits & ((1 << (Self::PRECISION - 1)) - 1)],
// Convert the exponent from its bias representation to a signed integer.
exp: (exponent as ExpInt) - Self::MAX_EXP,
category: Category::Zero,
sign: sign != 0,
marker: PhantomData,
};
if r.exp == Self::MIN_EXP - 1 && r.sig == [0] {
// Exponent, significand meaningless.
r.category = Category::Zero;
} else if r.exp == Self::MAX_EXP + 1 && r.sig == [1 << (Self::PRECISION - 1)] {
// Exponent, significand meaningless.
r.category = Category::Infinity;
} else if r.exp == Self::MAX_EXP + 1 && r.sig != [1 << (Self::PRECISION - 1)] {
// Sign, exponent, significand meaningless.
r.category = Category::NaN;
} else {
r.category = Category::Normal;
if r.exp == Self::MIN_EXP - 1 {
// Denormal.
r.exp = Self::MIN_EXP;
}
}
r
}
fn to_bits(x: IeeeFloat<Self>) -> u128 {
// Get integer bit from significand.
let integer_bit = sig::get_bit(&x.sig, Self::PRECISION - 1);
let mut significand = x.sig[0] & ((1 << Self::PRECISION) - 1);
let exponent = match x.category {
Category::Normal => {
if x.exp == Self::MIN_EXP && !integer_bit {
// Denormal.
Self::MIN_EXP - 1
} else {
x.exp
}
}
Category::Zero => {
// FIXME(eddyb) Maybe we should guarantee an invariant instead?
significand = 0;
Self::MIN_EXP - 1
}
Category::Infinity => {
// FIXME(eddyb) Maybe we should guarantee an invariant instead?
significand = 1 << (Self::PRECISION - 1);
Self::MAX_EXP + 1
}
Category::NaN => Self::MAX_EXP + 1,
};
// Convert the exponent from a signed integer to its bias representation.
let exponent = (exponent + Self::MAX_EXP) as u128;
((x.sign as u128) << (Self::BITS - 1)) | (exponent << Self::PRECISION) | significand
}
}
float_common_impls!(IeeeFloat<S>);
impl<S: Semantics> PartialEq for IeeeFloat<S> {
fn eq(&self, rhs: &Self) -> bool {
self.partial_cmp(rhs) == Some(Ordering::Equal)
}
}
impl<S: Semantics> PartialOrd for IeeeFloat<S> {
fn partial_cmp(&self, rhs: &Self) -> Option<Ordering> {
match (self.category, rhs.category) {
(Category::NaN, _) |
(_, Category::NaN) => None,
(Category::Infinity, Category::Infinity) => Some((!self.sign).cmp(&(!rhs.sign))),
(Category::Zero, Category::Zero) => Some(Ordering::Equal),
(Category::Infinity, _) |
(Category::Normal, Category::Zero) => Some((!self.sign).cmp(&self.sign)),
(_, Category::Infinity) |
(Category::Zero, Category::Normal) => Some(rhs.sign.cmp(&(!rhs.sign))),
(Category::Normal, Category::Normal) => {
// Two normal numbers. Do they have the same sign?
Some((!self.sign).cmp(&(!rhs.sign)).then_with(|| {
// Compare absolute values; invert result if negative.
let result = self.cmp_abs_normal(*rhs);
if self.sign { result.reverse() } else { result }
}))
}
}
}
}
impl<S> Neg for IeeeFloat<S> {
type Output = Self;
fn neg(mut self) -> Self {
self.sign = !self.sign;
self
}
}
/// Prints this value as a decimal string.
///
/// \param precision The maximum number of digits of
/// precision to output. If there are fewer digits available,
/// zero padding will not be used unless the value is
/// integral and small enough to be expressed in
/// precision digits. 0 means to use the natural
/// precision of the number.
/// \param width The maximum number of zeros to
/// consider inserting before falling back to scientific
/// notation. 0 means to always use scientific notation.
///
/// \param alternate Indicate whether to remove the trailing zero in
/// fraction part or not. Also setting this parameter to true forces
/// producing of output more similar to default printf behavior.
/// Specifically the lower e is used as exponent delimiter and exponent
/// always contains no less than two digits.
///
/// Number precision width Result
/// ------ --------- ----- ------
/// 1.01E+4 5 2 10100
/// 1.01E+4 4 2 1.01E+4
/// 1.01E+4 5 1 1.01E+4
/// 1.01E-2 5 2 0.0101
/// 1.01E-2 4 2 0.0101
/// 1.01E-2 4 1 1.01E-2
impl<S: Semantics> fmt::Display for IeeeFloat<S> {
fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
let width = f.width().unwrap_or(3);
let alternate = f.alternate();
match self.category {
Category::Infinity => {
if self.sign {
return f.write_str("-Inf");
} else {
return f.write_str("+Inf");
}
}
Category::NaN => return f.write_str("NaN"),
Category::Zero => {
if self.sign {
f.write_char('-')?;
}
if width == 0 {
if alternate {
f.write_str("0.0")?;
if let Some(n) = f.precision() {
for _ in 1..n {
f.write_char('0')?;
}
}
f.write_str("e+00")?;
} else {
f.write_str("0.0E+0")?;
}
} else {
f.write_char('0')?;
}
return Ok(());
}
Category::Normal => {}
}
if self.sign {
f.write_char('-')?;
}
// We use enough digits so the number can be round-tripped back to an
// APFloat. The formula comes from "How to Print Floating-Point Numbers
// Accurately" by Steele and White.
// FIXME: Using a formula based purely on the precision is conservative;
// we can print fewer digits depending on the actual value being printed.
// precision = 2 + floor(S::PRECISION / lg_2(10))
let precision = f.precision().unwrap_or(2 + S::PRECISION * 59 / 196);
// Decompose the number into an APInt and an exponent.
let mut exp = self.exp - (S::PRECISION as ExpInt - 1);
let mut sig = vec![self.sig[0]];
// Ignore trailing binary zeros.
let trailing_zeros = sig[0].trailing_zeros();
let _: Loss = sig::shift_right(&mut sig, &mut exp, trailing_zeros as usize);
// Change the exponent from 2^e to 10^e.
if exp == 0 {
// Nothing to do.
} else if exp > 0 {
// Just shift left.
let shift = exp as usize;
sig.resize(limbs_for_bits(S::PRECISION + shift), 0);
sig::shift_left(&mut sig, &mut exp, shift);
} else {
// exp < 0
let mut texp = -exp as usize;
// We transform this using the identity:
// (N)(2^-e) == (N)(5^e)(10^-e)
// Multiply significand by 5^e.
// N * 5^0101 == N * 5^(1*1) * 5^(0*2) * 5^(1*4) * 5^(0*8)
let mut sig_scratch = vec![];
let mut p5 = vec![];
let mut p5_scratch = vec![];
while texp != 0 {
if p5.is_empty() {
p5.push(5);
} else {
p5_scratch.resize(p5.len() * 2, 0);
let _: Loss =
sig::mul(&mut p5_scratch, &mut 0, &p5, &p5, p5.len() * 2 * LIMB_BITS);
while p5_scratch.last() == Some(&0) {
p5_scratch.pop();
}
mem::swap(&mut p5, &mut p5_scratch);
}
if texp & 1 != 0 {
sig_scratch.resize(sig.len() + p5.len(), 0);
let _: Loss = sig::mul(
&mut sig_scratch,
&mut 0,
&sig,
&p5,
(sig.len() + p5.len()) * LIMB_BITS,
);
while sig_scratch.last() == Some(&0) {
sig_scratch.pop();
}
mem::swap(&mut sig, &mut sig_scratch);
}
texp >>= 1;
}
}
// Fill the buffer.
let mut buffer = vec![];
// Ignore digits from the significand until it is no more
// precise than is required for the desired precision.
// 196/59 is a very slight overestimate of lg_2(10).
let required = (precision * 196 + 58) / 59;
let mut discard_digits = sig::omsb(&sig).saturating_sub(required) * 59 / 196;
let mut in_trail = true;
while !sig.is_empty() {
// Perform short division by 10 to extract the rightmost digit.
// rem <- sig % 10
// sig <- sig / 10
let mut rem = 0;
// Use 64-bit division and remainder, with 32-bit chunks from sig.
sig::each_chunk(&mut sig, 32, |chunk| {
let chunk = chunk as u32;
let combined = ((rem as u64) << 32) | (chunk as u64);
rem = (combined % 10) as u8;
(combined / 10) as u32 as Limb
});
// Reduce the sigificand to avoid wasting time dividing 0's.
while sig.last() == Some(&0) {
sig.pop();
}
let digit = rem;
// Ignore digits we don't need.
if discard_digits > 0 {
discard_digits -= 1;
exp += 1;
continue;
}
// Drop trailing zeros.
if in_trail && digit == 0 {
exp += 1;
} else {
in_trail = false;
buffer.push(b'0' + digit);
}
}
assert!(!buffer.is_empty(), "no characters in buffer!");
// Drop down to precision.
// FIXME: don't do more precise calculations above than are required.
if buffer.len() > precision {
// The most significant figures are the last ones in the buffer.
let mut first_sig = buffer.len() - precision;
// Round.
// FIXME: this probably shouldn't use 'round half up'.
// Rounding down is just a truncation, except we also want to drop
// trailing zeros from the new result.
if buffer[first_sig - 1] < b'5' {
while first_sig < buffer.len() && buffer[first_sig] == b'0' {
first_sig += 1;
}
} else {
// Rounding up requires a decimal add-with-carry. If we continue
// the carry, the newly-introduced zeros will just be truncated.
for x in &mut buffer[first_sig..] {
if *x == b'9' {
first_sig += 1;
} else {
*x += 1;
break;
}
}
}
exp += first_sig as ExpInt;
buffer.drain(..first_sig);
// If we carried through, we have exactly one digit of precision.
if buffer.is_empty() {
buffer.push(b'1');
}
}
let digits = buffer.len();
// Check whether we should use scientific notation.
let scientific = if width == 0 {
true
} else if exp >= 0 {
// 765e3 --> 765000
// ^^^
// But we shouldn't make the number look more precise than it is.
exp as usize > width || digits + exp as usize > precision
} else {
// Power of the most significant digit.
let msd = exp + (digits - 1) as ExpInt;
if msd >= 0 {
// 765e-2 == 7.65
false
} else {
// 765e-5 == 0.00765
// ^ ^^
-msd as usize > width
}
};
// Scientific formatting is pretty straightforward.
if scientific {
exp += digits as ExpInt - 1;
f.write_char(buffer[digits - 1] as char)?;
f.write_char('.')?;
let truncate_zero = !alternate;
if digits == 1 && truncate_zero {
f.write_char('0')?;
} else {
for &d in buffer[..digits - 1].iter().rev() {
f.write_char(d as char)?;
}
}
// Fill with zeros up to precision.
if !truncate_zero && precision > digits - 1 {
for _ in 0..=precision - digits {
f.write_char('0')?;
}
}
// For alternate we use lower 'e'.
f.write_char(if alternate { 'e' } else { 'E' })?;
// Exponent always at least two digits if we do not truncate zeros.
if truncate_zero {
write!(f, "{:+}", exp)?;
} else {
write!(f, "{:+03}", exp)?;
}
return Ok(());
}
// Non-scientific, positive exponents.
if exp >= 0 {
for &d in buffer.iter().rev() {
f.write_char(d as char)?;
}
for _ in 0..exp {
f.write_char('0')?;
}
return Ok(());
}
// Non-scientific, negative exponents.
let unit_place = -exp as usize;
if unit_place < digits {
for &d in buffer[unit_place..].iter().rev() {
f.write_char(d as char)?;
}
f.write_char('.')?;
for &d in buffer[..unit_place].iter().rev() {
f.write_char(d as char)?;
}
} else {
f.write_str("0.")?;
for _ in digits..unit_place {
f.write_char('0')?;
}
for &d in buffer.iter().rev() {
f.write_char(d as char)?;
}
}
Ok(())
}
}
impl<S: Semantics> fmt::Debug for IeeeFloat<S> {
fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
write!(f, "{}({:?} | {}{:?} * 2^{})",
self, self.category,
if self.sign { "-" } else { "+" },
self.sig,
self.exp)
}
}
impl<S: Semantics> Float for IeeeFloat<S> {
const BITS: usize = S::BITS;
const PRECISION: usize = S::PRECISION;
const MAX_EXP: ExpInt = S::MAX_EXP;
const MIN_EXP: ExpInt = S::MIN_EXP;
const ZERO: Self = IeeeFloat {
sig: [0],
exp: S::MIN_EXP - 1,
category: Category::Zero,
sign: false,
marker: PhantomData,
};
const INFINITY: Self = IeeeFloat {
sig: [0],
exp: S::MAX_EXP + 1,
category: Category::Infinity,
sign: false,
marker: PhantomData,
};
// FIXME(eddyb) remove when qnan becomes const fn.
const NAN: Self = IeeeFloat {
sig: [S::QNAN_SIGNIFICAND],
exp: S::MAX_EXP + 1,
category: Category::NaN,
sign: false,
marker: PhantomData,
};
fn qnan(payload: Option<u128>) -> Self {
IeeeFloat {
sig: [
S::QNAN_SIGNIFICAND |
payload.map_or(0, |payload| {
// Zero out the excess bits of the significand.
payload & ((1 << S::QNAN_BIT) - 1)
}),
],
exp: S::MAX_EXP + 1,
category: Category::NaN,
sign: false,
marker: PhantomData,
}
}
fn snan(payload: Option<u128>) -> Self {
let mut snan = Self::qnan(payload);
// We always have to clear the QNaN bit to make it an SNaN.
sig::clear_bit(&mut snan.sig, S::QNAN_BIT);
// If there are no bits set in the payload, we have to set
// *something* to make it a NaN instead of an infinity;
// conventionally, this is the next bit down from the QNaN bit.
if snan.sig[0] & !S::QNAN_SIGNIFICAND == 0 {
sig::set_bit(&mut snan.sig, S::QNAN_BIT - 1);
}
snan
}
fn largest() -> Self {
// We want (in interchange format):
// exponent = 1..10
// significand = 1..1
IeeeFloat {
sig: [(1 << S::PRECISION) - 1],
exp: S::MAX_EXP,
category: Category::Normal,
sign: false,
marker: PhantomData,
}
}
// We want (in interchange format):
// exponent = 0..0
// significand = 0..01
const SMALLEST: Self = IeeeFloat {
sig: [1],
exp: S::MIN_EXP,
category: Category::Normal,
sign: false,
marker: PhantomData,
};
fn smallest_normalized() -> Self {
// We want (in interchange format):
// exponent = 0..0
// significand = 10..0
IeeeFloat {
sig: [1 << (S::PRECISION - 1)],
exp: S::MIN_EXP,
category: Category::Normal,
sign: false,
marker: PhantomData,
}
}
fn add_r(mut self, rhs: Self, round: Round) -> StatusAnd<Self> {
let status = match (self.category, rhs.category) {
(Category::Infinity, Category::Infinity) => {
// Differently signed infinities can only be validly
// subtracted.
if self.sign != rhs.sign {
self = Self::NAN;
Status::INVALID_OP
} else {
Status::OK
}
}
// Sign may depend on rounding mode; handled below.
(_, Category::Zero) |
(Category::NaN, _) |
(Category::Infinity, Category::Normal) => Status::OK,
(Category::Zero, _) |
(_, Category::NaN) |
(_, Category::Infinity) => {
self = rhs;
Status::OK
}
// This return code means it was not a simple case.
(Category::Normal, Category::Normal) => {
let loss = sig::add_or_sub(
&mut self.sig,
&mut self.exp,
&mut self.sign,
&mut [rhs.sig[0]],
rhs.exp,
rhs.sign,
);
let status;
self = unpack!(status=, self.normalize(round, loss));
// Can only be zero if we lost no fraction.
assert!(self.category != Category::Zero || loss == Loss::ExactlyZero);
status
}
};
// If two numbers add (exactly) to zero, IEEE 754 decrees it is a
// positive zero unless rounding to minus infinity, except that
// adding two like-signed zeroes gives that zero.
if self.category == Category::Zero &&
(rhs.category != Category::Zero || self.sign != rhs.sign)
{
self.sign = round == Round::TowardNegative;
}
status.and(self)
}
fn mul_r(mut self, rhs: Self, round: Round) -> StatusAnd<Self> {
self.sign ^= rhs.sign;
match (self.category, rhs.category) {
(Category::NaN, _) => {
self.sign = false;
Status::OK.and(self)
}
(_, Category::NaN) => {
self.sign = false;
self.category = Category::NaN;
self.sig = rhs.sig;
Status::OK.and(self)
}
(Category::Zero, Category::Infinity) |
(Category::Infinity, Category::Zero) => Status::INVALID_OP.and(Self::NAN),
(_, Category::Infinity) |
(Category::Infinity, _) => {
self.category = Category::Infinity;
Status::OK.and(self)
}
(Category::Zero, _) |
(_, Category::Zero) => {
self.category = Category::Zero;
Status::OK.and(self)
}
(Category::Normal, Category::Normal) => {
self.exp += rhs.exp;
let mut wide_sig = [0; 2];
let loss = sig::mul(
&mut wide_sig,
&mut self.exp,
&self.sig,
&rhs.sig,
S::PRECISION,
);
self.sig = [wide_sig[0]];
let mut status;
self = unpack!(status=, self.normalize(round, loss));
if loss != Loss::ExactlyZero {
status |= Status::INEXACT;
}
status.and(self)
}
}
}
fn mul_add_r(mut self, multiplicand: Self, addend: Self, round: Round) -> StatusAnd<Self> {
// If and only if all arguments are normal do we need to do an
// extended-precision calculation.
if !self.is_finite_non_zero() || !multiplicand.is_finite_non_zero() || !addend.is_finite() {
let mut status;
self = unpack!(status=, self.mul_r(multiplicand, round));
// FS can only be Status::OK or Status::INVALID_OP. There is no more work
// to do in the latter case. The IEEE-754R standard says it is
// implementation-defined in this case whether, if ADDEND is a
// quiet NaN, we raise invalid op; this implementation does so.
//
// If we need to do the addition we can do so with normal
// precision.
if status == Status::OK {
self = unpack!(status=, self.add_r(addend, round));
}
return status.and(self);
}
// Post-multiplication sign, before addition.
self.sign ^= multiplicand.sign;
// Allocate space for twice as many bits as the original significand, plus one
// extra bit for the addition to overflow into.
assert!(limbs_for_bits(S::PRECISION * 2 + 1) <= 2);
let mut wide_sig = sig::widening_mul(self.sig[0], multiplicand.sig[0]);
let mut loss = Loss::ExactlyZero;
let mut omsb = sig::omsb(&wide_sig);
self.exp += multiplicand.exp;
// Assume the operands involved in the multiplication are single-precision
// FP, and the two multiplicants are:
// lhs = a23 . a22 ... a0 * 2^e1
// rhs = b23 . b22 ... b0 * 2^e2
// the result of multiplication is:
// lhs = c48 c47 c46 . c45 ... c0 * 2^(e1+e2)
// Note that there are three significant bits at the left-hand side of the
// radix point: two for the multiplication, and an overflow bit for the
// addition (that will always be zero at this point). Move the radix point
// toward left by two bits, and adjust exponent accordingly.
self.exp += 2;
if addend.is_non_zero() {
// Normalize our MSB to one below the top bit to allow for overflow.
let ext_precision = 2 * S::PRECISION + 1;
if omsb != ext_precision - 1 {
assert!(ext_precision > omsb);
sig::shift_left(&mut wide_sig, &mut self.exp, (ext_precision - 1) - omsb);
}
// The intermediate result of the multiplication has "2 * S::PRECISION"
// significant bit; adjust the addend to be consistent with mul result.
let mut ext_addend_sig = [addend.sig[0], 0];
// Extend the addend significand to ext_precision - 1. This guarantees
// that the high bit of the significand is zero (same as wide_sig),
// so the addition will overflow (if it does overflow at all) into the top bit.
sig::shift_left(
&mut ext_addend_sig,
&mut 0,
ext_precision - 1 - S::PRECISION,
);
loss = sig::add_or_sub(
&mut wide_sig,
&mut self.exp,
&mut self.sign,
&mut ext_addend_sig,
addend.exp + 1,
addend.sign,
);
omsb = sig::omsb(&wide_sig);
}
// Convert the result having "2 * S::PRECISION" significant-bits back to the one
// having "S::PRECISION" significant-bits. First, move the radix point from
// position "2*S::PRECISION - 1" to "S::PRECISION - 1". The exponent need to be
// adjusted by "2*S::PRECISION - 1" - "S::PRECISION - 1" = "S::PRECISION".
self.exp -= S::PRECISION as ExpInt + 1;
// In case MSB resides at the left-hand side of radix point, shift the
// mantissa right by some amount to make sure the MSB reside right before
// the radix point (i.e., "MSB . rest-significant-bits").
if omsb > S::PRECISION {
let bits = omsb - S::PRECISION;
loss = sig::shift_right(&mut wide_sig, &mut self.exp, bits).combine(loss);
}
self.sig[0] = wide_sig[0];
let mut status;
self = unpack!(status=, self.normalize(round, loss));
if loss != Loss::ExactlyZero {
status |= Status::INEXACT;
}
// If two numbers add (exactly) to zero, IEEE 754 decrees it is a
// positive zero unless rounding to minus infinity, except that
// adding two like-signed zeroes gives that zero.
if self.category == Category::Zero && !status.intersects(Status::UNDERFLOW) &&
self.sign != addend.sign
{
self.sign = round == Round::TowardNegative;
}
status.and(self)
}
fn div_r(mut self, rhs: Self, round: Round) -> StatusAnd<Self> {
self.sign ^= rhs.sign;
match (self.category, rhs.category) {
(Category::NaN, _) => {
self.sign = false;
Status::OK.and(self)
}
(_, Category::NaN) => {
self.category = Category::NaN;
self.sig = rhs.sig;
self.sign = false;
Status::OK.and(self)
}
(Category::Infinity, Category::Infinity) |
(Category::Zero, Category::Zero) => Status::INVALID_OP.and(Self::NAN),
(Category::Infinity, _) |
(Category::Zero, _) => Status::OK.and(self),
(Category::Normal, Category::Infinity) => {
self.category = Category::Zero;
Status::OK.and(self)
}
(Category::Normal, Category::Zero) => {
self.category = Category::Infinity;
Status::DIV_BY_ZERO.and(self)
}
(Category::Normal, Category::Normal) => {
self.exp -= rhs.exp;
let dividend = self.sig[0];
let loss = sig::div(
&mut self.sig,
&mut self.exp,
&mut [dividend],
&mut [rhs.sig[0]],
S::PRECISION,
);
let mut status;
self = unpack!(status=, self.normalize(round, loss));
if loss != Loss::ExactlyZero {
status |= Status::INEXACT;
}
status.and(self)
}
}
}
fn c_fmod(mut self, rhs: Self) -> StatusAnd<Self> {
match (self.category, rhs.category) {
(Category::NaN, _) |
(Category::Zero, Category::Infinity) |
(Category::Zero, Category::Normal) |
(Category::Normal, Category::Infinity) => Status::OK.and(self),
(_, Category::NaN) => {
self.sign = false;
self.category = Category::NaN;
self.sig = rhs.sig;
Status::OK.and(self)
}
(Category::Infinity, _) |
(_, Category::Zero) => Status::INVALID_OP.and(Self::NAN),
(Category::Normal, Category::Normal) => {
while self.is_finite_non_zero() && rhs.is_finite_non_zero() &&
self.cmp_abs_normal(rhs) != Ordering::Less
{
let mut v = rhs.scalbn(self.ilogb() - rhs.ilogb());
if self.cmp_abs_normal(v) == Ordering::Less {
v = v.scalbn(-1);
}
v.sign = self.sign;
let status;
self = unpack!(status=, self - v);
assert_eq!(status, Status::OK);
}
Status::OK.and(self)
}
}
}
fn round_to_integral(self, round: Round) -> StatusAnd<Self> {
// If the exponent is large enough, we know that this value is already
// integral, and the arithmetic below would potentially cause it to saturate
// to +/-Inf. Bail out early instead.
if self.is_finite_non_zero() && self.exp + 1 >= S::PRECISION as ExpInt {
return Status::OK.and(self);
}
// The algorithm here is quite simple: we add 2^(p-1), where p is the
// precision of our format, and then subtract it back off again. The choice
// of rounding modes for the addition/subtraction determines the rounding mode
// for our integral rounding as well.
// NOTE: When the input value is negative, we do subtraction followed by
// addition instead.
assert!(S::PRECISION <= 128);
let mut status;
let magic_const = unpack!(status=, Self::from_u128(1 << (S::PRECISION - 1)));
let magic_const = magic_const.copy_sign(self);
if status != Status::OK {
return status.and(self);
}
let mut r = self;
r = unpack!(status=, r.add_r(magic_const, round));
if status != Status::OK && status != Status::INEXACT {
return status.and(self);
}
// Restore the input sign to handle 0.0/-0.0 cases correctly.
r.sub_r(magic_const, round).map(|r| r.copy_sign(self))
}
fn next_up(mut self) -> StatusAnd<Self> {
// Compute nextUp(x), handling each float category separately.
match self.category {
Category::Infinity => {
if self.sign {
// nextUp(-inf) = -largest
Status::OK.and(-Self::largest())
} else {
// nextUp(+inf) = +inf
Status::OK.and(self)
}
}
Category::NaN => {
// IEEE-754R 2008 6.2 Par 2: nextUp(sNaN) = qNaN. Set Invalid flag.
// IEEE-754R 2008 6.2: nextUp(qNaN) = qNaN. Must be identity so we do not
// change the payload.
if self.is_signaling() {
// For consistency, propagate the sign of the sNaN to the qNaN.
Status::INVALID_OP.and(Self::NAN.copy_sign(self))
} else {
Status::OK.and(self)
}
}
Category::Zero => {
// nextUp(pm 0) = +smallest
Status::OK.and(Self::SMALLEST)
}
Category::Normal => {
// nextUp(-smallest) = -0
if self.is_smallest() && self.sign {
return Status::OK.and(-Self::ZERO);
}
// nextUp(largest) == INFINITY
if self.is_largest() && !self.sign {
return Status::OK.and(Self::INFINITY);
}
// Excluding the integral bit. This allows us to test for binade boundaries.
let sig_mask = (1 << (S::PRECISION - 1)) - 1;
// nextUp(normal) == normal + inc.
if self.sign {
// If we are negative, we need to decrement the significand.
// We only cross a binade boundary that requires adjusting the exponent
// if:
// 1. exponent != S::MIN_EXP. This implies we are not in the
// smallest binade or are dealing with denormals.
// 2. Our significand excluding the integral bit is all zeros.
let crossing_binade_boundary = self.exp != S::MIN_EXP &&
self.sig[0] & sig_mask == 0;
// Decrement the significand.
//
// We always do this since:
// 1. If we are dealing with a non-binade decrement, by definition we
// just decrement the significand.
// 2. If we are dealing with a normal -> normal binade decrement, since
// we have an explicit integral bit the fact that all bits but the
// integral bit are zero implies that subtracting one will yield a
// significand with 0 integral bit and 1 in all other spots. Thus we
// must just adjust the exponent and set the integral bit to 1.
// 3. If we are dealing with a normal -> denormal binade decrement,
// since we set the integral bit to 0 when we represent denormals, we
// just decrement the significand.
sig::decrement(&mut self.sig);
if crossing_binade_boundary {
// Our result is a normal number. Do the following:
// 1. Set the integral bit to 1.
// 2. Decrement the exponent.
sig::set_bit(&mut self.sig, S::PRECISION - 1);
self.exp -= 1;
}
} else {
// If we are positive, we need to increment the significand.
// We only cross a binade boundary that requires adjusting the exponent if
// the input is not a denormal and all of said input's significand bits
// are set. If all of said conditions are true: clear the significand, set
// the integral bit to 1, and increment the exponent. If we have a
// denormal always increment since moving denormals and the numbers in the
// smallest normal binade have the same exponent in our representation.
let crossing_binade_boundary = !self.is_denormal() &&
self.sig[0] & sig_mask == sig_mask;
if crossing_binade_boundary {
self.sig = [0];
sig::set_bit(&mut self.sig, S::PRECISION - 1);
assert_ne!(
self.exp,
S::MAX_EXP,
"We can not increment an exponent beyond the MAX_EXP \
allowed by the given floating point semantics."
);
self.exp += 1;
} else {
sig::increment(&mut self.sig);
}
}
Status::OK.and(self)
}
}
}
fn from_bits(input: u128) -> Self {
// Dispatch to semantics.
S::from_bits(input)
}
fn from_u128_r(input: u128, round: Round) -> StatusAnd<Self> {
IeeeFloat {
sig: [input],
exp: S::PRECISION as ExpInt - 1,
category: Category::Normal,
sign: false,
marker: PhantomData,
}.normalize(round, Loss::ExactlyZero)
}
fn from_str_r(mut s: &str, mut round: Round) -> Result<StatusAnd<Self>, ParseError> {
if s.is_empty() {
return Err(ParseError("Invalid string length"));
}
// Handle special cases.
match s {
"inf" | "INFINITY" => return Ok(Status::OK.and(Self::INFINITY)),
"-inf" | "-INFINITY" => return Ok(Status::OK.and(-Self::INFINITY)),
"nan" | "NaN" => return Ok(Status::OK.and(Self::NAN)),
"-nan" | "-NaN" => return Ok(Status::OK.and(-Self::NAN)),
_ => {}
}
// Handle a leading minus sign.
let minus = s.starts_with("-");
if minus || s.starts_with("+") {
s = &s[1..];
if s.is_empty() {
return Err(ParseError("String has no digits"));
}
}
// Adjust the rounding mode for the absolute value below.
if minus {
round = -round;
}
let r = if s.starts_with("0x") || s.starts_with("0X") {
s = &s[2..];
if s.is_empty() {
return Err(ParseError("Invalid string"));
}
Self::from_hexadecimal_string(s, round)?
} else {
Self::from_decimal_string(s, round)?
};
Ok(r.map(|r| if minus { -r } else { r }))
}
fn to_bits(self) -> u128 {
// Dispatch to semantics.
S::to_bits(self)
}
fn to_u128_r(self, width: usize, round: Round, is_exact: &mut bool) -> StatusAnd<u128> {
// The result of trying to convert a number too large.
let overflow = if self.sign {
// Negative numbers cannot be represented as unsigned.
0
} else {
// Largest unsigned integer of the given width.
!0 >> (128 - width)
};
*is_exact = false;
match self.category {
Category::NaN => Status::INVALID_OP.and(0),
Category::Infinity => Status::INVALID_OP.and(overflow),
Category::Zero => {
// Negative zero can't be represented as an int.
*is_exact = !self.sign;
Status::OK.and(0)
}
Category::Normal => {
let mut r = 0;
// Step 1: place our absolute value, with any fraction truncated, in
// the destination.
let truncated_bits = if self.exp < 0 {
// Our absolute value is less than one; truncate everything.
// For exponent -1 the integer bit represents .5, look at that.
// For smaller exponents leftmost truncated bit is 0.
S::PRECISION - 1 + (-self.exp) as usize
} else {
// We want the most significant (exponent + 1) bits; the rest are
// truncated.
let bits = self.exp as usize + 1;
// Hopelessly large in magnitude?
if bits > width {
return Status::INVALID_OP.and(overflow);
}
if bits < S::PRECISION {
// We truncate (S::PRECISION - bits) bits.
r = self.sig[0] >> (S::PRECISION - bits);
S::PRECISION - bits
} else {
// We want at least as many bits as are available.
r = self.sig[0] << (bits - S::PRECISION);
0
}
};
// Step 2: work out any lost fraction, and increment the absolute
// value if we would round away from zero.
let mut loss = Loss::ExactlyZero;
if truncated_bits > 0 {
loss = Loss::through_truncation(&self.sig, truncated_bits);
if loss != Loss::ExactlyZero &&
self.round_away_from_zero(round, loss, truncated_bits)
{
r = r.wrapping_add(1);
if r == 0 {
return Status::INVALID_OP.and(overflow); // Overflow.
}
}
}
// Step 3: check if we fit in the destination.
if r > overflow {
return Status::INVALID_OP.and(overflow);
}
if loss == Loss::ExactlyZero {
*is_exact = true;
Status::OK.and(r)
} else {
Status::INEXACT.and(r)
}
}
}
}
fn cmp_abs_normal(self, rhs: Self) -> Ordering {
assert!(self.is_finite_non_zero());
assert!(rhs.is_finite_non_zero());
// If exponents are equal, do an unsigned comparison of the significands.
self.exp.cmp(&rhs.exp).then_with(
|| sig::cmp(&self.sig, &rhs.sig),
)
}
fn bitwise_eq(self, rhs: Self) -> bool {
if self.category != rhs.category || self.sign != rhs.sign {
return false;
}
if self.category == Category::Zero || self.category == Category::Infinity {
return true;
}
if self.is_finite_non_zero() && self.exp != rhs.exp {
return false;
}
self.sig == rhs.sig
}
fn is_negative(self) -> bool {
self.sign
}
fn is_denormal(self) -> bool {
self.is_finite_non_zero() && self.exp == S::MIN_EXP &&
!sig::get_bit(&self.sig, S::PRECISION - 1)
}
fn is_signaling(self) -> bool {
// IEEE-754R 2008 6.2.1: A signaling NaN bit string should be encoded with the
// first bit of the trailing significand being 0.
self.is_nan() && !sig::get_bit(&self.sig, S::QNAN_BIT)
}
fn category(self) -> Category {
self.category
}
fn get_exact_inverse(self) -> Option<Self> {
// Special floats and denormals have no exact inverse.
if !self.is_finite_non_zero() {
return None;
}
// Check that the number is a power of two by making sure that only the
// integer bit is set in the significand.
if self.sig != [1 << (S::PRECISION - 1)] {
return None;
}
// Get the inverse.
let mut reciprocal = Self::from_u128(1).value;
let status;
reciprocal = unpack!(status=, reciprocal / self);
if status != Status::OK {
return None;
}
// Avoid multiplication with a denormal, it is not safe on all platforms and
// may be slower than a normal division.
if reciprocal.is_denormal() {
return None;
}
assert!(reciprocal.is_finite_non_zero());
assert_eq!(reciprocal.sig, [1 << (S::PRECISION - 1)]);
Some(reciprocal)
}
fn ilogb(mut self) -> ExpInt {
if self.is_nan() {
return IEK_NAN;
}
if self.is_zero() {
return IEK_ZERO;
}
if self.is_infinite() {
return IEK_INF;
}
if !self.is_denormal() {
return self.exp;
}
let sig_bits = (S::PRECISION - 1) as ExpInt;
self.exp += sig_bits;
self = self.normalize(Round::NearestTiesToEven, Loss::ExactlyZero)
.value;
self.exp - sig_bits
}
fn scalbn_r(mut self, exp: ExpInt, round: Round) -> Self {
// If exp is wildly out-of-scale, simply adding it to self.exp will
// overflow; clamp it to a safe range before adding, but ensure that the range
// is large enough that the clamp does not change the result. The range we
// need to support is the difference between the largest possible exponent and
// the normalized exponent of half the smallest denormal.
let sig_bits = (S::PRECISION - 1) as i32;
let max_change = S::MAX_EXP as i32 - (S::MIN_EXP as i32 - sig_bits) + 1;
// Clamp to one past the range ends to let normalize handle overflow.
let exp_change = cmp::min(cmp::max(exp as i32, -max_change - 1), max_change);
self.exp = self.exp.saturating_add(exp_change as ExpInt);
self = self.normalize(round, Loss::ExactlyZero).value;
if self.is_nan() {
sig::set_bit(&mut self.sig, S::QNAN_BIT);
}
self
}
fn frexp_r(mut self, exp: &mut ExpInt, round: Round) -> Self {
*exp = self.ilogb();
// Quiet signalling nans.
if *exp == IEK_NAN {
sig::set_bit(&mut self.sig, S::QNAN_BIT);
return self;
}
if *exp == IEK_INF {
return self;
}
// 1 is added because frexp is defined to return a normalized fraction in
// +/-[0.5, 1.0), rather than the usual +/-[1.0, 2.0).
if *exp == IEK_ZERO {
*exp = 0;
} else {
*exp += 1;
}
self.scalbn_r(-*exp, round)
}
}
impl<S: Semantics, T: Semantics> FloatConvert<IeeeFloat<T>> for IeeeFloat<S> {
fn convert_r(self, round: Round, loses_info: &mut bool) -> StatusAnd<IeeeFloat<T>> {
let mut r = IeeeFloat {
sig: self.sig,
exp: self.exp,
category: self.category,
sign: self.sign,
marker: PhantomData,
};
// x86 has some unusual NaNs which cannot be represented in any other
// format; note them here.
fn is_x87_double_extended<S: Semantics>() -> bool {
S::QNAN_SIGNIFICAND == X87DoubleExtendedS::QNAN_SIGNIFICAND
}
let x87_special_nan = is_x87_double_extended::<S>() && !is_x87_double_extended::<T>() &&
r.category == Category::NaN &&
(r.sig[0] & S::QNAN_SIGNIFICAND) != S::QNAN_SIGNIFICAND;
// If this is a truncation of a denormal number, and the target semantics
// has larger exponent range than the source semantics (this can happen
// when truncating from PowerPC double-double to double format), the
// right shift could lose result mantissa bits. Adjust exponent instead
// of performing excessive shift.
let mut shift = T::PRECISION as ExpInt - S::PRECISION as ExpInt;
if shift < 0 && r.is_finite_non_zero() {
let mut exp_change = sig::omsb(&r.sig) as ExpInt - S::PRECISION as ExpInt;
if r.exp + exp_change < T::MIN_EXP {
exp_change = T::MIN_EXP - r.exp;
}
if exp_change < shift {
exp_change = shift;
}
if exp_change < 0 {
shift -= exp_change;
r.exp += exp_change;
}
}
// If this is a truncation, perform the shift.
let loss = if shift < 0 && (r.is_finite_non_zero() || r.category == Category::NaN) {
sig::shift_right(&mut r.sig, &mut 0, -shift as usize)
} else {
Loss::ExactlyZero
};
// If this is an extension, perform the shift.
if shift > 0 && (r.is_finite_non_zero() || r.category == Category::NaN) {
sig::shift_left(&mut r.sig, &mut 0, shift as usize);
}
let status;
if r.is_finite_non_zero() {
r = unpack!(status=, r.normalize(round, loss));
*loses_info = status != Status::OK;
} else if r.category == Category::NaN {
*loses_info = loss != Loss::ExactlyZero || x87_special_nan;
// For x87 extended precision, we want to make a NaN, not a special NaN if
// the input wasn't special either.
if !x87_special_nan && is_x87_double_extended::<T>() {
sig::set_bit(&mut r.sig, T::PRECISION - 1);
}
// gcc forces the Quiet bit on, which means (float)(double)(float_sNan)
// does not give you back the same bits. This is dubious, and we
// don't currently do it. You're really supposed to get
// an invalid operation signal at runtime, but nobody does that.
status = Status::OK;
} else {
*loses_info = false;
status = Status::OK;
}
status.and(r)
}
}
impl<S: Semantics> IeeeFloat<S> {
/// Handle positive overflow. We either return infinity or
/// the largest finite number. For negative overflow,
/// negate the `round` argument before calling.
fn overflow_result(round: Round) -> StatusAnd<Self> {
match round {
// Infinity?
Round::NearestTiesToEven | Round::NearestTiesToAway | Round::TowardPositive => {
(Status::OVERFLOW | Status::INEXACT).and(Self::INFINITY)
}
// Otherwise we become the largest finite number.
Round::TowardNegative | Round::TowardZero => Status::INEXACT.and(Self::largest()),
}
}
/// Returns `true` if, when truncating the current number, with `bit` the
/// new LSB, with the given lost fraction and rounding mode, the result
/// would need to be rounded away from zero (i.e., by increasing the
/// signficand). This routine must work for `Category::Zero` of both signs, and
/// `Category::Normal` numbers.
fn round_away_from_zero(&self, round: Round, loss: Loss, bit: usize) -> bool {
// NaNs and infinities should not have lost fractions.
assert!(self.is_finite_non_zero() || self.is_zero());
// Current callers never pass this so we don't handle it.
assert_ne!(loss, Loss::ExactlyZero);
match round {
Round::NearestTiesToAway => loss == Loss::ExactlyHalf || loss == Loss::MoreThanHalf,
Round::NearestTiesToEven => {
if loss == Loss::MoreThanHalf {
return true;
}
// Our zeros don't have a significand to test.
if loss == Loss::ExactlyHalf && self.category != Category::Zero {
return sig::get_bit(&self.sig, bit);
}
false
}
Round::TowardZero => false,
Round::TowardPositive => !self.sign,
Round::TowardNegative => self.sign,
}
}
fn normalize(mut self, round: Round, mut loss: Loss) -> StatusAnd<Self> {
if !self.is_finite_non_zero() {
return Status::OK.and(self);
}
// Before rounding normalize the exponent of Category::Normal numbers.
let mut omsb = sig::omsb(&self.sig);
if omsb > 0 {
// OMSB is numbered from 1. We want to place it in the integer
// bit numbered PRECISION if possible, with a compensating change in
// the exponent.
let mut final_exp = self.exp.saturating_add(
omsb as ExpInt - S::PRECISION as ExpInt,
);
// If the resulting exponent is too high, overflow according to
// the rounding mode.
if final_exp > S::MAX_EXP {
let round = if self.sign { -round } else { round };
return Self::overflow_result(round).map(|r| r.copy_sign(self));
}
// Subnormal numbers have exponent MIN_EXP, and their MSB
// is forced based on that.
if final_exp < S::MIN_EXP {
final_exp = S::MIN_EXP;
}
// Shifting left is easy as we don't lose precision.
if final_exp < self.exp {
assert_eq!(loss, Loss::ExactlyZero);
let exp_change = (self.exp - final_exp) as usize;
sig::shift_left(&mut self.sig, &mut self.exp, exp_change);
return Status::OK.and(self);
}
// Shift right and capture any new lost fraction.
if final_exp > self.exp {
let exp_change = (final_exp - self.exp) as usize;
loss = sig::shift_right(&mut self.sig, &mut self.exp, exp_change).combine(loss);
// Keep OMSB up-to-date.
omsb = omsb.saturating_sub(exp_change);
}
}
// Now round the number according to round given the lost
// fraction.
// As specified in IEEE 754, since we do not trap we do not report
// underflow for exact results.
if loss == Loss::ExactlyZero {
// Canonicalize zeros.
if omsb == 0 {
self.category = Category::Zero;
}
return Status::OK.and(self);
}
// Increment the significand if we're rounding away from zero.
if self.round_away_from_zero(round, loss, 0) {
if omsb == 0 {
self.exp = S::MIN_EXP;
}
// We should never overflow.
assert_eq!(sig::increment(&mut self.sig), 0);
omsb = sig::omsb(&self.sig);
// Did the significand increment overflow?
if omsb == S::PRECISION + 1 {
// Renormalize by incrementing the exponent and shifting our
// significand right one. However if we already have the
// maximum exponent we overflow to infinity.
if self.exp == S::MAX_EXP {
self.category = Category::Infinity;
return (Status::OVERFLOW | Status::INEXACT).and(self);
}
let _: Loss = sig::shift_right(&mut self.sig, &mut self.exp, 1);
return Status::INEXACT.and(self);
}
}
// The normal case - we were and are not denormal, and any
// significand increment above didn't overflow.
if omsb == S::PRECISION {
return Status::INEXACT.and(self);
}
// We have a non-zero denormal.
assert!(omsb < S::PRECISION);
// Canonicalize zeros.
if omsb == 0 {
self.category = Category::Zero;
}
// The Category::Zero case is a denormal that underflowed to zero.
(Status::UNDERFLOW | Status::INEXACT).and(self)
}
fn from_hexadecimal_string(s: &str, round: Round) -> Result<StatusAnd<Self>, ParseError> {
let mut r = IeeeFloat {
sig: [0],
exp: 0,
category: Category::Normal,
sign: false,
marker: PhantomData,
};
let mut any_digits = false;
let mut has_exp = false;
let mut bit_pos = LIMB_BITS as isize;
let mut loss = None;
// Without leading or trailing zeros, irrespective of the dot.
let mut first_sig_digit = None;
let mut dot = s.len();
for (p, c) in s.char_indices() {
// Skip leading zeros and any (hexa)decimal point.
if c == '.' {
if dot != s.len() {
return Err(ParseError("String contains multiple dots"));
}
dot = p;
} else if let Some(hex_value) = c.to_digit(16) {
any_digits = true;
if first_sig_digit.is_none() {
if hex_value == 0 {
continue;
}
first_sig_digit = Some(p);
}
// Store the number while we have space.
bit_pos -= 4;
if bit_pos >= 0 {
r.sig[0] |= (hex_value as Limb) << bit_pos;
// If zero or one-half (the hexadecimal digit 8) are followed
// by non-zero, they're a little more than zero or one-half.
} else if let Some(ref mut loss) = loss {
if hex_value != 0 {
if *loss == Loss::ExactlyZero {
*loss = Loss::LessThanHalf;
}
if *loss == Loss::ExactlyHalf {
*loss = Loss::MoreThanHalf;
}
}
} else {
loss = Some(match hex_value {
0 => Loss::ExactlyZero,
1..=7 => Loss::LessThanHalf,
8 => Loss::ExactlyHalf,
9..=15 => Loss::MoreThanHalf,
_ => unreachable!(),
});
}
} else if c == 'p' || c == 'P' {
if !any_digits {
return Err(ParseError("Significand has no digits"));
}
if dot == s.len() {
dot = p;
}
let mut chars = s[p + 1..].chars().peekable();
// Adjust for the given exponent.
let exp_minus = chars.peek() == Some(&'-');
if exp_minus || chars.peek() == Some(&'+') {
chars.next();
}
for c in chars {
if let Some(value) = c.to_digit(10) {
has_exp = true;
r.exp = r.exp.saturating_mul(10).saturating_add(value as ExpInt);
} else {
return Err(ParseError("Invalid character in exponent"));
}
}
if !has_exp {
return Err(ParseError("Exponent has no digits"));
}
if exp_minus {
r.exp = -r.exp;
}
break;
} else {
return Err(ParseError("Invalid character in significand"));
}
}
if !any_digits {
return Err(ParseError("Significand has no digits"));
}
// Hex floats require an exponent but not a hexadecimal point.
if !has_exp {
return Err(ParseError("Hex strings require an exponent"));
}
// Ignore the exponent if we are zero.
let first_sig_digit = match first_sig_digit {
Some(p) => p,
None => return Ok(Status::OK.and(Self::ZERO)),
};
// Calculate the exponent adjustment implicit in the number of
// significant digits and adjust for writing the significand starting
// at the most significant nibble.
let exp_adjustment = if dot > first_sig_digit {
ExpInt::try_from(dot - first_sig_digit).unwrap()
} else {
-ExpInt::try_from(first_sig_digit - dot - 1).unwrap()
};
let exp_adjustment = exp_adjustment
.saturating_mul(4)
.saturating_sub(1)
.saturating_add(S::PRECISION as ExpInt)
.saturating_sub(LIMB_BITS as ExpInt);
r.exp = r.exp.saturating_add(exp_adjustment);
Ok(r.normalize(round, loss.unwrap_or(Loss::ExactlyZero)))
}
fn from_decimal_string(s: &str, round: Round) -> Result<StatusAnd<Self>, ParseError> {
// Given a normal decimal floating point number of the form
//
// dddd.dddd[eE][+-]ddd
//
// where the decimal point and exponent are optional, fill out the
// variables below. Exponent is appropriate if the significand is
// treated as an integer, and normalized_exp if the significand
// is taken to have the decimal point after a single leading
// non-zero digit.
//
// If the value is zero, first_sig_digit is None.
let mut any_digits = false;
let mut dec_exp = 0i32;
// Without leading or trailing zeros, irrespective of the dot.
let mut first_sig_digit = None;
let mut last_sig_digit = 0;
let mut dot = s.len();
for (p, c) in s.char_indices() {
if c == '.' {
if dot != s.len() {
return Err(ParseError("String contains multiple dots"));
}
dot = p;
} else if let Some(dec_value) = c.to_digit(10) {
any_digits = true;
if dec_value != 0 {
if first_sig_digit.is_none() {
first_sig_digit = Some(p);
}
last_sig_digit = p;
}
} else if c == 'e' || c == 'E' {
if !any_digits {
return Err(ParseError("Significand has no digits"));
}
if dot == s.len() {
dot = p;
}
let mut chars = s[p + 1..].chars().peekable();
// Adjust for the given exponent.
let exp_minus = chars.peek() == Some(&'-');
if exp_minus || chars.peek() == Some(&'+') {
chars.next();
}
any_digits = false;
for c in chars {
if let Some(value) = c.to_digit(10) {
any_digits = true;
dec_exp = dec_exp.saturating_mul(10).saturating_add(value as i32);
} else {
return Err(ParseError("Invalid character in exponent"));
}
}
if !any_digits {
return Err(ParseError("Exponent has no digits"));
}
if exp_minus {
dec_exp = -dec_exp;
}
break;
} else {
return Err(ParseError("Invalid character in significand"));
}
}
if !any_digits {
return Err(ParseError("Significand has no digits"));
}
// Test if we have a zero number allowing for non-zero exponents.
let first_sig_digit = match first_sig_digit {
Some(p) => p,
None => return Ok(Status::OK.and(Self::ZERO)),
};
// Adjust the exponents for any decimal point.
if dot > last_sig_digit {
dec_exp = dec_exp.saturating_add((dot - last_sig_digit - 1) as i32);
} else {
dec_exp = dec_exp.saturating_sub((last_sig_digit - dot) as i32);
}
let significand_digits = last_sig_digit - first_sig_digit + 1 -
(dot > first_sig_digit && dot < last_sig_digit) as usize;
let normalized_exp = dec_exp.saturating_add(significand_digits as i32 - 1);
// Handle the cases where exponents are obviously too large or too
// small. Writing L for log 10 / log 2, a number d.ddddd*10^dec_exp
// definitely overflows if
//
// (dec_exp - 1) * L >= MAX_EXP
//
// and definitely underflows to zero where
//
// (dec_exp + 1) * L <= MIN_EXP - PRECISION
//
// With integer arithmetic the tightest bounds for L are
//
// 93/28 < L < 196/59 [ numerator <= 256 ]
// 42039/12655 < L < 28738/8651 [ numerator <= 65536 ]
// Check for MAX_EXP.
if normalized_exp.saturating_sub(1).saturating_mul(42039) >= 12655 * S::MAX_EXP as i32 {
// Overflow and round.
return Ok(Self::overflow_result(round));
}
// Check for MIN_EXP.
if normalized_exp.saturating_add(1).saturating_mul(28738) <=
8651 * (S::MIN_EXP as i32 - S::PRECISION as i32)
{
// Underflow to zero and round.
let r = if round == Round::TowardPositive {
IeeeFloat::SMALLEST
} else {
IeeeFloat::ZERO
};
return Ok((Status::UNDERFLOW | Status::INEXACT).and(r));
}
// A tight upper bound on number of bits required to hold an
// N-digit decimal integer is N * 196 / 59. Allocate enough space
// to hold the full significand, and an extra limb required by
// tcMultiplyPart.
let max_limbs = limbs_for_bits(1 + 196 * significand_digits / 59);
let mut dec_sig: SmallVec<[Limb; 1]> = SmallVec::with_capacity(max_limbs);
// Convert to binary efficiently - we do almost all multiplication
// in a Limb. When this would overflow do we do a single
// bignum multiplication, and then revert again to multiplication
// in a Limb.
let mut chars = s[first_sig_digit..=last_sig_digit].chars();
loop {
let mut val = 0;
let mut multiplier = 1;
loop {
let dec_value = match chars.next() {
Some('.') => continue,
Some(c) => c.to_digit(10).unwrap(),
None => break,
};
multiplier *= 10;
val = val * 10 + dec_value as Limb;
// The maximum number that can be multiplied by ten with any
// digit added without overflowing a Limb.
if multiplier > (!0 - 9) / 10 {
break;
}
}
// If we've consumed no digits, we're done.
if multiplier == 1 {
break;
}
// Multiply out the current limb.
let mut carry = val;
for x in &mut dec_sig {
let [low, mut high] = sig::widening_mul(*x, multiplier);
// Now add carry.
let (low, overflow) = low.overflowing_add(carry);
high += overflow as Limb;
*x = low;
carry = high;
}
// If we had carry, we need another limb (likely but not guaranteed).
if carry > 0 {
dec_sig.push(carry);
}
}
// Calculate pow(5, abs(dec_exp)) into `pow5_full`.
// The *_calc Vec's are reused scratch space, as an optimization.
let (pow5_full, mut pow5_calc, mut sig_calc, mut sig_scratch_calc) = {
let mut power = dec_exp.abs() as usize;
const FIRST_EIGHT_POWERS: [Limb; 8] = [1, 5, 25, 125, 625, 3125, 15625, 78125];
let mut p5_scratch = smallvec![];
let mut p5: SmallVec<[Limb; 1]> = smallvec![FIRST_EIGHT_POWERS[4]];
let mut r_scratch = smallvec![];
let mut r: SmallVec<[Limb; 1]> = smallvec![FIRST_EIGHT_POWERS[power & 7]];
power >>= 3;
while power > 0 {
// Calculate pow(5,pow(2,n+3)).
p5_scratch.resize(p5.len() * 2, 0);
let _: Loss = sig::mul(&mut p5_scratch, &mut 0, &p5, &p5, p5.len() * 2 * LIMB_BITS);
while p5_scratch.last() == Some(&0) {
p5_scratch.pop();
}
mem::swap(&mut p5, &mut p5_scratch);
if power & 1 != 0 {
r_scratch.resize(r.len() + p5.len(), 0);
let _: Loss = sig::mul(
&mut r_scratch,
&mut 0,
&r,
&p5,
(r.len() + p5.len()) * LIMB_BITS,
);
while r_scratch.last() == Some(&0) {
r_scratch.pop();
}
mem::swap(&mut r, &mut r_scratch);
}
power >>= 1;
}
(r, r_scratch, p5, p5_scratch)
};
// Attempt dec_sig * 10^dec_exp with increasing precision.
let mut attempt = 0;
loop {
let calc_precision = (LIMB_BITS << attempt) - 1;
attempt += 1;
let calc_normal_from_limbs = |sig: &mut SmallVec<[Limb; 1]>,
limbs: &[Limb]|
-> StatusAnd<ExpInt> {
sig.resize(limbs_for_bits(calc_precision), 0);
let (mut loss, mut exp) = sig::from_limbs(sig, limbs, calc_precision);
// Before rounding normalize the exponent of Category::Normal numbers.
let mut omsb = sig::omsb(sig);
assert_ne!(omsb, 0);
// OMSB is numbered from 1. We want to place it in the integer
// bit numbered PRECISION if possible, with a compensating change in
// the exponent.
let final_exp = exp.saturating_add(omsb as ExpInt - calc_precision as ExpInt);
// Shifting left is easy as we don't lose precision.
if final_exp < exp {
assert_eq!(loss, Loss::ExactlyZero);
let exp_change = (exp - final_exp) as usize;
sig::shift_left(sig, &mut exp, exp_change);
return Status::OK.and(exp);
}
// Shift right and capture any new lost fraction.
if final_exp > exp {
let exp_change = (final_exp - exp) as usize;
loss = sig::shift_right(sig, &mut exp, exp_change).combine(loss);
// Keep OMSB up-to-date.
omsb = omsb.saturating_sub(exp_change);
}
assert_eq!(omsb, calc_precision);
// Now round the number according to round given the lost
// fraction.
// As specified in IEEE 754, since we do not trap we do not report
// underflow for exact results.
if loss == Loss::ExactlyZero {
return Status::OK.and(exp);
}
// Increment the significand if we're rounding away from zero.
if loss == Loss::MoreThanHalf || loss == Loss::ExactlyHalf && sig::get_bit(sig, 0) {
// We should never overflow.
assert_eq!(sig::increment(sig), 0);
omsb = sig::omsb(sig);
// Did the significand increment overflow?
if omsb == calc_precision + 1 {
let _: Loss = sig::shift_right(sig, &mut exp, 1);
return Status::INEXACT.and(exp);
}
}
// The normal case - we were and are not denormal, and any
// significand increment above didn't overflow.
Status::INEXACT.and(exp)
};
let status;
let mut exp = unpack!(status=,
calc_normal_from_limbs(&mut sig_calc, &dec_sig));
let pow5_status;
let pow5_exp = unpack!(pow5_status=,
calc_normal_from_limbs(&mut pow5_calc, &pow5_full));
// Add dec_exp, as 10^n = 5^n * 2^n.
exp += dec_exp as ExpInt;
let mut used_bits = S::PRECISION;
let mut truncated_bits = calc_precision - used_bits;
let half_ulp_err1 = (status != Status::OK) as Limb;
let (calc_loss, half_ulp_err2);
if dec_exp >= 0 {
exp += pow5_exp;
sig_scratch_calc.resize(sig_calc.len() + pow5_calc.len(), 0);
calc_loss = sig::mul(
&mut sig_scratch_calc,
&mut exp,
&sig_calc,
&pow5_calc,
calc_precision,
);
mem::swap(&mut sig_calc, &mut sig_scratch_calc);
half_ulp_err2 = (pow5_status != Status::OK) as Limb;
} else {
exp -= pow5_exp;
sig_scratch_calc.resize(sig_calc.len(), 0);
calc_loss = sig::div(
&mut sig_scratch_calc,
&mut exp,
&mut sig_calc,
&mut pow5_calc,
calc_precision,
);
mem::swap(&mut sig_calc, &mut sig_scratch_calc);
// Denormal numbers have less precision.
if exp < S::MIN_EXP {
truncated_bits += (S::MIN_EXP - exp) as usize;
used_bits = calc_precision.saturating_sub(truncated_bits);
}
// Extra half-ulp lost in reciprocal of exponent.
half_ulp_err2 = 2 *
(pow5_status != Status::OK || calc_loss != Loss::ExactlyZero) as Limb;
}
// Both sig::mul and sig::div return the
// result with the integer bit set.
assert!(sig::get_bit(&sig_calc, calc_precision - 1));
// The error from the true value, in half-ulps, on multiplying two
// floating point numbers, which differ from the value they
// approximate by at most half_ulp_err1 and half_ulp_err2 half-ulps, is strictly less
// than the returned value.
//
// See "How to Read Floating Point Numbers Accurately" by William D Clinger.
assert!(
half_ulp_err1 < 2 || half_ulp_err2 < 2 || (half_ulp_err1 + half_ulp_err2 < 8)
);
let inexact = (calc_loss != Loss::ExactlyZero) as Limb;
let half_ulp_err = if half_ulp_err1 + half_ulp_err2 == 0 {
inexact * 2 // <= inexact half-ulps.
} else {
inexact + 2 * (half_ulp_err1 + half_ulp_err2)
};
let ulps_from_boundary = {
let bits = calc_precision - used_bits - 1;
let i = bits / LIMB_BITS;
let limb = sig_calc[i] & (!0 >> (LIMB_BITS - 1 - bits % LIMB_BITS));
let boundary = match round {
Round::NearestTiesToEven | Round::NearestTiesToAway => 1 << (bits % LIMB_BITS),
_ => 0,
};
if i == 0 {
let delta = limb.wrapping_sub(boundary);
cmp::min(delta, delta.wrapping_neg())
} else if limb == boundary {
if !sig::is_all_zeros(&sig_calc[1..i]) {
!0 // A lot.
} else {
sig_calc[0]
}
} else if limb == boundary.wrapping_sub(1) {
if sig_calc[1..i].iter().any(|&x| x.wrapping_neg() != 1) {
!0 // A lot.
} else {
sig_calc[0].wrapping_neg()
}
} else {
!0 // A lot.
}
};
// Are we guaranteed to round correctly if we truncate?
if ulps_from_boundary.saturating_mul(2) >= half_ulp_err {
let mut r = IeeeFloat {
sig: [0],
exp,
category: Category::Normal,
sign: false,
marker: PhantomData,
};
sig::extract(&mut r.sig, &sig_calc, used_bits, calc_precision - used_bits);
// If we extracted less bits above we must adjust our exponent
// to compensate for the implicit right shift.
r.exp += (S::PRECISION - used_bits) as ExpInt;
let loss = Loss::through_truncation(&sig_calc, truncated_bits);
return Ok(r.normalize(round, loss));
}
}
}
}
impl Loss {
/// Combine the effect of two lost fractions.
fn combine(self, less_significant: Loss) -> Loss {
let mut more_significant = self;
if less_significant != Loss::ExactlyZero {
if more_significant == Loss::ExactlyZero {
more_significant = Loss::LessThanHalf;
} else if more_significant == Loss::ExactlyHalf {
more_significant = Loss::MoreThanHalf;
}
}
more_significant
}
/// Returns the fraction lost were a bignum truncated losing the least
/// significant `bits` bits.
fn through_truncation(limbs: &[Limb], bits: usize) -> Loss {
if bits == 0 {
return Loss::ExactlyZero;
}
let half_bit = bits - 1;
let half_limb = half_bit / LIMB_BITS;
let (half_limb, rest) = if half_limb < limbs.len() {
(limbs[half_limb], &limbs[..half_limb])
} else {
(0, limbs)
};
let half = 1 << (half_bit % LIMB_BITS);
let has_half = half_limb & half != 0;
let has_rest = half_limb & (half - 1) != 0 || !sig::is_all_zeros(rest);
match (has_half, has_rest) {
(false, false) => Loss::ExactlyZero,
(false, true) => Loss::LessThanHalf,
(true, false) => Loss::ExactlyHalf,
(true, true) => Loss::MoreThanHalf,
}
}
}
/// Implementation details of IeeeFloat significands, such as big integer arithmetic.
/// As a rule of thumb, no functions in this module should dynamically allocate.
mod sig {
use std::cmp::Ordering;
use std::mem;
use super::{ExpInt, Limb, LIMB_BITS, limbs_for_bits, Loss};
pub(super) fn is_all_zeros(limbs: &[Limb]) -> bool {
limbs.iter().all(|&l| l == 0)
}
/// One, not zero, based LSB. That is, returns 0 for a zeroed significand.
pub(super) fn olsb(limbs: &[Limb]) -> usize {
limbs.iter().enumerate().find(|(_, &limb)| limb != 0).map_or(0,
|(i, limb)| i * LIMB_BITS + limb.trailing_zeros() as usize + 1)
}
/// One, not zero, based MSB. That is, returns 0 for a zeroed significand.
pub(super) fn omsb(limbs: &[Limb]) -> usize {
limbs.iter().enumerate().rfind(|(_, &limb)| limb != 0).map_or(0,
|(i, limb)| (i + 1) * LIMB_BITS - limb.leading_zeros() as usize)
}
/// Comparison (unsigned) of two significands.
pub(super) fn cmp(a: &[Limb], b: &[Limb]) -> Ordering {
assert_eq!(a.len(), b.len());
for (a, b) in a.iter().zip(b).rev() {
match a.cmp(b) {
Ordering::Equal => {}
o => return o,
}
}
Ordering::Equal
}
/// Extracts the given bit.
pub(super) fn get_bit(limbs: &[Limb], bit: usize) -> bool {
limbs[bit / LIMB_BITS] & (1 << (bit % LIMB_BITS)) != 0
}
/// Sets the given bit.
pub(super) fn set_bit(limbs: &mut [Limb], bit: usize) {
limbs[bit / LIMB_BITS] |= 1 << (bit % LIMB_BITS);
}
/// Clear the given bit.
pub(super) fn clear_bit(limbs: &mut [Limb], bit: usize) {
limbs[bit / LIMB_BITS] &= !(1 << (bit % LIMB_BITS));
}
/// Shifts `dst` left `bits` bits, subtract `bits` from its exponent.
pub(super) fn shift_left(dst: &mut [Limb], exp: &mut ExpInt, bits: usize) {
if bits > 0 {
// Our exponent should not underflow.
*exp = exp.checked_sub(bits as ExpInt).unwrap();
// Jump is the inter-limb jump; shift is the intra-limb shift.
let jump = bits / LIMB_BITS;
let shift = bits % LIMB_BITS;
for i in (0..dst.len()).rev() {
let mut limb;
if i < jump {
limb = 0;
} else {
// dst[i] comes from the two limbs src[i - jump] and, if we have
// an intra-limb shift, src[i - jump - 1].
limb = dst[i - jump];
if shift > 0 {
limb <<= shift;
if i > jump {
limb |= dst[i - jump - 1] >> (LIMB_BITS - shift);
}
}
}
dst[i] = limb;
}
}
}
/// Shifts `dst` right `bits` bits noting lost fraction.
pub(super) fn shift_right(dst: &mut [Limb], exp: &mut ExpInt, bits: usize) -> Loss {
let loss = Loss::through_truncation(dst, bits);
if bits > 0 {
// Our exponent should not overflow.
*exp = exp.checked_add(bits as ExpInt).unwrap();
// Jump is the inter-limb jump; shift is the intra-limb shift.
let jump = bits / LIMB_BITS;
let shift = bits % LIMB_BITS;
// Perform the shift. This leaves the most significant `bits` bits
// of the result at zero.
for i in 0..dst.len() {
let mut limb;
if i + jump >= dst.len() {
limb = 0;
} else {
limb = dst[i + jump];
if shift > 0 {
limb >>= shift;
if i + jump + 1 < dst.len() {
limb |= dst[i + jump + 1] << (LIMB_BITS - shift);
}
}
}
dst[i] = limb;
}
}
loss
}
/// Copies the bit vector of width `src_bits` from `src`, starting at bit SRC_LSB,
/// to `dst`, such that the bit SRC_LSB becomes the least significant bit of `dst`.
/// All high bits above `src_bits` in `dst` are zero-filled.
pub(super) fn extract(dst: &mut [Limb], src: &[Limb], src_bits: usize, src_lsb: usize) {
if src_bits == 0 {
return;
}
let dst_limbs = limbs_for_bits(src_bits);
assert!(dst_limbs <= dst.len());
let src = &src[src_lsb / LIMB_BITS..];
dst[..dst_limbs].copy_from_slice(&src[..dst_limbs]);
let shift = src_lsb % LIMB_BITS;
let _: Loss = shift_right(&mut dst[..dst_limbs], &mut 0, shift);
// We now have (dst_limbs * LIMB_BITS - shift) bits from `src`
// in `dst`. If this is less that src_bits, append the rest, else
// clear the high bits.
let n = dst_limbs * LIMB_BITS - shift;
if n < src_bits {
let mask = (1 << (src_bits - n)) - 1;
dst[dst_limbs - 1] |= (src[dst_limbs] & mask) << (n % LIMB_BITS);
} else if n > src_bits && src_bits % LIMB_BITS > 0 {
dst[dst_limbs - 1] &= (1 << (src_bits % LIMB_BITS)) - 1;
}
// Clear high limbs.
for x in &mut dst[dst_limbs..] {
*x = 0;
}
}
/// We want the most significant PRECISION bits of `src`. There may not
/// be that many; extract what we can.
pub(super) fn from_limbs(dst: &mut [Limb], src: &[Limb], precision: usize) -> (Loss, ExpInt) {
let omsb = omsb(src);
if precision <= omsb {
extract(dst, src, precision, omsb - precision);
(
Loss::through_truncation(src, omsb - precision),
omsb as ExpInt - 1,
)
} else {
extract(dst, src, omsb, 0);
(Loss::ExactlyZero, precision as ExpInt - 1)
}
}
/// For every consecutive chunk of `bits` bits from `limbs`,
/// going from most significant to the least significant bits,
/// call `f` to transform those bits and store the result back.
pub(super) fn each_chunk<F: FnMut(Limb) -> Limb>(limbs: &mut [Limb], bits: usize, mut f: F) {
assert_eq!(LIMB_BITS % bits, 0);
for limb in limbs.iter_mut().rev() {
let mut r = 0;
for i in (0..LIMB_BITS / bits).rev() {
r |= f((*limb >> (i * bits)) & ((1 << bits) - 1)) << (i * bits);
}
*limb = r;
}
}
/// Increment in-place, return the carry flag.
pub(super) fn increment(dst: &mut [Limb]) -> Limb {
for x in dst {
*x = x.wrapping_add(1);
if *x != 0 {
return 0;
}
}
1
}
/// Decrement in-place, return the borrow flag.
pub(super) fn decrement(dst: &mut [Limb]) -> Limb {
for x in dst {
*x = x.wrapping_sub(1);
if *x != !0 {
return 0;
}
}
1
}
/// `a += b + c` where `c` is zero or one. Returns the carry flag.
pub(super) fn add(a: &mut [Limb], b: &[Limb], mut c: Limb) -> Limb {
assert!(c <= 1);
for (a, &b) in a.iter_mut().zip(b) {
let (r, overflow) = a.overflowing_add(b);
let (r, overflow2) = r.overflowing_add(c);
*a = r;
c = (overflow | overflow2) as Limb;
}
c
}
/// `a -= b + c` where `c` is zero or one. Returns the borrow flag.
pub(super) fn sub(a: &mut [Limb], b: &[Limb], mut c: Limb) -> Limb {
assert!(c <= 1);
for (a, &b) in a.iter_mut().zip(b) {
let (r, overflow) = a.overflowing_sub(b);
let (r, overflow2) = r.overflowing_sub(c);
*a = r;
c = (overflow | overflow2) as Limb;
}
c
}
/// `a += b` or `a -= b`. Does not preserve `b`.
pub(super) fn add_or_sub(
a_sig: &mut [Limb],
a_exp: &mut ExpInt,
a_sign: &mut bool,
b_sig: &mut [Limb],
b_exp: ExpInt,
b_sign: bool,
) -> Loss {
// Are we bigger exponent-wise than the RHS?
let bits = *a_exp - b_exp;
// Determine if the operation on the absolute values is effectively
// an addition or subtraction.
// Subtraction is more subtle than one might naively expect.
if *a_sign ^ b_sign {
let (reverse, loss);
if bits == 0 {
reverse = cmp(a_sig, b_sig) == Ordering::Less;
loss = Loss::ExactlyZero;
} else if bits > 0 {
loss = shift_right(b_sig, &mut 0, (bits - 1) as usize);
shift_left(a_sig, a_exp, 1);
reverse = false;
} else {
loss = shift_right(a_sig, a_exp, (-bits - 1) as usize);
shift_left(b_sig, &mut 0, 1);
reverse = true;
}
let borrow = (loss != Loss::ExactlyZero) as Limb;
if reverse {
// The code above is intended to ensure that no borrow is necessary.
assert_eq!(sub(b_sig, a_sig, borrow), 0);
a_sig.copy_from_slice(b_sig);
*a_sign = !*a_sign;
} else {
// The code above is intended to ensure that no borrow is necessary.
assert_eq!(sub(a_sig, b_sig, borrow), 0);
}
// Invert the lost fraction - it was on the RHS and subtracted.
match loss {
Loss::LessThanHalf => Loss::MoreThanHalf,
Loss::MoreThanHalf => Loss::LessThanHalf,
_ => loss,
}
} else {
let loss = if bits > 0 {
shift_right(b_sig, &mut 0, bits as usize)
} else {
shift_right(a_sig, a_exp, -bits as usize)
};
// We have a guard bit; generating a carry cannot happen.
assert_eq!(add(a_sig, b_sig, 0), 0);
loss
}
}
/// `[low, high] = a * b`.
///
/// This cannot overflow, because
///
/// `(n - 1) * (n - 1) + 2 * (n - 1) == (n - 1) * (n + 1)`
///
/// which is less than n<sup>2</sup>.
pub(super) fn widening_mul(a: Limb, b: Limb) -> [Limb; 2] {
let mut wide = [0, 0];
if a == 0 || b == 0 {
return wide;
}
const HALF_BITS: usize = LIMB_BITS / 2;
let select = |limb, i| (limb >> (i * HALF_BITS)) & ((1 << HALF_BITS) - 1);
for i in 0..2 {
for j in 0..2 {
let mut x = [select(a, i) * select(b, j), 0];
shift_left(&mut x, &mut 0, (i + j) * HALF_BITS);
assert_eq!(add(&mut wide, &x, 0), 0);
}
}
wide
}
/// `dst = a * b` (for normal `a` and `b`). Returns the lost fraction.
pub(super) fn mul<'a>(
dst: &mut [Limb],
exp: &mut ExpInt,
mut a: &'a [Limb],
mut b: &'a [Limb],
precision: usize,
) -> Loss {
// Put the narrower number on the `a` for less loops below.
if a.len() > b.len() {
mem::swap(&mut a, &mut b);
}
for x in &mut dst[..b.len()] {
*x = 0;
}
for i in 0..a.len() {
let mut carry = 0;
for j in 0..b.len() {
let [low, mut high] = widening_mul(a[i], b[j]);
// Now add carry.
let (low, overflow) = low.overflowing_add(carry);
high += overflow as Limb;
// And now `dst[i + j]`, and store the new low part there.
let (low, overflow) = low.overflowing_add(dst[i + j]);
high += overflow as Limb;
dst[i + j] = low;
carry = high;
}
dst[i + b.len()] = carry;
}
// Assume the operands involved in the multiplication are single-precision
// FP, and the two multiplicants are:
// a = a23 . a22 ... a0 * 2^e1
// b = b23 . b22 ... b0 * 2^e2
// the result of multiplication is:
// dst = c48 c47 c46 . c45 ... c0 * 2^(e1+e2)
// Note that there are three significant bits at the left-hand side of the
// radix point: two for the multiplication, and an overflow bit for the
// addition (that will always be zero at this point). Move the radix point
// toward left by two bits, and adjust exponent accordingly.
*exp += 2;
// Convert the result having "2 * precision" significant-bits back to the one
// having "precision" significant-bits. First, move the radix point from
// poision "2*precision - 1" to "precision - 1". The exponent need to be
// adjusted by "2*precision - 1" - "precision - 1" = "precision".
*exp -= precision as ExpInt + 1;
// In case MSB resides at the left-hand side of radix point, shift the
// mantissa right by some amount to make sure the MSB reside right before
// the radix point (i.e., "MSB . rest-significant-bits").
//
// Note that the result is not normalized when "omsb < precision". So, the
// caller needs to call IeeeFloat::normalize() if normalized value is
// expected.
let omsb = omsb(dst);
if omsb <= precision {
Loss::ExactlyZero
} else {
shift_right(dst, exp, omsb - precision)
}
}
/// `quotient = dividend / divisor`. Returns the lost fraction.
/// Does not preserve `dividend` or `divisor`.
pub(super) fn div(
quotient: &mut [Limb],
exp: &mut ExpInt,
dividend: &mut [Limb],
divisor: &mut [Limb],
precision: usize,
) -> Loss {
// Normalize the divisor.
let bits = precision - omsb(divisor);
shift_left(divisor, &mut 0, bits);
*exp += bits as ExpInt;
// Normalize the dividend.
let bits = precision - omsb(dividend);
shift_left(dividend, exp, bits);
// Division by 1.
let olsb_divisor = olsb(divisor);
if olsb_divisor == precision {
quotient.copy_from_slice(dividend);
return Loss::ExactlyZero;
}
// Ensure the dividend >= divisor initially for the loop below.
// Incidentally, this means that the division loop below is
// guaranteed to set the integer bit to one.
if cmp(dividend, divisor) == Ordering::Less {
shift_left(dividend, exp, 1);
assert_ne!(cmp(dividend, divisor), Ordering::Less)
}
// Helper for figuring out the lost fraction.
let lost_fraction = |dividend: &[Limb], divisor: &[Limb]| {
match cmp(dividend, divisor) {
Ordering::Greater => Loss::MoreThanHalf,
Ordering::Equal => Loss::ExactlyHalf,
Ordering::Less => {
if is_all_zeros(dividend) {
Loss::ExactlyZero
} else {
Loss::LessThanHalf
}
}
}
};
// Try to perform a (much faster) short division for small divisors.
let divisor_bits = precision - (olsb_divisor - 1);
macro_rules! try_short_div {
($W:ty, $H:ty, $half:expr) => {
if divisor_bits * 2 <= $half {
// Extract the small divisor.
let _: Loss = shift_right(divisor, &mut 0, olsb_divisor - 1);
let divisor = divisor[0] as $H as $W;
// Shift the dividend to produce a quotient with the unit bit set.
let top_limb = *dividend.last().unwrap();
let mut rem = (top_limb >> (LIMB_BITS - (divisor_bits - 1))) as $H;
shift_left(dividend, &mut 0, divisor_bits - 1);
// Apply short division in place on $H (of $half bits) chunks.
each_chunk(dividend, $half, |chunk| {
let chunk = chunk as $H;
let combined = ((rem as $W) << $half) | (chunk as $W);
rem = (combined % divisor) as $H;
(combined / divisor) as $H as Limb
});
quotient.copy_from_slice(dividend);
return lost_fraction(&[(rem as Limb) << 1], &[divisor as Limb]);
}
}
}
try_short_div!(u32, u16, 16);
try_short_div!(u64, u32, 32);
try_short_div!(u128, u64, 64);
// Zero the quotient before setting bits in it.
for x in &mut quotient[..limbs_for_bits(precision)] {
*x = 0;
}
// Long division.
for bit in (0..precision).rev() {
if cmp(dividend, divisor) != Ordering::Less {
sub(dividend, divisor, 0);
set_bit(quotient, bit);
}
shift_left(dividend, &mut 0, 1);
}
lost_fraction(dividend, divisor)
}
}
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