//! Slow, fallback cases where we cannot unambiguously round a float.
//!
//! This occurs when we cannot determine the exact representation using
//! both the fast path (native) cases nor the Lemire/Bellerophon algorithms,
//! and therefore must fallback to a slow, arbitrary-precision representation.
#![doc(hidden)]
use core::cmp;
#[cfg(not(feature = "compact"))]
use lexical_parse_integer::algorithm;
use lexical_util::digit::char_to_valid_digit_const;
#[cfg(feature = "radix")]
use lexical_util::digit::digit_to_char_const;
use lexical_util::format::NumberFormat;
use lexical_util::iterator::{AsBytes, DigitsIter, Iter};
use lexical_util::num::{AsPrimitive, Integer};
#[cfg(feature = "radix")]
use crate::bigint::Bigfloat;
use crate::bigint::{Bigint, Limb};
use crate::float::{extended_to_float, ExtendedFloat80, RawFloat};
use crate::limits::{u32_power_limit, u64_power_limit};
use crate::number::Number;
use crate::shared;
// ALGORITHM
// ---------
/// Parse the significant digits and biased, binary exponent of a float.
///
/// This is a fallback algorithm that uses a big-integer representation
/// of the float, and therefore is considerably slower than faster
/// approximations. However, it will always determine how to round
/// the significant digits to the nearest machine float, allowing
/// use to handle near half-way cases.
///
/// Near half-way cases are halfway between two consecutive machine floats.
/// For example, the float `16777217.0` has a bitwise representation of
/// `100000000000000000000000 1`. Rounding to a single-precision float,
/// the trailing `1` is truncated. Using round-nearest, tie-even, any
/// value above `16777217.0` must be rounded up to `16777218.0`, while
/// any value before or equal to `16777217.0` must be rounded down
/// to `16777216.0`. These near-halfway conversions therefore may require
/// a large number of digits to unambiguously determine how to round.
#[must_use]
#[allow(clippy::unwrap_used)] // reason = "none is a developer error"
pub fn slow_radix<F: RawFloat, const FORMAT: u128>(
num: Number,
fp: ExtendedFloat80,
) -> ExtendedFloat80 {
// Ensure our preconditions are valid:
// 1. The significant digits are not shifted into place.
debug_assert!(fp.mant & (1 << 63) != 0, "number must be normalized");
let format = NumberFormat::<{ FORMAT }> {};
// This assumes the sign bit has already been parsed, and we're
// starting with the integer digits, and the float format has been
// correctly validated.
let sci_exp = scientific_exponent::<FORMAT>(&num);
// We have 3 major algorithms we use for this:
// 1. An algorithm with a finite number of digits and a positive exponent.
// 2. An algorithm with a finite number of digits and a negative exponent.
// 3. A fallback algorithm with a non-finite number of digits.
// In order for a float in radix `b` with a finite number of digits
// to have a finite representation in radix `y`, `b` should divide
// an integer power of `y`. This means for binary, all even radixes
// have finite representations, and all odd ones do not.
#[cfg(feature = "radix")]
{
if let Some(max_digits) = F::max_digits(format.radix()) {
// Can use our finite number of digit algorithm.
digit_comp::<F, FORMAT>(num, fp, sci_exp, max_digits)
} else {
// Fallback to infinite digits.
byte_comp::<F, FORMAT>(num, fp, sci_exp)
}
}
#[cfg(not(feature = "radix"))]
{
// Can use our finite number of digit algorithm.
let max_digits = F::max_digits(format.radix()).unwrap();
digit_comp::<F, FORMAT>(num, fp, sci_exp, max_digits)
}
}
/// Algorithm that generates the mantissa for a finite representation.
///
/// For a positive exponent relative to the significant digits, this
/// is just a multiplication by an exponent power. For a negative
/// exponent relative to the significant digits, we scale the real
/// digits to the theoretical digits for `b` and determine if we
/// need to round-up.
#[must_use]
#[inline(always)]
#[allow(clippy::cast_possible_wrap)] // reason = "the value range is [-324, 308]"
pub fn digit_comp<F: RawFloat, const FORMAT: u128>(
num: Number,
fp: ExtendedFloat80,
sci_exp: i32,
max_digits: usize,
) -> ExtendedFloat80 {
let (bigmant, digits) = parse_mantissa::<FORMAT>(num, max_digits);
// This can't underflow, since `digits` is at most `max_digits`.
let exponent = sci_exp + 1 - digits as i32;
if exponent >= 0 {
positive_digit_comp::<F, FORMAT>(bigmant, exponent)
} else {
negative_digit_comp::<F, FORMAT>(bigmant, fp, exponent)
}
}
/// Generate the significant digits with a positive exponent relative to
/// mantissa.
#[must_use]
#[inline(always)]
#[allow(clippy::unwrap_used)] // reason = "none is a developer error"
#[allow(clippy::cast_possible_wrap)] // reason = "can't wrap in practice: max is ~1000 limbs"
#[allow(clippy::missing_inline_in_public_items)] // reason = "only public for testing"
pub fn positive_digit_comp<F: RawFloat, const FORMAT: u128>(
mut bigmant: Bigint,
exponent: i32,
) -> ExtendedFloat80 {
let format = NumberFormat::<{ FORMAT }> {};
// Simple, we just need to multiply by the power of the radix.
// Now, we can calculate the mantissa and the exponent from this.
// The binary exponent is the binary exponent for the mantissa
// shifted to the hidden bit.
bigmant.pow(format.radix(), exponent as u32).unwrap();
// Get the exact representation of the float from the big integer.
// hi64 checks **all** the remaining bits after the mantissa,
// so it will check if **any** truncated digits exist.
let (mant, is_truncated) = bigmant.hi64();
let exp = bigmant.bit_length() as i32 - 64 + F::EXPONENT_BIAS;
let mut fp = ExtendedFloat80 {
mant,
exp,
};
// Shift the digits into position and determine if we need to round-up.
shared::round::<F, _>(&mut fp, |f, s| {
shared::round_nearest_tie_even(f, s, |is_odd, is_halfway, is_above| {
is_above || (is_halfway && is_truncated) || (is_odd && is_halfway)
});
});
fp
}
/// Generate the significant digits with a negative exponent relative to
/// mantissa.
///
/// This algorithm is quite simple: we have the significant digits `m1 * b^N1`,
/// where `m1` is the bigint mantissa, `b` is the radix, and `N1` is the radix
/// exponent. We then calculate the theoretical representation of `b+h`, which
/// is `m2 * 2^N2`, where `m2` is the bigint mantissa and `N2` is the binary
/// exponent. If we had infinite, efficient floating precision, this would be
/// equal to `m1 / b^-N1` and then compare it to `m2 * 2^N2`.
///
/// Since we cannot divide and keep precision, we must multiply the other:
/// if we want to do `m1 / b^-N1 >= m2 * 2^N2`, we can do
/// `m1 >= m2 * b^-N1 * 2^N2` Going to the decimal case, we can show and example
/// and simplify this further: `m1 >= m2 * 2^N2 * 10^-N1`. Since we can remove
/// a power-of-two, this is `m1 >= m2 * 2^(N2 - N1) * 5^-N1`. Therefore, if
/// `N2 - N1 > 0`, we need have `m1 >= m2 * 2^(N2 - N1) * 5^-N1`, otherwise,
/// we have `m1 * 2^(N1 - N2) >= m2 * 5^-N1`, where the resulting exponents
/// are all positive.
///
/// This allows us to compare both floats using integers efficiently
/// without any loss of precision.
#[must_use]
#[inline(always)]
#[allow(clippy::match_bool)] // reason = "simplifies documentation"
#[allow(clippy::unwrap_used)] // reason = "unwrap panics if a developer error"
#[allow(clippy::comparison_chain)] // reason = "logically different conditions for algorithm"
#[allow(clippy::missing_inline_in_public_items)] // reason = "only exposed for unittesting"
pub fn negative_digit_comp<F: RawFloat, const FORMAT: u128>(
bigmant: Bigint,
mut fp: ExtendedFloat80,
exponent: i32,
) -> ExtendedFloat80 {
// Ensure our preconditions are valid:
// 1. The significant digits are not shifted into place.
debug_assert!(fp.mant & (1 << 63) != 0, "the significant digits must be normalized");
let format = NumberFormat::<FORMAT> {};
let radix = format.radix();
// Get the significant digits and radix exponent for the real digits.
let mut real_digits = bigmant;
let real_exp = exponent;
debug_assert!(real_exp < 0, "algorithm only works with negative numbers");
// Round down our extended-precision float and calculate `b`.
let mut b = fp;
shared::round::<F, _>(&mut b, shared::round_down);
let b = extended_to_float::<F>(b);
// Get the significant digits and the binary exponent for `b+h`.
let theor = bh(b);
let mut theor_digits = Bigint::from_u64(theor.mant);
let theor_exp = theor.exp;
// We need to scale the real digits and `b+h` digits to be the same
// order. We currently have `real_exp`, in `radix`, that needs to be
// shifted to `theor_digits` (since it is negative), and `theor_exp`
// to either `theor_digits` or `real_digits` as a power of 2 (since it
// may be positive or negative). Try to remove as many powers of 2
// as possible. All values are relative to `theor_digits`, that is,
// reflect the power you need to multiply `theor_digits` by.
let (binary_exp, halfradix_exp, radix_exp) = match radix.is_even() {
// Can remove a power-of-two.
// Both are on opposite-sides of equation, can factor out a
// power of two.
//
// Example: 10^-10, 2^-10 -> ( 0, 10, 0)
// Example: 10^-10, 2^-15 -> (-5, 10, 0)
// Example: 10^-10, 2^-5 -> ( 5, 10, 0)
// Example: 10^-10, 2^5 -> (15, 10, 0)
true => (theor_exp - real_exp, -real_exp, 0),
// Cannot remove a power-of-two.
false => (theor_exp, 0, -real_exp),
};
if halfradix_exp != 0 {
theor_digits.pow(radix / 2, halfradix_exp as u32).unwrap();
}
if radix_exp != 0 {
theor_digits.pow(radix, radix_exp as u32).unwrap();
}
if binary_exp > 0 {
theor_digits.pow(2, binary_exp as u32).unwrap();
} else if binary_exp < 0 {
real_digits.pow(2, (-binary_exp) as u32).unwrap();
}
// Compare our theoretical and real digits and round nearest, tie even.
let ord = real_digits.data.cmp(&theor_digits.data);
shared::round::<F, _>(&mut fp, |f, s| {
shared::round_nearest_tie_even(f, s, |is_odd, _, _| {
// Can ignore `is_halfway` and `is_above`, since those were
// calculates using less significant digits.
match ord {
cmp::Ordering::Greater => true,
cmp::Ordering::Less => false,
cmp::Ordering::Equal if is_odd => true,
cmp::Ordering::Equal => false,
}
});
});
fp
}
/// Try to parse 8 digits at a time.
///
/// - `format` - The numerical format specification as a packed 128-bit integer
/// - `iter` - An iterator over all bytes in the buffer
/// - `value` - The currently parsed value.
/// - `count` - The total number of parsed digits
/// - `counter` - The number of parsed digits since creating the current u32
/// - `step` - The maximum number of digits for the radix that can fit in a u32.
/// - `max_digits` - The maximum number of digits that can affect floating-point
/// rounding.
#[cfg(not(feature = "compact"))]
macro_rules! try_parse_8digits {
(
$format:ident,
$iter:ident,
$value:ident,
$count:ident,
$counter:ident,
$step:ident,
$max_digits:ident
) => {{
let format = NumberFormat::<$format> {};
let radix = format.radix() as Limb;
// Try 8-digit optimizations.
if can_try_parse_multidigit!($iter, radix) {
debug_assert!(radix < 16);
let radix8 = format.radix8() as Limb;
while $step - $counter >= 8 && $max_digits - $count >= 8 {
if let Some(v) = algorithm::try_parse_8digits::<Limb, _, FORMAT>(&mut $iter) {
$value = $value.wrapping_mul(radix8).wrapping_add(v);
$counter += 8;
$count += 8;
} else {
break;
}
}
}
}};
}
/// Add a digit to the temporary value.
///
/// - `c` - The character to convert to a digit.
/// - `value` - The currently parsed value.
/// - `count` - The total number of parsed digits
/// - `counter` - The number of parsed digits since creating the current u32
macro_rules! add_digit {
($c:ident, $radix:ident, $value:ident, $counter:ident, $count:ident) => {{
let digit = char_to_valid_digit_const($c, $radix);
$value *= $radix as Limb;
$value += digit as Limb;
// Increment our counters.
$counter += 1;
$count += 1;
}};
}
/// Add a temporary value to our mantissa.
///
/// - `format` - The numerical format specification as a packed 128-bit integer
/// - `result` - The big integer,
/// - `power` - The power to scale the big integer by.
/// - `value` - The value to add to the big integer,
/// - `counter` - The number of parsed digits since creating the current u32
macro_rules! add_temporary {
// Multiply by the small power and add the native value.
(@mul $result:ident, $power:expr, $value:expr) => {
$result.data.mul_small($power).unwrap();
$result.data.add_small($value).unwrap();
};
// Add a temporary where we won't read the counter results internally.
(@end $format:ident, $result:ident, $counter:ident, $value:ident) => {
if $counter != 0 {
let small_power = f64::int_pow_fast_path($counter, $format.radix());
add_temporary!(@mul $result, small_power as Limb, $value);
}
};
// Add the maximum native value.
(@max $format:ident, $result:ident, $counter:ident, $value:ident, $max:ident) => {
add_temporary!(@mul $result, $max, $value);
$counter = 0;
$value = 0;
};
}
/// Round-up a truncated value.
///
/// - `format` - The numerical format specification as a packed 128-bit integer
/// - `result` - The big integer,
/// - `count` - The total number of parsed digits
macro_rules! round_up_truncated {
($format:ident, $result:ident, $count:ident) => {{
// Need to round-up.
// Can't just add 1, since this can accidentally round-up
// values to a halfway point, which can cause invalid results.
add_temporary!(@mul $result, $format.radix() as Limb, 1);
$count += 1;
}};
}
/// Check and round-up the fraction if any non-zero digits exist.
///
/// - `format` - The numerical format specification as a packed 128-bit integer
/// - `iter` - An iterator over all bytes in the buffer
/// - `result` - The big integer,
/// - `count` - The total number of parsed digits
macro_rules! round_up_nonzero {
($format:ident, $iter:expr, $result:ident, $count:ident) => {{
// NOTE: All digits must already be valid.
let mut iter = $iter;
// First try reading 8-digits at a time.
if iter.is_contiguous() {
while let Some(value) = iter.peek_u64() {
// SAFETY: safe since we have at least 8 bytes in the buffer.
unsafe { iter.step_by_unchecked(8) };
if value != 0x3030_3030_3030_3030 {
// Have non-zero digits, exit early.
round_up_truncated!($format, $result, $count);
return ($result, $count);
}
}
}
for &digit in iter {
if digit != b'0' {
round_up_truncated!($format, $result, $count);
return ($result, $count);
}
}
}};
}
/// Parse the full mantissa into a big integer.
///
/// Returns the parsed mantissa and the number of digits in the mantissa.
/// The max digits is the maximum number of digits plus one.
#[must_use]
#[allow(clippy::cognitive_complexity)] // reason = "complexity broken into macros"
#[allow(clippy::missing_inline_in_public_items)] // reason = "only public for testing"
pub fn parse_mantissa<const FORMAT: u128>(num: Number, max_digits: usize) -> (Bigint, usize) {
let format = NumberFormat::<FORMAT> {};
let radix = format.radix();
// Iteratively process all the data in the mantissa.
// We do this via small, intermediate values which once we reach
// the maximum number of digits we can process without overflow,
// we add the temporary to the big integer.
let mut counter: usize = 0;
let mut count: usize = 0;
let mut value: Limb = 0;
let mut result = Bigint::new();
// Now use our pre-computed small powers iteratively.
let step = if Limb::BITS == 32 {
u32_power_limit(format.radix())
} else {
u64_power_limit(format.radix())
} as usize;
let max_native = (format.radix() as Limb).pow(step as u32);
// Process the integer digits.
let mut integer = num.integer.bytes::<FORMAT>();
let mut integer_iter = integer.integer_iter();
integer_iter.skip_zeros();
'integer: loop {
#[cfg(not(feature = "compact"))]
try_parse_8digits!(FORMAT, integer_iter, value, count, counter, step, max_digits);
// Parse a digit at a time, until we reach step.
while counter < step && count < max_digits {
if let Some(&c) = integer_iter.next() {
add_digit!(c, radix, value, counter, count);
} else {
break 'integer;
}
}
// Check if we've exhausted our max digits.
if count == max_digits {
// Need to check if we're truncated, and round-up accordingly.
// SAFETY: safe since `counter <= step`.
add_temporary!(@end format, result, counter, value);
round_up_nonzero!(format, integer_iter, result, count);
if let Some(fraction) = num.fraction {
let mut fraction = fraction.bytes::<FORMAT>();
round_up_nonzero!(format, fraction.fraction_iter(), result, count);
}
return (result, count);
} else {
// Add our temporary from the loop.
// SAFETY: safe since `counter <= step`.
add_temporary!(@max format, result, counter, value, max_native);
}
}
// Process the fraction digits.
if let Some(fraction) = num.fraction {
let mut fraction = fraction.bytes::<FORMAT>();
let mut fraction_iter = fraction.integer_iter();
if count == 0 {
// No digits added yet, can skip leading fraction zeros too.
fraction_iter.skip_zeros();
}
'fraction: loop {
#[cfg(not(feature = "compact"))]
try_parse_8digits!(FORMAT, fraction_iter, value, count, counter, step, max_digits);
// Parse a digit at a time, until we reach step.
while counter < step && count < max_digits {
if let Some(&c) = fraction_iter.next() {
add_digit!(c, radix, value, counter, count);
} else {
break 'fraction;
}
}
// Check if we've exhausted our max digits.
if count == max_digits {
// SAFETY: safe since `counter <= step`.
add_temporary!(@end format, result, counter, value);
round_up_nonzero!(format, fraction_iter, result, count);
return (result, count);
} else {
// Add our temporary from the loop.
// SAFETY: safe since `counter <= step`.
add_temporary!(@max format, result, counter, value, max_native);
}
}
}
// We will always have a remainder, as long as we entered the loop
// once, or counter % step is 0.
// SAFETY: safe since `counter <= step`.
add_temporary!(@end format, result, counter, value);
(result, count)
}
/// Compare actual integer digits to the theoretical digits.
///
/// - `iter` - An iterator over all bytes in the buffer
/// - `num` - The actual digits of the real floating point number.
/// - `den` - The theoretical digits created by `b+h` to determine if `b` or
/// `b+1`
#[cfg(feature = "radix")]
macro_rules! integer_compare {
($iter:ident, $num:ident, $den:ident, $radix:ident) => {{
// Compare the integer digits.
while !$num.data.is_empty() {
// All digits **must** be valid.
let actual = match $iter.next() {
Some(&v) => v,
// Could have hit the decimal point.
_ => break,
};
let rem = $num.data.quorem(&$den.data) as u32;
let expected = digit_to_char_const(rem, $radix);
$num.data.mul_small($radix as Limb).unwrap();
if actual < expected {
return cmp::Ordering::Less;
} else if actual > expected {
return cmp::Ordering::Greater;
}
}
// Still have integer digits, check if any are non-zero.
if $num.data.is_empty() {
for &digit in $iter {
if digit != b'0' {
return cmp::Ordering::Greater;
}
}
}
}};
}
/// Compare actual fraction digits to the theoretical digits.
///
/// - `iter` - An iterator over all bytes in the buffer
/// - `num` - The actual digits of the real floating point number.
/// - `den` - The theoretical digits created by `b+h` to determine if `b` or
/// `b+1`
#[cfg(feature = "radix")]
macro_rules! fraction_compare {
($iter:ident, $num:ident, $den:ident, $radix:ident) => {{
// Compare the fraction digits.
// We can only be here if we hit a decimal point.
while !$num.data.is_empty() {
// All digits **must** be valid.
let actual = match $iter.next() {
Some(&v) => v,
// No more actual digits, or hit the exponent.
_ => return cmp::Ordering::Less,
};
let rem = $num.data.quorem(&$den.data) as u32;
let expected = digit_to_char_const(rem, $radix);
$num.data.mul_small($radix as Limb).unwrap();
if actual < expected {
return cmp::Ordering::Less;
} else if actual > expected {
return cmp::Ordering::Greater;
}
}
// Still have fraction digits, check if any are non-zero.
for &digit in $iter {
if digit != b'0' {
return cmp::Ordering::Greater;
}
}
}};
}
/// Compare theoretical digits to halfway point from theoretical digits.
///
/// Generates a float representing the halfway point, and generates
/// theoretical digits as bytes, and compares the generated digits to
/// the actual input.
///
/// Compares the known string to theoretical digits generated on the
/// fly for `b+h`, where a string representation of a float is between
/// `b` and `b+u`, where `b+u` is 1 unit in the least-precision. Therefore,
/// the string must be close to `b+h`.
///
/// Adapted from "Bigcomp: Deciding Truncated, Near Halfway Conversions",
/// available [here](https://www.exploringbinary.com/bigcomp-deciding-truncated-near-halfway-conversions/).
#[cfg(feature = "radix")]
#[allow(clippy::unwrap_used)] // reason = "none is a developer error due to shl overflow"
#[allow(clippy::comparison_chain)] // reason = "logically different conditions for algorithm"
pub fn byte_comp<F: RawFloat, const FORMAT: u128>(
number: Number,
mut fp: ExtendedFloat80,
sci_exp: i32,
) -> ExtendedFloat80 {
// Ensure our preconditions are valid:
// 1. The significant digits are not shifted into place.
debug_assert!(fp.mant & (1 << 63) != 0);
let format = NumberFormat::<FORMAT> {};
// Round down our extended-precision float and calculate `b`.
let mut b = fp;
shared::round::<F, _>(&mut b, shared::round_down);
let b = extended_to_float::<F>(b);
// Calculate `b+h` to create a ratio for our theoretical digits.
let theor = Bigfloat::from_float(bh::<F>(b));
// Now, create a scaling factor for the digit count.
let mut factor = Bigfloat::from_u32(1);
factor.pow(format.radix(), sci_exp.unsigned_abs()).unwrap();
let mut num: Bigfloat;
let mut den: Bigfloat;
if sci_exp < 0 {
// Need to have the basen factor be the numerator, and the `fp`
// be the denominator. Since we assumed that `theor` was the numerator,
// if it's the denominator, we need to multiply it into the numerator.
num = factor;
num.data *= &theor.data;
den = Bigfloat::from_u32(1);
den.exp = -theor.exp;
} else {
num = theor;
den = factor;
}
// Scale the denominator so it has the number of bits
// in the radix as the number of leading zeros.
let wlz = integral_binary_factor(format.radix());
let nlz = den.leading_zeros().wrapping_sub(wlz) & (32 - 1);
if nlz != 0 {
den.shl_bits(nlz as usize).unwrap();
den.exp -= nlz as i32;
}
// Need to scale the numerator or denominator to the same value.
// We don't want to shift the denominator, so...
let diff = den.exp - num.exp;
let shift = diff.unsigned_abs() as usize;
if diff < 0 {
// Need to shift the numerator left.
num.shl(shift).unwrap();
num.exp -= shift as i32;
} else if diff > 0 {
// Need to shift denominator left, go by a power of Limb::BITS.
// After this, the numerator will be non-normalized, and the
// denominator will be normalized. We need to add one to the
// quotient,since we're calculating the ceiling of the divmod.
let (q, r) = shift.ceil_divmod(Limb::BITS as usize);
let r = -r;
if r != 0 {
num.shl_bits(r as usize).unwrap();
num.exp -= r;
}
if q != 0 {
den.shl_limbs(q).unwrap();
den.exp -= Limb::BITS as i32 * q as i32;
}
}
// Compare our theoretical and real digits and round nearest, tie even.
let ord = compare_bytes::<FORMAT>(number, num, den);
shared::round::<F, _>(&mut fp, |f, s| {
shared::round_nearest_tie_even(f, s, |is_odd, _, _| {
// Can ignore `is_halfway` and `is_above`, since those were
// calculates using less significant digits.
match ord {
cmp::Ordering::Greater => true,
cmp::Ordering::Less => false,
cmp::Ordering::Equal if is_odd => true,
cmp::Ordering::Equal => false,
}
});
});
fp
}
/// Compare digits between the generated values the ratio and the actual view.
///
/// - `number` - The representation of the float as a big number, with the
/// parsed digits.
/// - `num` - The actual digits of the real floating point number.
/// - `den` - The theoretical digits created by `b+h` to determine if `b` or
/// `b+1`
#[cfg(feature = "radix")]
#[allow(clippy::unwrap_used)] // reason = "none is a developer error due to a missing fraction"
pub fn compare_bytes<const FORMAT: u128>(
number: Number,
mut num: Bigfloat,
den: Bigfloat,
) -> cmp::Ordering {
let format = NumberFormat::<FORMAT> {};
let radix = format.radix();
// Now need to compare the theoretical digits. First, I need to trim
// any leading zeros, and will also need to ignore trailing ones.
let mut integer = number.integer.bytes::<{ FORMAT }>();
let mut integer_iter = integer.integer_iter();
integer_iter.skip_zeros();
if integer_iter.is_buffer_empty() {
// Cannot be empty, since we must have at least **some** significant digits.
let mut fraction = number.fraction.unwrap().bytes::<{ FORMAT }>();
let mut fraction_iter = fraction.fraction_iter();
fraction_iter.skip_zeros();
fraction_compare!(fraction_iter, num, den, radix);
} else {
integer_compare!(integer_iter, num, den, radix);
if let Some(fraction) = number.fraction {
let mut fraction = fraction.bytes::<{ FORMAT }>();
let mut fraction_iter = fraction.fraction_iter();
fraction_compare!(fraction_iter, num, den, radix);
} else if !num.data.is_empty() {
// We had more theoretical digits, but no more actual digits.
return cmp::Ordering::Less;
}
}
// Exhausted both, must be equal.
cmp::Ordering::Equal
}
// SCALING
// -------
/// Calculate the scientific exponent from a `Number` value.
/// Any other attempts would require slowdowns for faster algorithms.
#[must_use]
#[inline(always)]
pub fn scientific_exponent<const FORMAT: u128>(num: &Number) -> i32 {
// This has the significant digits and exponent relative to those
// digits: therefore, we just need to scale to mantissa to `[1, radix)`.
// This doesn't need to be very fast.
let format = NumberFormat::<FORMAT> {};
// Use power reduction to make this faster: we need at least
// `F::MANTISSA_SIZE` bits, so we must have at least radix^4 digits.
// IF we're using base 3, we can have at most 11 divisions, and
// base 36, at most ~4. So, this is reasonably efficient.
let radix = format.radix() as u64;
let radix2 = radix * radix;
let radix4 = radix2 * radix2;
let mut mantissa = num.mantissa;
let mut exponent = num.exponent;
while mantissa >= radix4 {
mantissa /= radix4;
exponent += 4;
}
while mantissa >= radix2 {
mantissa /= radix2;
exponent += 2;
}
while mantissa >= radix {
mantissa /= radix;
exponent += 1;
}
exponent as i32
}
/// Calculate `b` from a a representation of `b` as a float.
#[must_use]
#[inline(always)]
pub fn b<F: RawFloat>(float: F) -> ExtendedFloat80 {
ExtendedFloat80 {
mant: float.mantissa().as_u64(),
exp: float.exponent(),
}
}
/// Calculate `b+h` from a a representation of `b` as a float.
#[must_use]
#[inline(always)]
pub fn bh<F: RawFloat>(float: F) -> ExtendedFloat80 {
let fp = b(float);
ExtendedFloat80 {
mant: (fp.mant << 1) + 1,
exp: fp.exp - 1,
}
}
// NOTE: There will never be binary factors here.
/// Calculate the integral ceiling of the binary factor from a basen number.
#[must_use]
#[inline(always)]
#[cfg(feature = "radix")]
pub const fn integral_binary_factor(radix: u32) -> u32 {
match radix {
3 => 2,
5 => 3,
6 => 3,
7 => 3,
9 => 4,
10 => 4,
11 => 4,
12 => 4,
13 => 4,
14 => 4,
15 => 4,
17 => 5,
18 => 5,
19 => 5,
20 => 5,
21 => 5,
22 => 5,
23 => 5,
24 => 5,
25 => 5,
26 => 5,
27 => 5,
28 => 5,
29 => 5,
30 => 5,
31 => 5,
33 => 6,
34 => 6,
35 => 6,
36 => 6,
// Invalid radix
_ => 0,
}
}
/// Calculate the integral ceiling of the binary factor from a basen number.
#[must_use]
#[inline(always)]
#[cfg(not(feature = "radix"))]
pub const fn integral_binary_factor(radix: u32) -> u32 {
match radix {
10 => 4,
// Invalid radix
_ => 0,
}
}
