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//! Algorithms to efficiently convert strings to floats.
use super::bhcomp::*;
use super::cached::*;
use super::errors::*;
use super::float::ExtendedFloat;
use super::num::*;
use super::small_powers::*;
// FAST
// ----
/// Convert mantissa to exact value for a non-base2 power.
///
/// Returns the resulting float and if the value can be represented exactly.
pub(crate) fn fast_path<F>(mantissa: u64, exponent: i32) -> Option<F>
where
F: Float,
{
// `mantissa >> (F::MANTISSA_SIZE+1) != 0` effectively checks if the
// value has a no bits above the hidden bit, which is what we want.
let (min_exp, max_exp) = F::exponent_limit();
let shift_exp = F::mantissa_limit();
let mantissa_size = F::MANTISSA_SIZE + 1;
if mantissa == 0 {
Some(F::ZERO)
} else if mantissa >> mantissa_size != 0 {
// Would require truncation of the mantissa.
None
} else if exponent == 0 {
// 0 exponent, same as value, exact representation.
let float = F::as_cast(mantissa);
Some(float)
} else if exponent >= min_exp && exponent <= max_exp {
// Value can be exactly represented, return the value.
// Do not use powi, since powi can incrementally introduce
// error.
let float = F::as_cast(mantissa);
Some(float.pow10(exponent))
} else if exponent >= 0 && exponent <= max_exp + shift_exp {
// Check to see if we have a disguised fast-path, where the
// number of digits in the mantissa is very small, but and
// so digits can be shifted from the exponent to the mantissa.
let small_powers = POW10_64;
let shift = exponent - max_exp;
let power = small_powers[shift as usize];
// Compute the product of the power, if it overflows,
// prematurely return early, otherwise, if we didn't overshoot,
// we can get an exact value.
let value = match mantissa.checked_mul(power) {
None => return None,
Some(value) => value,
};
if value >> mantissa_size != 0 {
None
} else {
// Use powi, since it's correct, and faster on
// the fast-path.
let float = F::as_cast(value);
Some(float.pow10(max_exp))
}
} else {
// Cannot be exactly represented, exponent too small or too big,
// would require truncation.
None
}
}
// MODERATE
// --------
/// Multiply the floating-point by the exponent.
///
/// Multiply by pre-calculated powers of the base, modify the extended-
/// float, and return if new value and if the value can be represented
/// accurately.
fn multiply_exponent_extended<F>(fp: &mut ExtendedFloat, exponent: i32, truncated: bool) -> bool
where
F: Float,
{
let powers = ExtendedFloat::get_powers();
let exponent = exponent.saturating_add(powers.bias);
let small_index = exponent % powers.step;
let large_index = exponent / powers.step;
if exponent < 0 {
// Guaranteed underflow (assign 0).
fp.mant = 0;
true
} else if large_index as usize >= powers.large.len() {
// Overflow (assign infinity)
fp.mant = 1 << 63;
fp.exp = 0x7FF;
true
} else {
// Within the valid exponent range, multiply by the large and small
// exponents and return the resulting value.
// Track errors to as a factor of unit in last-precision.
let mut errors: u32 = 0;
if truncated {
errors += u64::error_halfscale();
}
// Multiply by the small power.
// Check if we can directly multiply by an integer, if not,
// use extended-precision multiplication.
match fp
.mant
.overflowing_mul(powers.get_small_int(small_index as usize))
{
// Overflow, multiplication unsuccessful, go slow path.
(_, true) => {
fp.normalize();
fp.imul(&powers.get_small(small_index as usize));
errors += u64::error_halfscale();
}
// No overflow, multiplication successful.
(mant, false) => {
fp.mant = mant;
fp.normalize();
}
}
// Multiply by the large power
fp.imul(&powers.get_large(large_index as usize));
if errors > 0 {
errors += 1;
}
errors += u64::error_halfscale();
// Normalize the floating point (and the errors).
let shift = fp.normalize();
errors <<= shift;
u64::error_is_accurate::<F>(errors, fp)
}
}
/// Create a precise native float using an intermediate extended-precision float.
///
/// Return the float approximation and if the value can be accurately
/// represented with mantissa bits of precision.
#[inline]
pub(crate) fn moderate_path<F>(
mantissa: u64,
exponent: i32,
truncated: bool,
) -> (ExtendedFloat, bool)
where
F: Float,
{
let mut fp = ExtendedFloat {
mant: mantissa,
exp: 0,
};
let valid = multiply_exponent_extended::<F>(&mut fp, exponent, truncated);
(fp, valid)
}
// FALLBACK
// --------
/// Fallback path when the fast path does not work.
///
/// Uses the moderate path, if applicable, otherwise, uses the slow path
/// as required.
pub(crate) fn fallback_path<F>(
integer: &[u8],
fraction: &[u8],
mantissa: u64,
exponent: i32,
mantissa_exponent: i32,
truncated: bool,
) -> F
where
F: Float,
{
// Moderate path (use an extended 80-bit representation).
let (fp, valid) = moderate_path::<F>(mantissa, mantissa_exponent, truncated);
if valid {
return fp.into_float::<F>();
}
// Slow path, fast path didn't work.
let b = fp.into_downward_float::<F>();
if b.is_special() {
// We have a non-finite number, we get to leave early.
b
} else {
bhcomp(b, integer, fraction, exponent)
}
}