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//! Utilities for Rust numbers.
use core::ops;
/// Precalculated values of radix**i for i in range [0, arr.len()-1].
/// Each value can be **exactly** represented as that type.
const F32_POW10: [f32; 11] = [
1.0,
10.0,
100.0,
1000.0,
10000.0,
100000.0,
1000000.0,
10000000.0,
100000000.0,
1000000000.0,
10000000000.0,
];
/// Precalculated values of radix**i for i in range [0, arr.len()-1].
/// Each value can be **exactly** represented as that type.
const F64_POW10: [f64; 23] = [
1.0,
10.0,
100.0,
1000.0,
10000.0,
100000.0,
1000000.0,
10000000.0,
100000000.0,
1000000000.0,
10000000000.0,
100000000000.0,
1000000000000.0,
10000000000000.0,
100000000000000.0,
1000000000000000.0,
10000000000000000.0,
100000000000000000.0,
1000000000000000000.0,
10000000000000000000.0,
100000000000000000000.0,
1000000000000000000000.0,
10000000000000000000000.0,
];
/// Type that can be converted to primitive with `as`.
pub trait AsPrimitive: Sized + Copy + PartialOrd {
fn as_u32(self) -> u32;
fn as_u64(self) -> u64;
fn as_u128(self) -> u128;
fn as_usize(self) -> usize;
fn as_f32(self) -> f32;
fn as_f64(self) -> f64;
}
macro_rules! as_primitive_impl {
($($ty:ident)*) => {
$(
impl AsPrimitive for $ty {
#[inline]
fn as_u32(self) -> u32 {
self as u32
}
#[inline]
fn as_u64(self) -> u64 {
self as u64
}
#[inline]
fn as_u128(self) -> u128 {
self as u128
}
#[inline]
fn as_usize(self) -> usize {
self as usize
}
#[inline]
fn as_f32(self) -> f32 {
self as f32
}
#[inline]
fn as_f64(self) -> f64 {
self as f64
}
}
)*
};
}
as_primitive_impl! { u32 u64 u128 usize f32 f64 }
/// An interface for casting between machine scalars.
pub trait AsCast: AsPrimitive {
/// Creates a number from another value that can be converted into
/// a primitive via the `AsPrimitive` trait.
fn as_cast<N: AsPrimitive>(n: N) -> Self;
}
macro_rules! as_cast_impl {
($ty:ident, $method:ident) => {
impl AsCast for $ty {
#[inline]
fn as_cast<N: AsPrimitive>(n: N) -> Self {
n.$method()
}
}
};
}
as_cast_impl!(u32, as_u32);
as_cast_impl!(u64, as_u64);
as_cast_impl!(u128, as_u128);
as_cast_impl!(usize, as_usize);
as_cast_impl!(f32, as_f32);
as_cast_impl!(f64, as_f64);
/// Numerical type trait.
pub trait Number: AsCast + ops::Add<Output = Self> {}
macro_rules! number_impl {
($($ty:ident)*) => {
$(
impl Number for $ty {}
)*
};
}
number_impl! { u32 u64 u128 usize f32 f64 }
/// Defines a trait that supports integral operations.
pub trait Integer: Number + ops::BitAnd<Output = Self> + ops::Shr<i32, Output = Self> {
const ZERO: Self;
}
macro_rules! integer_impl {
($($ty:tt)*) => {
$(
impl Integer for $ty {
const ZERO: Self = 0;
}
)*
};
}
integer_impl! { u32 u64 u128 usize }
/// Type trait for the mantissa type.
pub trait Mantissa: Integer {
/// Mask to extract the high bits from the integer.
const HIMASK: Self;
/// Mask to extract the low bits from the integer.
const LOMASK: Self;
/// Full size of the integer, in bits.
const FULL: i32;
/// Half size of the integer, in bits.
const HALF: i32 = Self::FULL / 2;
}
impl Mantissa for u64 {
const HIMASK: u64 = 0xFFFFFFFF00000000;
const LOMASK: u64 = 0x00000000FFFFFFFF;
const FULL: i32 = 64;
}
/// Get exact exponent limit for radix.
pub trait Float: Number {
/// Unsigned type of the same size.
type Unsigned: Integer;
/// Literal zero.
const ZERO: Self;
/// Maximum number of digits that can contribute in the mantissa.
///
/// We can exactly represent a float in radix `b` from radix 2 if
/// `b` is divisible by 2. This function calculates the exact number of
/// digits required to exactly represent that float.
///
/// According to the "Handbook of Floating Point Arithmetic",
/// for IEEE754, with emin being the min exponent, p2 being the
/// precision, and b being the radix, the number of digits follows as:
///
/// `−emin + p2 + ⌊(emin + 1) log(2, b) − log(1 − 2^(−p2), b)⌋`
///
/// For f32, this follows as:
/// emin = -126
/// p2 = 24
///
/// For f64, this follows as:
/// emin = -1022
/// p2 = 53
///
/// In Python:
/// `-emin + p2 + math.floor((emin+1)*math.log(2, b) - math.log(1-2**(-p2), b))`
///
/// This was used to calculate the maximum number of digits for [2, 36].
const MAX_DIGITS: usize;
// MASKS
/// Bitmask for the sign bit.
const SIGN_MASK: Self::Unsigned;
/// Bitmask for the exponent, including the hidden bit.
const EXPONENT_MASK: Self::Unsigned;
/// Bitmask for the hidden bit in exponent, which is an implicit 1 in the fraction.
const HIDDEN_BIT_MASK: Self::Unsigned;
/// Bitmask for the mantissa (fraction), excluding the hidden bit.
const MANTISSA_MASK: Self::Unsigned;
// PROPERTIES
/// Positive infinity as bits.
const INFINITY_BITS: Self::Unsigned;
/// Positive infinity as bits.
const NEGATIVE_INFINITY_BITS: Self::Unsigned;
/// Size of the significand (mantissa) without hidden bit.
const MANTISSA_SIZE: i32;
/// Bias of the exponent
const EXPONENT_BIAS: i32;
/// Exponent portion of a denormal float.
const DENORMAL_EXPONENT: i32;
/// Maximum exponent value in float.
const MAX_EXPONENT: i32;
// ROUNDING
/// Default number of bits to shift (or 64 - mantissa size - 1).
const DEFAULT_SHIFT: i32;
/// Mask to determine if a full-carry occurred (1 in bit above hidden bit).
const CARRY_MASK: u64;
/// Get min and max exponent limits (exact) from radix.
fn exponent_limit() -> (i32, i32);
/// Get the number of digits that can be shifted from exponent to mantissa.
fn mantissa_limit() -> i32;
// Re-exported methods from std.
fn pow10(self, n: i32) -> Self;
fn from_bits(u: Self::Unsigned) -> Self;
fn to_bits(self) -> Self::Unsigned;
fn is_sign_positive(self) -> bool;
/// Returns true if the float is a denormal.
#[inline]
fn is_denormal(self) -> bool {
self.to_bits() & Self::EXPONENT_MASK == Self::Unsigned::ZERO
}
/// Returns true if the float is a NaN or Infinite.
#[inline]
fn is_special(self) -> bool {
self.to_bits() & Self::EXPONENT_MASK == Self::EXPONENT_MASK
}
/// Returns true if the float is infinite.
#[inline]
fn is_inf(self) -> bool {
self.is_special() && (self.to_bits() & Self::MANTISSA_MASK) == Self::Unsigned::ZERO
}
/// Get exponent component from the float.
#[inline]
fn exponent(self) -> i32 {
if self.is_denormal() {
return Self::DENORMAL_EXPONENT;
}
let bits = self.to_bits();
let biased_e = ((bits & Self::EXPONENT_MASK) >> Self::MANTISSA_SIZE).as_u32();
biased_e as i32 - Self::EXPONENT_BIAS
}
/// Get mantissa (significand) component from float.
#[inline]
fn mantissa(self) -> Self::Unsigned {
let bits = self.to_bits();
let s = bits & Self::MANTISSA_MASK;
if !self.is_denormal() {
s + Self::HIDDEN_BIT_MASK
} else {
s
}
}
/// Get next greater float for a positive float.
/// Value must be >= 0.0 and < INFINITY.
#[inline]
fn next_positive(self) -> Self {
debug_assert!(self.is_sign_positive() && !self.is_inf());
Self::from_bits(self.to_bits() + Self::Unsigned::as_cast(1u32))
}
/// Round a positive number to even.
#[inline]
fn round_positive_even(self) -> Self {
if self.mantissa() & Self::Unsigned::as_cast(1u32) == Self::Unsigned::as_cast(1u32) {
self.next_positive()
} else {
self
}
}
}
impl Float for f32 {
type Unsigned = u32;
const ZERO: f32 = 0.0;
const MAX_DIGITS: usize = 114;
const SIGN_MASK: u32 = 0x80000000;
const EXPONENT_MASK: u32 = 0x7F800000;
const HIDDEN_BIT_MASK: u32 = 0x00800000;
const MANTISSA_MASK: u32 = 0x007FFFFF;
const INFINITY_BITS: u32 = 0x7F800000;
const NEGATIVE_INFINITY_BITS: u32 = Self::INFINITY_BITS | Self::SIGN_MASK;
const MANTISSA_SIZE: i32 = 23;
const EXPONENT_BIAS: i32 = 127 + Self::MANTISSA_SIZE;
const DENORMAL_EXPONENT: i32 = 1 - Self::EXPONENT_BIAS;
const MAX_EXPONENT: i32 = 0xFF - Self::EXPONENT_BIAS;
const DEFAULT_SHIFT: i32 = u64::FULL - f32::MANTISSA_SIZE - 1;
const CARRY_MASK: u64 = 0x1000000;
#[inline]
fn exponent_limit() -> (i32, i32) {
(-10, 10)
}
#[inline]
fn mantissa_limit() -> i32 {
7
}
#[inline]
fn pow10(self, n: i32) -> f32 {
// Check the exponent is within bounds in debug builds.
debug_assert!({
let (min, max) = Self::exponent_limit();
n >= min && n <= max
});
if n > 0 {
self * F32_POW10[n as usize]
} else {
self / F32_POW10[-n as usize]
}
}
#[inline]
fn from_bits(u: u32) -> f32 {
f32::from_bits(u)
}
#[inline]
fn to_bits(self) -> u32 {
f32::to_bits(self)
}
#[inline]
fn is_sign_positive(self) -> bool {
f32::is_sign_positive(self)
}
}
impl Float for f64 {
type Unsigned = u64;
const ZERO: f64 = 0.0;
const MAX_DIGITS: usize = 769;
const SIGN_MASK: u64 = 0x8000000000000000;
const EXPONENT_MASK: u64 = 0x7FF0000000000000;
const HIDDEN_BIT_MASK: u64 = 0x0010000000000000;
const MANTISSA_MASK: u64 = 0x000FFFFFFFFFFFFF;
const INFINITY_BITS: u64 = 0x7FF0000000000000;
const NEGATIVE_INFINITY_BITS: u64 = Self::INFINITY_BITS | Self::SIGN_MASK;
const MANTISSA_SIZE: i32 = 52;
const EXPONENT_BIAS: i32 = 1023 + Self::MANTISSA_SIZE;
const DENORMAL_EXPONENT: i32 = 1 - Self::EXPONENT_BIAS;
const MAX_EXPONENT: i32 = 0x7FF - Self::EXPONENT_BIAS;
const DEFAULT_SHIFT: i32 = u64::FULL - f64::MANTISSA_SIZE - 1;
const CARRY_MASK: u64 = 0x20000000000000;
#[inline]
fn exponent_limit() -> (i32, i32) {
(-22, 22)
}
#[inline]
fn mantissa_limit() -> i32 {
15
}
#[inline]
fn pow10(self, n: i32) -> f64 {
// Check the exponent is within bounds in debug builds.
debug_assert!({
let (min, max) = Self::exponent_limit();
n >= min && n <= max
});
if n > 0 {
self * F64_POW10[n as usize]
} else {
self / F64_POW10[-n as usize]
}
}
#[inline]
fn from_bits(u: u64) -> f64 {
f64::from_bits(u)
}
#[inline]
fn to_bits(self) -> u64 {
f64::to_bits(self)
}
#[inline]
fn is_sign_positive(self) -> bool {
f64::is_sign_positive(self)
}
}