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//! Logarithm
use crate::{ibig::IBig, ops::EstimatedLog2, ubig::UBig};
impl UBig {
/// Calculate the (truncated) logarithm of the [UBig]
///
/// This function could takes a long time when the integer is very large.
/// In applications where an exact result is not necessary,
/// [log2_bounds][UBig::log2_bounds] could be used.
///
/// # Panics
///
/// Panics if the number is 0, or the base is 0 or 1
///
/// # Examples
///
/// ```
/// # use dashu_int::UBig;
/// let base = UBig::from(3u8);
/// assert_eq!(UBig::from(81u8).ilog(&base), 4);
/// assert_eq!(UBig::from(1000u16).ilog(&base), 6);
/// ```
#[inline]
pub fn ilog(&self, base: &UBig) -> usize {
self.repr().log(base.repr()).0
}
}
impl EstimatedLog2 for UBig {
#[inline]
fn log2_bounds(&self) -> (f32, f32) {
self.repr().log2_bounds()
}
}
impl IBig {
/// Calculate the (truncated) logarithm of the magnitude of [IBig]
///
/// This function could takes a long time when the integer is very large.
/// In applications where an exact result is not necessary,
/// [log2_bounds][IBig::log2_bounds] could be used.
///
/// # Panics
///
/// Panics if the number is 0, or the base is 0 or 1
///
/// # Examples
///
/// ```
/// # use dashu_int::{UBig, IBig};
/// let base = UBig::from(3u8);
/// assert_eq!(IBig::from(-81).ilog(&base), 4);
/// assert_eq!(IBig::from(-1000).ilog(&base), 6);
/// ```
#[inline]
pub fn ilog(&self, base: &UBig) -> usize {
self.as_sign_repr().1.log(base.repr()).0
}
}
impl EstimatedLog2 for IBig {
#[inline]
fn log2_bounds(&self) -> (f32, f32) {
self.as_sign_repr().1.log2_bounds()
}
}
pub(crate) mod repr {
use core::cmp::Ordering;
use dashu_base::EstimatedLog2;
use crate::{
arch::word::{DoubleWord, Word},
buffer::Buffer,
cmp::cmp_in_place,
div,
error::panic_invalid_log_oprand,
helper_macros::debug_assert_zero,
math::max_exp_in_word,
mul, mul_ops, pow,
primitive::{extend_word, highest_dword, shrink_dword, split_dword, WORD_BITS_USIZE},
radix,
repr::{
Repr,
TypedReprRef::{self, *},
},
};
impl TypedReprRef<'_> {
/// Floor logarithm, returns (log(self), base^log(self))
pub fn log(self, base: TypedReprRef<'_>) -> (usize, Repr) {
// shortcuts
if let RefSmall(dw) = base {
match dw {
0 | 1 => panic_invalid_log_oprand(),
2 => {
return (
self.bit_len() - 1,
Repr::zero().into_typed().set_bit(self.bit_len()),
)
}
b if b.is_power_of_two() => {
let base_bits = b.trailing_zeros() as usize;
let exp = (self.bit_len() - 1) / base_bits;
return (exp, Repr::zero().into_typed().set_bit(exp * base_bits));
}
_ => {}
}
}
match (self, base) {
(RefSmall(dword), RefSmall(base_dword)) => log_dword(dword, base_dword),
(RefSmall(_), RefLarge(_)) => (0, Repr::one()),
(RefLarge(words), RefSmall(base_dword)) => {
if let Some(base_word) = shrink_dword(base_dword) {
log_word_base(words, base_word)
} else {
let mut buffer: [Word; 2] = [0; 2];
let (lo, hi) = split_dword(base_dword);
buffer[0] = lo;
buffer[1] = hi;
log_large(words, &buffer)
}
}
(RefLarge(words), RefLarge(base_words)) => match cmp_in_place(words, base_words) {
Ordering::Less => (0, Repr::one()),
Ordering::Equal => (1, Repr::from_buffer(Buffer::from(words))),
Ordering::Greater => log_large(words, base_words),
},
}
}
pub fn log2_bounds(self) -> (f32, f32) {
match self {
RefSmall(dword) => dword.log2_bounds(),
RefLarge(words) => log2_bounds_large(words),
}
}
}
fn log_dword(target: DoubleWord, base: DoubleWord) -> (usize, Repr) {
debug_assert!(base > 1);
// shortcuts
match target {
0 => panic_invalid_log_oprand(),
1 => return (0, Repr::one()),
i if i < base => return (0, Repr::one()),
i if i == base => return (1, Repr::from_dword(base)),
_ => {}
}
let log2_self = target.log2_bounds().0;
let log2_base = base.log2_bounds().1;
let mut est = (log2_self / log2_base) as u32; // float to int is underestimate
let mut est_pow = base.pow(est);
assert!(est_pow <= target);
while let Some(next_pow) = est_pow.checked_mul(base) {
let cmp = next_pow.cmp(&target);
if cmp.is_le() {
est_pow = next_pow;
est += 1;
}
if cmp.is_ge() {
break;
}
}
(est as usize, Repr::from_dword(est_pow))
}
pub(crate) fn log_word_base(target: &[Word], base: Word) -> (usize, Repr) {
let log2_self = log2_bounds_large(target).0;
let (wexp, wbase) = if base == 10 {
// specialize for base 10, which is cached in radix_info
(radix::RADIX10_INFO.digits_per_word, radix::RADIX10_INFO.range_per_word)
} else {
max_exp_in_word(base)
};
let log2_wbase = wbase.log2_bounds().1;
let mut est = (log2_self * wexp as f32 / log2_wbase) as usize; // est >= 1
let mut est_pow = if est == 1 {
Repr::from_word(base)
} else {
pow::repr::pow_word_base(base, est)
}
.into_buffer();
assert!(cmp_in_place(&est_pow, target).is_le());
// first proceed by multiplying wbase, which should happen very rarely
while est_pow.len() < target.len() {
if est_pow.len() == target.len() - 1 {
let target_hi = highest_dword(target);
let next_hi = (extend_word(*est_pow.last().unwrap()) + 1) * extend_word(wbase); // overestimate
if next_hi > target_hi {
break;
}
}
let carry = mul::mul_word_in_place(&mut est_pow, wbase);
est_pow.push_resizing(carry);
est += wexp;
}
// then proceed by multiplying base, which can require a few steps
loop {
match cmp_in_place(&est_pow, target) {
Ordering::Less => {
let carry = mul::mul_word_in_place(&mut est_pow, base);
est_pow.push_resizing(carry);
est += 1;
}
Ordering::Equal => break,
Ordering::Greater => {
// recover the over estimate
debug_assert_zero!(div::div_by_word_in_place(&mut est_pow, base));
est -= 1;
break;
}
}
}
(est, Repr::from_buffer(est_pow))
}
fn log_large(target: &[Word], base: &[Word]) -> (usize, Repr) {
debug_assert!(cmp_in_place(target, base).is_ge()); // this ensures est >= 1
// first estimates the result
let log2_self = log2_bounds_large(target).0;
let log2_base = log2_bounds_large(base).1;
let mut est = (log2_self / log2_base) as usize; // float to int is underestimate
est = est.max(1); // sometimes est can be zero due to estimation error
let mut est_pow = if est == 1 {
Repr::from_buffer(Buffer::from(base))
} else if base.len() == 2 {
let base_dword = highest_dword(base);
pow::repr::pow_dword_base(base_dword, est)
} else {
pow::repr::pow_large_base(base, est)
};
assert!(cmp_in_place(est_pow.as_slice(), target).is_le());
// then fix the error by trials
loop {
let next_pow = mul_ops::repr::mul_large(est_pow.as_slice(), base);
let cmp = cmp_in_place(next_pow.as_slice(), target);
if cmp.is_le() {
est_pow = next_pow;
est += 1;
}
if cmp.is_ge() {
break;
}
}
(est, est_pow)
}
#[inline]
fn log2_bounds_large(words: &[Word]) -> (f32, f32) {
// notice that the bit length can be larger than 2^24, so the result
// cannot be exact even if the input is a power of two
let hi = highest_dword(words);
let rem_bits = (words.len() - 2) * WORD_BITS_USIZE;
let (hi_lb, hi_ub) = hi.log2_bounds();
/// Adjustment required to ensure floor or ceil operation
const ADJUST: f32 = 2. * f32::EPSILON;
let est_lb = (hi_lb + rem_bits as f32) * (1. - ADJUST);
let est_ub = (hi_ub + rem_bits as f32) * (1. + ADJUST);
(est_lb, est_ub)
}
}