rust_decimal/maths.rs
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use crate::prelude::*;
use num_traits::pow::Pow;
// Tolerance for inaccuracies when calculating exp
const EXP_TOLERANCE: Decimal = Decimal::from_parts(2, 0, 0, false, 7);
// Approximation of 1/ln(10) = 0.4342944819032518276511289189
const LN10_INVERSE: Decimal = Decimal::from_parts_raw(1763037029, 1670682625, 235431510, 1835008);
// Total iterations of taylor series for Trig.
const TRIG_SERIES_UPPER_BOUND: usize = 6;
// PI / 8
const EIGHTH_PI: Decimal = Decimal::from_parts_raw(2822163429, 3244459792, 212882598, 1835008);
// Table representing {index}!
const FACTORIAL: [Decimal; 28] = [
Decimal::from_parts(1, 0, 0, false, 0),
Decimal::from_parts(1, 0, 0, false, 0),
Decimal::from_parts(2, 0, 0, false, 0),
Decimal::from_parts(6, 0, 0, false, 0),
Decimal::from_parts(24, 0, 0, false, 0),
// 5!
Decimal::from_parts(120, 0, 0, false, 0),
Decimal::from_parts(720, 0, 0, false, 0),
Decimal::from_parts(5040, 0, 0, false, 0),
Decimal::from_parts(40320, 0, 0, false, 0),
Decimal::from_parts(362880, 0, 0, false, 0),
// 10!
Decimal::from_parts(3628800, 0, 0, false, 0),
Decimal::from_parts(39916800, 0, 0, false, 0),
Decimal::from_parts(479001600, 0, 0, false, 0),
Decimal::from_parts(1932053504, 1, 0, false, 0),
Decimal::from_parts(1278945280, 20, 0, false, 0),
// 15!
Decimal::from_parts(2004310016, 304, 0, false, 0),
Decimal::from_parts(2004189184, 4871, 0, false, 0),
Decimal::from_parts(4006445056, 82814, 0, false, 0),
Decimal::from_parts(3396534272, 1490668, 0, false, 0),
Decimal::from_parts(109641728, 28322707, 0, false, 0),
// 20!
Decimal::from_parts(2192834560, 566454140, 0, false, 0),
Decimal::from_parts(3099852800, 3305602358, 2, false, 0),
Decimal::from_parts(3772252160, 4003775155, 60, false, 0),
Decimal::from_parts(862453760, 1892515369, 1401, false, 0),
Decimal::from_parts(3519021056, 2470695900, 33634, false, 0),
// 25!
Decimal::from_parts(2076180480, 1637855376, 840864, false, 0),
Decimal::from_parts(2441084928, 3929534124, 21862473, false, 0),
Decimal::from_parts(1484783616, 3018206259, 590286795, false, 0),
];
/// Trait exposing various mathematical operations that can be applied using a Decimal. This is only
/// present when the `maths` feature has been enabled.
pub trait MathematicalOps {
/// The estimated exponential function, e<sup>x</sup>. Stops calculating when it is within
/// tolerance of roughly `0.0000002`.
fn exp(&self) -> Decimal;
/// The estimated exponential function, e<sup>x</sup>. Stops calculating when it is within
/// tolerance of roughly `0.0000002`. Returns `None` on overflow.
fn checked_exp(&self) -> Option<Decimal>;
/// The estimated exponential function, e<sup>x</sup> using the `tolerance` provided as a hint
/// as to when to stop calculating. A larger tolerance will cause the number to stop calculating
/// sooner at the potential cost of a slightly less accurate result.
fn exp_with_tolerance(&self, tolerance: Decimal) -> Decimal;
/// The estimated exponential function, e<sup>x</sup> using the `tolerance` provided as a hint
/// as to when to stop calculating. A larger tolerance will cause the number to stop calculating
/// sooner at the potential cost of a slightly less accurate result.
/// Returns `None` on overflow.
fn checked_exp_with_tolerance(&self, tolerance: Decimal) -> Option<Decimal>;
/// Raise self to the given integer exponent: x<sup>y</sup>
fn powi(&self, exp: i64) -> Decimal;
/// Raise self to the given integer exponent x<sup>y</sup> returning `None` on overflow.
fn checked_powi(&self, exp: i64) -> Option<Decimal>;
/// Raise self to the given unsigned integer exponent: x<sup>y</sup>
fn powu(&self, exp: u64) -> Decimal;
/// Raise self to the given unsigned integer exponent x<sup>y</sup> returning `None` on overflow.
fn checked_powu(&self, exp: u64) -> Option<Decimal>;
/// Raise self to the given floating point exponent: x<sup>y</sup>
fn powf(&self, exp: f64) -> Decimal;
/// Raise self to the given floating point exponent x<sup>y</sup> returning `None` on overflow.
fn checked_powf(&self, exp: f64) -> Option<Decimal>;
/// Raise self to the given Decimal exponent: x<sup>y</sup>. If `exp` is not whole then the approximation
/// e<sup>y*ln(x)</sup> is used.
fn powd(&self, exp: Decimal) -> Decimal;
/// Raise self to the given Decimal exponent x<sup>y</sup> returning `None` on overflow.
/// If `exp` is not whole then the approximation e<sup>y*ln(x)</sup> is used.
fn checked_powd(&self, exp: Decimal) -> Option<Decimal>;
/// The square root of a Decimal. Uses a standard Babylonian method.
fn sqrt(&self) -> Option<Decimal>;
/// Calculates the natural logarithm for a Decimal calculated using Taylor's series.
fn ln(&self) -> Decimal;
/// Calculates the checked natural logarithm for a Decimal calculated using Taylor's series.
/// Returns `None` for negative numbers or zero.
fn checked_ln(&self) -> Option<Decimal>;
/// Calculates the base 10 logarithm of a specified Decimal number.
fn log10(&self) -> Decimal;
/// Calculates the checked base 10 logarithm of a specified Decimal number.
/// Returns `None` for negative numbers or zero.
fn checked_log10(&self) -> Option<Decimal>;
/// Abramowitz Approximation of Error Function from [wikipedia](https://en.wikipedia.org/wiki/Error_function#Numerical_approximations)
fn erf(&self) -> Decimal;
/// The Cumulative distribution function for a Normal distribution
fn norm_cdf(&self) -> Decimal;
/// The Probability density function for a Normal distribution.
fn norm_pdf(&self) -> Decimal;
/// The Probability density function for a Normal distribution returning `None` on overflow.
fn checked_norm_pdf(&self) -> Option<Decimal>;
/// Computes the sine of a number (in radians).
/// Panics upon overflow.
fn sin(&self) -> Decimal;
/// Computes the checked sine of a number (in radians).
fn checked_sin(&self) -> Option<Decimal>;
/// Computes the cosine of a number (in radians).
/// Panics upon overflow.
fn cos(&self) -> Decimal;
/// Computes the checked cosine of a number (in radians).
fn checked_cos(&self) -> Option<Decimal>;
/// Computes the tangent of a number (in radians).
/// Panics upon overflow or upon approaching a limit.
fn tan(&self) -> Decimal;
/// Computes the checked tangent of a number (in radians).
/// Returns None on limit.
fn checked_tan(&self) -> Option<Decimal>;
}
impl MathematicalOps for Decimal {
fn exp(&self) -> Decimal {
self.exp_with_tolerance(EXP_TOLERANCE)
}
fn checked_exp(&self) -> Option<Decimal> {
self.checked_exp_with_tolerance(EXP_TOLERANCE)
}
fn exp_with_tolerance(&self, tolerance: Decimal) -> Decimal {
match self.checked_exp_with_tolerance(tolerance) {
Some(d) => d,
None => {
if self.is_sign_negative() {
panic!("Exp underflowed")
} else {
panic!("Exp overflowed")
}
}
}
}
fn checked_exp_with_tolerance(&self, tolerance: Decimal) -> Option<Decimal> {
if self.is_zero() {
return Some(Decimal::ONE);
}
if self.is_sign_negative() {
let mut flipped = *self;
flipped.set_sign_positive(true);
let exp = flipped.checked_exp_with_tolerance(tolerance)?;
return Decimal::ONE.checked_div(exp);
}
let mut term = *self;
let mut result = self.checked_add(Decimal::ONE)?;
for factorial in FACTORIAL.iter().skip(2) {
term = self.checked_mul(term)?;
let next = result + (term / factorial);
let diff = (next - result).abs();
result = next;
if diff <= tolerance {
break;
}
}
Some(result)
}
fn powi(&self, exp: i64) -> Decimal {
match self.checked_powi(exp) {
Some(result) => result,
None => panic!("Pow overflowed"),
}
}
fn checked_powi(&self, exp: i64) -> Option<Decimal> {
// For negative exponents we change x^-y into 1 / x^y.
// Otherwise, we calculate a standard unsigned exponent
if exp >= 0 {
return self.checked_powu(exp as u64);
}
// Get the unsigned exponent
let exp = exp.unsigned_abs();
let pow = match self.checked_powu(exp) {
Some(v) => v,
None => return None,
};
Decimal::ONE.checked_div(pow)
}
fn powu(&self, exp: u64) -> Decimal {
match self.checked_powu(exp) {
Some(result) => result,
None => panic!("Pow overflowed"),
}
}
fn checked_powu(&self, exp: u64) -> Option<Decimal> {
if exp == 0 {
return Some(Decimal::ONE);
}
if self.is_zero() {
return Some(Decimal::ZERO);
}
if self.is_one() {
return Some(Decimal::ONE);
}
match exp {
0 => unreachable!(),
1 => Some(*self),
2 => self.checked_mul(*self),
// Do the exponentiation by multiplying squares:
// y = Sum (for each 1 bit in binary representation) of (2 ^ bit)
// x ^ y = Sum (for each 1 bit in y) of (x ^ (2 ^ bit))
// See: https://en.wikipedia.org/wiki/Exponentiation_by_squaring
_ => {
let mut product = Decimal::ONE;
let mut mask = exp;
let mut power = *self;
// Run through just enough 1 bits
for n in 0..(64 - exp.leading_zeros()) {
if n > 0 {
power = power.checked_mul(power)?;
mask >>= 1;
}
if mask & 0x01 > 0 {
match product.checked_mul(power) {
Some(r) => product = r,
None => return None,
};
}
}
product.normalize_assign();
Some(product)
}
}
}
fn powf(&self, exp: f64) -> Decimal {
match self.checked_powf(exp) {
Some(result) => result,
None => panic!("Pow overflowed"),
}
}
fn checked_powf(&self, exp: f64) -> Option<Decimal> {
let exp = match Decimal::from_f64(exp) {
Some(f) => f,
None => return None,
};
self.checked_powd(exp)
}
fn powd(&self, exp: Decimal) -> Decimal {
match self.checked_powd(exp) {
Some(result) => result,
None => panic!("Pow overflowed"),
}
}
fn checked_powd(&self, exp: Decimal) -> Option<Decimal> {
if exp.is_zero() {
return Some(Decimal::ONE);
}
if self.is_zero() {
return Some(Decimal::ZERO);
}
if self.is_one() {
return Some(Decimal::ONE);
}
if exp.is_one() {
return Some(*self);
}
// If the scale is 0 then it's a trivial calculation
let exp = exp.normalize();
if exp.scale() == 0 {
if exp.mid() != 0 || exp.hi() != 0 {
// Exponent way too big
return None;
}
return if exp.is_sign_negative() {
self.checked_powi(-(exp.lo() as i64))
} else {
self.checked_powu(exp.lo() as u64)
};
}
// We do some approximations since we've got a decimal exponent.
// For positive bases: a^b = exp(b*ln(a))
let negative = self.is_sign_negative();
let e = match self.abs().ln().checked_mul(exp) {
Some(e) => e,
None => return None,
};
let mut result = e.checked_exp()?;
result.set_sign_negative(negative);
Some(result)
}
fn sqrt(&self) -> Option<Decimal> {
if self.is_sign_negative() {
return None;
}
if self.is_zero() {
return Some(Decimal::ZERO);
}
// Start with an arbitrary number as the first guess
let mut result = self / Decimal::TWO;
// Too small to represent, so we start with self
// Future iterations could actually avoid using a decimal altogether and use a buffered
// vector, only combining back into a decimal on return
if result.is_zero() {
result = *self;
}
let mut last = result + Decimal::ONE;
// Keep going while the difference is larger than the tolerance
let mut circuit_breaker = 0;
while last != result {
circuit_breaker += 1;
assert!(circuit_breaker < 1000, "geo mean circuit breaker");
last = result;
result = (result + self / result) / Decimal::TWO;
}
Some(result)
}
#[cfg(feature = "maths-nopanic")]
fn ln(&self) -> Decimal {
match self.checked_ln() {
Some(result) => result,
None => Decimal::ZERO,
}
}
#[cfg(not(feature = "maths-nopanic"))]
fn ln(&self) -> Decimal {
match self.checked_ln() {
Some(result) => result,
None => {
if self.is_sign_negative() {
panic!("Unable to calculate ln for negative numbers")
} else if self.is_zero() {
panic!("Unable to calculate ln for zero")
} else {
panic!("Calculation of ln failed for unknown reasons")
}
}
}
}
fn checked_ln(&self) -> Option<Decimal> {
if self.is_sign_negative() || self.is_zero() {
return None;
}
if self.is_one() {
return Some(Decimal::ZERO);
}
// Approximate using Taylor Series
let mut x = *self;
let mut count = 0;
while x >= Decimal::ONE {
x *= Decimal::E_INVERSE;
count += 1;
}
while x <= Decimal::E_INVERSE {
x *= Decimal::E;
count -= 1;
}
x -= Decimal::ONE;
if x.is_zero() {
return Some(Decimal::new(count, 0));
}
let mut result = Decimal::ZERO;
let mut iteration = 0;
let mut y = Decimal::ONE;
let mut last = Decimal::ONE;
while last != result && iteration < 100 {
iteration += 1;
last = result;
y *= -x;
result += y / Decimal::new(iteration, 0);
}
Some(Decimal::new(count, 0) - result)
}
#[cfg(feature = "maths-nopanic")]
fn log10(&self) -> Decimal {
match self.checked_log10() {
Some(result) => result,
None => Decimal::ZERO,
}
}
#[cfg(not(feature = "maths-nopanic"))]
fn log10(&self) -> Decimal {
match self.checked_log10() {
Some(result) => result,
None => {
if self.is_sign_negative() {
panic!("Unable to calculate log10 for negative numbers")
} else if self.is_zero() {
panic!("Unable to calculate log10 for zero")
} else {
panic!("Calculation of log10 failed for unknown reasons")
}
}
}
}
fn checked_log10(&self) -> Option<Decimal> {
use crate::ops::array::{div_by_u32, is_all_zero};
// Early exits
if self.is_sign_negative() || self.is_zero() {
return None;
}
if self.is_one() {
return Some(Decimal::ZERO);
}
// This uses a very basic method for calculating log10. We know the following is true:
// log10(n) = ln(n) / ln(10)
// From this we can perform some small optimizations:
// 1. ln(10) is a constant
// 2. Multiplication is faster than division, so we can pre-calculate the constant 1/ln(10)
// This allows us to then simplify log10(n) to:
// log10(n) = C * ln(n)
// Before doing all of this however, we see if there are simple calculations to be made.
let scale = self.scale();
let mut working = self.mantissa_array3();
// Check for scales less than 1 as an early exit
if scale > 0 && working[2] == 0 && working[1] == 0 && working[0] == 1 {
return Some(Decimal::from_parts(scale, 0, 0, true, 0));
}
// Loop for detecting bordering base 10 values
let mut result = 0;
let mut base10 = true;
while !is_all_zero(&working) {
let remainder = div_by_u32(&mut working, 10u32);
if remainder != 0 {
base10 = false;
break;
}
result += 1;
if working[2] == 0 && working[1] == 0 && working[0] == 1 {
break;
}
}
if base10 {
return Some((result - scale as i32).into());
}
self.checked_ln().map(|result| LN10_INVERSE * result)
}
fn erf(&self) -> Decimal {
if self.is_sign_positive() {
let one = &Decimal::ONE;
let xa1 = self * Decimal::from_parts(705230784, 0, 0, false, 10);
let xa2 = self.powi(2) * Decimal::from_parts(422820123, 0, 0, false, 10);
let xa3 = self.powi(3) * Decimal::from_parts(92705272, 0, 0, false, 10);
let xa4 = self.powi(4) * Decimal::from_parts(1520143, 0, 0, false, 10);
let xa5 = self.powi(5) * Decimal::from_parts(2765672, 0, 0, false, 10);
let xa6 = self.powi(6) * Decimal::from_parts(430638, 0, 0, false, 10);
let sum = one + xa1 + xa2 + xa3 + xa4 + xa5 + xa6;
one - (one / sum.powi(16))
} else {
-self.abs().erf()
}
}
fn norm_cdf(&self) -> Decimal {
(Decimal::ONE + (self / Decimal::from_parts(2318911239, 3292722, 0, false, 16)).erf()) / Decimal::TWO
}
fn norm_pdf(&self) -> Decimal {
match self.checked_norm_pdf() {
Some(d) => d,
None => panic!("Norm Pdf overflowed"),
}
}
fn checked_norm_pdf(&self) -> Option<Decimal> {
let sqrt2pi = Decimal::from_parts_raw(2133383024, 2079885984, 1358845910, 1835008);
let factor = -self.checked_powi(2)?;
let factor = factor.checked_div(Decimal::TWO)?;
factor.checked_exp()?.checked_div(sqrt2pi)
}
fn sin(&self) -> Decimal {
match self.checked_sin() {
Some(x) => x,
None => panic!("Sin overflowed"),
}
}
fn checked_sin(&self) -> Option<Decimal> {
if self.is_zero() {
return Some(Decimal::ZERO);
}
if self.is_sign_negative() {
// -Sin(-x)
return (-self).checked_sin().map(|x| -x);
}
if self >= &Decimal::TWO_PI {
// Reduce large numbers early - we can do this using rem to constrain to a range
let adjusted = self.checked_rem(Decimal::TWO_PI)?;
return adjusted.checked_sin();
}
if self >= &Decimal::PI {
// -Sin(x-π)
return (self - Decimal::PI).checked_sin().map(|x| -x);
}
if self > &Decimal::QUARTER_PI {
// Cos(π2-x)
return (Decimal::HALF_PI - self).checked_cos();
}
// Taylor series:
// ∑(n=0 to ∞) : ((−1)^n / (2n + 1)!) * x^(2n + 1) , x∈R
// First few expansions:
// x^1/1! - x^3/3! + x^5/5! - x^7/7! + x^9/9!
let mut result = Decimal::ZERO;
for n in 0..TRIG_SERIES_UPPER_BOUND {
let x = 2 * n + 1;
let element = self.checked_powi(x as i64)?.checked_div(FACTORIAL[x])?;
if n & 0x1 == 0 {
result += element;
} else {
result -= element;
}
}
Some(result)
}
fn cos(&self) -> Decimal {
match self.checked_cos() {
Some(x) => x,
None => panic!("Cos overflowed"),
}
}
fn checked_cos(&self) -> Option<Decimal> {
if self.is_zero() {
return Some(Decimal::ONE);
}
if self.is_sign_negative() {
// Cos(-x)
return (-self).checked_cos();
}
if self >= &Decimal::TWO_PI {
// Reduce large numbers early - we can do this using rem to constrain to a range
let adjusted = self.checked_rem(Decimal::TWO_PI)?;
return adjusted.checked_cos();
}
if self >= &Decimal::PI {
// -Cos(x-π)
return (self - Decimal::PI).checked_cos().map(|x| -x);
}
if self > &Decimal::QUARTER_PI {
// Sin(π2-x)
return (Decimal::HALF_PI - self).checked_sin();
}
// Taylor series:
// ∑(n=0 to ∞) : ((−1)^n / (2n)!) * x^(2n) , x∈R
// First few expansions:
// x^0/0! - x^2/2! + x^4/4! - x^6/6! + x^8/8!
let mut result = Decimal::ZERO;
for n in 0..TRIG_SERIES_UPPER_BOUND {
let x = 2 * n;
let element = self.checked_powi(x as i64)?.checked_div(FACTORIAL[x])?;
if n & 0x1 == 0 {
result += element;
} else {
result -= element;
}
}
Some(result)
}
fn tan(&self) -> Decimal {
match self.checked_tan() {
Some(x) => x,
None => panic!("Tan overflowed"),
}
}
fn checked_tan(&self) -> Option<Decimal> {
if self.is_zero() {
return Some(Decimal::ZERO);
}
if self.is_sign_negative() {
// -Tan(-x)
return (-self).checked_tan().map(|x| -x);
}
if self >= &Decimal::TWO_PI {
// Reduce large numbers early - we can do this using rem to constrain to a range
let adjusted = self.checked_rem(Decimal::TWO_PI)?;
return adjusted.checked_tan();
}
// Reduce to 0 <= x <= PI
if self >= &Decimal::PI {
// Tan(x-π)
return (self - Decimal::PI).checked_tan();
}
// Reduce to 0 <= x <= PI/2
if self > &Decimal::HALF_PI {
// We can use the symmetrical function inside the first quadrant
// e.g. tan(x) = -tan((PI/2 - x) + PI/2)
return ((Decimal::HALF_PI - self) + Decimal::HALF_PI).checked_tan().map(|x| -x);
}
// It has now been reduced to 0 <= x <= PI/2. If it is >= PI/4 we can make it even smaller
// by calculating tan(PI/2 - x) and taking the reciprocal
if self > &Decimal::QUARTER_PI {
return match (Decimal::HALF_PI - self).checked_tan() {
Some(x) => Decimal::ONE.checked_div(x),
None => None,
};
}
// Due the way that tan(x) sharply tends towards infinity, we try to optimize
// the resulting accuracy by using Trigonometric identity when > PI/8. We do this by
// replacing the angle with one that is half as big.
if self > &EIGHTH_PI {
// Work out tan(x/2)
let tan_half = (self / Decimal::TWO).checked_tan()?;
// Work out the dividend i.e. 2tan(x/2)
let dividend = Decimal::TWO.checked_mul(tan_half)?;
// Work out the divisor i.e. 1 - tan^2(x/2)
let squared = tan_half.checked_mul(tan_half)?;
let divisor = Decimal::ONE - squared;
// Treat this as infinity
if divisor.is_zero() {
return None;
}
return dividend.checked_div(divisor);
}
// Do a polynomial approximation based upon the Maclaurin series.
// This can be simplified to something like:
//
// ∑(n=1,3,5,7,9)(f(n)(0)/n!)x^n
//
// First few expansions (which we leverage):
// (f'(0)/1!)x^1 + (f'''(0)/3!)x^3 + (f'''''(0)/5!)x^5 + (f'''''''/7!)x^7
//
// x + (1/3)x^3 + (2/15)x^5 + (17/315)x^7 + (62/2835)x^9 + (1382/155925)x^11
//
// (Generated by https://www.wolframalpha.com/widgets/view.jsp?id=fe1ad8d4f5dbb3cb866d0c89beb527a6)
// The more terms, the better the accuracy. This generates accuracy within approx 10^-8 for angles
// less than PI/8.
const SERIES: [(Decimal, u64); 6] = [
// 1 / 3
(Decimal::from_parts_raw(89478485, 347537611, 180700362, 1835008), 3),
// 2 / 15
(Decimal::from_parts_raw(894784853, 3574988881, 72280144, 1835008), 5),
// 17 / 315
(Decimal::from_parts_raw(905437054, 3907911371, 2925624, 1769472), 7),
// 62 / 2835
(Decimal::from_parts_raw(3191872741, 2108928381, 11855473, 1835008), 9),
// 1382 / 155925
(Decimal::from_parts_raw(3482645539, 2612995122, 4804769, 1835008), 11),
// 21844 / 6081075
(Decimal::from_parts_raw(4189029078, 2192791200, 1947296, 1835008), 13),
];
let mut result = *self;
for (fraction, pow) in SERIES {
result += fraction * self.powu(pow);
}
Some(result)
}
}
impl Pow<Decimal> for Decimal {
type Output = Decimal;
fn pow(self, rhs: Decimal) -> Self::Output {
MathematicalOps::powd(&self, rhs)
}
}
impl Pow<u64> for Decimal {
type Output = Decimal;
fn pow(self, rhs: u64) -> Self::Output {
MathematicalOps::powu(&self, rhs)
}
}
impl Pow<i64> for Decimal {
type Output = Decimal;
fn pow(self, rhs: i64) -> Self::Output {
MathematicalOps::powi(&self, rhs)
}
}
impl Pow<f64> for Decimal {
type Output = Decimal;
fn pow(self, rhs: f64) -> Self::Output {
MathematicalOps::powf(&self, rhs)
}
}
#[cfg(test)]
mod test {
use super::*;
#[cfg(not(feature = "std"))]
use alloc::string::ToString;
#[test]
fn test_factorials() {
assert_eq!("1", FACTORIAL[0].to_string(), "0!");
assert_eq!("1", FACTORIAL[1].to_string(), "1!");
assert_eq!("2", FACTORIAL[2].to_string(), "2!");
assert_eq!("6", FACTORIAL[3].to_string(), "3!");
assert_eq!("24", FACTORIAL[4].to_string(), "4!");
assert_eq!("120", FACTORIAL[5].to_string(), "5!");
assert_eq!("720", FACTORIAL[6].to_string(), "6!");
assert_eq!("5040", FACTORIAL[7].to_string(), "7!");
assert_eq!("40320", FACTORIAL[8].to_string(), "8!");
assert_eq!("362880", FACTORIAL[9].to_string(), "9!");
assert_eq!("3628800", FACTORIAL[10].to_string(), "10!");
assert_eq!("39916800", FACTORIAL[11].to_string(), "11!");
assert_eq!("479001600", FACTORIAL[12].to_string(), "12!");
assert_eq!("6227020800", FACTORIAL[13].to_string(), "13!");
assert_eq!("87178291200", FACTORIAL[14].to_string(), "14!");
assert_eq!("1307674368000", FACTORIAL[15].to_string(), "15!");
assert_eq!("20922789888000", FACTORIAL[16].to_string(), "16!");
assert_eq!("355687428096000", FACTORIAL[17].to_string(), "17!");
assert_eq!("6402373705728000", FACTORIAL[18].to_string(), "18!");
assert_eq!("121645100408832000", FACTORIAL[19].to_string(), "19!");
assert_eq!("2432902008176640000", FACTORIAL[20].to_string(), "20!");
assert_eq!("51090942171709440000", FACTORIAL[21].to_string(), "21!");
assert_eq!("1124000727777607680000", FACTORIAL[22].to_string(), "22!");
assert_eq!("25852016738884976640000", FACTORIAL[23].to_string(), "23!");
assert_eq!("620448401733239439360000", FACTORIAL[24].to_string(), "24!");
assert_eq!("15511210043330985984000000", FACTORIAL[25].to_string(), "25!");
assert_eq!("403291461126605635584000000", FACTORIAL[26].to_string(), "26!");
assert_eq!("10888869450418352160768000000", FACTORIAL[27].to_string(), "27!");
}
}