#
#
# Nim's Runtime Library
# (c) Copyright 2015 Andreas Rumpf
#
# See the file "copying.txt", included in this
# distribution, for details about the copyright.
#
## *Constructive mathematics is naturally typed.* -- Simon Thompson
##
## Basic math routines for Nim.
##
## Note that the trigonometric functions naturally operate on radians.
## The helper functions `degToRad <#degToRad,T>`_ and `radToDeg <#radToDeg,T>`_
## provide conversion between radians and degrees.
runnableExamples:
from std/fenv import epsilon
from std/random import rand
proc generateGaussianNoise(mu: float = 0.0, sigma: float = 1.0): (float, float) =
# Generates values from a normal distribution.
# Translated from https://en.wikipedia.org/wiki/Box%E2%80%93Muller_transform#Implementation.
var u1: float
var u2: float
while true:
u1 = rand(1.0)
u2 = rand(1.0)
if u1 > epsilon(float): break
let mag = sigma * sqrt(-2 * ln(u1))
let z0 = mag * cos(2 * PI * u2) + mu
let z1 = mag * sin(2 * PI * u2) + mu
(z0, z1)
echo generateGaussianNoise()
## This module is available for the `JavaScript target
## <backends.html#backends-the-javascript-target>`_.
##
## See also
## ========
## * `complex module <complex.html>`_ for complex numbers and their
## mathematical operations
## * `rationals module <rationals.html>`_ for rational numbers and their
## mathematical operations
## * `fenv module <fenv.html>`_ for handling of floating-point rounding
## and exceptions (overflow, zero-divide, etc.)
## * `random module <random.html>`_ for a fast and tiny random number generator
## * `stats module <stats.html>`_ for statistical analysis
## * `strformat module <strformat.html>`_ for formatting floats for printing
## * `system module <system.html>`_ for some very basic and trivial math operators
## (`shr`, `shl`, `xor`, `clamp`, etc.)
import std/private/since
{.push debugger: off.} # the user does not want to trace a part
# of the standard library!
import bitops, fenv
when defined(nimPreviewSlimSystem):
import std/assertions
when defined(c) or defined(cpp):
proc c_isnan(x: float): bool {.importc: "isnan", header: "<math.h>".}
# a generic like `x: SomeFloat` might work too if this is implemented via a C macro.
proc c_copysign(x, y: cfloat): cfloat {.importc: "copysignf", header: "<math.h>".}
proc c_copysign(x, y: cdouble): cdouble {.importc: "copysign", header: "<math.h>".}
proc c_signbit(x: SomeFloat): cint {.importc: "signbit", header: "<math.h>".}
# don't export `c_frexp` in the future and remove `c_frexp2`.
func c_frexp2(x: cfloat, exponent: var cint): cfloat {.
importc: "frexpf", header: "<math.h>".}
func c_frexp2(x: cdouble, exponent: var cint): cdouble {.
importc: "frexp", header: "<math.h>".}
func binom*(n, k: int): int =
## Computes the [binomial coefficient](https://en.wikipedia.org/wiki/Binomial_coefficient).
runnableExamples:
doAssert binom(6, 2) == 15
doAssert binom(-6, 2) == 1
doAssert binom(6, 0) == 1
if k <= 0: return 1
if 2 * k > n: return binom(n, n - k)
result = n
for i in countup(2, k):
result = (result * (n + 1 - i)) div i
func createFactTable[N: static[int]]: array[N, int] =
result[0] = 1
for i in 1 ..< N:
result[i] = result[i - 1] * i
func fac*(n: int): int =
## Computes the [factorial](https://en.wikipedia.org/wiki/Factorial) of
## a non-negative integer `n`.
##
## **See also:**
## * `prod func <#prod,openArray[T]>`_
runnableExamples:
doAssert fac(0) == 1
doAssert fac(4) == 24
doAssert fac(10) == 3628800
const factTable =
when sizeof(int) == 2:
createFactTable[5]()
elif sizeof(int) == 4:
createFactTable[13]()
else:
createFactTable[21]()
assert(n >= 0, $n & " must not be negative.")
assert(n < factTable.len, $n & " is too large to look up in the table")
factTable[n]
{.push checks: off, line_dir: off, stack_trace: off.}
when defined(posix) and not defined(genode):
{.passl: "-lm".}
const
PI* = 3.1415926535897932384626433 ## The circle constant PI (Ludolph's number).
TAU* = 2.0 * PI ## The circle constant TAU (= 2 * PI).
E* = 2.71828182845904523536028747 ## Euler's number.
MaxFloat64Precision* = 16 ## Maximum number of meaningful digits
## after the decimal point for Nim's
## `float64` type.
MaxFloat32Precision* = 8 ## Maximum number of meaningful digits
## after the decimal point for Nim's
## `float32` type.
MaxFloatPrecision* = MaxFloat64Precision ## Maximum number of
## meaningful digits
## after the decimal point
## for Nim's `float` type.
MinFloatNormal* = 2.225073858507201e-308 ## Smallest normal number for Nim's
## `float` type (= 2^-1022).
RadPerDeg = PI / 180.0 ## Number of radians per degree.
type
FloatClass* = enum ## Describes the class a floating point value belongs to.
## This is the type that is returned by the
## `classify func <#classify,float>`_.
fcNormal, ## value is an ordinary nonzero floating point value
fcSubnormal, ## value is a subnormal (a very small) floating point value
fcZero, ## value is zero
fcNegZero, ## value is the negative zero
fcNan, ## value is Not a Number (NaN)
fcInf, ## value is positive infinity
fcNegInf ## value is negative infinity
func isNaN*(x: SomeFloat): bool {.inline, since: (1,5,1).} =
## Returns whether `x` is a `NaN`, more efficiently than via `classify(x) == fcNan`.
## Works even with `--passc:-ffast-math`.
runnableExamples:
doAssert NaN.isNaN
doAssert not Inf.isNaN
doAssert not isNaN(3.1415926)
template fn: untyped = result = x != x
when nimvm: fn()
else:
when defined(js): fn()
else: result = c_isnan(x)
when defined(js):
import std/private/jsutils
proc toBitsImpl(x: float): array[2, uint32] =
let buffer = newArrayBuffer(8)
let a = newFloat64Array(buffer)
let b = newUint32Array(buffer)
a[0] = x
{.emit: "`result` = `b`;".}
# result = cast[array[2, uint32]](b)
proc jsSetSign(x: float, sgn: bool): float =
let buffer = newArrayBuffer(8)
let a = newFloat64Array(buffer)
let b = newUint32Array(buffer)
a[0] = x
asm """
function updateBit(num, bitPos, bitVal) {
return (num & ~(1 << bitPos)) | (bitVal << bitPos);
}
`b`[1] = updateBit(`b`[1], 31, `sgn`);
`result` = `a`[0]
"""
proc signbit*(x: SomeFloat): bool {.inline, since: (1, 5, 1).} =
## Returns true if `x` is negative, false otherwise.
runnableExamples:
doAssert not signbit(0.0)
doAssert signbit(-0.0)
doAssert signbit(-0.1)
doAssert not signbit(0.1)
when defined(js):
let uintBuffer = toBitsImpl(x)
result = (uintBuffer[1] shr 31) != 0
else:
result = c_signbit(x) != 0
func copySign*[T: SomeFloat](x, y: T): T {.inline, since: (1, 5, 1).} =
## Returns a value with the magnitude of `x` and the sign of `y`;
## this works even if x or y are NaN, infinity or zero, all of which can carry a sign.
runnableExamples:
doAssert copySign(10.0, 1.0) == 10.0
doAssert copySign(10.0, -1.0) == -10.0
doAssert copySign(-Inf, -0.0) == -Inf
doAssert copySign(NaN, 1.0).isNaN
doAssert copySign(1.0, copySign(NaN, -1.0)) == -1.0
# TODO: use signbit for examples
when defined(js):
let uintBuffer = toBitsImpl(y)
let sgn = (uintBuffer[1] shr 31) != 0
result = jsSetSign(x, sgn)
else:
when nimvm: # not exact but we have a vmops for recent enough nim
if y > 0.0 or (y == 0.0 and 1.0 / y > 0.0):
result = abs(x)
elif y <= 0.0:
result = -abs(x)
else: # must be NaN
result = abs(x)
else: result = c_copysign(x, y)
func classify*(x: float): FloatClass =
## Classifies a floating point value.
##
## Returns `x`'s class as specified by the `FloatClass enum<#FloatClass>`_.
## Doesn't work with `--passc:-ffast-math`.
runnableExamples:
doAssert classify(0.3) == fcNormal
doAssert classify(0.0) == fcZero
doAssert classify(0.3 / 0.0) == fcInf
doAssert classify(-0.3 / 0.0) == fcNegInf
doAssert classify(5.0e-324) == fcSubnormal
# JavaScript and most C compilers have no classify:
if x == 0.0:
if 1.0 / x == Inf:
return fcZero
else:
return fcNegZero
if x * 0.5 == x:
if x > 0.0: return fcInf
else: return fcNegInf
if x != x: return fcNan
if abs(x) < MinFloatNormal:
return fcSubnormal
return fcNormal
func almostEqual*[T: SomeFloat](x, y: T; unitsInLastPlace: Natural = 4): bool {.
since: (1, 5), inline.} =
## Checks if two float values are almost equal, using the
## [machine epsilon](https://en.wikipedia.org/wiki/Machine_epsilon).
##
## `unitsInLastPlace` is the max number of
## [units in the last place](https://en.wikipedia.org/wiki/Unit_in_the_last_place)
## difference tolerated when comparing two numbers. The larger the value, the
## more error is allowed. A `0` value means that two numbers must be exactly the
## same to be considered equal.
##
## The machine epsilon has to be scaled to the magnitude of the values used
## and multiplied by the desired precision in ULPs unless the difference is
## subnormal.
##
# taken from: https://en.cppreference.com/w/cpp/types/numeric_limits/epsilon
runnableExamples:
doAssert almostEqual(PI, 3.14159265358979)
doAssert almostEqual(Inf, Inf)
doAssert not almostEqual(NaN, NaN)
if x == y:
# short circuit exact equality -- needed to catch two infinities of
# the same sign. And perhaps speeds things up a bit sometimes.
return true
let diff = abs(x - y)
result = diff <= epsilon(T) * abs(x + y) * T(unitsInLastPlace) or
diff < minimumPositiveValue(T)
func isPowerOfTwo*(x: int): bool =
## Returns `true`, if `x` is a power of two, `false` otherwise.
##
## Zero and negative numbers are not a power of two.
##
## **See also:**
## * `nextPowerOfTwo func <#nextPowerOfTwo,int>`_
runnableExamples:
doAssert isPowerOfTwo(16)
doAssert not isPowerOfTwo(5)
doAssert not isPowerOfTwo(0)
doAssert not isPowerOfTwo(-16)
return (x > 0) and ((x and (x - 1)) == 0)
func nextPowerOfTwo*(x: int): int =
## Returns `x` rounded up to the nearest power of two.
##
## Zero and negative numbers get rounded up to 1.
##
## **See also:**
## * `isPowerOfTwo func <#isPowerOfTwo,int>`_
runnableExamples:
doAssert nextPowerOfTwo(16) == 16
doAssert nextPowerOfTwo(5) == 8
doAssert nextPowerOfTwo(0) == 1
doAssert nextPowerOfTwo(-16) == 1
result = x - 1
when defined(cpu64):
result = result or (result shr 32)
when sizeof(int) > 2:
result = result or (result shr 16)
when sizeof(int) > 1:
result = result or (result shr 8)
result = result or (result shr 4)
result = result or (result shr 2)
result = result or (result shr 1)
result += 1 + ord(x <= 0)
func sum*[T](x: openArray[T]): T =
## Computes the sum of the elements in `x`.
##
## If `x` is empty, 0 is returned.
##
## **See also:**
## * `prod func <#prod,openArray[T]>`_
## * `cumsum func <#cumsum,openArray[T]>`_
## * `cumsummed func <#cumsummed,openArray[T]>`_
runnableExamples:
doAssert sum([1, 2, 3, 4]) == 10
doAssert sum([-4, 3, 5]) == 4
for i in items(x): result = result + i
func prod*[T](x: openArray[T]): T =
## Computes the product of the elements in `x`.
##
## If `x` is empty, 1 is returned.
##
## **See also:**
## * `sum func <#sum,openArray[T]>`_
## * `fac func <#fac,int>`_
runnableExamples:
doAssert prod([1, 2, 3, 4]) == 24
doAssert prod([-4, 3, 5]) == -60
result = T(1)
for i in items(x): result = result * i
func cumsummed*[T](x: openArray[T]): seq[T] =
## Returns the cumulative (aka prefix) summation of `x`.
##
## If `x` is empty, `@[]` is returned.
##
## **See also:**
## * `sum func <#sum,openArray[T]>`_
## * `cumsum func <#cumsum,openArray[T]>`_ for the in-place version
runnableExamples:
doAssert cumsummed([1, 2, 3, 4]) == @[1, 3, 6, 10]
let xLen = x.len
if xLen == 0:
return @[]
result.setLen(xLen)
result[0] = x[0]
for i in 1 ..< xLen: result[i] = result[i - 1] + x[i]
func cumsum*[T](x: var openArray[T]) =
## Transforms `x` in-place (must be declared as `var`) into its
## cumulative (aka prefix) summation.
##
## **See also:**
## * `sum func <#sum,openArray[T]>`_
## * `cumsummed func <#cumsummed,openArray[T]>`_ for a version which
## returns a cumsummed sequence
runnableExamples:
var a = [1, 2, 3, 4]
cumsum(a)
doAssert a == @[1, 3, 6, 10]
for i in 1 ..< x.len: x[i] = x[i - 1] + x[i]
when not defined(js): # C
func sqrt*(x: float32): float32 {.importc: "sqrtf", header: "<math.h>".}
func sqrt*(x: float64): float64 {.importc: "sqrt", header: "<math.h>".} =
## Computes the square root of `x`.
##
## **See also:**
## * `cbrt func <#cbrt,float64>`_ for the cube root
runnableExamples:
doAssert almostEqual(sqrt(4.0), 2.0)
doAssert almostEqual(sqrt(1.44), 1.2)
func cbrt*(x: float32): float32 {.importc: "cbrtf", header: "<math.h>".}
func cbrt*(x: float64): float64 {.importc: "cbrt", header: "<math.h>".} =
## Computes the cube root of `x`.
##
## **See also:**
## * `sqrt func <#sqrt,float64>`_ for the square root
runnableExamples:
doAssert almostEqual(cbrt(8.0), 2.0)
doAssert almostEqual(cbrt(2.197), 1.3)
doAssert almostEqual(cbrt(-27.0), -3.0)
func ln*(x: float32): float32 {.importc: "logf", header: "<math.h>".}
func ln*(x: float64): float64 {.importc: "log", header: "<math.h>".} =
## Computes the [natural logarithm](https://en.wikipedia.org/wiki/Natural_logarithm)
## of `x`.
##
## **See also:**
## * `log func <#log,T,T>`_
## * `log10 func <#log10,float64>`_
## * `log2 func <#log2,float64>`_
## * `exp func <#exp,float64>`_
runnableExamples:
doAssert almostEqual(ln(exp(4.0)), 4.0)
doAssert almostEqual(ln(1.0), 0.0)
doAssert almostEqual(ln(0.0), -Inf)
doAssert ln(-7.0).isNaN
else: # JS
func sqrt*(x: float32): float32 {.importc: "Math.sqrt", nodecl.}
func sqrt*(x: float64): float64 {.importc: "Math.sqrt", nodecl.}
func cbrt*(x: float32): float32 {.importc: "Math.cbrt", nodecl.}
func cbrt*(x: float64): float64 {.importc: "Math.cbrt", nodecl.}
func ln*(x: float32): float32 {.importc: "Math.log", nodecl.}
func ln*(x: float64): float64 {.importc: "Math.log", nodecl.}
func log*[T: SomeFloat](x, base: T): T =
## Computes the logarithm of `x` to base `base`.
##
## **See also:**
## * `ln func <#ln,float64>`_
## * `log10 func <#log10,float64>`_
## * `log2 func <#log2,float64>`_
runnableExamples:
doAssert almostEqual(log(9.0, 3.0), 2.0)
doAssert almostEqual(log(0.0, 2.0), -Inf)
doAssert log(-7.0, 4.0).isNaN
doAssert log(8.0, -2.0).isNaN
ln(x) / ln(base)
when not defined(js): # C
func log10*(x: float32): float32 {.importc: "log10f", header: "<math.h>".}
func log10*(x: float64): float64 {.importc: "log10", header: "<math.h>".} =
## Computes the common logarithm (base 10) of `x`.
##
## **See also:**
## * `ln func <#ln,float64>`_
## * `log func <#log,T,T>`_
## * `log2 func <#log2,float64>`_
runnableExamples:
doAssert almostEqual(log10(100.0) , 2.0)
doAssert almostEqual(log10(0.0), -Inf)
doAssert log10(-100.0).isNaN
func exp*(x: float32): float32 {.importc: "expf", header: "<math.h>".}
func exp*(x: float64): float64 {.importc: "exp", header: "<math.h>".} =
## Computes the exponential function of `x` (`e^x`).
##
## **See also:**
## * `ln func <#ln,float64>`_
runnableExamples:
doAssert almostEqual(exp(1.0), E)
doAssert almostEqual(ln(exp(4.0)), 4.0)
doAssert almostEqual(exp(0.0), 1.0)
func sin*(x: float32): float32 {.importc: "sinf", header: "<math.h>".}
func sin*(x: float64): float64 {.importc: "sin", header: "<math.h>".} =
## Computes the sine of `x`.
##
## **See also:**
## * `arcsin func <#arcsin,float64>`_
runnableExamples:
doAssert almostEqual(sin(PI / 6), 0.5)
doAssert almostEqual(sin(degToRad(90.0)), 1.0)
func cos*(x: float32): float32 {.importc: "cosf", header: "<math.h>".}
func cos*(x: float64): float64 {.importc: "cos", header: "<math.h>".} =
## Computes the cosine of `x`.
##
## **See also:**
## * `arccos func <#arccos,float64>`_
runnableExamples:
doAssert almostEqual(cos(2 * PI), 1.0)
doAssert almostEqual(cos(degToRad(60.0)), 0.5)
func tan*(x: float32): float32 {.importc: "tanf", header: "<math.h>".}
func tan*(x: float64): float64 {.importc: "tan", header: "<math.h>".} =
## Computes the tangent of `x`.
##
## **See also:**
## * `arctan func <#arctan,float64>`_
runnableExamples:
doAssert almostEqual(tan(degToRad(45.0)), 1.0)
doAssert almostEqual(tan(PI / 4), 1.0)
func sinh*(x: float32): float32 {.importc: "sinhf", header: "<math.h>".}
func sinh*(x: float64): float64 {.importc: "sinh", header: "<math.h>".} =
## Computes the [hyperbolic sine](https://en.wikipedia.org/wiki/Hyperbolic_function#Definitions) of `x`.
##
## **See also:**
## * `arcsinh func <#arcsinh,float64>`_
runnableExamples:
doAssert almostEqual(sinh(0.0), 0.0)
doAssert almostEqual(sinh(1.0), 1.175201193643801)
func cosh*(x: float32): float32 {.importc: "coshf", header: "<math.h>".}
func cosh*(x: float64): float64 {.importc: "cosh", header: "<math.h>".} =
## Computes the [hyperbolic cosine](https://en.wikipedia.org/wiki/Hyperbolic_function#Definitions) of `x`.
##
## **See also:**
## * `arccosh func <#arccosh,float64>`_
runnableExamples:
doAssert almostEqual(cosh(0.0), 1.0)
doAssert almostEqual(cosh(1.0), 1.543080634815244)
func tanh*(x: float32): float32 {.importc: "tanhf", header: "<math.h>".}
func tanh*(x: float64): float64 {.importc: "tanh", header: "<math.h>".} =
## Computes the [hyperbolic tangent](https://en.wikipedia.org/wiki/Hyperbolic_function#Definitions) of `x`.
##
## **See also:**
## * `arctanh func <#arctanh,float64>`_
runnableExamples:
doAssert almostEqual(tanh(0.0), 0.0)
doAssert almostEqual(tanh(1.0), 0.7615941559557649)
func arcsin*(x: float32): float32 {.importc: "asinf", header: "<math.h>".}
func arcsin*(x: float64): float64 {.importc: "asin", header: "<math.h>".} =
## Computes the arc sine of `x`.
##
## **See also:**
## * `sin func <#sin,float64>`_
runnableExamples:
doAssert almostEqual(radToDeg(arcsin(0.0)), 0.0)
doAssert almostEqual(radToDeg(arcsin(1.0)), 90.0)
func arccos*(x: float32): float32 {.importc: "acosf", header: "<math.h>".}
func arccos*(x: float64): float64 {.importc: "acos", header: "<math.h>".} =
## Computes the arc cosine of `x`.
##
## **See also:**
## * `cos func <#cos,float64>`_
runnableExamples:
doAssert almostEqual(radToDeg(arccos(0.0)), 90.0)
doAssert almostEqual(radToDeg(arccos(1.0)), 0.0)
func arctan*(x: float32): float32 {.importc: "atanf", header: "<math.h>".}
func arctan*(x: float64): float64 {.importc: "atan", header: "<math.h>".} =
## Calculate the arc tangent of `x`.
##
## **See also:**
## * `arctan2 func <#arctan2,float64,float64>`_
## * `tan func <#tan,float64>`_
runnableExamples:
doAssert almostEqual(arctan(1.0), 0.7853981633974483)
doAssert almostEqual(radToDeg(arctan(1.0)), 45.0)
func arctan2*(y, x: float32): float32 {.importc: "atan2f", header: "<math.h>".}
func arctan2*(y, x: float64): float64 {.importc: "atan2", header: "<math.h>".} =
## Calculate the arc tangent of `y/x`.
##
## It produces correct results even when the resulting angle is near
## `PI/2` or `-PI/2` (`x` near 0).
##
## **See also:**
## * `arctan func <#arctan,float64>`_
runnableExamples:
doAssert almostEqual(arctan2(1.0, 0.0), PI / 2.0)
doAssert almostEqual(radToDeg(arctan2(1.0, 0.0)), 90.0)
func arcsinh*(x: float32): float32 {.importc: "asinhf", header: "<math.h>".}
func arcsinh*(x: float64): float64 {.importc: "asinh", header: "<math.h>".}
## Computes the inverse hyperbolic sine of `x`.
##
## **See also:**
## * `sinh func <#sinh,float64>`_
func arccosh*(x: float32): float32 {.importc: "acoshf", header: "<math.h>".}
func arccosh*(x: float64): float64 {.importc: "acosh", header: "<math.h>".}
## Computes the inverse hyperbolic cosine of `x`.
##
## **See also:**
## * `cosh func <#cosh,float64>`_
func arctanh*(x: float32): float32 {.importc: "atanhf", header: "<math.h>".}
func arctanh*(x: float64): float64 {.importc: "atanh", header: "<math.h>".}
## Computes the inverse hyperbolic tangent of `x`.
##
## **See also:**
## * `tanh func <#tanh,float64>`_
else: # JS
func log10*(x: float32): float32 {.importc: "Math.log10", nodecl.}
func log10*(x: float64): float64 {.importc: "Math.log10", nodecl.}
func log2*(x: float32): float32 {.importc: "Math.log2", nodecl.}
func log2*(x: float64): float64 {.importc: "Math.log2", nodecl.}
func exp*(x: float32): float32 {.importc: "Math.exp", nodecl.}
func exp*(x: float64): float64 {.importc: "Math.exp", nodecl.}
func sin*[T: float32|float64](x: T): T {.importc: "Math.sin", nodecl.}
func cos*[T: float32|float64](x: T): T {.importc: "Math.cos", nodecl.}
func tan*[T: float32|float64](x: T): T {.importc: "Math.tan", nodecl.}
func sinh*[T: float32|float64](x: T): T {.importc: "Math.sinh", nodecl.}
func cosh*[T: float32|float64](x: T): T {.importc: "Math.cosh", nodecl.}
func tanh*[T: float32|float64](x: T): T {.importc: "Math.tanh", nodecl.}
func arcsin*[T: float32|float64](x: T): T {.importc: "Math.asin", nodecl.}
# keep this as generic or update test in `tvmops.nim` to make sure we
# keep testing that generic importc procs work
func arccos*[T: float32|float64](x: T): T {.importc: "Math.acos", nodecl.}
func arctan*[T: float32|float64](x: T): T {.importc: "Math.atan", nodecl.}
func arctan2*[T: float32|float64](y, x: T): T {.importc: "Math.atan2", nodecl.}
func arcsinh*[T: float32|float64](x: T): T {.importc: "Math.asinh", nodecl.}
func arccosh*[T: float32|float64](x: T): T {.importc: "Math.acosh", nodecl.}
func arctanh*[T: float32|float64](x: T): T {.importc: "Math.atanh", nodecl.}
func cot*[T: float32|float64](x: T): T = 1.0 / tan(x)
## Computes the cotangent of `x` (`1/tan(x)`).
func sec*[T: float32|float64](x: T): T = 1.0 / cos(x)
## Computes the secant of `x` (`1/cos(x)`).
func csc*[T: float32|float64](x: T): T = 1.0 / sin(x)
## Computes the cosecant of `x` (`1/sin(x)`).
func coth*[T: float32|float64](x: T): T = 1.0 / tanh(x)
## Computes the hyperbolic cotangent of `x` (`1/tanh(x)`).
func sech*[T: float32|float64](x: T): T = 1.0 / cosh(x)
## Computes the hyperbolic secant of `x` (`1/cosh(x)`).
func csch*[T: float32|float64](x: T): T = 1.0 / sinh(x)
## Computes the hyperbolic cosecant of `x` (`1/sinh(x)`).
func arccot*[T: float32|float64](x: T): T = arctan(1.0 / x)
## Computes the inverse cotangent of `x` (`arctan(1/x)`).
func arcsec*[T: float32|float64](x: T): T = arccos(1.0 / x)
## Computes the inverse secant of `x` (`arccos(1/x)`).
func arccsc*[T: float32|float64](x: T): T = arcsin(1.0 / x)
## Computes the inverse cosecant of `x` (`arcsin(1/x)`).
func arccoth*[T: float32|float64](x: T): T = arctanh(1.0 / x)
## Computes the inverse hyperbolic cotangent of `x` (`arctanh(1/x)`).
func arcsech*[T: float32|float64](x: T): T = arccosh(1.0 / x)
## Computes the inverse hyperbolic secant of `x` (`arccosh(1/x)`).
func arccsch*[T: float32|float64](x: T): T = arcsinh(1.0 / x)
## Computes the inverse hyperbolic cosecant of `x` (`arcsinh(1/x)`).
const windowsCC89 = defined(windows) and defined(bcc)
when not defined(js): # C
func hypot*(x, y: float32): float32 {.importc: "hypotf", header: "<math.h>".}
func hypot*(x, y: float64): float64 {.importc: "hypot", header: "<math.h>".} =
## Computes the length of the hypotenuse of a right-angle triangle with
## `x` as its base and `y` as its height. Equivalent to `sqrt(x*x + y*y)`.
runnableExamples:
doAssert almostEqual(hypot(3.0, 4.0), 5.0)
func pow*(x, y: float32): float32 {.importc: "powf", header: "<math.h>".}
func pow*(x, y: float64): float64 {.importc: "pow", header: "<math.h>".} =
## Computes `x` raised to the power of `y`.
##
## To compute the power between integers (e.g. 2^6),
## use the `^ func <#^,T,Natural>`_.
##
## **See also:**
## * `^ func <#^,T,Natural>`_
## * `sqrt func <#sqrt,float64>`_
## * `cbrt func <#cbrt,float64>`_
runnableExamples:
doAssert almostEqual(pow(100, 1.5), 1000.0)
doAssert almostEqual(pow(16.0, 0.5), 4.0)
# TODO: add C89 version on windows
when not windowsCC89:
func erf*(x: float32): float32 {.importc: "erff", header: "<math.h>".}
func erf*(x: float64): float64 {.importc: "erf", header: "<math.h>".}
## Computes the [error function](https://en.wikipedia.org/wiki/Error_function) for `x`.
##
## **Note:** Not available for the JS backend.
func erfc*(x: float32): float32 {.importc: "erfcf", header: "<math.h>".}
func erfc*(x: float64): float64 {.importc: "erfc", header: "<math.h>".}
## Computes the [complementary error function](https://en.wikipedia.org/wiki/Error_function#Complementary_error_function) for `x`.
##
## **Note:** Not available for the JS backend.
func gamma*(x: float32): float32 {.importc: "tgammaf", header: "<math.h>".}
func gamma*(x: float64): float64 {.importc: "tgamma", header: "<math.h>".} =
## Computes the [gamma function](https://en.wikipedia.org/wiki/Gamma_function) for `x`.
##
## **Note:** Not available for the JS backend.
##
## **See also:**
## * `lgamma func <#lgamma,float64>`_ for the natural logarithm of the gamma function
runnableExamples:
doAssert almostEqual(gamma(1.0), 1.0)
doAssert almostEqual(gamma(4.0), 6.0)
doAssert almostEqual(gamma(11.0), 3628800.0)
func lgamma*(x: float32): float32 {.importc: "lgammaf", header: "<math.h>".}
func lgamma*(x: float64): float64 {.importc: "lgamma", header: "<math.h>".} =
## Computes the natural logarithm of the gamma function for `x`.
##
## **Note:** Not available for the JS backend.
##
## **See also:**
## * `gamma func <#gamma,float64>`_ for gamma function
func floor*(x: float32): float32 {.importc: "floorf", header: "<math.h>".}
func floor*(x: float64): float64 {.importc: "floor", header: "<math.h>".} =
## Computes the floor function (i.e. the largest integer not greater than `x`).
##
## **See also:**
## * `ceil func <#ceil,float64>`_
## * `round func <#round,float64>`_
## * `trunc func <#trunc,float64>`_
runnableExamples:
doAssert floor(2.1) == 2.0
doAssert floor(2.9) == 2.0
doAssert floor(-3.5) == -4.0
func ceil*(x: float32): float32 {.importc: "ceilf", header: "<math.h>".}
func ceil*(x: float64): float64 {.importc: "ceil", header: "<math.h>".} =
## Computes the ceiling function (i.e. the smallest integer not smaller
## than `x`).
##
## **See also:**
## * `floor func <#floor,float64>`_
## * `round func <#round,float64>`_
## * `trunc func <#trunc,float64>`_
runnableExamples:
doAssert ceil(2.1) == 3.0
doAssert ceil(2.9) == 3.0
doAssert ceil(-2.1) == -2.0
when windowsCC89:
# MSVC 2010 don't have trunc/truncf
# this implementation was inspired by Go-lang Math.Trunc
func truncImpl(f: float64): float64 =
const
mask: uint64 = 0x7FF
shift: uint64 = 64 - 12
bias: uint64 = 0x3FF
if f < 1:
if f < 0: return -truncImpl(-f)
elif f == 0: return f # Return -0 when f == -0
else: return 0
var x = cast[uint64](f)
let e = (x shr shift) and mask - bias
# Keep the top 12+e bits, the integer part; clear the rest.
if e < 64 - 12:
x = x and (not (1'u64 shl (64'u64 - 12'u64 - e) - 1'u64))
result = cast[float64](x)
func truncImpl(f: float32): float32 =
const
mask: uint32 = 0xFF
shift: uint32 = 32 - 9
bias: uint32 = 0x7F
if f < 1:
if f < 0: return -truncImpl(-f)
elif f == 0: return f # Return -0 when f == -0
else: return 0
var x = cast[uint32](f)
let e = (x shr shift) and mask - bias
# Keep the top 9+e bits, the integer part; clear the rest.
if e < 32 - 9:
x = x and (not (1'u32 shl (32'u32 - 9'u32 - e) - 1'u32))
result = cast[float32](x)
func trunc*(x: float64): float64 =
if classify(x) in {fcZero, fcNegZero, fcNan, fcInf, fcNegInf}: return x
result = truncImpl(x)
func trunc*(x: float32): float32 =
if classify(x) in {fcZero, fcNegZero, fcNan, fcInf, fcNegInf}: return x
result = truncImpl(x)
func round*[T: float32|float64](x: T): T =
## Windows compilers prior to MSVC 2012 do not implement 'round',
## 'roundl' or 'roundf'.
result = if x < 0.0: ceil(x - T(0.5)) else: floor(x + T(0.5))
else:
func round*(x: float32): float32 {.importc: "roundf", header: "<math.h>".}
func round*(x: float64): float64 {.importc: "round", header: "<math.h>".} =
## Rounds a float to zero decimal places.
##
## Used internally by the `round func <#round,T,int>`_
## when the specified number of places is 0.
##
## **See also:**
## * `round func <#round,T,int>`_ for rounding to the specific
## number of decimal places
## * `floor func <#floor,float64>`_
## * `ceil func <#ceil,float64>`_
## * `trunc func <#trunc,float64>`_
runnableExamples:
doAssert round(3.4) == 3.0
doAssert round(3.5) == 4.0
doAssert round(4.5) == 5.0
func trunc*(x: float32): float32 {.importc: "truncf", header: "<math.h>".}
func trunc*(x: float64): float64 {.importc: "trunc", header: "<math.h>".} =
## Truncates `x` to the decimal point.
##
## **See also:**
## * `floor func <#floor,float64>`_
## * `ceil func <#ceil,float64>`_
## * `round func <#round,float64>`_
runnableExamples:
doAssert trunc(PI) == 3.0
doAssert trunc(-1.85) == -1.0
func `mod`*(x, y: float32): float32 {.importc: "fmodf", header: "<math.h>".}
func `mod`*(x, y: float64): float64 {.importc: "fmod", header: "<math.h>".} =
## Computes the modulo operation for float values (the remainder of `x` divided by `y`).
##
## **See also:**
## * `floorMod func <#floorMod,T,T>`_ for Python-like (`%` operator) behavior
runnableExamples:
doAssert 6.5 mod 2.5 == 1.5
doAssert -6.5 mod 2.5 == -1.5
doAssert 6.5 mod -2.5 == 1.5
doAssert -6.5 mod -2.5 == -1.5
else: # JS
func hypot*(x, y: float32): float32 {.importc: "Math.hypot", varargs, nodecl.}
func hypot*(x, y: float64): float64 {.importc: "Math.hypot", varargs, nodecl.}
func pow*(x, y: float32): float32 {.importc: "Math.pow", nodecl.}
func pow*(x, y: float64): float64 {.importc: "Math.pow", nodecl.}
func floor*(x: float32): float32 {.importc: "Math.floor", nodecl.}
func floor*(x: float64): float64 {.importc: "Math.floor", nodecl.}
func ceil*(x: float32): float32 {.importc: "Math.ceil", nodecl.}
func ceil*(x: float64): float64 {.importc: "Math.ceil", nodecl.}
when (NimMajor, NimMinor) < (1, 5) or defined(nimLegacyJsRound):
func round*(x: float): float {.importc: "Math.round", nodecl.}
else:
func jsRound(x: float): float {.importc: "Math.round", nodecl.}
func round*[T: float64 | float32](x: T): T =
if x >= 0: result = jsRound(x)
else:
result = ceil(x)
if result - x >= T(0.5):
result -= T(1.0)
func trunc*(x: float32): float32 {.importc: "Math.trunc", nodecl.}
func trunc*(x: float64): float64 {.importc: "Math.trunc", nodecl.}
func `mod`*(x, y: float32): float32 {.importjs: "(# % #)".}
func `mod`*(x, y: float64): float64 {.importjs: "(# % #)".} =
## Computes the modulo operation for float values (the remainder of `x` divided by `y`).
runnableExamples:
doAssert 6.5 mod 2.5 == 1.5
doAssert -6.5 mod 2.5 == -1.5
doAssert 6.5 mod -2.5 == 1.5
doAssert -6.5 mod -2.5 == -1.5
func round*[T: float32|float64](x: T, places: int): T =
## Decimal rounding on a binary floating point number.
##
## This function is NOT reliable. Floating point numbers cannot hold
## non integer decimals precisely. If `places` is 0 (or omitted),
## round to the nearest integral value following normal mathematical
## rounding rules (e.g. `round(54.5) -> 55.0`). If `places` is
## greater than 0, round to the given number of decimal places,
## e.g. `round(54.346, 2) -> 54.350000000000001421…`. If `places` is negative, round
## to the left of the decimal place, e.g. `round(537.345, -1) -> 540.0`.
runnableExamples:
doAssert round(PI, 2) == 3.14
doAssert round(PI, 4) == 3.1416
if places == 0:
result = round(x)
else:
var mult = pow(10.0, T(places))
result = round(x * mult) / mult
func floorDiv*[T: SomeInteger](x, y: T): T =
## Floor division is conceptually defined as `floor(x / y)`.
##
## This is different from the `system.div <system.html#div,int,int>`_
## operator, which is defined as `trunc(x / y)`.
## That is, `div` rounds towards `0` and `floorDiv` rounds down.
##
## **See also:**
## * `system.div proc <system.html#div,int,int>`_ for integer division
## * `floorMod func <#floorMod,T,T>`_ for Python-like (`%` operator) behavior
runnableExamples:
doAssert floorDiv( 13, 3) == 4
doAssert floorDiv(-13, 3) == -5
doAssert floorDiv( 13, -3) == -5
doAssert floorDiv(-13, -3) == 4
result = x div y
let r = x mod y
if (r > 0 and y < 0) or (r < 0 and y > 0): result.dec 1
func floorMod*[T: SomeNumber](x, y: T): T =
## Floor modulo is conceptually defined as `x - (floorDiv(x, y) * y)`.
##
## This func behaves the same as the `%` operator in Python.
##
## **See also:**
## * `mod func <#mod,float64,float64>`_
## * `floorDiv func <#floorDiv,T,T>`_
runnableExamples:
doAssert floorMod( 13, 3) == 1
doAssert floorMod(-13, 3) == 2
doAssert floorMod( 13, -3) == -2
doAssert floorMod(-13, -3) == -1
result = x mod y
if (result > 0 and y < 0) or (result < 0 and y > 0): result += y
func euclDiv*[T: SomeInteger](x, y: T): T {.since: (1, 5, 1).} =
## Returns euclidean division of `x` by `y`.
runnableExamples:
doAssert euclDiv(13, 3) == 4
doAssert euclDiv(-13, 3) == -5
doAssert euclDiv(13, -3) == -4
doAssert euclDiv(-13, -3) == 5
result = x div y
if x mod y < 0:
if y > 0:
dec result
else:
inc result
func euclMod*[T: SomeNumber](x, y: T): T {.since: (1, 5, 1).} =
## Returns euclidean modulo of `x` by `y`.
## `euclMod(x, y)` is non-negative.
runnableExamples:
doAssert euclMod(13, 3) == 1
doAssert euclMod(-13, 3) == 2
doAssert euclMod(13, -3) == 1
doAssert euclMod(-13, -3) == 2
result = x mod y
if result < 0:
result += abs(y)
func ceilDiv*[T: SomeInteger](x, y: T): T {.inline, since: (1, 5, 1).} =
## Ceil division is conceptually defined as `ceil(x / y)`.
##
## Assumes `x >= 0` and `y > 0` (and `x + y - 1 <= high(T)` if T is SomeUnsignedInt).
##
## This is different from the `system.div <system.html#div,int,int>`_
## operator, which works like `trunc(x / y)`.
## That is, `div` rounds towards `0` and `ceilDiv` rounds up.
##
## This function has the above input limitation, because that allows the
## compiler to generate faster code and it is rarely used with
## negative values or unsigned integers close to `high(T)/2`.
## If you need a `ceilDiv` that works with any input, see:
## https://github.com/demotomohiro/divmath.
##
## **See also:**
## * `system.div proc <system.html#div,int,int>`_ for integer division
## * `floorDiv func <#floorDiv,T,T>`_ for integer division which rounds down.
runnableExamples:
assert ceilDiv(12, 3) == 4
assert ceilDiv(13, 3) == 5
when sizeof(T) == 8:
type UT = uint64
elif sizeof(T) == 4:
type UT = uint32
elif sizeof(T) == 2:
type UT = uint16
elif sizeof(T) == 1:
type UT = uint8
else:
{.fatal: "Unsupported int type".}
assert x >= 0 and y > 0
when T is SomeUnsignedInt:
assert x + y - 1 >= x
# If the divisor is const, the backend C/C++ compiler generates code without a `div`
# instruction, as it is slow on most CPUs.
# If the divisor is a power of 2 and a const unsigned integer type, the
# compiler generates faster code.
# If the divisor is const and a signed integer, generated code becomes slower
# than the code with unsigned integers, because division with signed integers
# need to works for both positive and negative value without `idiv`/`sdiv`.
# That is why this code convert parameters to unsigned.
# This post contains a comparison of the performance of signed/unsigned integers:
# https://github.com/nim-lang/Nim/pull/18596#issuecomment-894420984.
# If signed integer arguments were not converted to unsigned integers,
# `ceilDiv` wouldn't work for any positive signed integer value, because
# `x + (y - 1)` can overflow.
((x.UT + (y.UT - 1.UT)) div y.UT).T
func frexp*[T: float32|float64](x: T): tuple[frac: T, exp: int] {.inline.} =
## Splits `x` into a normalized fraction `frac` and an integral power of 2 `exp`,
## such that `abs(frac) in 0.5..<1` and `x == frac * 2 ^ exp`, except for special
## cases shown below.
runnableExamples:
doAssert frexp(8.0) == (0.5, 4)
doAssert frexp(-8.0) == (-0.5, 4)
doAssert frexp(0.0) == (0.0, 0)
# special cases:
when sizeof(int) == 8:
doAssert frexp(-0.0).frac.signbit # signbit preserved for +-0
doAssert frexp(Inf).frac == Inf # +- Inf preserved
doAssert frexp(NaN).frac.isNaN
when not defined(js):
var exp: cint
result.frac = c_frexp2(x, exp)
result.exp = exp
else:
if x == 0.0:
# reuse signbit implementation
let uintBuffer = toBitsImpl(x)
if (uintBuffer[1] shr 31) != 0:
# x is -0.0
result = (-0.0, 0)
else:
result = (0.0, 0)
elif x < 0.0:
result = frexp(-x)
result.frac = -result.frac
else:
var ex = trunc(log2(x))
result.exp = int(ex)
result.frac = x / pow(2.0, ex)
if abs(result.frac) >= 1:
inc(result.exp)
result.frac = result.frac / 2
if result.exp == 1024 and result.frac == 0.0:
result.frac = 0.99999999999999988898
func frexp*[T: float32|float64](x: T, exponent: var int): T {.inline.} =
## Overload of `frexp` that calls `(result, exponent) = frexp(x)`.
runnableExamples:
var x: int
doAssert frexp(5.0, x) == 0.625
doAssert x == 3
(result, exponent) = frexp(x)
when not defined(js):
when windowsCC89:
# taken from Go-lang Math.Log2
const ln2 = 0.693147180559945309417232121458176568075500134360255254120680009
template log2Impl[T](x: T): T =
var exp: int
var frac = frexp(x, exp)
# Make sure exact powers of two give an exact answer.
# Don't depend on Log(0.5)*(1/Ln2)+exp being exactly exp-1.
if frac == 0.5: return T(exp - 1)
log10(frac) * (1 / ln2) + T(exp)
func log2*(x: float32): float32 = log2Impl(x)
func log2*(x: float64): float64 = log2Impl(x)
## Log2 returns the binary logarithm of x.
## The special cases are the same as for Log.
else:
func log2*(x: float32): float32 {.importc: "log2f", header: "<math.h>".}
func log2*(x: float64): float64 {.importc: "log2", header: "<math.h>".} =
## Computes the binary logarithm (base 2) of `x`.
##
## **See also:**
## * `log func <#log,T,T>`_
## * `log10 func <#log10,float64>`_
## * `ln func <#ln,float64>`_
runnableExamples:
doAssert almostEqual(log2(8.0), 3.0)
doAssert almostEqual(log2(1.0), 0.0)
doAssert almostEqual(log2(0.0), -Inf)
doAssert log2(-2.0).isNaN
func splitDecimal*[T: float32|float64](x: T): tuple[intpart: T, floatpart: T] =
## Breaks `x` into an integer and a fractional part.
##
## Returns a tuple containing `intpart` and `floatpart`, representing
## the integer part and the fractional part, respectively.
##
## Both parts have the same sign as `x`. Analogous to the `modf`
## function in C.
runnableExamples:
doAssert splitDecimal(5.25) == (intpart: 5.0, floatpart: 0.25)
doAssert splitDecimal(-2.73) == (intpart: -2.0, floatpart: -0.73)
var
absolute: T
absolute = abs(x)
result.intpart = floor(absolute)
result.floatpart = absolute - result.intpart
if x < 0:
result.intpart = -result.intpart
result.floatpart = -result.floatpart
func degToRad*[T: float32|float64](d: T): T {.inline.} =
## Converts from degrees to radians.
##
## **See also:**
## * `radToDeg func <#radToDeg,T>`_
runnableExamples:
doAssert almostEqual(degToRad(180.0), PI)
result = d * T(RadPerDeg)
func radToDeg*[T: float32|float64](d: T): T {.inline.} =
## Converts from radians to degrees.
##
## **See also:**
## * `degToRad func <#degToRad,T>`_
runnableExamples:
doAssert almostEqual(radToDeg(2 * PI), 360.0)
result = d / T(RadPerDeg)
func sgn*[T: SomeNumber](x: T): int {.inline.} =
## Sign function.
##
## Returns:
## * `-1` for negative numbers and `NegInf`,
## * `1` for positive numbers and `Inf`,
## * `0` for positive zero, negative zero and `NaN`
runnableExamples:
doAssert sgn(5) == 1
doAssert sgn(0) == 0
doAssert sgn(-4.1) == -1
ord(T(0) < x) - ord(x < T(0))
{.pop.}
{.pop.}
func `^`*[T: SomeNumber](x: T, y: Natural): T =
## Computes `x` to the power of `y`.
##
## The exponent `y` must be non-negative, use
## `pow <#pow,float64,float64>`_ for negative exponents.
##
## **See also:**
## * `pow func <#pow,float64,float64>`_ for negative exponent or
## floats
## * `sqrt func <#sqrt,float64>`_
## * `cbrt func <#cbrt,float64>`_
runnableExamples:
doAssert -3 ^ 0 == 1
doAssert -3 ^ 1 == -3
doAssert -3 ^ 2 == 9
case y
of 0: result = 1
of 1: result = x
of 2: result = x * x
of 3: result = x * x * x
else:
var (x, y) = (x, y)
result = 1
while true:
if (y and 1) != 0:
result *= x
y = y shr 1
if y == 0:
break
x *= x
func gcd*[T](x, y: T): T =
## Computes the greatest common (positive) divisor of `x` and `y`.
##
## Note that for floats, the result cannot always be interpreted as
## "greatest decimal `z` such that `z*N == x and z*M == y`
## where N and M are positive integers".
##
## **See also:**
## * `gcd func <#gcd,SomeInteger,SomeInteger>`_ for an integer version
## * `lcm func <#lcm,T,T>`_
runnableExamples:
doAssert gcd(13.5, 9.0) == 4.5
var (x, y) = (x, y)
while y != 0:
x = x mod y
swap x, y
abs x
func gcd*(x, y: SomeInteger): SomeInteger =
## Computes the greatest common (positive) divisor of `x` and `y`,
## using the binary GCD (aka Stein's) algorithm.
##
## **See also:**
## * `gcd func <#gcd,T,T>`_ for a float version
## * `lcm func <#lcm,T,T>`_
runnableExamples:
doAssert gcd(12, 8) == 4
doAssert gcd(17, 63) == 1
when x is SomeSignedInt:
var x = abs(x)
else:
var x = x
when y is SomeSignedInt:
var y = abs(y)
else:
var y = y
if x == 0:
return y
if y == 0:
return x
let shift = countTrailingZeroBits(x or y)
y = y shr countTrailingZeroBits(y)
while x != 0:
x = x shr countTrailingZeroBits(x)
if y > x:
swap y, x
x -= y
y shl shift
func gcd*[T](x: openArray[T]): T {.since: (1, 1).} =
## Computes the greatest common (positive) divisor of the elements of `x`.
##
## **See also:**
## * `gcd func <#gcd,T,T>`_ for a version with two arguments
runnableExamples:
doAssert gcd(@[13.5, 9.0]) == 4.5
result = x[0]
for i in 1 ..< x.len:
result = gcd(result, x[i])
func lcm*[T](x, y: T): T =
## Computes the least common multiple of `x` and `y`.
##
## **See also:**
## * `gcd func <#gcd,T,T>`_
runnableExamples:
doAssert lcm(24, 30) == 120
doAssert lcm(13, 39) == 39
x div gcd(x, y) * y
func clamp*[T](val: T, bounds: Slice[T]): T {.since: (1, 5), inline.} =
## Like `system.clamp`, but takes a slice, so you can easily clamp within a range.
runnableExamples:
assert clamp(10, 1 .. 5) == 5
assert clamp(1, 1 .. 3) == 1
type A = enum a0, a1, a2, a3, a4, a5
assert a1.clamp(a2..a4) == a2
assert clamp((3, 0), (1, 0) .. (2, 9)) == (2, 9)
doAssertRaises(AssertionDefect): discard clamp(1, 3..2) # invalid bounds
assert bounds.a <= bounds.b, $(bounds.a, bounds.b)
clamp(val, bounds.a, bounds.b)
func lcm*[T](x: openArray[T]): T {.since: (1, 1).} =
## Computes the least common multiple of the elements of `x`.
##
## **See also:**
## * `lcm func <#lcm,T,T>`_ for a version with two arguments
runnableExamples:
doAssert lcm(@[24, 30]) == 120
result = x[0]
for i in 1 ..< x.len:
result = lcm(result, x[i])