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Nim/lib/pure/math.nim
flywind bab0aa6ecf add math.signbit (#16592)
2021-01-07 08:48:02 +01:00

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#
#
# 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.
##
## .. code-block::
##
## import math
## from sequtils import map
##
## let a = [0.0, PI/6, PI/4, PI/3, PI/2]
##
## echo a.map(sin)
## # @[0.0, 0.499…, 0.707…, 0.866…, 1.0]
##
## echo a.map(tan)
## # @[0.0, 0.577…, 0.999…, 1.732…, 1.633…e+16]
##
## echo cos(degToRad(180.0))
## # -1.0
##
## echo sqrt(-1.0)
## # nan (use `complex` module)
##
## 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 fast and tiny random number generator
## * `mersenne module<mersenne.html>`_ for Mersenne twister random number generator
## * `stats module<stats.html>`_ for statistical analysis
## * `strformat module<strformat.html>`_ for formatting floats for print
## * `system module<system.html>`_ Some very basic and trivial math operators
## are on system directly, to name a few ``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 std/[bitops, fenv]
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>".}
func binom*(n, k: int): int =
## Computes the `binomial coefficient <https://en.wikipedia.org/wiki/Binomial_coefficient>`_.
runnableExamples:
doAssert binom(6, 2) == binom(6, 4)
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(3) == 6
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
## `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 isNaN(Inf - Inf)
doAssert not isNaN(3.1415926)
doAssert not isNaN(0'f32)
template fn: untyped = result = x != x
when nimvm: fn()
else:
when defined(js): fn()
else: result = c_isnan(x)
when defined(js):
proc toBitsImpl(x: float): array[2, uint32] =
asm """
const buffer = new ArrayBuffer(8);
const floatBuffer = new Float64Array(buffer);
const uintBuffer = new Uint32Array(buffer);
floatBuffer[0] = `x`;
`result` = uintBuffer
"""
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 or zero, both of which can carry a sign.
runnableExamples:
doAssert copySign(1.0, -0.0) == -1.0
doAssert copySign(0.0, -0.0) == -0.0
doAssert copySign(-1.0, 0.0) == 1.0
doAssert copySign(10.0, 0.0) == 10.0
doAssert copySign(Inf, -1.0) == -Inf
doAssert copySign(-Inf, 1.0) == Inf
doAssert copySign(-1.0, NaN) == 1.0
doAssert copySign(10.0, NaN) == 10.0
doAssert copySign(NaN, 0.0).isNaN
doAssert copySign(NaN, -0.0).isNaN
# fails in VM and JS backend
doAssert copySign(1.0, copySign(NaN, -1.0)) == -1.0
# TODO use signbit for examples
template impl() =
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)
when defined(js): impl()
else:
when nimvm: impl()
else: result = c_copysign(x, y)
func classify*(x: float): FloatClass =
## Classifies a floating point value.
##
## Returns ``x``'s class as specified by `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
## `machine epsilon <https://en.wikipedia.org/wiki/Machine_epsilon>`_.
##
## `unitsInLastPlace` is the max number of
## `units in 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(3.141592653589793, 3.1415926535897936)
doAssert almostEqual(1.6777215e7'f32, 1.6777216e7'f32)
doAssert almostEqual(Inf, Inf)
doAssert almostEqual(-Inf, -Inf)
doAssert almostEqual(Inf, -Inf) == false
doAssert almostEqual(-Inf, Inf) == false
doAssert almostEqual(Inf, NaN) == false
doAssert almostEqual(NaN, NaN) == false
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) == true
doAssert isPowerOfTwo(5) == false
doAssert isPowerOfTwo(0) == false
doAssert isPowerOfTwo(-16) == false
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([-1.5, 2.7, -0.1]) == 1.1
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 = 1.T
for i in items(x): result = result * i
func cumsummed*[T](x: openArray[T]): seq[T] =
## Return cumulative (aka prefix) summation of ``x``.
##
## See also:
## * `sum func <#sum,openArray[T]>`_
## * `cumsum func <#cumsum,openArray[T]>`_ for the in-place version
runnableExamples:
let a = [1, 2, 3, 4]
doAssert cumsummed(a) == @[1, 3, 6, 10]
result.setLen(x.len)
result[0] = x[0]
for i in 1 ..< x.len: 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 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 cubic root
##
## .. code-block:: nim
## echo sqrt(4.0) ## 2.0
## echo sqrt(1.44) ## 1.2
## echo sqrt(-4.0) ## nan
func cbrt*(x: float32): float32 {.importc: "cbrtf", header: "<math.h>".}
func cbrt*(x: float64): float64 {.importc: "cbrt", header: "<math.h>".}
## Computes the cubic root of ``x``.
##
## See also:
## * `sqrt func <#sqrt,float64>`_ for square root
##
## .. code-block:: nim
## echo cbrt(8.0) ## 2.0
## echo cbrt(2.197) ## 1.3
## echo 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>`_
##
## .. code-block:: nim
## echo ln(exp(4.0)) ## 4.0
## echo ln(1.0)) ## 0.0
## echo ln(0.0) ## -inf
## echo ln(-7.0) ## nan
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>`_
## * `exp func <#exp,float64>`_
##
## .. code-block:: nim
## echo log(9.0, 3.0) ## 2.0
## echo log(32.0, 2.0) ## 5.0
## echo log(0.0, 2.0) ## -inf
## echo log(-7.0, 4.0) ## nan
## echo log(8.0, -2.0) ## nan
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>`_
## * `exp func <#exp,float64>`_
##
## .. code-block:: nim
## echo log10(100.0) ## 2.0
## echo log10(0.0) ## nan
## echo log10(-100.0) ## -inf
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>`_
## * `log func <#log,T,T>`_
## * `log10 func <#log10,float64>`_
## * `log2 func <#log2,float64>`_
##
## .. code-block:: nim
## echo exp(1.0) ## 2.718281828459045
## echo ln(exp(4.0)) ## 4.0
## echo exp(0.0) ## 1.0
## echo exp(-1.0) ## 0.3678794411714423
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:
## * `cos func <#cos,float64>`_
## * `tan func <#tan,float64>`_
## * `arcsin func <#arcsin,float64>`_
## * `sinh func <#sinh,float64>`_
##
## .. code-block:: nim
## echo sin(PI / 6) ## 0.4999999999999999
## echo 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:
## * `sin func <#sin,float64>`_
## * `tan func <#tan,float64>`_
## * `arccos func <#arccos,float64>`_
## * `cosh func <#cosh,float64>`_
##
## .. code-block:: nim
## echo cos(2 * PI) ## 1.0
## echo cos(degToRad(60.0)) ## 0.5000000000000001
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:
## * `sin func <#sin,float64>`_
## * `cos func <#cos,float64>`_
## * `arctan func <#arctan,float64>`_
## * `tanh func <#tanh,float64>`_
##
## .. code-block:: nim
## echo tan(degToRad(45.0)) ## 0.9999999999999999
## echo tan(PI / 4) ## 0.9999999999999999
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:
## * `cosh func <#cosh,float64>`_
## * `tanh func <#tanh,float64>`_
## * `arcsinh func <#arcsinh,float64>`_
## * `sin func <#sin,float64>`_
##
## .. code-block:: nim
## echo sinh(0.0) ## 0.0
## echo sinh(1.0) ## 1.175201193643801
## echo sinh(degToRad(90.0)) ## 2.301298902307295
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:
## * `sinh func <#sinh,float64>`_
## * `tanh func <#tanh,float64>`_
## * `arccosh func <#arccosh,float64>`_
## * `cos func <#cos,float64>`_
##
## .. code-block:: nim
## echo cosh(0.0) ## 1.0
## echo cosh(1.0) ## 1.543080634815244
## echo cosh(degToRad(90.0)) ## 2.509178478658057
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:
## * `sinh func <#sinh,float64>`_
## * `cosh func <#cosh,float64>`_
## * `arctanh func <#arctanh,float64>`_
## * `tan func <#tan,float64>`_
##
## .. code-block:: nim
## echo tanh(0.0) ## 0.0
## echo tanh(1.0) ## 0.7615941559557649
## echo tanh(degToRad(90.0)) ## 0.9171523356672744
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:
## * `arcsin func <#arcsin,float64>`_
## * `arctan func <#arctan,float64>`_
## * `arctan2 func <#arctan2,float64,float64>`_
## * `cos func <#cos,float64>`_
##
## .. code-block:: nim
## echo radToDeg(arccos(0.0)) ## 90.0
## echo radToDeg(arccos(1.0)) ## 0.0
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:
## * `arccos func <#arccos,float64>`_
## * `arctan func <#arctan,float64>`_
## * `arctan2 func <#arctan2,float64,float64>`_
## * `sin func <#sin,float64>`_
##
## .. code-block:: nim
## echo radToDeg(arcsin(0.0)) ## 0.0
## echo radToDeg(arcsin(1.0)) ## 90.0
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:
## * `arcsin func <#arcsin,float64>`_
## * `arccos func <#arccos,float64>`_
## * `arctan2 func <#arctan2,float64,float64>`_
## * `tan func <#tan,float64>`_
##
## .. code-block:: nim
## echo arctan(1.0) ## 0.7853981633974483
## echo 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:
## * `arcsin func <#arcsin,float64>`_
## * `arccos func <#arccos,float64>`_
## * `arctan func <#arctan,float64>`_
## * `tan func <#tan,float64>`_
##
## .. code-block:: nim
## echo arctan2(1.0, 0.0) ## 1.570796326794897
## echo 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``.
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``.
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``.
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``.
func arcsec*[T: float32|float64](x: T): T = arccos(1.0 / x)
## Computes the inverse secant of ``x``.
func arccsc*[T: float32|float64](x: T): T = arcsin(1.0 / x)
## Computes the inverse cosecant of ``x``.
func arccoth*[T: float32|float64](x: T): T = arctanh(1.0 / x)
## Computes the inverse hyperbolic cotangent of ``x``.
func arcsech*[T: float32|float64](x: T): T = arccosh(1.0 / x)
## Computes the inverse hyperbolic secant of ``x``.
func arccsch*[T: float32|float64](x: T): T = arcsinh(1.0 / x)
## Computes the inverse hyperbolic cosecant of ``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 hypotenuse of a right-angle triangle with ``x`` and
## ``y`` as its base and height. Equivalent to ``sqrt(x*x + y*y)``.
##
## .. code-block:: nim
## echo hypot(4.0, 3.0) ## 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 to power raised of y.
##
## To compute power between integers (e.g. 2^6), use `^ func<#^,T,Natural>`_.
##
## See also:
## * `^ func<#^,T,Natural>`_
## * `sqrt func <#sqrt,float64>`_
## * `cbrt func <#cbrt,float64>`_
##
## .. code-block:: nim
## echo pow(100, 1.5) ## 1000.0
## echo 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 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 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 JS backend.
##
## See also:
## * `lgamma func <#lgamma,float64>`_ for a natural log of gamma function
##
## .. code-block:: Nim
## echo gamma(1.0) # 1.0
## echo gamma(4.0) # 6.0
## echo gamma(11.0) # 3628800.0
## echo gamma(-1.0) # nan
func lgamma*(x: float32): float32 {.importc: "lgammaf", header: "<math.h>".}
func lgamma*(x: float64): float64 {.importc: "lgamma", header: "<math.h>".}
## Computes the natural log of the gamma function for ``x``.
##
## Note: Not available for JS backend.
##
## See also:
## * `gamma func <#gamma,float64>`_ for gamma function
##
## .. code-block:: Nim
## echo lgamma(1.0) # 1.0
## echo lgamma(4.0) # 1.791759469228055
## echo lgamma(11.0) # 15.10441257307552
## echo lgamma(-1.0) # inf
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>`_
##
## .. code-block:: nim
## echo floor(2.1) ## 2.0
## echo floor(2.9) ## 2.0
## echo 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>`_
##
## .. code-block:: nim
## echo ceil(2.1) ## 3.0
## echo ceil(2.9) ## 3.0
## echo 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>`_
##
## .. code-block:: nim
## echo round(3.4) ## 3.0
## echo round(3.5) ## 4.0
## echo 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>`_
##
## .. code-block:: nim
## echo trunc(PI) # 3.0
## echo 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
##
## .. code-block:: nim
## ( 6.5 mod 2.5) == 1.5
## (-6.5 mod 2.5) == -1.5
## ( 6.5 mod -2.5) == 1.5
## (-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 {.importcpp: "# % #".}
func `mod`*(x, y: float64): float64 {.importcpp: "# % #".}
## Computes the modulo operation for float values (the remainder of ``x`` divided by ``y``).
##
## .. code-block:: nim
## ( 6.5 mod 2.5) == 1.5
## (-6.5 mod 2.5) == -1.5
## ( 6.5 mod -2.5) == 1.5
## (-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``
##
## .. code-block:: Nim
## echo round(PI, 2) ## 3.14
## echo round(PI, 4) ## 3.1416
if places == 0:
result = round(x)
else:
var mult = pow(10.0, places.T)
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
##
## .. code-block:: nim
## echo floorDiv( 13, 3) # 4
## echo floorDiv(-13, 3) # -5
## echo floorDiv( 13, -3) # -5
## echo 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 modulus 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>`_
##
## .. code-block:: nim
## echo floorMod( 13, 3) # 1
## echo floorMod(-13, 3) # 2
## echo floorMod( 13, -3) # -2
## echo 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:
assert euclDiv(13, 3) == 4
assert euclDiv(-13, 3) == -5
assert euclDiv(13, -3) == -4
assert 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:
assert euclMod(13, 3) == 1
assert euclMod(-13, 3) == 2
assert euclMod(13, -3) == 1
assert euclMod(-13, -3) == 2
result = x mod y
if result < 0:
result += abs(y)
when not defined(js):
func c_frexp*(x: float32, exponent: var int32): float32 {.
importc: "frexp", header: "<math.h>".}
func c_frexp*(x: float64, exponent: var int32): float64 {.
importc: "frexp", header: "<math.h>".}
func frexp*[T, U](x: T, exponent: var U): T =
## Split a number into mantissa and exponent.
##
## ``frexp`` calculates the mantissa m (a float greater than or equal to 0.5
## and less than 1) and the integer value n such that ``x`` (the original
## float value) equals ``m * 2**n``. frexp stores n in `exponent` and returns
## m.
##
runnableExamples:
var x: int
doAssert frexp(5.0, x) == 0.625
doAssert x == 3
var exp: int32
result = c_frexp(x, exp)
exponent = exp
when windowsCC89:
# taken from Go-lang Math.Log2
const ln2 = 0.693147180559945309417232121458176568075500134360255254120680009
template log2Impl[T](x: T): T =
var exp: int32
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>`_
## * `exp func <#exp,float64>`_
##
## .. code-block:: Nim
## echo log2(8.0) # 3.0
## echo log2(1.0) # 0.0
## echo log2(0.0) # -inf
## echo log2(-2.0) # nan
else:
func frexp*[T: float32|float64](x: T, exponent: var int): T =
if x == 0.0:
exponent = 0
result = 0.0
elif x < 0.0:
result = -frexp(-x, exponent)
else:
var ex = trunc(log2(x))
exponent = int(ex)
result = x / pow(2.0, ex)
if abs(result) >= 1:
inc(exponent)
result = result / 2
if exponent == 1024 and result == 0.0:
result = 0.99999999999999988898
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.} =
## Convert from degrees to radians.
##
## See also:
## * `radToDeg func <#radToDeg,T>`_
##
runnableExamples:
doAssert degToRad(180.0) == 3.141592653589793
result = T(d) * RadPerDeg
func radToDeg*[T: float32|float64](d: T): T {.inline.} =
## Convert from radians to degrees.
##
## See also:
## * `degToRad func <#degToRad,T>`_
##
runnableExamples:
doAssert radToDeg(2 * PI) == 360.0
result = T(d) / 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 ``y``.
##
## Exponent ``y`` must be non-negative, use
## `pow func <#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:
assert -3.0^0 == 1.0
assert -3^1 == -3
assert -3^2 == 9
assert -3.0^3 == -27.0
assert -3.0^4 == 81.0
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 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 binary GCD (aka Stein's) algorithm.
##
## See also:
## * `gcd func <#gcd,T,T>`_ for floats 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 integer version
runnableExamples:
doAssert gcd(@[13.5, 9.0]) == 4.5
result = x[0]
var i = 1
while i < x.len:
result = gcd(result, x[i])
inc(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 lcm*[T](x: openArray[T]): T {.since: (1, 1).} =
## Computes the least common multiple of the elements of ``x``.
##
## See also:
## * `gcd func <#gcd,T,T>`_ for integer version
runnableExamples:
doAssert lcm(@[24, 30]) == 120
result = x[0]
var i = 1
while i < x.len:
result = lcm(result, x[i])
inc(i)
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