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Diffstat (limited to 'lib/pure/math.nim')
| -rw-r--r-- | lib/pure/math.nim | 1556 |
1 files changed, 1249 insertions, 307 deletions
diff --git a/lib/pure/math.nim b/lib/pure/math.nim index c902af381..ed7d2382f 100644 --- a/lib/pure/math.nim +++ b/lib/pure/math.nim @@ -7,90 +7,349 @@ # distribution, for details about the copyright. # -## Constructive mathematics is naturally typed. -- Simon Thompson -## +## *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#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.) + -include "system/inclrtl" -{.push debugger:off .} # the user does not want to trace a part +import std/private/since +{.push debugger: off.} # the user does not want to trace a part # of the standard library! -{.push checks:off, line_dir:off, stack_trace:off.} +import std/[bitops, fenv] +import system/countbits_impl + +when defined(nimPreviewSlimSystem): + import std/assertions + + +when not defined(js) and not defined(nimscript): # C + 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>".} + + type + div_t {.importc, header: "<stdlib.h>".} = object + quot: cint + rem: cint + ldiv_t {.importc, header: "<stdlib.h>".} = object + quot: clong + rem: clong + lldiv_t {.importc, header: "<stdlib.h>".} = object + quot: clonglong + rem: clonglong + + when cint isnot clong: + func divmod_c(x, y: cint): div_t {.importc: "div", header: "<stdlib.h>".} + when clong isnot clonglong: + func divmod_c(x, y: clonglong): lldiv_t {.importc: "lldiv", header: "<stdlib.h>".} + func divmod_c(x, y: clong): ldiv_t {.importc: "ldiv", header: "<stdlib.h>".} + func divmod*[T: SomeInteger](x, y: T): (T, T) {.inline.} = + ## Specialized instructions for computing both division and modulus. + ## Return structure is: (quotient, remainder) + runnableExamples: + doAssert divmod(5, 2) == (2, 1) + doAssert divmod(5, -3) == (-1, 2) + when T is cint | clong | clonglong: + when compileOption("overflowChecks"): + if y == 0: + raise new(DivByZeroDefect) + elif (x == T.low and y == -1.T): + raise new(OverflowDefect) + let res = divmod_c(x, y) + result[0] = res.quot + result[1] = res.rem + else: + result[0] = x div y + result[1] = x mod y + +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 -when defined(Posix) and not defined(haiku): +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) and not defined(macosx): {.passl: "-lm".} -when not defined(js): - import times const - PI* = 3.1415926535897932384626433 ## the circle constant PI (Ludolph's number) - 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. + 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`. - 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 - -proc classify*(x: float): FloatClass = - ## classifies a floating point value. Returns `x`'s class as specified by - ## `FloatClass`. - + 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) or defined(nimscript): 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 + {.emit: """ + 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>`_. + 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 isNan(x): return fcNan if x == 0.0: - if 1.0/x == Inf: + if 1.0 / x == Inf: return fcZero else: return fcNegZero - if x*0.5 == x: + 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 - # XXX: fcSubnormal is not detected! +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) -proc binom*(n, k: int): int {.noSideEffect.} = - ## computes the binomial coefficient - 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 - -proc fac*(n: int): int {.noSideEffect.} = - ## computes the faculty/factorial function. - result = 1 - for i in countup(2, n): - result = result * i - -proc isPowerOfTwo*(x: int): bool {.noSideEffect.} = - ## returns true, if `x` is a power of two, false otherwise. + 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. - return (x != 0) and ((x and (x - 1)) == 0) + ## + ## **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) -proc nextPowerOfTwo*(x: int): int {.noSideEffect.} = - ## returns `x` rounded up to the nearest power of two. +func nextPowerOfTwo*(x: int): int = + ## Returns `x` rounded up to the nearest power of two. + ## ## Zero and negative numbers get rounded up to 1. - result = x - 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: @@ -100,273 +359,956 @@ proc nextPowerOfTwo*(x: int): int {.noSideEffect.} = result = result or (result shr 4) result = result or (result shr 2) result = result or (result shr 1) - result += 1 + ord(x<=0) + result += 1 + ord(x <= 0) -proc countBits32*(n: int32): int {.noSideEffect.} = - ## counts the set bits in `n`. - var v = n - v = v -% ((v shr 1'i32) and 0x55555555'i32) - v = (v and 0x33333333'i32) +% ((v shr 2'i32) and 0x33333333'i32) - result = ((v +% (v shr 4'i32) and 0xF0F0F0F'i32) *% 0x1010101'i32) shr 24'i32 -proc sum*[T](x: openArray[T]): T {.noSideEffect.} = - ## computes the sum of the elements in `x`. - ## If `x` is empty, 0 is returned. - for i in items(x): result = result + i -proc mean*(x: openArray[float]): float {.noSideEffect.} = - ## computes the mean of the elements in `x`. - ## If `x` is empty, NaN is returned. - result = sum(x) / toFloat(len(x)) - -proc variance*(x: openArray[float]): float {.noSideEffect.} = - ## computes the variance of the elements in `x`. - ## If `x` is empty, NaN is returned. - result = 0.0 - var m = mean(x) - for i in 0 .. high(x): - var diff = x[i] - m - result = result + diff*diff - result = result / toFloat(len(x)) - -proc random*(max: int): int {.benign.} - ## returns a random number in the range 0..max-1. The sequence of - ## random number is always the same, unless `randomize` is called - ## which initializes the random number generator with a "random" - ## number, i.e. a tickcount. - -proc random*(max: float): float {.benign.} - ## returns a random number in the range 0..<max. The sequence of - ## random number is always the same, unless `randomize` is called - ## which initializes the random number generator with a "random" - ## number, i.e. a tickcount. This has a 16-bit resolution on windows - ## and a 48-bit resolution on other platforms. - -proc randomize*() {.benign.} - ## initializes the random number generator with a "random" - ## number, i.e. a tickcount. Note: Does nothing for the JavaScript target, - ## as JavaScript does not support this. - -proc randomize*(seed: int) {.benign.} - ## initializes the random number generator with a specific seed. - ## Note: Does nothing for the JavaScript target, - ## as JavaScript does not support this. - -when not defined(JS): - proc sqrt*(x: float): float {.importc: "sqrt", header: "<math.h>".} - ## computes the square root of `x`. - - proc ln*(x: float): float {.importc: "log", header: "<math.h>".} - ## computes ln(x). - proc log10*(x: float): float {.importc: "log10", header: "<math.h>".} - proc log2*(x: float): float = return ln(x) / ln(2.0) - proc exp*(x: float): float {.importc: "exp", header: "<math.h>".} - ## computes e**x. - - proc frexp*(x: float, exponent: var int): float {. - importc: "frexp", header: "<math.h>".} - ## 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. - - proc round*(x: float): int {.importc: "lrint", header: "<math.h>".} - ## converts a float to an int by rounding. - - proc arccos*(x: float): float {.importc: "acos", header: "<math.h>".} - proc arcsin*(x: float): float {.importc: "asin", header: "<math.h>".} - proc arctan*(x: float): float {.importc: "atan", header: "<math.h>".} - proc arctan2*(y, x: float): float {.importc: "atan2", header: "<math.h>".} - ## Calculate the arc tangent of `y` / `x`. - ## `atan2` returns 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). - - proc cos*(x: float): float {.importc: "cos", header: "<math.h>".} - proc cosh*(x: float): float {.importc: "cosh", header: "<math.h>".} - proc hypot*(x, y: float): float {.importc: "hypot", header: "<math.h>".} - ## same as ``sqrt(x*x + y*y)``. - - proc sinh*(x: float): float {.importc: "sinh", header: "<math.h>".} - proc sin*(x: float): float {.importc: "sin", header: "<math.h>".} - proc tan*(x: float): float {.importc: "tan", header: "<math.h>".} - proc tanh*(x: float): float {.importc: "tanh", header: "<math.h>".} - proc pow*(x, y: float): float {.importc: "pow", header: "<math.h>".} - ## computes x to power raised of y. - - # C procs: - proc srand(seed: cint) {.importc: "srand", header: "<stdlib.h>".} - proc rand(): cint {.importc: "rand", header: "<stdlib.h>".} - - when not defined(windows): - proc srand48(seed: clong) {.importc: "srand48", header: "<stdlib.h>".} - proc drand48(): float {.importc: "drand48", header: "<stdlib.h>".} - proc random(max: float): float = - result = drand48() * max - when defined(windows): - proc random(max: float): float = - # we are hardcodeing this because - # importcing macros is extremely problematic - # and because the value is publicly documented - # on MSDN and very unlikely to change - const rand_max = 32767 - result = (float(rand()) / float(rand_max)) * max - proc randomize() = - randomize(cast[int](epochTime())) - - proc randomize(seed: int) = - srand(cint(seed)) - when declared(srand48): srand48(seed) - proc random(max: int): int = - result = int(rand()) mod max - - proc trunc*(x: float): float {.importc: "trunc", header: "<math.h>".} - proc floor*(x: float): float {.importc: "floor", header: "<math.h>".} - proc ceil*(x: float): float {.importc: "ceil", header: "<math.h>".} - - proc fmod*(x, y: float): float {.importc: "fmod", header: "<math.h>".} - -else: - proc mathrandom(): float {.importc: "Math.random", nodecl.} - proc floor*(x: float): float {.importc: "Math.floor", nodecl.} - proc ceil*(x: float): float {.importc: "Math.ceil", nodecl.} - proc random(max: int): int = - result = int(floor(mathrandom() * float(max))) - proc random(max: float): float = - result = float(mathrandom() * float(max)) - proc randomize() = discard - proc randomize(seed: int) = discard + +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 - proc sqrt*(x: float): float {.importc: "Math.sqrt", nodecl.} - proc ln*(x: float): float {.importc: "Math.log", nodecl.} - proc log10*(x: float): float = return ln(x) / ln(10.0) - proc log2*(x: float): float = return ln(x) / ln(2.0) - - proc exp*(x: float): float {.importc: "Math.exp", nodecl.} - proc round*(x: float): int {.importc: "Math.round", nodecl.} - proc pow*(x, y: float): float {.importc: "Math.pow", nodecl.} + func divmod*[T:SomeInteger](num, denom: T): (T, T) = + runnableExamples: + doAssert divmod(5, 2) == (2, 1) + doAssert divmod(5, -3) == (-1, 2) + result[0] = num div denom + result[1] = num mod denom - proc frexp*(x: float, exponent: var int): float = + +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: - exponent = 0 - result = 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, exponent) + result = frexp(-x) + result.frac = -result.frac else: - var ex = floor(log2(x)) - exponent = round(ex) - result = x / pow(2.0, ex) - - proc arccos*(x: float): float {.importc: "Math.acos", nodecl.} - proc arcsin*(x: float): float {.importc: "Math.asin", nodecl.} - proc arctan*(x: float): float {.importc: "Math.atan", nodecl.} - proc arctan2*(y, x: float): float {.importc: "Math.atan2", nodecl.} - - proc cos*(x: float): float {.importc: "Math.cos", nodecl.} - proc cosh*(x: float): float = return (exp(x)+exp(-x))*0.5 - proc hypot*(x, y: float): float = return sqrt(x*x + y*y) - proc sinh*(x: float): float = return (exp(x)-exp(-x))*0.5 - proc sin*(x: float): float {.importc: "Math.sin", nodecl.} - proc tan*(x: float): float {.importc: "Math.tan", nodecl.} - proc tanh*(x: float): float = - var y = exp(2.0*x) - return (y-1.0)/(y+1.0) - -proc `mod`*(x, y: float): float = - result = if y == 0.0: x else: x - y * (x/y).floor - -proc random*[T](x: Slice[T]): T = - ## For a slice `a .. b` returns a value in the range `a .. b-1`. - result = random(x.b - x.a) + x.a - -proc random*[T](a: openArray[T]): T = - ## returns a random element from the openarray `a`. - result = a[random(a.low..a.len)] + 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. -type - RunningStat* = object ## an accumulator for statistical data - n*: int ## number of pushed data - sum*, min*, max*, mean*: float ## self-explaining - oldM, oldS, newS: float - -{.deprecated: [TFloatClass: FloatClass, TRunningStat: RunningStat].} - -proc push*(s: var RunningStat, x: float) = - ## pushes a value `x` for processing - inc(s.n) - # See Knuth TAOCP vol 2, 3rd edition, page 232 - if s.n == 1: - s.min = x - s.max = x - s.oldM = x - s.mean = x - s.oldS = 0.0 else: - if s.min > x: s.min = x - if s.max < x: s.max = x - s.mean = s.oldM + (x - s.oldM)/toFloat(s.n) - s.newS = s.oldS + (x - s.oldM)*(x - s.mean) - - # set up for next iteration: - s.oldM = s.mean - s.oldS = s.newS - s.sum = s.sum + x - -proc push*(s: var RunningStat, x: int) = - ## pushes a value `x` for processing. `x` is simply converted to ``float`` - ## and the other push operation is called. - push(s, toFloat(x)) - -proc variance*(s: RunningStat): float = - ## computes the current variance of `s` - if s.n > 1: result = s.newS / (toFloat(s.n - 1)) + 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) -proc standardDeviation*(s: RunningStat): float = - ## computes the current standard deviation of `s` - result = sqrt(variance(s)) +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.} -proc `^`*[T](x, y: T): T = - ## Computes ``x`` to the power ``y`. ``x`` must be non-negative, use - ## `pow <#pow,float,float>` for negative exponents. - assert y >= 0 - var (x, y) = (x, y) - result = 1 +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 - while y != 0: - if (y and 1) != 0: - result *= x - y = y shr 1 - x *= x - -proc gcd*[T](x, y: T): T = - ## Computes the greatest common divisor of ``x`` and ``y``. - var (x,y) = (x,y) + 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] + +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 -proc lcm*[T](x, y: T): T = - ## Computes the least common multiple of ``x`` and ``y``. +when useBuiltins: + ## this func uses bitwise comparisons from C compilers, which are not always available. + 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 -when isMainModule and not defined(JS): - proc gettime(dummy: ptr cint): cint {.importc: "time", header: "<time.h>".} - - # Verifies random seed initialization. - let seed = gettime(nil) - randomize(seed) - const SIZE = 10 - var buf : array[0..SIZE, int] - # Fill the buffer with random values - for i in 0..SIZE-1: - buf[i] = random(high(int)) - # Check that the second random calls are the same for each position. - randomize(seed) - for i in 0..SIZE-1: - assert buf[i] == random(high(int)), "non deterministic random seeding" - echo "random values equal after reseeding" +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]) |