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304 lines
11 KiB
Nim
Executable File
304 lines
11 KiB
Nim
Executable File
#
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#
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# Nimrod's Runtime Library
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# (c) Copyright 2012 Andreas Rumpf
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#
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# See the file "copying.txt", included in this
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# distribution, for details about the copyright.
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#
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## Constructive mathematics is naturally typed. -- Simon Thompson
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##
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## Basic math routines for Nimrod.
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## This module is available for the ECMAScript target.
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{.push debugger:off .} # the user does not want to trace a part
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# of the standard library!
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{.push checks:off, line_dir:off, stack_trace:off.}
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when defined(Posix):
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{.passl: "-lm".}
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const
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PI* = 3.1415926535897932384626433 ## the circle constant PI (Ludolph's number)
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E* = 2.71828182845904523536028747 ## Euler's number
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MaxFloat64Precision* = 16 ## maximum number of meaningful digits
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## after the decimal point for Nimrod's
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## ``float64`` type.
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MaxFloat32Precision* = 8 ## maximum number of meaningful digits
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## after the decimal point for Nimrod's
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## ``float32`` type.
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MaxFloatPrecision* = MaxFloat64Precision ## maximum number of
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## meaningful digits
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## after the decimal point
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## for Nimrod's ``float`` type.
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type
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TFloatClass* = enum ## describes the class a floating point value belongs to.
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## This is the type that is returned by `classify`.
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fcNormal, ## value is an ordinary nonzero floating point value
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fcSubnormal, ## value is a subnormal (a very small) floating point value
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fcZero, ## value is zero
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fcNegZero, ## value is the negative zero
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fcNan, ## value is Not-A-Number (NAN)
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fcInf, ## value is positive infinity
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fcNegInf ## value is negative infinity
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proc classify*(x: float): TFloatClass =
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## classifies a floating point value. Returns `x`'s class as specified by
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## `TFloatClass`.
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# ECMAScript and most C compilers have no classify:
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if x == 0.0:
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if 1.0/x == Inf:
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return fcZero
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else:
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return fcNegZero
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if x*0.5 == x:
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if x > 0.0: return fcInf
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else: return fcNegInf
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if x != x: return fcNan
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return fcNormal
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# XXX: fcSubnormal is not detected!
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proc binom*(n, k: int): int {.noSideEffect.} =
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## computes the binomial coefficient
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if k <= 0: return 1
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if 2*k > n: return binom(n, n-k)
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result = n
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for i in countup(2, k):
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result = (result * (n + 1 - i)) div i
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proc fac*(n: int): int {.noSideEffect.} =
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## computes the faculty function
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result = 1
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for i in countup(2, n):
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result = result * i
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proc isPowerOfTwo*(x: int): bool {.noSideEffect.} =
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## returns true, if x is a power of two, false otherwise.
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## Negative numbers are not a power of two.
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return (x and -x) == x
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proc nextPowerOfTwo*(x: int): int =
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## returns the nearest power of two, so that
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## result**2 >= x > (result-1)**2.
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result = x - 1
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when defined(cpu64):
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result = result or (result shr 32)
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result = result or (result shr 16)
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result = result or (result shr 8)
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result = result or (result shr 4)
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result = result or (result shr 2)
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result = result or (result shr 1)
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Inc(result)
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proc countBits32*(n: int32): int {.noSideEffect.} =
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## counts the set bits in `n`.
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var v = n
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v = v -% ((v shr 1'i32) and 0x55555555'i32)
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v = (v and 0x33333333'i32) +% ((v shr 2'i32) and 0x33333333'i32)
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result = ((v +% (v shr 4'i32) and 0xF0F0F0F'i32) *% 0x1010101'i32) shr 24'i32
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proc sum*[T](x: openarray[T]): T {.noSideEffect.} =
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## computes the sum of the elements in `x`.
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## If `x` is empty, 0 is returned.
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for i in items(x): result = result + i
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proc mean*(x: openarray[float]): float {.noSideEffect.} =
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## computes the mean of the elements in `x`.
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## If `x` is empty, NaN is returned.
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result = sum(x) / toFloat(len(x))
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proc variance*(x: openarray[float]): float {.noSideEffect.} =
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## computes the variance of the elements in `x`.
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## If `x` is empty, NaN is returned.
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result = 0.0
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var m = mean(x)
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for i in 0 .. high(x):
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var diff = x[i] - m
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result = result + diff*diff
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result = result / toFloat(len(x))
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proc random*(max: int): int
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## returns a random number in the range 0..max-1. The sequence of
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## random number is always the same, unless `randomize` is called
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## which initializes the random number generator with a "random"
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## number, i.e. a tickcount.
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when not defined(windows):
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proc random*(max: float): float
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## returns a random number in the range 0..<max. The sequence of
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## random number is always the same, unless `randomize` is called
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## which initializes the random number generator with a "random"
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## number, i.e. a tickcount. This is currently not supported for windows.
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proc randomize*()
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## initializes the random number generator with a "random"
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## number, i.e. a tickcount. Note: Does nothing for the ECMAScript target,
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## as ECMAScript does not support this.
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when not defined(ECMAScript):
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proc sqrt*(x: float): float {.importc: "sqrt", header: "<math.h>".}
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## computes the square root of `x`.
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proc ln*(x: float): float {.importc: "log", header: "<math.h>".}
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## computes ln(x).
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proc log10*(x: float): float {.importc: "log10", header: "<math.h>".}
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proc log2*(x: float): float = return ln(x) / ln(2.0)
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proc exp*(x: float): float {.importc: "exp", header: "<math.h>".}
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## computes e**x.
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proc frexp*(x: float, exponent: var int): float {.
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importc: "frexp", header: "<math.h>".}
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## Split a number into mantissa and exponent.
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## `frexp` calculates the mantissa m (a float greater than or equal to 0.5
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## and less than 1) and the integer value n such that `x` (the original
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## float value) equals m * 2**n. frexp stores n in `exponent` and returns
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## m.
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proc round*(x: float): int {.importc: "lrint", nodecl.}
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## converts a float to an int by rounding.
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proc arccos*(x: float): float {.importc: "acos", header: "<math.h>".}
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proc arcsin*(x: float): float {.importc: "asin", header: "<math.h>".}
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proc arctan*(x: float): float {.importc: "atan", header: "<math.h>".}
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proc arctan2*(y, x: float): float {.importc: "atan2", header: "<math.h>".}
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## Calculate the arc tangent of `y` / `x`.
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## `atan2` returns the arc tangent of `y` / `x`; it produces correct
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## results even when the resulting angle is near pi/2 or -pi/2
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## (`x` near 0).
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proc cos*(x: float): float {.importc: "cos", header: "<math.h>".}
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proc cosh*(x: float): float {.importc: "cosh", header: "<math.h>".}
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proc hypot*(x, y: float): float {.importc: "hypot", header: "<math.h>".}
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## same as ``sqrt(x*x + y*y)``.
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proc sinh*(x: float): float {.importc: "sinh", header: "<math.h>".}
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proc sin*(x: float): float {.importc: "sin", header: "<math.h>".}
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proc tan*(x: float): float {.importc: "tan", header: "<math.h>".}
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proc tanh*(x: float): float {.importc: "tanh", header: "<math.h>".}
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proc pow*(x, y: float): float {.importc: "pow", header: "<math.h>".}
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## computes x to power raised of y.
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# C procs:
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proc gettime(dummy: ptr cint): cint {.importc: "time", header: "<time.h>".}
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proc srand(seed: cint) {.importc: "srand", nodecl.}
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proc rand(): cint {.importc: "rand", nodecl.}
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when not defined(windows):
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proc srand48(seed: cint) {.importc: "srand48", nodecl.}
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proc drand48(): float {.importc: "drand48", nodecl.}
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proc random(max: float): float =
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result = drand48() * max
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proc randomize() =
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let x = gettime(nil)
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srand(x)
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when defined(srand48): srand48(x)
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proc random(max: int): int =
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result = int(rand()) mod max
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proc trunc*(x: float): float {.importc: "trunc", nodecl.}
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proc floor*(x: float): float {.importc: "floor", nodecl.}
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proc ceil*(x: float): float {.importc: "ceil", nodecl.}
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proc fmod*(x, y: float): float {.importc: "fmod", header: "<math.h>".}
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else:
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proc mathrandom(): float {.importc: "Math.random", nodecl.}
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proc floor*(x: float): float {.importc: "Math.floor", nodecl.}
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proc ceil*(x: float): float {.importc: "Math.ceil", nodecl.}
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proc random(max: int): int =
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result = int(floor(mathrandom() * float(max)))
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proc random(max: float): float =
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result = float(mathrandom() * float(max))
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proc randomize() = nil
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proc sqrt*(x: float): float {.importc: "Math.sqrt", nodecl.}
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proc ln*(x: float): float {.importc: "Math.log", nodecl.}
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proc log10*(x: float): float = return ln(x) / ln(10.0)
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proc log2*(x: float): float = return ln(x) / ln(2.0)
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proc exp*(x: float): float {.importc: "Math.exp", nodecl.}
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proc round*(x: float): int {.importc: "Math.round", nodecl.}
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proc pow*(x, y: float): float {.importc: "Math.pow", nodecl.}
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proc frexp*(x: float, exponent: var int): float =
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if x == 0.0:
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exponent = 0
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result = 0.0
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elif x < 0.0:
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result = -frexp(-x, exponent)
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else:
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var ex = floor(log2(x))
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exponent = round(ex)
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result = x / pow(2.0, ex)
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proc arccos*(x: float): float {.importc: "Math.acos", nodecl.}
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proc arcsin*(x: float): float {.importc: "Math.asin", nodecl.}
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proc arctan*(x: float): float {.importc: "Math.atan", nodecl.}
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proc arctan2*(y, x: float): float {.importc: "Math.atan2", nodecl.}
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proc cos*(x: float): float {.importc: "Math.cos", nodecl.}
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proc cosh*(x: float): float = return (exp(x)+exp(-x))*0.5
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proc hypot*(x, y: float): float = return sqrt(x*x + y*y)
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proc sinh*(x: float): float = return (exp(x)-exp(-x))*0.5
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proc sin*(x: float): float {.importc: "Math.sin", nodecl.}
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proc tan*(x: float): float {.importc: "Math.tan", nodecl.}
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proc tanh*(x: float): float =
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var y = exp(2.0*x)
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return (y-1.0)/(y+1.0)
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proc `mod`*(x, y: float): float =
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result = if y == 0.0: x else: x - y * (x/y).floor
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proc random*[T](x: TSlice[T]): T =
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result = random(x.b - x.a) + x.a
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type
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TRunningStat* {.pure,final.} = object ## an accumulator for statistical data
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n*: int ## number of pushed data
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sum*, min*, max*, mean*: float ## self-explaining
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oldM, oldS, newS: float
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proc push*(s: var TRunningStat, x: float) =
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## pushes a value `x` for processing
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inc(s.n)
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# See Knuth TAOCP vol 2, 3rd edition, page 232
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if s.n == 1:
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s.min = x
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s.max = x
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s.oldM = x
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s.mean = x
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s.oldS = 0.0
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else:
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if s.min > x: s.min = x
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if s.max < x: s.max = x
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s.mean = s.oldM + (x - s.oldM)/toFloat(s.n)
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s.newS = s.oldS + (x - s.oldM)*(x - s.mean)
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# set up for next iteration:
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s.oldM = s.mean
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s.oldS = s.newS
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s.sum = s.sum + x
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proc push*(s: var TRunningStat, x: int) =
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## pushes a value `x` for processing. `x` is simply converted to ``float``
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## and the other push operation is called.
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push(s, toFloat(x))
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proc variance*(s: TRunningStat): float =
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## computes the current variance of `s`
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if s.n > 1: result = s.newS / (toFloat(s.n - 1))
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proc standardDeviation*(s: TRunningStat): float =
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## computes the current standard deviation of `s`
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result = sqrt(variance(s))
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{.pop.}
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{.pop.}
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