System.Random.MWC.Distributions:truncatedExp from mwc-random-0.13.3.2

Percentage Accurate: 60.9% → 93.1%

Time: 8.5s

Alternatives: 7

Speedup: 3.1×

• could not determine a ground truth

  1. x = -3.4010628291505833e-161
  2. y = 1.183971236926438e-57
  3. z = 1.0838215949654154e+142
  4. t = -4.640143542400253e-262

Specification

Math

\[\begin{array}{l} \\ x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- x (/ (log (+ (- 1.0 y) (* y (exp z)))) t)))

C

double code(double x, double y, double z, double t) {
	return x - (log(((1.0 - y) + (y * exp(z)))) / t);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x - (log(((1.0d0 - y) + (y * exp(z)))) / t)
end function

Java

public static double code(double x, double y, double z, double t) {
	return x - (Math.log(((1.0 - y) + (y * Math.exp(z)))) / t);
}

Python

def code(x, y, z, t):
	return x - (math.log(((1.0 - y) + (y * math.exp(z)))) / t)

Julia

function code(x, y, z, t)
	return Float64(x - Float64(log(Float64(Float64(1.0 - y) + Float64(y * exp(z)))) / t))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x - (log(((1.0 - y) + (y * exp(z)))) / t);
end

Wolfram

code[x_, y_, z_, t_] := N[(x - N[(N[Log[N[(N[(1.0 - y), $MachinePrecision] + N[(y * N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Initial Program: 60.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- x (/ (log (+ (- 1.0 y) (* y (exp z)))) t)))

C

double code(double x, double y, double z, double t) {
	return x - (log(((1.0 - y) + (y * exp(z)))) / t);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x - (log(((1.0d0 - y) + (y * exp(z)))) / t)
end function

Java

public static double code(double x, double y, double z, double t) {
	return x - (Math.log(((1.0 - y) + (y * Math.exp(z)))) / t);
}

Python

def code(x, y, z, t):
	return x - (math.log(((1.0 - y) + (y * math.exp(z)))) / t)

Julia

function code(x, y, z, t)
	return Float64(x - Float64(log(Float64(Float64(1.0 - y) + Float64(y * exp(z)))) / t))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x - (log(((1.0 - y) + (y * exp(z)))) / t);
end

Wolfram

code[x_, y_, z_, t_] := N[(x - N[(N[Log[N[(N[(1.0 - y), $MachinePrecision] + N[(y * N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Alternative 1: 93.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \mathsf{expm1}\left(z\right)\\ \mathbf{if}\;y \leq -4 \cdot 10^{+16}:\\ \;\;\;\;x - \frac{\log \left(1 + t\_1\right)}{t}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+82}:\\ \;\;\;\;x - \frac{t\_1}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot \left(y + z \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot z, 0.5 \cdot y\right)\right)\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (expm1 z))))
   (if (<= y -4e+16)
     (- x (/ (log (+ 1.0 t_1)) t))
     (if (<= y 8e+82)
       (- x (/ t_1 t))
       (-
        x
        (/
         (log
          (+
           1.0
           (* z (+ y (* z (fma 0.16666666666666666 (* y z) (* 0.5 y)))))))
         t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * expm1(z);
	double tmp;
	if (y <= -4e+16) {
		tmp = x - (log((1.0 + t_1)) / t);
	} else if (y <= 8e+82) {
		tmp = x - (t_1 / t);
	} else {
		tmp = x - (log((1.0 + (z * (y + (z * fma(0.16666666666666666, (y * z), (0.5 * y))))))) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y * expm1(z))
	tmp = 0.0
	if (y <= -4e+16)
		tmp = Float64(x - Float64(log(Float64(1.0 + t_1)) / t));
	elseif (y <= 8e+82)
		tmp = Float64(x - Float64(t_1 / t));
	else
		tmp = Float64(x - Float64(log(Float64(1.0 + Float64(z * Float64(y + Float64(z * fma(0.16666666666666666, Float64(y * z), Float64(0.5 * y))))))) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4e+16], N[(x - N[(N[Log[N[(1.0 + t$95$1), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8e+82], N[(x - N[(t$95$1 / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[N[(1.0 + N[(z * N[(y + N[(z * N[(0.16666666666666666 * N[(y * z), $MachinePrecision] + N[(0.5 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4e16

   1. Initial program 60.9%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]

   2. Taylor expanded in y around 0

      \[\leadsto x - \frac{\log \color{blue}{\left(1 + y \cdot \left(e^{z} - 1\right)\right)}}{t} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto x - \frac{\log \left(1 + \color{blue}{y \cdot \left(e^{z} - 1\right)}\right)}{t} \]

      2. lower-*.f64 N/A

         \[\leadsto x - \frac{\log \left(1 + y \cdot \color{blue}{\left(e^{z} - 1\right)}\right)}{t} \]

      3. lower-expm1.f64 81.6

         \[\leadsto x - \frac{\log \left(1 + y \cdot \mathsf{expm1}\left(z\right)\right)}{t} \]

   4. Applied rewrites 81.6%

      \[\leadsto x - \frac{\log \color{blue}{\left(1 + y \cdot \mathsf{expm1}\left(z\right)\right)}}{t} \]

   if -4e16 < y < 7.9999999999999997e82

   1. Initial program 60.9%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]

   2. Taylor expanded in y around 0

      \[\leadsto x - \frac{\color{blue}{y \cdot \left(e^{z} - 1\right)}}{t} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x - \frac{y \cdot \color{blue}{\left(e^{z} - 1\right)}}{t} \]

      2. lower-expm1.f64 85.4

         \[\leadsto x - \frac{y \cdot \mathsf{expm1}\left(z\right)}{t} \]

   4. Applied rewrites 85.4%

      \[\leadsto x - \frac{\color{blue}{y \cdot \mathsf{expm1}\left(z\right)}}{t} \]

   if 7.9999999999999997e82 < y

   1. Initial program 60.9%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]

   2. Taylor expanded in z around 0

      \[\leadsto x - \frac{\log \color{blue}{\left(1 + z \cdot \left(y + z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right)\right)\right)}}{t} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto x - \frac{\log \left(1 + \color{blue}{z \cdot \left(y + z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right)\right)}\right)}{t} \]

      2. lower-*.f64 N/A

         \[\leadsto x - \frac{\log \left(1 + z \cdot \color{blue}{\left(y + z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right)\right)}\right)}{t} \]

      3. lower-+.f64 N/A

         \[\leadsto x - \frac{\log \left(1 + z \cdot \left(y + \color{blue}{z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right)}\right)\right)}{t} \]

      4. lower-*.f64 N/A

         \[\leadsto x - \frac{\log \left(1 + z \cdot \left(y + z \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right)}\right)\right)}{t} \]

      5. lower-fma.f64 N/A

         \[\leadsto x - \frac{\log \left(1 + z \cdot \left(y + z \cdot \mathsf{fma}\left(\frac{1}{6}, \color{blue}{y \cdot z}, \frac{1}{2} \cdot y\right)\right)\right)}{t} \]

      6. lower-*.f64 N/A

         \[\leadsto x - \frac{\log \left(1 + z \cdot \left(y + z \cdot \mathsf{fma}\left(\frac{1}{6}, y \cdot \color{blue}{z}, \frac{1}{2} \cdot y\right)\right)\right)}{t} \]

      7. lower-*.f64 63.0

         \[\leadsto x - \frac{\log \left(1 + z \cdot \left(y + z \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot z, 0.5 \cdot y\right)\right)\right)}{t} \]

   4. Applied rewrites 63.0%

      \[\leadsto x - \frac{\log \color{blue}{\left(1 + z \cdot \left(y + z \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot z, 0.5 \cdot y\right)\right)\right)}}{t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 93.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \mathsf{expm1}\left(z\right)\\ t_2 := x - \frac{\log \left(1 + t\_1\right)}{t}\\ \mathbf{if}\;y \leq -4 \cdot 10^{+16}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+82}:\\ \;\;\;\;x - \frac{t\_1}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (expm1 z))) (t_2 (- x (/ (log (+ 1.0 t_1)) t))))
   (if (<= y -4e+16) t_2 (if (<= y 8e+82) (- x (/ t_1 t)) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * expm1(z);
	double t_2 = x - (log((1.0 + t_1)) / t);
	double tmp;
	if (y <= -4e+16) {
		tmp = t_2;
	} else if (y <= 8e+82) {
		tmp = x - (t_1 / t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = y * Math.expm1(z);
	double t_2 = x - (Math.log((1.0 + t_1)) / t);
	double tmp;
	if (y <= -4e+16) {
		tmp = t_2;
	} else if (y <= 8e+82) {
		tmp = x - (t_1 / t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = y * math.expm1(z)
	t_2 = x - (math.log((1.0 + t_1)) / t)
	tmp = 0
	if y <= -4e+16:
		tmp = t_2
	elif y <= 8e+82:
		tmp = x - (t_1 / t)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(y * expm1(z))
	t_2 = Float64(x - Float64(log(Float64(1.0 + t_1)) / t))
	tmp = 0.0
	if (y <= -4e+16)
		tmp = t_2;
	elseif (y <= 8e+82)
		tmp = Float64(x - Float64(t_1 / t));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x - N[(N[Log[N[(1.0 + t$95$1), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4e+16], t$95$2, If[LessEqual[y, 8e+82], N[(x - N[(t$95$1 / t), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if y < -4e16 or 7.9999999999999997e82 < y

   1. Initial program 60.9%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]

   2. Taylor expanded in y around 0

      \[\leadsto x - \frac{\log \color{blue}{\left(1 + y \cdot \left(e^{z} - 1\right)\right)}}{t} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto x - \frac{\log \left(1 + \color{blue}{y \cdot \left(e^{z} - 1\right)}\right)}{t} \]

      2. lower-*.f64 N/A

         \[\leadsto x - \frac{\log \left(1 + y \cdot \color{blue}{\left(e^{z} - 1\right)}\right)}{t} \]

      3. lower-expm1.f64 81.6

         \[\leadsto x - \frac{\log \left(1 + y \cdot \mathsf{expm1}\left(z\right)\right)}{t} \]

   4. Applied rewrites 81.6%

      \[\leadsto x - \frac{\log \color{blue}{\left(1 + y \cdot \mathsf{expm1}\left(z\right)\right)}}{t} \]

   if -4e16 < y < 7.9999999999999997e82

   1. Initial program 60.9%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]

   2. Taylor expanded in y around 0

      \[\leadsto x - \frac{\color{blue}{y \cdot \left(e^{z} - 1\right)}}{t} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x - \frac{y \cdot \color{blue}{\left(e^{z} - 1\right)}}{t} \]

      2. lower-expm1.f64 85.4

         \[\leadsto x - \frac{y \cdot \mathsf{expm1}\left(z\right)}{t} \]

   4. Applied rewrites 85.4%

      \[\leadsto x - \frac{\color{blue}{y \cdot \mathsf{expm1}\left(z\right)}}{t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 87.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{\log \left(\mathsf{fma}\left(z, y, 1\right)\right)}{t}\\ \mathbf{if}\;y \leq -7 \cdot 10^{+179}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+82}:\\ \;\;\;\;x - \frac{y \cdot \mathsf{expm1}\left(z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ (log (fma z y 1.0)) t))))
   (if (<= y -7e+179) t_1 (if (<= y 8e+82) (- x (/ (* y (expm1 z)) t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (log(fma(z, y, 1.0)) / t);
	double tmp;
	if (y <= -7e+179) {
		tmp = t_1;
	} else if (y <= 8e+82) {
		tmp = x - ((y * expm1(z)) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(log(fma(z, y, 1.0)) / t))
	tmp = 0.0
	if (y <= -7e+179)
		tmp = t_1;
	elseif (y <= 8e+82)
		tmp = Float64(x - Float64(Float64(y * expm1(z)) / t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(N[Log[N[(z * y + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7e+179], t$95$1, If[LessEqual[y, 8e+82], N[(x - N[(N[(y * N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.0000000000000003e179 or 7.9999999999999997e82 < y

   1. Initial program 60.9%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]

   2. Taylor expanded in z around 0

      \[\leadsto x - \frac{\log \color{blue}{\left(1 + y \cdot z\right)}}{t} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto x - \frac{\log \left(1 + \color{blue}{y \cdot z}\right)}{t} \]

      2. lower-*.f64 70.0

         \[\leadsto x - \frac{\log \left(1 + y \cdot \color{blue}{z}\right)}{t} \]

   4. Applied rewrites 70.0%

      \[\leadsto x - \frac{\log \color{blue}{\left(1 + y \cdot z\right)}}{t} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto x - \frac{\log \left(1 + \color{blue}{y \cdot z}\right)}{t} \]

      2. +-commutative N/A

         \[\leadsto x - \frac{\log \left(y \cdot z + \color{blue}{1}\right)}{t} \]

      3. lift-*.f64 N/A

         \[\leadsto x - \frac{\log \left(y \cdot z + 1\right)}{t} \]

      4. *-commutative N/A

         \[\leadsto x - \frac{\log \left(z \cdot y + 1\right)}{t} \]

      5. lower-fma.f64 70.0

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(z, \color{blue}{y}, 1\right)\right)}{t} \]

   6. Applied rewrites 70.0%

      \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(z, \color{blue}{y}, 1\right)\right)}{t} \]

   if -7.0000000000000003e179 < y < 7.9999999999999997e82

   1. Initial program 60.9%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]

   2. Taylor expanded in y around 0

      \[\leadsto x - \frac{\color{blue}{y \cdot \left(e^{z} - 1\right)}}{t} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x - \frac{y \cdot \color{blue}{\left(e^{z} - 1\right)}}{t} \]

      2. lower-expm1.f64 85.4

         \[\leadsto x - \frac{y \cdot \mathsf{expm1}\left(z\right)}{t} \]

   4. Applied rewrites 85.4%

      \[\leadsto x - \frac{\color{blue}{y \cdot \mathsf{expm1}\left(z\right)}}{t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 86.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+169}:\\ \;\;\;\;x - \frac{\log 1}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot \mathsf{expm1}\left(z\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -3.2e+169) (- x (/ (log 1.0) t)) (- x (/ (* y (expm1 z)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -3.2e+169) {
		tmp = x - (log(1.0) / t);
	} else {
		tmp = x - ((y * expm1(z)) / t);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -3.2e+169) {
		tmp = x - (Math.log(1.0) / t);
	} else {
		tmp = x - ((y * Math.expm1(z)) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -3.2e+169:
		tmp = x - (math.log(1.0) / t)
	else:
		tmp = x - ((y * math.expm1(z)) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -3.2e+169)
		tmp = Float64(x - Float64(log(1.0) / t));
	else
		tmp = Float64(x - Float64(Float64(y * expm1(z)) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -3.2e+169], N[(x - N[(N[Log[1.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(y * N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.1999999999999998e169

   1. Initial program 60.9%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]

   2. Taylor expanded in y around 0

      \[\leadsto x - \frac{\log \color{blue}{1}}{t} \]
   3. Step-by-step derivation

   4. Applied rewrites 71.1%

      \[\leadsto x - \frac{\log \color{blue}{1}}{t} \]

   if -3.1999999999999998e169 < y

   1. Initial program 60.9%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]

   2. Taylor expanded in y around 0

      \[\leadsto x - \frac{\color{blue}{y \cdot \left(e^{z} - 1\right)}}{t} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x - \frac{y \cdot \color{blue}{\left(e^{z} - 1\right)}}{t} \]

      2. lower-expm1.f64 85.4

         \[\leadsto x - \frac{y \cdot \mathsf{expm1}\left(z\right)}{t} \]

   4. Applied rewrites 85.4%

      \[\leadsto x - \frac{\color{blue}{y \cdot \mathsf{expm1}\left(z\right)}}{t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 80.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -40000000:\\ \;\;\;\;x - \frac{\log 1}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -40000000.0) (- x (/ (log 1.0) t)) (- x (/ (* y z) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -40000000.0) {
		tmp = x - (log(1.0) / t);
	} else {
		tmp = x - ((y * z) / t);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-40000000.0d0)) then
        tmp = x - (log(1.0d0) / t)
    else
        tmp = x - ((y * z) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -40000000.0) {
		tmp = x - (Math.log(1.0) / t);
	} else {
		tmp = x - ((y * z) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -40000000.0:
		tmp = x - (math.log(1.0) / t)
	else:
		tmp = x - ((y * z) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -40000000.0)
		tmp = Float64(x - Float64(log(1.0) / t));
	else
		tmp = Float64(x - Float64(Float64(y * z) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -40000000.0)
		tmp = x - (log(1.0) / t);
	else
		tmp = x - ((y * z) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -40000000.0], N[(x - N[(N[Log[1.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4e7

   1. Initial program 60.9%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]

   2. Taylor expanded in y around 0

      \[\leadsto x - \frac{\log \color{blue}{1}}{t} \]
   3. Step-by-step derivation

   4. Applied rewrites 71.1%

      \[\leadsto x - \frac{\log \color{blue}{1}}{t} \]

   if -4e7 < z

   1. Initial program 60.9%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]

   2. Taylor expanded in z around 0

      \[\leadsto x - \frac{\color{blue}{y \cdot z}}{t} \]
   3. Step-by-step derivation

      1. lower-*.f64 73.3

         \[\leadsto x - \frac{y \cdot \color{blue}{z}}{t} \]

   4. Applied rewrites 73.3%

      \[\leadsto x - \frac{\color{blue}{y \cdot z}}{t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 73.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ x - \frac{y \cdot z}{t} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- x (/ (* y z) t)))

C

double code(double x, double y, double z, double t) {
	return x - ((y * z) / t);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x - ((y * z) / t)
end function

Java

public static double code(double x, double y, double z, double t) {
	return x - ((y * z) / t);
}

Python

def code(x, y, z, t):
	return x - ((y * z) / t)

Julia

function code(x, y, z, t)
	return Float64(x - Float64(Float64(y * z) / t))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x - ((y * z) / t);
end

Wolfram

code[x_, y_, z_, t_] := N[(x - N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.9%

   \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]

2. Taylor expanded in z around 0

   \[\leadsto x - \frac{\color{blue}{y \cdot z}}{t} \]
3. Step-by-step derivation

   1. lower-*.f64 73.3

      \[\leadsto x - \frac{y \cdot \color{blue}{z}}{t} \]

4. Applied rewrites 73.3%

   \[\leadsto x - \frac{\color{blue}{y \cdot z}}{t} \]
5. Add Preprocessing

Alternative 7: 13.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ -z \cdot \frac{y}{t} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (* z (/ y t))))

C

double code(double x, double y, double z, double t) {
	return -(z * (y / t));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = -(z * (y / t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return -(z * (y / t));
}

Python

def code(x, y, z, t):
	return -(z * (y / t))

Julia

function code(x, y, z, t)
	return Float64(-Float64(z * Float64(y / t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = -(z * (y / t));
end

Wolfram

code[x_, y_, z_, t_] := (-N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision])

Derivation

1. Initial program 60.9%

   \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{t}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{t}} \]

   2. lower-/.f64 N/A

      \[\leadsto -1 \cdot \frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{\color{blue}{t}} \]

   3. lower-log.f64 N/A

      \[\leadsto -1 \cdot \frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{t} \]

   4. lower--.f64 N/A

      \[\leadsto -1 \cdot \frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{t} \]

   5. lower-+.f64 N/A

      \[\leadsto -1 \cdot \frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{t} \]

   6. lower-*.f64 N/A

      \[\leadsto -1 \cdot \frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{t} \]

   7. lower-exp.f64 8.4

      \[\leadsto -1 \cdot \frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{t} \]

4. Applied rewrites 8.4%

   \[\leadsto \color{blue}{-1 \cdot \frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{t}} \]

5. Taylor expanded in z around 0

   \[\leadsto -1 \cdot \color{blue}{\frac{y \cdot z}{t}} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto -1 \cdot \frac{y \cdot z}{\color{blue}{t}} \]

   2. lower-/.f64 N/A

      \[\leadsto -1 \cdot \frac{y \cdot z}{t} \]

   3. lower-*.f64 14.4

      \[\leadsto -1 \cdot \frac{y \cdot z}{t} \]

7. Applied rewrites 14.4%

   \[\leadsto -1 \cdot \color{blue}{\frac{y \cdot z}{t}} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto -1 \cdot \frac{y \cdot z}{\color{blue}{t}} \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{y \cdot z}{t}\right) \]

   3. lower-neg.f64 14.4

      \[\leadsto -\frac{y \cdot z}{t} \]

   4. lift-*.f64 N/A

      \[\leadsto -\frac{y \cdot z}{t} \]

   5. *-commutative N/A

      \[\leadsto -\frac{z \cdot y}{t} \]

   6. lower-*.f64 14.4

      \[\leadsto -\frac{z \cdot y}{t} \]

9. Applied rewrites 14.4%

   \[\leadsto \color{blue}{-\frac{z \cdot y}{t}} \]
10. Step-by-step derivation

    1. lift-/.f64 N/A

       \[\leadsto -\frac{z \cdot y}{t} \]

    2. lift-*.f64 N/A

       \[\leadsto -\frac{z \cdot y}{t} \]

    3. associate-/l* N/A

       \[\leadsto -z \cdot \frac{y}{t} \]

    4. lower-*.f64 N/A

       \[\leadsto -z \cdot \frac{y}{t} \]

    5. lower-/.f64 13.1

       \[\leadsto -z \cdot \frac{y}{t} \]

11. Applied rewrites 13.1%

    \[\leadsto -z \cdot \frac{y}{t} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025156 
(FPCore (x y z t)
  :name "System.Random.MWC.Distributions:truncatedExp from mwc-random-0.13.3.2"
  :precision binary64
  (- x (/ (log (+ (- 1.0 y) (* y (exp z)))) t)))