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

Percentage Accurate: 62.0% → 94.1%

Time: 9.9s

Alternatives: 8

Speedup: 226.0×

• could not determine a ground truth

  1. x = -5.750491798705331e+263
  2. y = 1.3709645790309053e-128
  3. z = 2.0322717092821274e+156
  4. t = 3.127484231333784e-166

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: 62.0% 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: 94.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+71} \lor \neg \left(t\_1 \leq 10^{-185}\right):\\ \;\;\;\;\frac{t \cdot x - \mathsf{log1p}\left(\mathsf{expm1}\left(z\right) \cdot y\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (log (+ (- 1.0 y) (* y (exp z)))) t)))
   (if (or (<= t_1 -2e+71) (not (<= t_1 1e-185)))
     (/ (- (* t x) (log1p (* (expm1 z) y))) t)
     (- x (* (/ (expm1 z) t) y)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(((1.0 - y) + (y * exp(z)))) / t;
	double tmp;
	if ((t_1 <= -2e+71) || !(t_1 <= 1e-185)) {
		tmp = ((t * x) - log1p((expm1(z) * y))) / t;
	} else {
		tmp = x - ((expm1(z) / t) * y);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(((1.0 - y) + (y * Math.exp(z)))) / t;
	double tmp;
	if ((t_1 <= -2e+71) || !(t_1 <= 1e-185)) {
		tmp = ((t * x) - Math.log1p((Math.expm1(z) * y))) / t;
	} else {
		tmp = x - ((Math.expm1(z) / t) * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(((1.0 - y) + (y * math.exp(z)))) / t
	tmp = 0
	if (t_1 <= -2e+71) or not (t_1 <= 1e-185):
		tmp = ((t * x) - math.log1p((math.expm1(z) * y))) / t
	else:
		tmp = x - ((math.expm1(z) / t) * y)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(Float64(Float64(1.0 - y) + Float64(y * exp(z)))) / t)
	tmp = 0.0
	if ((t_1 <= -2e+71) || !(t_1 <= 1e-185))
		tmp = Float64(Float64(Float64(t * x) - log1p(Float64(expm1(z) * y))) / t);
	else
		tmp = Float64(x - Float64(Float64(expm1(z) / t) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[N[(N[(1.0 - y), $MachinePrecision] + N[(y * N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+71], N[Not[LessEqual[t$95$1, 1e-185]], $MachinePrecision]], N[(N[(N[(t * x), $MachinePrecision] - N[Log[1 + N[(N[(Exp[z] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (log.f64 (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))) t) < -2.0000000000000001e71 or 9.9999999999999999e-186 < (/.f64 (log.f64 (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))) t)

   1. Initial program 30.8%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate--l+ N/A

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

      5. lower-log1p.f64 N/A

         \[\leadsto \frac{t \cdot x - \mathsf{log1p}\left(y \cdot e^{z} - y\right)}{t} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{t \cdot x - \mathsf{log1p}\left(y \cdot e^{z} - y\right)}{t} \]

      7. *-commutative N/A

         \[\leadsto \frac{t \cdot x - \mathsf{log1p}\left(e^{z} \cdot y - y\right)}{t} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{t \cdot x - \mathsf{log1p}\left(e^{z} \cdot y - y\right)}{t} \]

      9. lift-exp.f64 67.2

         \[\leadsto \frac{t \cdot x - \mathsf{log1p}\left(e^{z} \cdot y - y\right)}{t} \]

   5. Applied rewrites 67.2%

      \[\leadsto \color{blue}{\frac{t \cdot x - \mathsf{log1p}\left(e^{z} \cdot y - y\right)}{t}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{t \cdot x - \mathsf{log1p}\left(y \cdot \left(e^{z} - 1\right)\right)}{t} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{t \cdot x - \mathsf{log1p}\left(\left(e^{z} - 1\right) \cdot y\right)}{t} \]

      2. lift-expm1.f64 N/A

         \[\leadsto \frac{t \cdot x - \mathsf{log1p}\left(\mathsf{expm1}\left(z\right) \cdot y\right)}{t} \]

      3. lift-*.f64 95.0

         \[\leadsto \frac{t \cdot x - \mathsf{log1p}\left(\mathsf{expm1}\left(z\right) \cdot y\right)}{t} \]

   8. Applied rewrites 95.0%

      \[\leadsto \frac{t \cdot x - \mathsf{log1p}\left(\mathsf{expm1}\left(z\right) \cdot y\right)}{t} \]

   if -2.0000000000000001e71 < (/.f64 (log.f64 (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))) t) < 9.9999999999999999e-186

   1. Initial program 87.5%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

         \[\leadsto x - \left(\frac{-1}{2} \cdot \frac{y \cdot {\left(e^{z} - 1\right)}^{2}}{t} + \left(\frac{e^{z}}{t} - \frac{1}{t}\right)\right) \cdot y \]

      4. lower-fma.f64 N/A

         \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{y \cdot {\left(e^{z} - 1\right)}^{2}}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

      5. lower-/.f64 N/A

         \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{y \cdot {\left(e^{z} - 1\right)}^{2}}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

      6. *-commutative N/A

         \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(e^{z} - 1\right)}^{2} \cdot y}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

      7. lower-*.f64 N/A

         \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(e^{z} - 1\right)}^{2} \cdot y}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

      8. lower-pow.f64 N/A

         \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(e^{z} - 1\right)}^{2} \cdot y}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

      9. lower-expm1.f64 N/A

         \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

      10. sub-div N/A

          \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{e^{z} - 1}{t}\right) \cdot y \]

      11. lower-/.f64 N/A

          \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{e^{z} - 1}{t}\right) \cdot y \]

      12. lower-expm1.f64 98.1

          \[\leadsto x - \mathsf{fma}\left(-0.5, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{\mathsf{expm1}\left(z\right)}{t}\right) \cdot y \]

   5. Applied rewrites 98.1%

      \[\leadsto x - \color{blue}{\mathsf{fma}\left(-0.5, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{\mathsf{expm1}\left(z\right)}{t}\right) \cdot y} \]

   6. Taylor expanded in y around 0

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

      1. sub-div N/A

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

      2. lift-expm1.f64 N/A

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

      3. lift-/.f64 100.0

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

   8. Applied rewrites 100.0%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \leq -2 \cdot 10^{+71} \lor \neg \left(\frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \leq 10^{-185}\right):\\ \;\;\;\;\frac{t \cdot x - \mathsf{log1p}\left(\mathsf{expm1}\left(z\right) \cdot y\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(\left(1 - y\right) + y \cdot e^{z}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{t \cdot x - \mathsf{log1p}\left(z \cdot y\right)}{t}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (log (+ (- 1.0 y) (* y (exp z))))))
   (if (<= t_1 (- INFINITY))
     (/ (- (* t x) (log1p (* z y))) t)
     (if (<= t_1 0.0)
       (- x (* (/ (expm1 z) t) y))
       (- x (/ (log (fma (expm1 z) y 1.0)) t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(((1.0 - y) + (y * exp(z))));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((t * x) - log1p((z * y))) / t;
	} else if (t_1 <= 0.0) {
		tmp = x - ((expm1(z) / t) * y);
	} else {
		tmp = x - (log(fma(expm1(z), y, 1.0)) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = log(Float64(Float64(1.0 - y) + Float64(y * exp(z))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(t * x) - log1p(Float64(z * y))) / t);
	elseif (t_1 <= 0.0)
		tmp = Float64(x - Float64(Float64(expm1(z) / t) * y));
	else
		tmp = Float64(x - Float64(log(fma(expm1(z), y, 1.0)) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Log[N[(N[(1.0 - y), $MachinePrecision] + N[(y * N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(t * x), $MachinePrecision] - N[Log[1 + N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[N[(N[(Exp[z] - 1), $MachinePrecision] * y + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (log.f64 (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))) < -inf.0

   1. Initial program 2.0%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate--l+ N/A

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

      5. lower-log1p.f64 N/A

         \[\leadsto \frac{t \cdot x - \mathsf{log1p}\left(y \cdot e^{z} - y\right)}{t} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{t \cdot x - \mathsf{log1p}\left(y \cdot e^{z} - y\right)}{t} \]

      7. *-commutative N/A

         \[\leadsto \frac{t \cdot x - \mathsf{log1p}\left(e^{z} \cdot y - y\right)}{t} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{t \cdot x - \mathsf{log1p}\left(e^{z} \cdot y - y\right)}{t} \]

      9. lift-exp.f64 52.8

         \[\leadsto \frac{t \cdot x - \mathsf{log1p}\left(e^{z} \cdot y - y\right)}{t} \]

   5. Applied rewrites 52.8%

      \[\leadsto \color{blue}{\frac{t \cdot x - \mathsf{log1p}\left(e^{z} \cdot y - y\right)}{t}} \]

   6. Taylor expanded in z around 0

      \[\leadsto \frac{t \cdot x - \mathsf{log1p}\left(y \cdot z\right)}{t} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{t \cdot x - \mathsf{log1p}\left(z \cdot y\right)}{t} \]

      2. lift-*.f64 92.9

         \[\leadsto \frac{t \cdot x - \mathsf{log1p}\left(z \cdot y\right)}{t} \]

   8. Applied rewrites 92.9%

      \[\leadsto \frac{t \cdot x - \mathsf{log1p}\left(z \cdot y\right)}{t} \]

   if -inf.0 < (log.f64 (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))) < 0.0

   1. Initial program 86.2%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

         \[\leadsto x - \left(\frac{-1}{2} \cdot \frac{y \cdot {\left(e^{z} - 1\right)}^{2}}{t} + \left(\frac{e^{z}}{t} - \frac{1}{t}\right)\right) \cdot y \]

      4. lower-fma.f64 N/A

         \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{y \cdot {\left(e^{z} - 1\right)}^{2}}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

      5. lower-/.f64 N/A

         \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{y \cdot {\left(e^{z} - 1\right)}^{2}}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

      6. *-commutative N/A

         \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(e^{z} - 1\right)}^{2} \cdot y}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

      7. lower-*.f64 N/A

         \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(e^{z} - 1\right)}^{2} \cdot y}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

      8. lower-pow.f64 N/A

         \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(e^{z} - 1\right)}^{2} \cdot y}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

      9. lower-expm1.f64 N/A

         \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

      10. sub-div N/A

          \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{e^{z} - 1}{t}\right) \cdot y \]

      11. lower-/.f64 N/A

          \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{e^{z} - 1}{t}\right) \cdot y \]

      12. lower-expm1.f64 99.9

          \[\leadsto x - \mathsf{fma}\left(-0.5, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{\mathsf{expm1}\left(z\right)}{t}\right) \cdot y \]

   5. Applied rewrites 99.9%

      \[\leadsto x - \color{blue}{\mathsf{fma}\left(-0.5, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{\mathsf{expm1}\left(z\right)}{t}\right) \cdot y} \]

   6. Taylor expanded in y around 0

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

      1. sub-div N/A

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

      2. lift-expm1.f64 N/A

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

      3. lift-/.f64 99.5

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

   8. Applied rewrites 99.5%

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

   if 0.0 < (log.f64 (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z))))

   1. Initial program 98.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-expm1.f64 99.1

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

   5. Applied rewrites 99.1%

      \[\leadsto x - \frac{\log \color{blue}{\left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)\right)}}{t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 96.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(\left(1 - y\right) + y \cdot e^{z}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{t \cdot x - \mathsf{log1p}\left(z \cdot y\right)}{t}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{expm1}\left(z\right) \cdot y\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (log (+ (- 1.0 y) (* y (exp z))))))
   (if (<= t_1 (- INFINITY))
     (/ (- (* t x) (log1p (* z y))) t)
     (if (<= t_1 5e-6)
       (- x (* (/ (expm1 z) t) y))
       (- x (/ (log (* (expm1 z) y)) t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(((1.0 - y) + (y * exp(z))));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((t * x) - log1p((z * y))) / t;
	} else if (t_1 <= 5e-6) {
		tmp = x - ((expm1(z) / t) * y);
	} else {
		tmp = x - (log((expm1(z) * y)) / t);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(((1.0 - y) + (y * Math.exp(z))));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = ((t * x) - Math.log1p((z * y))) / t;
	} else if (t_1 <= 5e-6) {
		tmp = x - ((Math.expm1(z) / t) * y);
	} else {
		tmp = x - (Math.log((Math.expm1(z) * y)) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(((1.0 - y) + (y * math.exp(z))))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = ((t * x) - math.log1p((z * y))) / t
	elif t_1 <= 5e-6:
		tmp = x - ((math.expm1(z) / t) * y)
	else:
		tmp = x - (math.log((math.expm1(z) * y)) / t)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = log(Float64(Float64(1.0 - y) + Float64(y * exp(z))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(t * x) - log1p(Float64(z * y))) / t);
	elseif (t_1 <= 5e-6)
		tmp = Float64(x - Float64(Float64(expm1(z) / t) * y));
	else
		tmp = Float64(x - Float64(log(Float64(expm1(z) * y)) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Log[N[(N[(1.0 - y), $MachinePrecision] + N[(y * N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(t * x), $MachinePrecision] - N[Log[1 + N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[t$95$1, 5e-6], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[N[(N[(Exp[z] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (log.f64 (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))) < -inf.0

   1. Initial program 2.0%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate--l+ N/A

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

      5. lower-log1p.f64 N/A

         \[\leadsto \frac{t \cdot x - \mathsf{log1p}\left(y \cdot e^{z} - y\right)}{t} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{t \cdot x - \mathsf{log1p}\left(y \cdot e^{z} - y\right)}{t} \]

      7. *-commutative N/A

         \[\leadsto \frac{t \cdot x - \mathsf{log1p}\left(e^{z} \cdot y - y\right)}{t} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{t \cdot x - \mathsf{log1p}\left(e^{z} \cdot y - y\right)}{t} \]

      9. lift-exp.f64 52.8

         \[\leadsto \frac{t \cdot x - \mathsf{log1p}\left(e^{z} \cdot y - y\right)}{t} \]

   5. Applied rewrites 52.8%

      \[\leadsto \color{blue}{\frac{t \cdot x - \mathsf{log1p}\left(e^{z} \cdot y - y\right)}{t}} \]

   6. Taylor expanded in z around 0

      \[\leadsto \frac{t \cdot x - \mathsf{log1p}\left(y \cdot z\right)}{t} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{t \cdot x - \mathsf{log1p}\left(z \cdot y\right)}{t} \]

      2. lift-*.f64 92.9

         \[\leadsto \frac{t \cdot x - \mathsf{log1p}\left(z \cdot y\right)}{t} \]

   8. Applied rewrites 92.9%

      \[\leadsto \frac{t \cdot x - \mathsf{log1p}\left(z \cdot y\right)}{t} \]

   if -inf.0 < (log.f64 (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))) < 5.00000000000000041e-6

   1. Initial program 86.3%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

         \[\leadsto x - \left(\frac{-1}{2} \cdot \frac{y \cdot {\left(e^{z} - 1\right)}^{2}}{t} + \left(\frac{e^{z}}{t} - \frac{1}{t}\right)\right) \cdot y \]

      4. lower-fma.f64 N/A

         \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{y \cdot {\left(e^{z} - 1\right)}^{2}}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

      5. lower-/.f64 N/A

         \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{y \cdot {\left(e^{z} - 1\right)}^{2}}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

      6. *-commutative N/A

         \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(e^{z} - 1\right)}^{2} \cdot y}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

      7. lower-*.f64 N/A

         \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(e^{z} - 1\right)}^{2} \cdot y}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

      8. lower-pow.f64 N/A

         \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(e^{z} - 1\right)}^{2} \cdot y}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

      9. lower-expm1.f64 N/A

         \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

      10. sub-div N/A

          \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{e^{z} - 1}{t}\right) \cdot y \]

      11. lower-/.f64 N/A

          \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{e^{z} - 1}{t}\right) \cdot y \]

      12. lower-expm1.f64 99.7

          \[\leadsto x - \mathsf{fma}\left(-0.5, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{\mathsf{expm1}\left(z\right)}{t}\right) \cdot y \]

   5. Applied rewrites 99.7%

      \[\leadsto x - \color{blue}{\mathsf{fma}\left(-0.5, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{\mathsf{expm1}\left(z\right)}{t}\right) \cdot y} \]

   6. Taylor expanded in y around 0

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

      1. sub-div N/A

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

      2. lift-expm1.f64 N/A

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

      3. lift-/.f64 99.1

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

   8. Applied rewrites 99.1%

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

   if 5.00000000000000041e-6 < (log.f64 (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z))))

   1. Initial program 98.6%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 99.7

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

   5. Applied rewrites 99.7%

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

Alternative 4: 89.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(\left(1 - y\right) + y \cdot e^{z}\right) \leq -\infty:\\ \;\;\;\;\frac{t \cdot x - \mathsf{log1p}\left(z \cdot y\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (log (+ (- 1.0 y) (* y (exp z)))) (- INFINITY))
   (/ (- (* t x) (log1p (* z y))) t)
   (- x (* (/ (expm1 z) t) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (log(((1.0 - y) + (y * exp(z)))) <= -((double) INFINITY)) {
		tmp = ((t * x) - log1p((z * y))) / t;
	} else {
		tmp = x - ((expm1(z) / t) * y);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (Math.log(((1.0 - y) + (y * Math.exp(z)))) <= -Double.POSITIVE_INFINITY) {
		tmp = ((t * x) - Math.log1p((z * y))) / t;
	} else {
		tmp = x - ((Math.expm1(z) / t) * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if math.log(((1.0 - y) + (y * math.exp(z)))) <= -math.inf:
		tmp = ((t * x) - math.log1p((z * y))) / t
	else:
		tmp = x - ((math.expm1(z) / t) * y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (log(Float64(Float64(1.0 - y) + Float64(y * exp(z)))) <= Float64(-Inf))
		tmp = Float64(Float64(Float64(t * x) - log1p(Float64(z * y))) / t);
	else
		tmp = Float64(x - Float64(Float64(expm1(z) / t) * y));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (log.f64 (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))) < -inf.0

   1. Initial program 2.0%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate--l+ N/A

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

      5. lower-log1p.f64 N/A

         \[\leadsto \frac{t \cdot x - \mathsf{log1p}\left(y \cdot e^{z} - y\right)}{t} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{t \cdot x - \mathsf{log1p}\left(y \cdot e^{z} - y\right)}{t} \]

      7. *-commutative N/A

         \[\leadsto \frac{t \cdot x - \mathsf{log1p}\left(e^{z} \cdot y - y\right)}{t} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{t \cdot x - \mathsf{log1p}\left(e^{z} \cdot y - y\right)}{t} \]

      9. lift-exp.f64 52.8

         \[\leadsto \frac{t \cdot x - \mathsf{log1p}\left(e^{z} \cdot y - y\right)}{t} \]

   5. Applied rewrites 52.8%

      \[\leadsto \color{blue}{\frac{t \cdot x - \mathsf{log1p}\left(e^{z} \cdot y - y\right)}{t}} \]

   6. Taylor expanded in z around 0

      \[\leadsto \frac{t \cdot x - \mathsf{log1p}\left(y \cdot z\right)}{t} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{t \cdot x - \mathsf{log1p}\left(z \cdot y\right)}{t} \]

      2. lift-*.f64 92.9

         \[\leadsto \frac{t \cdot x - \mathsf{log1p}\left(z \cdot y\right)}{t} \]

   8. Applied rewrites 92.9%

      \[\leadsto \frac{t \cdot x - \mathsf{log1p}\left(z \cdot y\right)}{t} \]

   if -inf.0 < (log.f64 (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z))))

   1. Initial program 88.4%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

         \[\leadsto x - \left(\frac{-1}{2} \cdot \frac{y \cdot {\left(e^{z} - 1\right)}^{2}}{t} + \left(\frac{e^{z}}{t} - \frac{1}{t}\right)\right) \cdot y \]

      4. lower-fma.f64 N/A

         \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{y \cdot {\left(e^{z} - 1\right)}^{2}}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

      5. lower-/.f64 N/A

         \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{y \cdot {\left(e^{z} - 1\right)}^{2}}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

      6. *-commutative N/A

         \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(e^{z} - 1\right)}^{2} \cdot y}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

      7. lower-*.f64 N/A

         \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(e^{z} - 1\right)}^{2} \cdot y}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

      8. lower-pow.f64 N/A

         \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(e^{z} - 1\right)}^{2} \cdot y}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

      9. lower-expm1.f64 N/A

         \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

      10. sub-div N/A

          \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{e^{z} - 1}{t}\right) \cdot y \]

      11. lower-/.f64 N/A

          \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{e^{z} - 1}{t}\right) \cdot y \]

      12. lower-expm1.f64 85.7

          \[\leadsto x - \mathsf{fma}\left(-0.5, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{\mathsf{expm1}\left(z\right)}{t}\right) \cdot y \]

   5. Applied rewrites 85.7%

      \[\leadsto x - \color{blue}{\mathsf{fma}\left(-0.5, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{\mathsf{expm1}\left(z\right)}{t}\right) \cdot y} \]

   6. Taylor expanded in y around 0

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

      1. sub-div N/A

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

      2. lift-expm1.f64 N/A

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

      3. lift-/.f64 87.6

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

   8. Applied rewrites 87.6%

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

Alternative 5: 86.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+195}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(z, y, 1\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -5.5e+195)
   (- x (/ (log (fma z y 1.0)) t))
   (- x (* (/ (expm1 z) t) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -5.5e+195) {
		tmp = x - (log(fma(z, y, 1.0)) / t);
	} else {
		tmp = x - ((expm1(z) / t) * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -5.5e+195)
		tmp = Float64(x - Float64(log(fma(z, y, 1.0)) / t));
	else
		tmp = Float64(x - Float64(Float64(expm1(z) / t) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -5.5e+195], N[(x - N[(N[Log[N[(z * y + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.49999999999999994e195

   1. Initial program 57.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 56.5

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

   5. Applied rewrites 56.5%

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

   if -5.49999999999999994e195 < y

   1. Initial program 66.6%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

         \[\leadsto x - \left(\frac{-1}{2} \cdot \frac{y \cdot {\left(e^{z} - 1\right)}^{2}}{t} + \left(\frac{e^{z}}{t} - \frac{1}{t}\right)\right) \cdot y \]

      4. lower-fma.f64 N/A

         \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{y \cdot {\left(e^{z} - 1\right)}^{2}}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

      5. lower-/.f64 N/A

         \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{y \cdot {\left(e^{z} - 1\right)}^{2}}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

      6. *-commutative N/A

         \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(e^{z} - 1\right)}^{2} \cdot y}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

      7. lower-*.f64 N/A

         \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(e^{z} - 1\right)}^{2} \cdot y}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

      8. lower-pow.f64 N/A

         \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(e^{z} - 1\right)}^{2} \cdot y}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

      9. lower-expm1.f64 N/A

         \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

      10. sub-div N/A

          \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{e^{z} - 1}{t}\right) \cdot y \]

      11. lower-/.f64 N/A

          \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{e^{z} - 1}{t}\right) \cdot y \]

      12. lower-expm1.f64 89.1

          \[\leadsto x - \mathsf{fma}\left(-0.5, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{\mathsf{expm1}\left(z\right)}{t}\right) \cdot y \]

   5. Applied rewrites 89.1%

      \[\leadsto x - \color{blue}{\mathsf{fma}\left(-0.5, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{\mathsf{expm1}\left(z\right)}{t}\right) \cdot y} \]

   6. Taylor expanded in y around 0

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

      1. sub-div N/A

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

      2. lift-expm1.f64 N/A

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

      3. lift-/.f64 91.8

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

   8. Applied rewrites 91.8%

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

Alternative 6: 85.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- x (* (/ (expm1 z) t) y)))

C

double code(double x, double y, double z, double t) {
	return x - ((expm1(z) / t) * y);
}

Java

public static double code(double x, double y, double z, double t) {
	return x - ((Math.expm1(z) / t) * y);
}

Python

def code(x, y, z, t):
	return x - ((math.expm1(z) / t) * y)

Julia

function code(x, y, z, t)
	return Float64(x - Float64(Float64(expm1(z) / t) * y))
end

Wolfram

code[x_, y_, z_, t_] := N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.8%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. associate--l+ N/A

      \[\leadsto x - \left(\frac{-1}{2} \cdot \frac{y \cdot {\left(e^{z} - 1\right)}^{2}}{t} + \left(\frac{e^{z}}{t} - \frac{1}{t}\right)\right) \cdot y \]

   4. lower-fma.f64 N/A

      \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{y \cdot {\left(e^{z} - 1\right)}^{2}}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

   5. lower-/.f64 N/A

      \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{y \cdot {\left(e^{z} - 1\right)}^{2}}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

   6. *-commutative N/A

      \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(e^{z} - 1\right)}^{2} \cdot y}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

   7. lower-*.f64 N/A

      \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(e^{z} - 1\right)}^{2} \cdot y}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

   8. lower-pow.f64 N/A

      \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(e^{z} - 1\right)}^{2} \cdot y}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

   9. lower-expm1.f64 N/A

      \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

   10. sub-div N/A

       \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{e^{z} - 1}{t}\right) \cdot y \]

   11. lower-/.f64 N/A

       \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{e^{z} - 1}{t}\right) \cdot y \]

   12. lower-expm1.f64 83.6

       \[\leadsto x - \mathsf{fma}\left(-0.5, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{\mathsf{expm1}\left(z\right)}{t}\right) \cdot y \]

5. Applied rewrites 83.6%

   \[\leadsto x - \color{blue}{\mathsf{fma}\left(-0.5, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{\mathsf{expm1}\left(z\right)}{t}\right) \cdot y} \]

6. Taylor expanded in y around 0

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

   1. sub-div N/A

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

   2. lift-expm1.f64 N/A

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

   3. lift-/.f64 86.7

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

8. Applied rewrites 86.7%

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

Alternative 7: 81.7% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2400:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z}{t} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2400.0) x (- x (* (/ z t) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2400.0) {
		tmp = x;
	} else {
		tmp = x - ((z / t) * y);
	}
	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 <= (-2400.0d0)) then
        tmp = x
    else
        tmp = x - ((z / t) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2400.0) {
		tmp = x;
	} else {
		tmp = x - ((z / t) * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -2400.0:
		tmp = x
	else:
		tmp = x - ((z / t) * y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2400.0)
		tmp = x;
	else
		tmp = Float64(x - Float64(Float64(z / t) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -2400.0)
		tmp = x;
	else
		tmp = x - ((z / t) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2400.0], x, N[(x - N[(N[(z / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2400

   1. Initial program 88.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Applied rewrites 67.4%

      \[\leadsto \color{blue}{x} \]

   if -2400 < z

   1. Initial program 53.6%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

         \[\leadsto x - \left(\frac{-1}{2} \cdot \frac{y \cdot {\left(e^{z} - 1\right)}^{2}}{t} + \left(\frac{e^{z}}{t} - \frac{1}{t}\right)\right) \cdot y \]

      4. lower-fma.f64 N/A

         \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{y \cdot {\left(e^{z} - 1\right)}^{2}}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

      5. lower-/.f64 N/A

         \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{y \cdot {\left(e^{z} - 1\right)}^{2}}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

      6. *-commutative N/A

         \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(e^{z} - 1\right)}^{2} \cdot y}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

      7. lower-*.f64 N/A

         \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(e^{z} - 1\right)}^{2} \cdot y}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

      8. lower-pow.f64 N/A

         \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(e^{z} - 1\right)}^{2} \cdot y}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

      9. lower-expm1.f64 N/A

         \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

      10. sub-div N/A

          \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{e^{z} - 1}{t}\right) \cdot y \]

      11. lower-/.f64 N/A

          \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{e^{z} - 1}{t}\right) \cdot y \]

      12. lower-expm1.f64 89.8

          \[\leadsto x - \mathsf{fma}\left(-0.5, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{\mathsf{expm1}\left(z\right)}{t}\right) \cdot y \]

   5. Applied rewrites 89.8%

      \[\leadsto x - \color{blue}{\mathsf{fma}\left(-0.5, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{\mathsf{expm1}\left(z\right)}{t}\right) \cdot y} \]

   6. Taylor expanded in z around 0

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

      1. lower-/.f64 92.4

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

   8. Applied rewrites 92.4%

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

Alternative 8: 70.3% accurate, 226.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 x)

C

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

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
end function

Java

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

Python

def code(x, y, z, t):
	return x

Julia

function code(x, y, z, t)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := x

Derivation

1. Initial program 65.8%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Applied rewrites 72.9%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Developer Target 1: 73.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-0.5}{y \cdot t}\\ \mathbf{if}\;z < -2.8874623088207947 \cdot 10^{+119}:\\ \;\;\;\;\left(x - \frac{t\_1}{z \cdot z}\right) - t\_1 \cdot \frac{\frac{2}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot y\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- 0.5) (* y t))))
   (if (< z -2.8874623088207947e+119)
     (- (- x (/ t_1 (* z z))) (* t_1 (/ (/ 2.0 z) (* z z))))
     (- x (/ (log (+ 1.0 (* z y))) t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -0.5 / (y * t);
	double tmp;
	if (z < -2.8874623088207947e+119) {
		tmp = (x - (t_1 / (z * z))) - (t_1 * ((2.0 / z) / (z * z)));
	} else {
		tmp = x - (log((1.0 + (z * y))) / 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) :: t_1
    real(8) :: tmp
    t_1 = -0.5d0 / (y * t)
    if (z < (-2.8874623088207947d+119)) then
        tmp = (x - (t_1 / (z * z))) - (t_1 * ((2.0d0 / z) / (z * z)))
    else
        tmp = x - (log((1.0d0 + (z * y))) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = -0.5 / (y * t);
	double tmp;
	if (z < -2.8874623088207947e+119) {
		tmp = (x - (t_1 / (z * z))) - (t_1 * ((2.0 / z) / (z * z)));
	} else {
		tmp = x - (Math.log((1.0 + (z * y))) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = -0.5 / (y * t)
	tmp = 0
	if z < -2.8874623088207947e+119:
		tmp = (x - (t_1 / (z * z))) - (t_1 * ((2.0 / z) / (z * z)))
	else:
		tmp = x - (math.log((1.0 + (z * y))) / t)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(-0.5) / Float64(y * t))
	tmp = 0.0
	if (z < -2.8874623088207947e+119)
		tmp = Float64(Float64(x - Float64(t_1 / Float64(z * z))) - Float64(t_1 * Float64(Float64(2.0 / z) / Float64(z * z))));
	else
		tmp = Float64(x - Float64(log(Float64(1.0 + Float64(z * y))) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = -0.5 / (y * t);
	tmp = 0.0;
	if (z < -2.8874623088207947e+119)
		tmp = (x - (t_1 / (z * z))) - (t_1 * ((2.0 / z) / (z * z)));
	else
		tmp = x - (log((1.0 + (z * y))) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[((-0.5) / N[(y * t), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -2.8874623088207947e+119], N[(N[(x - N[(t$95$1 / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[(N[(2.0 / z), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[N[(1.0 + N[(z * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2025064 
(FPCore (x y z t)
  :name "System.Random.MWC.Distributions:truncatedExp from mwc-random-0.13.3.2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -288746230882079470000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (- x (/ (/ (- 1/2) (* y t)) (* z z))) (* (/ (- 1/2) (* y t)) (/ (/ 2 z) (* z z)))) (- x (/ (log (+ 1 (* z y))) t))))

  (- x (/ (log (+ (- 1.0 y) (* y (exp z)))) t)))