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

Percentage Accurate: 60.6% → 89.8%

Time: 5.7s

Alternatives: 13

Speedup: 37.7×

• could not determine a ground truth

  1. x = 2.560560322249414e-286
  2. y = 2.6280768679968732e-210
  3. z = 3.4544562976955696e+232
  4. t = 3.2169384387712626e+237

Specification

Math

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

FPCore

(FPCore (x y z t)
  :precision binary64
  (- x (/ (log (+ (- 1 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 - y), $MachinePrecision] + N[(y * N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Initial Program: 60.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t)
  :precision binary64
  (- x (/ (log (+ (- 1 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 - y), $MachinePrecision] + N[(y * N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Alternative 1: 89.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -22500000:\\ \;\;\;\;x - \frac{1}{\frac{1}{2} \cdot t + \frac{t}{\left(e^{z} - 1\right) \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{t + z \cdot \left(\frac{1}{2} \cdot \left(t \cdot y\right) - \frac{1}{2} \cdot t\right)}{z}}{y}}\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (if (<= z -22500000)
  (- x (/ 1 (+ (* 1/2 t) (/ t (* (- (exp z) 1) y)))))
  (- x (/ 1 (/ (/ (+ t (* z (- (* 1/2 (* t y)) (* 1/2 t)))) z) y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -22500000.0) {
		tmp = x - (1.0 / ((0.5 * t) + (t / ((exp(z) - 1.0) * y))));
	} else {
		tmp = x - (1.0 / (((t + (z * ((0.5 * (t * y)) - (0.5 * t)))) / z) / 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 <= (-22500000.0d0)) then
        tmp = x - (1.0d0 / ((0.5d0 * t) + (t / ((exp(z) - 1.0d0) * y))))
    else
        tmp = x - (1.0d0 / (((t + (z * ((0.5d0 * (t * y)) - (0.5d0 * t)))) / z) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -22500000.0) {
		tmp = x - (1.0 / ((0.5 * t) + (t / ((Math.exp(z) - 1.0) * y))));
	} else {
		tmp = x - (1.0 / (((t + (z * ((0.5 * (t * y)) - (0.5 * t)))) / z) / y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -22500000.0:
		tmp = x - (1.0 / ((0.5 * t) + (t / ((math.exp(z) - 1.0) * y))))
	else:
		tmp = x - (1.0 / (((t + (z * ((0.5 * (t * y)) - (0.5 * t)))) / z) / y))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -22500000.0)
		tmp = Float64(x - Float64(1.0 / Float64(Float64(0.5 * t) + Float64(t / Float64(Float64(exp(z) - 1.0) * y)))));
	else
		tmp = Float64(x - Float64(1.0 / Float64(Float64(Float64(t + Float64(z * Float64(Float64(0.5 * Float64(t * y)) - Float64(0.5 * t)))) / z) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -22500000.0)
		tmp = x - (1.0 / ((0.5 * t) + (t / ((exp(z) - 1.0) * y))));
	else
		tmp = x - (1.0 / (((t + (z * ((0.5 * (t * y)) - (0.5 * t)))) / z) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -22500000], N[(x - N[(1 / N[(N[(1/2 * t), $MachinePrecision] + N[(t / N[(N[(N[Exp[z], $MachinePrecision] - 1), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(1 / N[(N[(N[(t + N[(z * N[(N[(1/2 * N[(t * y), $MachinePrecision]), $MachinePrecision] - N[(1/2 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.25e7

   1. Initial program 60.6%

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

   2. Taylor expanded in z around 0

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

      1. lower-*.f64 73.1%

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

   4. Applied rewrites 73.1%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 73.1%

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 73.1%

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

   6. Applied rewrites 73.1%

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

   7. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-exp.f64 75.9%

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

   9. Applied rewrites 75.9%

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

       1. lift-/.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*r* N/A

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

       6. add-to-fraction-rev N/A

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

       7. lower-+.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lift-/.f64 N/A

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

       10. associate-/l/ N/A

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

       11. *-commutative N/A

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

       12. lift--.f64 N/A

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

       13. lift-exp.f64 N/A

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

       14. lower-/.f64 N/A

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

       15. lift-exp.f64 N/A

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

       16. lift--.f64 N/A

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

       17. *-commutative N/A

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

       18. lower-*.f64 78.9%

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

   11. Applied rewrites 78.9%

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

   if -2.25e7 < z

   1. Initial program 60.6%

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

   2. Taylor expanded in z around 0

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

      1. lower-*.f64 73.1%

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

   4. Applied rewrites 73.1%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 73.1%

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 73.1%

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

   6. Applied rewrites 73.1%

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

   7. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-exp.f64 75.9%

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

   9. Applied rewrites 75.9%

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

   10. Taylor expanded in z around 0

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower--.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 84.9%

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

   12. Applied rewrites 84.9%

       \[\leadsto x - \frac{1}{\frac{\frac{t + z \cdot \left(\frac{1}{2} \cdot \left(t \cdot y\right) - \frac{1}{2} \cdot t\right)}{z}}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 87.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -410000000:\\ \;\;\;\;x - \frac{y \cdot \left(e^{z} - 1\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{t + z \cdot \left(\frac{1}{2} \cdot \left(t \cdot y\right) - \frac{1}{2} \cdot t\right)}{z}}{y}}\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (if (<= z -410000000)
  (- x (/ (* y (- (exp z) 1)) t))
  (- x (/ 1 (/ (/ (+ t (* z (- (* 1/2 (* t y)) (* 1/2 t)))) z) y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -410000000.0) {
		tmp = x - ((y * (exp(z) - 1.0)) / t);
	} else {
		tmp = x - (1.0 / (((t + (z * ((0.5 * (t * y)) - (0.5 * t)))) / z) / 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 <= (-410000000.0d0)) then
        tmp = x - ((y * (exp(z) - 1.0d0)) / t)
    else
        tmp = x - (1.0d0 / (((t + (z * ((0.5d0 * (t * y)) - (0.5d0 * t)))) / z) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -410000000.0) {
		tmp = x - ((y * (Math.exp(z) - 1.0)) / t);
	} else {
		tmp = x - (1.0 / (((t + (z * ((0.5 * (t * y)) - (0.5 * t)))) / z) / y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -410000000.0:
		tmp = x - ((y * (math.exp(z) - 1.0)) / t)
	else:
		tmp = x - (1.0 / (((t + (z * ((0.5 * (t * y)) - (0.5 * t)))) / z) / y))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -410000000.0)
		tmp = Float64(x - Float64(Float64(y * Float64(exp(z) - 1.0)) / t));
	else
		tmp = Float64(x - Float64(1.0 / Float64(Float64(Float64(t + Float64(z * Float64(Float64(0.5 * Float64(t * y)) - Float64(0.5 * t)))) / z) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -410000000.0)
		tmp = x - ((y * (exp(z) - 1.0)) / t);
	else
		tmp = x - (1.0 / (((t + (z * ((0.5 * (t * y)) - (0.5 * t)))) / z) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -410000000], N[(x - N[(N[(y * N[(N[Exp[z], $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(1 / N[(N[(N[(t + N[(z * N[(N[(1/2 * N[(t * y), $MachinePrecision]), $MachinePrecision] - N[(1/2 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.1e8

   1. Initial program 60.6%

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

   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--.f64 N/A

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

      3. lower-exp.f64 75.9%

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

   4. Applied rewrites 75.9%

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

   if -4.1e8 < z

   1. Initial program 60.6%

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

   2. Taylor expanded in z around 0

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

      1. lower-*.f64 73.1%

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

   4. Applied rewrites 73.1%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 73.1%

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 73.1%

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

   6. Applied rewrites 73.1%

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

   7. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-exp.f64 75.9%

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

   9. Applied rewrites 75.9%

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

   10. Taylor expanded in z around 0

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower--.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 84.9%

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

   12. Applied rewrites 84.9%

       \[\leadsto x - \frac{1}{\frac{\frac{t + z \cdot \left(\frac{1}{2} \cdot \left(t \cdot y\right) - \frac{1}{2} \cdot t\right)}{z}}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 87.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -410000000:\\ \;\;\;\;x - \frac{e^{z} - 1}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{t + z \cdot \left(\frac{1}{2} \cdot \left(t \cdot y\right) - \frac{1}{2} \cdot t\right)}{z}}{y}}\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (if (<= z -410000000)
  (- x (* (/ (- (exp z) 1) t) y))
  (- x (/ 1 (/ (/ (+ t (* z (- (* 1/2 (* t y)) (* 1/2 t)))) z) y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -410000000.0) {
		tmp = x - (((exp(z) - 1.0) / t) * y);
	} else {
		tmp = x - (1.0 / (((t + (z * ((0.5 * (t * y)) - (0.5 * t)))) / z) / 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 <= (-410000000.0d0)) then
        tmp = x - (((exp(z) - 1.0d0) / t) * y)
    else
        tmp = x - (1.0d0 / (((t + (z * ((0.5d0 * (t * y)) - (0.5d0 * t)))) / z) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -410000000.0) {
		tmp = x - (((Math.exp(z) - 1.0) / t) * y);
	} else {
		tmp = x - (1.0 / (((t + (z * ((0.5 * (t * y)) - (0.5 * t)))) / z) / y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -410000000.0:
		tmp = x - (((math.exp(z) - 1.0) / t) * y)
	else:
		tmp = x - (1.0 / (((t + (z * ((0.5 * (t * y)) - (0.5 * t)))) / z) / y))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -410000000.0)
		tmp = Float64(x - Float64(Float64(Float64(exp(z) - 1.0) / t) * y));
	else
		tmp = Float64(x - Float64(1.0 / Float64(Float64(Float64(t + Float64(z * Float64(Float64(0.5 * Float64(t * y)) - Float64(0.5 * t)))) / z) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -410000000.0)
		tmp = x - (((exp(z) - 1.0) / t) * y);
	else
		tmp = x - (1.0 / (((t + (z * ((0.5 * (t * y)) - (0.5 * t)))) / z) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -410000000], N[(x - N[(N[(N[(N[Exp[z], $MachinePrecision] - 1), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x - N[(1 / N[(N[(N[(t + N[(z * N[(N[(1/2 * N[(t * y), $MachinePrecision]), $MachinePrecision] - N[(1/2 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.1e8

   1. Initial program 60.6%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-exp.f64 75.8%

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

   4. Applied rewrites 75.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. lower--.f64 N/A

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

      5. distribute-lft-neg-out N/A

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

      6. *-commutative N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. sub-div N/A

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

      14. lift--.f64 N/A

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

      15. lower-/.f64 75.9%

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

   6. Applied rewrites 75.9%

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

   if -4.1e8 < z

   1. Initial program 60.6%

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

   2. Taylor expanded in z around 0

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

      1. lower-*.f64 73.1%

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

   4. Applied rewrites 73.1%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 73.1%

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 73.1%

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

   6. Applied rewrites 73.1%

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

   7. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-exp.f64 75.9%

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

   9. Applied rewrites 75.9%

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

   10. Taylor expanded in z around 0

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower--.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 84.9%

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

   12. Applied rewrites 84.9%

       \[\leadsto x - \frac{1}{\frac{\frac{t + z \cdot \left(\frac{1}{2} \cdot \left(t \cdot y\right) - \frac{1}{2} \cdot t\right)}{z}}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 84.9% accurate, 3.6× speedup

Math

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

FPCore

(FPCore (x y z t)
  :precision binary64
  (- x (/ 1 (/ (/ (+ t (* z (- (* 1/2 (* t y)) (* 1/2 t)))) z) y))))

C

double code(double x, double y, double z, double t) {
	return x - (1.0 / (((t + (z * ((0.5 * (t * y)) - (0.5 * t)))) / z) / y));
}

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 - (1.0d0 / (((t + (z * ((0.5d0 * (t * y)) - (0.5d0 * t)))) / z) / y))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x - (1.0 / (((t + (z * ((0.5 * (t * y)) - (0.5 * t)))) / z) / y));
}

Python

def code(x, y, z, t):
	return x - (1.0 / (((t + (z * ((0.5 * (t * y)) - (0.5 * t)))) / z) / y))

Julia

function code(x, y, z, t)
	return Float64(x - Float64(1.0 / Float64(Float64(Float64(t + Float64(z * Float64(Float64(0.5 * Float64(t * y)) - Float64(0.5 * t)))) / z) / y)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x - (1.0 / (((t + (z * ((0.5 * (t * y)) - (0.5 * t)))) / z) / y));
end

Wolfram

code[x_, y_, z_, t_] := N[(x - N[(1 / N[(N[(N[(t + N[(z * N[(N[(1/2 * N[(t * y), $MachinePrecision]), $MachinePrecision] - N[(1/2 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.6%

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

2. Taylor expanded in z around 0

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

   1. lower-*.f64 73.1%

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

4. Applied rewrites 73.1%

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

   1. lift-/.f64 N/A

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

   2. div-flip N/A

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

   3. lower-unsound-/.f64 N/A

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

   4. lower-unsound-/.f64 73.1%

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 73.1%

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

6. Applied rewrites 73.1%

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

7. Taylor expanded in y around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-exp.f64 75.9%

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

9. Applied rewrites 75.9%

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

10. Taylor expanded in z around 0

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

    1. lower-/.f64 N/A

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

    2. lower-+.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower--.f64 N/A

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

    5. lower-*.f64 N/A

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

    6. lower-*.f64 N/A

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

    7. lower-*.f64 84.9%

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

12. Applied rewrites 84.9%

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

Alternative 5: 81.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -130000000000000005723571947544438292554521954144678769275685764875122985731145995914860441978030839927144448:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;z \leq \frac{-5791434876257087}{5515652263101987298728728207430913795608113109085112352897269396216198887424215820128660001943808587833784893551335930816647064191168732319583111500951066614122648616177179922993422016587311577585463592732098692120576}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{1}{2} \cdot \left(t \cdot y\right) + \frac{t}{z}}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot \left(z \cdot \left(1 + z \cdot \left(\frac{1}{2} + z \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot z\right)\right)\right)\right)}{t}\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (if (<=
     z
     -130000000000000005723571947544438292554521954144678769275685764875122985731145995914860441978030839927144448)
  (* 1 x)
  (if (<=
       z
       -5791434876257087/5515652263101987298728728207430913795608113109085112352897269396216198887424215820128660001943808587833784893551335930816647064191168732319583111500951066614122648616177179922993422016587311577585463592732098692120576)
    (- x (/ 1 (/ (+ (* 1/2 (* t y)) (/ t z)) y)))
    (-
     x
     (/ (* y (* z (+ 1 (* z (+ 1/2 (* z (+ 1/6 (* 1/24 z)))))))) t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.3e+107) {
		tmp = 1.0 * x;
	} else if (z <= -1.05e-201) {
		tmp = x - (1.0 / (((0.5 * (t * y)) + (t / z)) / y));
	} else {
		tmp = x - ((y * (z * (1.0 + (z * (0.5 + (z * (0.16666666666666666 + (0.041666666666666664 * 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 <= (-1.3d+107)) then
        tmp = 1.0d0 * x
    else if (z <= (-1.05d-201)) then
        tmp = x - (1.0d0 / (((0.5d0 * (t * y)) + (t / z)) / y))
    else
        tmp = x - ((y * (z * (1.0d0 + (z * (0.5d0 + (z * (0.16666666666666666d0 + (0.041666666666666664d0 * 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 <= -1.3e+107) {
		tmp = 1.0 * x;
	} else if (z <= -1.05e-201) {
		tmp = x - (1.0 / (((0.5 * (t * y)) + (t / z)) / y));
	} else {
		tmp = x - ((y * (z * (1.0 + (z * (0.5 + (z * (0.16666666666666666 + (0.041666666666666664 * z)))))))) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.3e+107:
		tmp = 1.0 * x
	elif z <= -1.05e-201:
		tmp = x - (1.0 / (((0.5 * (t * y)) + (t / z)) / y))
	else:
		tmp = x - ((y * (z * (1.0 + (z * (0.5 + (z * (0.16666666666666666 + (0.041666666666666664 * z)))))))) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.3e+107)
		tmp = Float64(1.0 * x);
	elseif (z <= -1.05e-201)
		tmp = Float64(x - Float64(1.0 / Float64(Float64(Float64(0.5 * Float64(t * y)) + Float64(t / z)) / y)));
	else
		tmp = Float64(x - Float64(Float64(y * Float64(z * Float64(1.0 + Float64(z * Float64(0.5 + Float64(z * Float64(0.16666666666666666 + Float64(0.041666666666666664 * z)))))))) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.3e+107)
		tmp = 1.0 * x;
	elseif (z <= -1.05e-201)
		tmp = x - (1.0 / (((0.5 * (t * y)) + (t / z)) / y));
	else
		tmp = x - ((y * (z * (1.0 + (z * (0.5 + (z * (0.16666666666666666 + (0.041666666666666664 * z)))))))) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -130000000000000005723571947544438292554521954144678769275685764875122985731145995914860441978030839927144448], N[(1 * x), $MachinePrecision], If[LessEqual[z, -5791434876257087/5515652263101987298728728207430913795608113109085112352897269396216198887424215820128660001943808587833784893551335930816647064191168732319583111500951066614122648616177179922993422016587311577585463592732098692120576], N[(x - N[(1 / N[(N[(N[(1/2 * N[(t * y), $MachinePrecision]), $MachinePrecision] + N[(t / z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(y * N[(z * N[(1 + N[(z * N[(1/2 + N[(z * N[(1/6 + N[(1/24 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.3000000000000001e107

   1. Initial program 60.6%

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

   2. Taylor expanded in z around 0

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

      1. lower-*.f64 73.1%

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

   4. Applied rewrites 73.1%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. lower-unsound-*.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-/.f64 71.6%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 71.6%

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

   6. Applied rewrites 71.6%

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

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \cdot x \]
   8. Step-by-step derivation

   9. Applied rewrites 71.9%

      \[\leadsto \color{blue}{1} \cdot x \]

   if -1.3000000000000001e107 < z < -1.0500000000000001e-201

   1. Initial program 60.6%

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

   2. Taylor expanded in z around 0

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

      1. lower-*.f64 73.1%

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

   4. Applied rewrites 73.1%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 73.1%

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 73.1%

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

   6. Applied rewrites 73.1%

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

   7. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-exp.f64 75.9%

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

   9. Applied rewrites 75.9%

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

   10. Taylor expanded in z around 0

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

       1. lower-/.f64 79.2%

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

   12. Applied rewrites 79.2%

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

   if -1.0500000000000001e-201 < z

   1. Initial program 60.6%

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

   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--.f64 N/A

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

      3. lower-exp.f64 75.9%

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

   4. Applied rewrites 75.9%

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

   5. Taylor expanded in z around 0

      \[\leadsto x - \frac{y \cdot \left(z \cdot \color{blue}{\left(1 + z \cdot \left(\frac{1}{2} + z \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot z\right)\right)\right)}\right)}{t} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x - \frac{y \cdot \left(z \cdot \left(1 + \color{blue}{z \cdot \left(\frac{1}{2} + z \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot z\right)\right)}\right)\right)}{t} \]

      2. lower-+.f64 N/A

         \[\leadsto x - \frac{y \cdot \left(z \cdot \left(1 + z \cdot \color{blue}{\left(\frac{1}{2} + z \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot z\right)\right)}\right)\right)}{t} \]

      3. lower-*.f64 N/A

         \[\leadsto x - \frac{y \cdot \left(z \cdot \left(1 + z \cdot \left(\frac{1}{2} + \color{blue}{z \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot z\right)}\right)\right)\right)}{t} \]

      4. lower-+.f64 N/A

         \[\leadsto x - \frac{y \cdot \left(z \cdot \left(1 + z \cdot \left(\frac{1}{2} + z \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{24} \cdot z\right)}\right)\right)\right)}{t} \]

      5. lower-*.f64 N/A

         \[\leadsto x - \frac{y \cdot \left(z \cdot \left(1 + z \cdot \left(\frac{1}{2} + z \cdot \left(\frac{1}{6} + \color{blue}{\frac{1}{24} \cdot z}\right)\right)\right)\right)}{t} \]

      6. lower-+.f64 N/A

         \[\leadsto x - \frac{y \cdot \left(z \cdot \left(1 + z \cdot \left(\frac{1}{2} + z \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot \color{blue}{z}\right)\right)\right)\right)}{t} \]

      7. lower-*.f64 65.1%

         \[\leadsto x - \frac{y \cdot \left(z \cdot \left(1 + z \cdot \left(\frac{1}{2} + z \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot z\right)\right)\right)\right)}{t} \]

   7. Applied rewrites 65.1%

      \[\leadsto x - \frac{y \cdot \left(z \cdot \color{blue}{\left(1 + z \cdot \left(\frac{1}{2} + z \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot z\right)\right)\right)}\right)}{t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 81.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -130000000000000005723571947544438292554521954144678769275685764875122985731145995914860441978030839927144448:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;z \leq \frac{-5791434876257087}{5515652263101987298728728207430913795608113109085112352897269396216198887424215820128660001943808587833784893551335930816647064191168732319583111500951066614122648616177179922993422016587311577585463592732098692120576}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{1}{2} \cdot \left(t \cdot y\right) + \frac{t}{z}}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot \left(y + z \cdot \left(\frac{1}{2} \cdot y + z \cdot \left(\frac{1}{24} \cdot \left(y \cdot z\right) + \frac{1}{6} \cdot y\right)\right)\right)}{t}\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (if (<=
     z
     -130000000000000005723571947544438292554521954144678769275685764875122985731145995914860441978030839927144448)
  (* 1 x)
  (if (<=
       z
       -5791434876257087/5515652263101987298728728207430913795608113109085112352897269396216198887424215820128660001943808587833784893551335930816647064191168732319583111500951066614122648616177179922993422016587311577585463592732098692120576)
    (- x (/ 1 (/ (+ (* 1/2 (* t y)) (/ t z)) y)))
    (-
     x
     (/
      (*
       z
       (+ y (* z (+ (* 1/2 y) (* z (+ (* 1/24 (* y z)) (* 1/6 y)))))))
      t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.3e+107) {
		tmp = 1.0 * x;
	} else if (z <= -1.05e-201) {
		tmp = x - (1.0 / (((0.5 * (t * y)) + (t / z)) / y));
	} else {
		tmp = x - ((z * (y + (z * ((0.5 * y) + (z * ((0.041666666666666664 * (y * z)) + (0.16666666666666666 * 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) :: tmp
    if (z <= (-1.3d+107)) then
        tmp = 1.0d0 * x
    else if (z <= (-1.05d-201)) then
        tmp = x - (1.0d0 / (((0.5d0 * (t * y)) + (t / z)) / y))
    else
        tmp = x - ((z * (y + (z * ((0.5d0 * y) + (z * ((0.041666666666666664d0 * (y * z)) + (0.16666666666666666d0 * y))))))) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.3e+107) {
		tmp = 1.0 * x;
	} else if (z <= -1.05e-201) {
		tmp = x - (1.0 / (((0.5 * (t * y)) + (t / z)) / y));
	} else {
		tmp = x - ((z * (y + (z * ((0.5 * y) + (z * ((0.041666666666666664 * (y * z)) + (0.16666666666666666 * y))))))) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.3e+107:
		tmp = 1.0 * x
	elif z <= -1.05e-201:
		tmp = x - (1.0 / (((0.5 * (t * y)) + (t / z)) / y))
	else:
		tmp = x - ((z * (y + (z * ((0.5 * y) + (z * ((0.041666666666666664 * (y * z)) + (0.16666666666666666 * y))))))) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.3e+107)
		tmp = Float64(1.0 * x);
	elseif (z <= -1.05e-201)
		tmp = Float64(x - Float64(1.0 / Float64(Float64(Float64(0.5 * Float64(t * y)) + Float64(t / z)) / y)));
	else
		tmp = Float64(x - Float64(Float64(z * Float64(y + Float64(z * Float64(Float64(0.5 * y) + Float64(z * Float64(Float64(0.041666666666666664 * Float64(y * z)) + Float64(0.16666666666666666 * y))))))) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.3e+107)
		tmp = 1.0 * x;
	elseif (z <= -1.05e-201)
		tmp = x - (1.0 / (((0.5 * (t * y)) + (t / z)) / y));
	else
		tmp = x - ((z * (y + (z * ((0.5 * y) + (z * ((0.041666666666666664 * (y * z)) + (0.16666666666666666 * y))))))) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -130000000000000005723571947544438292554521954144678769275685764875122985731145995914860441978030839927144448], N[(1 * x), $MachinePrecision], If[LessEqual[z, -5791434876257087/5515652263101987298728728207430913795608113109085112352897269396216198887424215820128660001943808587833784893551335930816647064191168732319583111500951066614122648616177179922993422016587311577585463592732098692120576], N[(x - N[(1 / N[(N[(N[(1/2 * N[(t * y), $MachinePrecision]), $MachinePrecision] + N[(t / z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(z * N[(y + N[(z * N[(N[(1/2 * y), $MachinePrecision] + N[(z * N[(N[(1/24 * N[(y * z), $MachinePrecision]), $MachinePrecision] + N[(1/6 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.3000000000000001e107

   1. Initial program 60.6%

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

   2. Taylor expanded in z around 0

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

      1. lower-*.f64 73.1%

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

   4. Applied rewrites 73.1%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. lower-unsound-*.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-/.f64 71.6%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 71.6%

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

   6. Applied rewrites 71.6%

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

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \cdot x \]
   8. Step-by-step derivation

   9. Applied rewrites 71.9%

      \[\leadsto \color{blue}{1} \cdot x \]

   if -1.3000000000000001e107 < z < -1.0500000000000001e-201

   1. Initial program 60.6%

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

   2. Taylor expanded in z around 0

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

      1. lower-*.f64 73.1%

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

   4. Applied rewrites 73.1%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 73.1%

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 73.1%

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

   6. Applied rewrites 73.1%

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

   7. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-exp.f64 75.9%

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

   9. Applied rewrites 75.9%

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

   10. Taylor expanded in z around 0

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

       1. lower-/.f64 79.2%

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

   12. Applied rewrites 79.2%

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

   if -1.0500000000000001e-201 < z

   1. Initial program 60.6%

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

   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--.f64 N/A

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

      3. lower-exp.f64 75.9%

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

   4. Applied rewrites 75.9%

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

   5. Taylor expanded in z around 0

      \[\leadsto x - \frac{z \cdot \color{blue}{\left(y + z \cdot \left(\frac{1}{2} \cdot y + z \cdot \left(\frac{1}{24} \cdot \left(y \cdot z\right) + \frac{1}{6} \cdot y\right)\right)\right)}}{t} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x - \frac{z \cdot \left(y + \color{blue}{z \cdot \left(\frac{1}{2} \cdot y + z \cdot \left(\frac{1}{24} \cdot \left(y \cdot z\right) + \frac{1}{6} \cdot y\right)\right)}\right)}{t} \]

      2. lower-+.f64 N/A

         \[\leadsto x - \frac{z \cdot \left(y + z \cdot \color{blue}{\left(\frac{1}{2} \cdot y + z \cdot \left(\frac{1}{24} \cdot \left(y \cdot z\right) + \frac{1}{6} \cdot y\right)\right)}\right)}{t} \]

      3. lower-*.f64 N/A

         \[\leadsto x - \frac{z \cdot \left(y + z \cdot \left(\frac{1}{2} \cdot y + \color{blue}{z \cdot \left(\frac{1}{24} \cdot \left(y \cdot z\right) + \frac{1}{6} \cdot y\right)}\right)\right)}{t} \]

      4. lower-+.f64 N/A

         \[\leadsto x - \frac{z \cdot \left(y + z \cdot \left(\frac{1}{2} \cdot y + z \cdot \color{blue}{\left(\frac{1}{24} \cdot \left(y \cdot z\right) + \frac{1}{6} \cdot y\right)}\right)\right)}{t} \]

      5. lower-*.f64 N/A

         \[\leadsto x - \frac{z \cdot \left(y + z \cdot \left(\frac{1}{2} \cdot y + z \cdot \left(\color{blue}{\frac{1}{24} \cdot \left(y \cdot z\right)} + \frac{1}{6} \cdot y\right)\right)\right)}{t} \]

      6. lower-*.f64 N/A

         \[\leadsto x - \frac{z \cdot \left(y + z \cdot \left(\frac{1}{2} \cdot y + z \cdot \left(\frac{1}{24} \cdot \left(y \cdot z\right) + \color{blue}{\frac{1}{6} \cdot y}\right)\right)\right)}{t} \]

      7. lower-+.f64 N/A

         \[\leadsto x - \frac{z \cdot \left(y + z \cdot \left(\frac{1}{2} \cdot y + z \cdot \left(\frac{1}{24} \cdot \left(y \cdot z\right) + \frac{1}{6} \cdot \color{blue}{y}\right)\right)\right)}{t} \]

      8. lower-*.f64 N/A

         \[\leadsto x - \frac{z \cdot \left(y + z \cdot \left(\frac{1}{2} \cdot y + z \cdot \left(\frac{1}{24} \cdot \left(y \cdot z\right) + \frac{1}{6} \cdot y\right)\right)\right)}{t} \]

      9. lower-*.f64 N/A

         \[\leadsto x - \frac{z \cdot \left(y + z \cdot \left(\frac{1}{2} \cdot y + z \cdot \left(\frac{1}{24} \cdot \left(y \cdot z\right) + \frac{1}{6} \cdot y\right)\right)\right)}{t} \]

      10. lower-*.f64 65.8%

          \[\leadsto x - \frac{z \cdot \left(y + z \cdot \left(\frac{1}{2} \cdot y + z \cdot \left(\frac{1}{24} \cdot \left(y \cdot z\right) + \frac{1}{6} \cdot y\right)\right)\right)}{t} \]

   7. Applied rewrites 65.8%

      \[\leadsto x - \frac{z \cdot \color{blue}{\left(y + z \cdot \left(\frac{1}{2} \cdot y + z \cdot \left(\frac{1}{24} \cdot \left(y \cdot z\right) + \frac{1}{6} \cdot y\right)\right)\right)}}{t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 81.4% accurate, 4.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq \frac{-6860092517126119}{21778071482940061661655974875633165533184}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot \left(z \cdot \left(1 + z \cdot \left(\frac{1}{2} + z \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot z\right)\right)\right)\right)}{t}\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (if (<= z -6860092517126119/21778071482940061661655974875633165533184)
  (* 1 x)
  (-
   x
   (/ (* y (* z (+ 1 (* z (+ 1/2 (* z (+ 1/6 (* 1/24 z)))))))) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.15e-25) {
		tmp = 1.0 * x;
	} else {
		tmp = x - ((y * (z * (1.0 + (z * (0.5 + (z * (0.16666666666666666 + (0.041666666666666664 * 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 <= (-3.15d-25)) then
        tmp = 1.0d0 * x
    else
        tmp = x - ((y * (z * (1.0d0 + (z * (0.5d0 + (z * (0.16666666666666666d0 + (0.041666666666666664d0 * 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 <= -3.15e-25) {
		tmp = 1.0 * x;
	} else {
		tmp = x - ((y * (z * (1.0 + (z * (0.5 + (z * (0.16666666666666666 + (0.041666666666666664 * z)))))))) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -3.15e-25:
		tmp = 1.0 * x
	else:
		tmp = x - ((y * (z * (1.0 + (z * (0.5 + (z * (0.16666666666666666 + (0.041666666666666664 * z)))))))) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -3.15e-25)
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(x - Float64(Float64(y * Float64(z * Float64(1.0 + Float64(z * Float64(0.5 + Float64(z * Float64(0.16666666666666666 + Float64(0.041666666666666664 * z)))))))) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -3.15e-25)
		tmp = 1.0 * x;
	else
		tmp = x - ((y * (z * (1.0 + (z * (0.5 + (z * (0.16666666666666666 + (0.041666666666666664 * z)))))))) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -6860092517126119/21778071482940061661655974875633165533184], N[(1 * x), $MachinePrecision], N[(x - N[(N[(y * N[(z * N[(1 + N[(z * N[(1/2 + N[(z * N[(1/6 + N[(1/24 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.1499999999999998e-25

   1. Initial program 60.6%

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

   2. Taylor expanded in z around 0

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

      1. lower-*.f64 73.1%

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

   4. Applied rewrites 73.1%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. lower-unsound-*.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-/.f64 71.6%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 71.6%

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

   6. Applied rewrites 71.6%

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

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \cdot x \]
   8. Step-by-step derivation

   9. Applied rewrites 71.9%

      \[\leadsto \color{blue}{1} \cdot x \]

   if -3.1499999999999998e-25 < z

   1. Initial program 60.6%

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

   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--.f64 N/A

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

      3. lower-exp.f64 75.9%

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

   4. Applied rewrites 75.9%

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

   5. Taylor expanded in z around 0

      \[\leadsto x - \frac{y \cdot \left(z \cdot \color{blue}{\left(1 + z \cdot \left(\frac{1}{2} + z \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot z\right)\right)\right)}\right)}{t} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x - \frac{y \cdot \left(z \cdot \left(1 + \color{blue}{z \cdot \left(\frac{1}{2} + z \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot z\right)\right)}\right)\right)}{t} \]

      2. lower-+.f64 N/A

         \[\leadsto x - \frac{y \cdot \left(z \cdot \left(1 + z \cdot \color{blue}{\left(\frac{1}{2} + z \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot z\right)\right)}\right)\right)}{t} \]

      3. lower-*.f64 N/A

         \[\leadsto x - \frac{y \cdot \left(z \cdot \left(1 + z \cdot \left(\frac{1}{2} + \color{blue}{z \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot z\right)}\right)\right)\right)}{t} \]

      4. lower-+.f64 N/A

         \[\leadsto x - \frac{y \cdot \left(z \cdot \left(1 + z \cdot \left(\frac{1}{2} + z \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{24} \cdot z\right)}\right)\right)\right)}{t} \]

      5. lower-*.f64 N/A

         \[\leadsto x - \frac{y \cdot \left(z \cdot \left(1 + z \cdot \left(\frac{1}{2} + z \cdot \left(\frac{1}{6} + \color{blue}{\frac{1}{24} \cdot z}\right)\right)\right)\right)}{t} \]

      6. lower-+.f64 N/A

         \[\leadsto x - \frac{y \cdot \left(z \cdot \left(1 + z \cdot \left(\frac{1}{2} + z \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot \color{blue}{z}\right)\right)\right)\right)}{t} \]

      7. lower-*.f64 65.1%

         \[\leadsto x - \frac{y \cdot \left(z \cdot \left(1 + z \cdot \left(\frac{1}{2} + z \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot z\right)\right)\right)\right)}{t} \]

   7. Applied rewrites 65.1%

      \[\leadsto x - \frac{y \cdot \left(z \cdot \color{blue}{\left(1 + z \cdot \left(\frac{1}{2} + z \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot z\right)\right)\right)}\right)}{t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 80.5% accurate, 4.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq \frac{-6860092517126119}{21778071482940061661655974875633165533184}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot \left(z \cdot \left(1 + z \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot z\right)\right)\right)}{t}\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (if (<= z -6860092517126119/21778071482940061661655974875633165533184)
  (* 1 x)
  (- x (/ (* y (* z (+ 1 (* z (+ 1/2 (* 1/6 z)))))) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.15e-25) {
		tmp = 1.0 * x;
	} else {
		tmp = x - ((y * (z * (1.0 + (z * (0.5 + (0.16666666666666666 * 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 <= (-3.15d-25)) then
        tmp = 1.0d0 * x
    else
        tmp = x - ((y * (z * (1.0d0 + (z * (0.5d0 + (0.16666666666666666d0 * 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 <= -3.15e-25) {
		tmp = 1.0 * x;
	} else {
		tmp = x - ((y * (z * (1.0 + (z * (0.5 + (0.16666666666666666 * z)))))) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -3.15e-25:
		tmp = 1.0 * x
	else:
		tmp = x - ((y * (z * (1.0 + (z * (0.5 + (0.16666666666666666 * z)))))) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -3.15e-25)
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(x - Float64(Float64(y * Float64(z * Float64(1.0 + Float64(z * Float64(0.5 + Float64(0.16666666666666666 * z)))))) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -3.15e-25)
		tmp = 1.0 * x;
	else
		tmp = x - ((y * (z * (1.0 + (z * (0.5 + (0.16666666666666666 * z)))))) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -6860092517126119/21778071482940061661655974875633165533184], N[(1 * x), $MachinePrecision], N[(x - N[(N[(y * N[(z * N[(1 + N[(z * N[(1/2 + N[(1/6 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.1499999999999998e-25

   1. Initial program 60.6%

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

   2. Taylor expanded in z around 0

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

      1. lower-*.f64 73.1%

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

   4. Applied rewrites 73.1%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. lower-unsound-*.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-/.f64 71.6%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 71.6%

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

   6. Applied rewrites 71.6%

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

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \cdot x \]
   8. Step-by-step derivation

   9. Applied rewrites 71.9%

      \[\leadsto \color{blue}{1} \cdot x \]

   if -3.1499999999999998e-25 < z

   1. Initial program 60.6%

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

   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--.f64 N/A

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

      3. lower-exp.f64 75.9%

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

   4. Applied rewrites 75.9%

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

   5. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 66.6%

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

   7. Applied rewrites 66.6%

      \[\leadsto x - \frac{y \cdot \left(z \cdot \color{blue}{\left(1 + z \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot z\right)\right)}\right)}{t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 80.5% accurate, 5.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq \frac{-6860092517126119}{21778071482940061661655974875633165533184}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot \left(y \cdot \left(1 + \frac{1}{2} \cdot z\right)\right)}{t}\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (if (<= z -6860092517126119/21778071482940061661655974875633165533184)
  (* 1 x)
  (- x (/ (* z (* y (+ 1 (* 1/2 z)))) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.15e-25) {
		tmp = 1.0 * x;
	} else {
		tmp = x - ((z * (y * (1.0 + (0.5 * 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 <= (-3.15d-25)) then
        tmp = 1.0d0 * x
    else
        tmp = x - ((z * (y * (1.0d0 + (0.5d0 * 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 <= -3.15e-25) {
		tmp = 1.0 * x;
	} else {
		tmp = x - ((z * (y * (1.0 + (0.5 * z)))) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -3.15e-25:
		tmp = 1.0 * x
	else:
		tmp = x - ((z * (y * (1.0 + (0.5 * z)))) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -3.15e-25)
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(x - Float64(Float64(z * Float64(y * Float64(1.0 + Float64(0.5 * z)))) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -3.15e-25)
		tmp = 1.0 * x;
	else
		tmp = x - ((z * (y * (1.0 + (0.5 * z)))) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -6860092517126119/21778071482940061661655974875633165533184], N[(1 * x), $MachinePrecision], N[(x - N[(N[(z * N[(y * N[(1 + N[(1/2 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.1499999999999998e-25

   1. Initial program 60.6%

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

   2. Taylor expanded in z around 0

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

      1. lower-*.f64 73.1%

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

   4. Applied rewrites 73.1%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. lower-unsound-*.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-/.f64 71.6%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 71.6%

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

   6. Applied rewrites 71.6%

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

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \cdot x \]
   8. Step-by-step derivation

   9. Applied rewrites 71.9%

      \[\leadsto \color{blue}{1} \cdot x \]

   if -3.1499999999999998e-25 < z

   1. Initial program 60.6%

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

   2. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 60.6%

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

   4. Applied rewrites 60.6%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 68.9%

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

   7. Applied rewrites 68.9%

      \[\leadsto x - \frac{z \cdot \left(y \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot z\right)}\right)}{t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 80.5% accurate, 7.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq \frac{-6860092517126119}{21778071482940061661655974875633165533184}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{t} \cdot \left(z \cdot y\right)\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (if (<= z -6860092517126119/21778071482940061661655974875633165533184)
  (* 1 x)
  (- x (* (/ 1 t) (* z y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.15e-25) {
		tmp = 1.0 * x;
	} else {
		tmp = x - ((1.0 / t) * (z * 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 <= (-3.15d-25)) then
        tmp = 1.0d0 * x
    else
        tmp = x - ((1.0d0 / t) * (z * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.15e-25) {
		tmp = 1.0 * x;
	} else {
		tmp = x - ((1.0 / t) * (z * y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -3.15e-25:
		tmp = 1.0 * x
	else:
		tmp = x - ((1.0 / t) * (z * y))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -3.15e-25)
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(x - Float64(Float64(1.0 / t) * Float64(z * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -3.15e-25)
		tmp = 1.0 * x;
	else
		tmp = x - ((1.0 / t) * (z * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -6860092517126119/21778071482940061661655974875633165533184], N[(1 * x), $MachinePrecision], N[(x - N[(N[(1 / t), $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.1499999999999998e-25

   1. Initial program 60.6%

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

   2. Taylor expanded in z around 0

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

      1. lower-*.f64 73.1%

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

   4. Applied rewrites 73.1%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. lower-unsound-*.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-/.f64 71.6%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 71.6%

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

   6. Applied rewrites 71.6%

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

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \cdot x \]
   8. Step-by-step derivation

   9. Applied rewrites 71.9%

      \[\leadsto \color{blue}{1} \cdot x \]

   if -3.1499999999999998e-25 < z

   1. Initial program 60.6%

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

   2. Taylor expanded in z around 0

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

      1. lower-*.f64 73.1%

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

   4. Applied rewrites 73.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 73.1%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 73.1%

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

   6. Applied rewrites 73.1%

      \[\leadsto x - \color{blue}{\frac{1}{t} \cdot \left(z \cdot y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 80.5% accurate, 8.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq \frac{-6860092517126119}{21778071482940061661655974875633165533184}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot z}{t}\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (if (<= z -6860092517126119/21778071482940061661655974875633165533184)
  (* 1 x)
  (- x (/ (* y z) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.15e-25) {
		tmp = 1.0 * x;
	} 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 <= (-3.15d-25)) then
        tmp = 1.0d0 * x
    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 <= -3.15e-25) {
		tmp = 1.0 * x;
	} else {
		tmp = x - ((y * z) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -3.15e-25:
		tmp = 1.0 * x
	else:
		tmp = x - ((y * z) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -3.15e-25)
		tmp = Float64(1.0 * x);
	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 <= -3.15e-25)
		tmp = 1.0 * x;
	else
		tmp = x - ((y * z) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -6860092517126119/21778071482940061661655974875633165533184], N[(1 * x), $MachinePrecision], N[(x - N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.1499999999999998e-25

   1. Initial program 60.6%

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

   2. Taylor expanded in z around 0

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

      1. lower-*.f64 73.1%

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

   4. Applied rewrites 73.1%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. lower-unsound-*.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-/.f64 71.6%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 71.6%

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

   6. Applied rewrites 71.6%

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

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \cdot x \]
   8. Step-by-step derivation

   9. Applied rewrites 71.9%

      \[\leadsto \color{blue}{1} \cdot x \]

   if -3.1499999999999998e-25 < z

   1. Initial program 60.6%

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

   2. Taylor expanded in z around 0

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

      1. lower-*.f64 73.1%

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

   4. Applied rewrites 73.1%

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

Alternative 12: 80.5% accurate, 8.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq \frac{-6860092517126119}{21778071482940061661655974875633165533184}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z}{t} \cdot y\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (if (<= z -6860092517126119/21778071482940061661655974875633165533184)
  (* 1 x)
  (- x (* (/ z t) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.15e-25) {
		tmp = 1.0 * 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 <= (-3.15d-25)) then
        tmp = 1.0d0 * 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 <= -3.15e-25) {
		tmp = 1.0 * x;
	} else {
		tmp = x - ((z / t) * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -3.15e-25:
		tmp = 1.0 * x
	else:
		tmp = x - ((z / t) * y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -3.15e-25)
		tmp = Float64(1.0 * 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 <= -3.15e-25)
		tmp = 1.0 * x;
	else
		tmp = x - ((z / t) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -6860092517126119/21778071482940061661655974875633165533184], N[(1 * x), $MachinePrecision], N[(x - N[(N[(z / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.1499999999999998e-25

   1. Initial program 60.6%

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

   2. Taylor expanded in z around 0

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

      1. lower-*.f64 73.1%

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

   4. Applied rewrites 73.1%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. lower-unsound-*.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-/.f64 71.6%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 71.6%

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

   6. Applied rewrites 71.6%

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

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \cdot x \]
   8. Step-by-step derivation

   9. Applied rewrites 71.9%

      \[\leadsto \color{blue}{1} \cdot x \]

   if -3.1499999999999998e-25 < z

   1. Initial program 60.6%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-exp.f64 75.8%

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

   4. Applied rewrites 75.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. lower--.f64 N/A

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

      5. distribute-lft-neg-out N/A

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

      6. *-commutative N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. sub-div N/A

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

      14. lift--.f64 N/A

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

      15. lower-/.f64 75.9%

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

   6. Applied rewrites 75.9%

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

   7. Taylor expanded in z around 0

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

      1. lower-/.f64 73.7%

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

   9. Applied rewrites 73.7%

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

Alternative 13: 71.9% accurate, 37.7× speedup

Math

\[1 \cdot x \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return 1.0 * 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 = 1.0d0 * x
end function

Java

public static double code(double x, double y, double z, double t) {
	return 1.0 * x;
}

Python

def code(x, y, z, t):
	return 1.0 * x

Julia

function code(x, y, z, t)
	return Float64(1.0 * x)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 1.0 * x;
end

Wolfram

code[x_, y_, z_, t_] := N[(1 * x), $MachinePrecision]

Derivation

1. Initial program 60.6%

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

2. Taylor expanded in z around 0

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

   1. lower-*.f64 73.1%

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

4. Applied rewrites 73.1%

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

   1. lift--.f64 N/A

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

   2. sub-to-mult N/A

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

   3. lower-unsound-*.f64 N/A

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

   4. lower-unsound--.f64 N/A

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

   5. lower-unsound-/.f64 71.6%

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 71.6%

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

6. Applied rewrites 71.6%

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

7. Taylor expanded in x around inf

   \[\leadsto \color{blue}{1} \cdot x \]
8. Step-by-step derivation

9. Applied rewrites 71.9%

   \[\leadsto \color{blue}{1} \cdot x \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025285 -o generate:evaluate
(FPCore (x y z t)
  :name "System.Random.MWC.Distributions:truncatedExp from mwc-random-0.13.3.2"
  :precision binary64
  (- x (/ (log (+ (- 1 y) (* y (exp z)))) t)))