Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, C

Percentage Accurate: 94.6% → 94.6%

Time: 4.0s

Alternatives: 9

Speedup: 1.0×

• 21.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.6%

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

Alternative 2: 63.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-t\right) \cdot x\\ \mathbf{if}\;t \leq -1.5 \cdot 10^{+136}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{+73}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+206}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- t) x)))
   (if (<= t -1.5e+136)
     t_1
     (if (<= t 1.9e+73) (* y (/ x z)) (if (<= t 5e+206) (* t (/ x z)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -t * x;
	double tmp;
	if (t <= -1.5e+136) {
		tmp = t_1;
	} else if (t <= 1.9e+73) {
		tmp = y * (x / z);
	} else if (t <= 5e+206) {
		tmp = t * (x / z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -t * x
    if (t <= (-1.5d+136)) then
        tmp = t_1
    else if (t <= 1.9d+73) then
        tmp = y * (x / z)
    else if (t <= 5d+206) then
        tmp = t * (x / z)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = -t * x;
	double tmp;
	if (t <= -1.5e+136) {
		tmp = t_1;
	} else if (t <= 1.9e+73) {
		tmp = y * (x / z);
	} else if (t <= 5e+206) {
		tmp = t * (x / z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = -t * x
	tmp = 0
	if t <= -1.5e+136:
		tmp = t_1
	elif t <= 1.9e+73:
		tmp = y * (x / z)
	elif t <= 5e+206:
		tmp = t * (x / z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(-t) * x)
	tmp = 0.0
	if (t <= -1.5e+136)
		tmp = t_1;
	elseif (t <= 1.9e+73)
		tmp = Float64(y * Float64(x / z));
	elseif (t <= 5e+206)
		tmp = Float64(t * Float64(x / z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = -t * x;
	tmp = 0.0;
	if (t <= -1.5e+136)
		tmp = t_1;
	elseif (t <= 1.9e+73)
		tmp = y * (x / z);
	elseif (t <= 5e+206)
		tmp = t * (x / z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[((-t) * x), $MachinePrecision]}, If[LessEqual[t, -1.5e+136], t$95$1, If[LessEqual[t, 1.9e+73], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5e+206], N[(t * N[(x / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.49999999999999989e136 or 5.0000000000000002e206 < t

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 48.8

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

   5. Applied rewrites 48.8%

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

   6. Taylor expanded in y around 0

      \[\leadsto -1 \cdot \color{blue}{\left(t \cdot x\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot t\right) \cdot x \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot t\right) \cdot x \]

      3. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(t\right)\right) \cdot x \]

      4. lift-neg.f64 46.9

         \[\leadsto \left(-t\right) \cdot x \]

   8. Applied rewrites 46.9%

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

   if -1.49999999999999989e136 < t < 1.90000000000000011e73

   1. Initial program 95.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 76.6

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

   5. Applied rewrites 76.6%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 75.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 78.6

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

   10. Applied rewrites 78.6%

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

   if 1.90000000000000011e73 < t < 5.0000000000000002e206

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 64.3

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

   5. Applied rewrites 64.3%

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

   6. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 61.2

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

   8. Applied rewrites 61.2%

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

Alternative 3: 88.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -31 \lor \neg \left(z \leq 3.5 \cdot 10^{+22}\right):\\ \;\;\;\;\frac{\left(t + y\right) \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -31.0) (not (<= z 3.5e+22)))
   (/ (* (+ t y) x) z)
   (* x (- (/ y z) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -31.0) || !(z <= 3.5e+22)) {
		tmp = ((t + y) * x) / z;
	} 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 <= (-31.0d0)) .or. (.not. (z <= 3.5d+22))) then
        tmp = ((t + y) * x) / z
    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 <= -31.0) || !(z <= 3.5e+22)) {
		tmp = ((t + y) * x) / z;
	} else {
		tmp = x * ((y / z) - t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -31.0) or not (z <= 3.5e+22):
		tmp = ((t + y) * x) / z
	else:
		tmp = x * ((y / z) - t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -31.0) || !(z <= 3.5e+22))
		tmp = Float64(Float64(Float64(t + y) * x) / z);
	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 <= -31.0) || ~((z <= 3.5e+22)))
		tmp = ((t + y) * x) / z;
	else
		tmp = x * ((y / z) - t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -31.0], N[Not[LessEqual[z, 3.5e+22]], $MachinePrecision]], N[(N[(N[(t + y), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision], N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -31 or 3.5e22 < z

   1. Initial program 96.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 86.5

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

   5. Applied rewrites 86.5%

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

   6. Taylor expanded in y around 0

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

      1. lower-+.f64 86.5

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

   8. Applied rewrites 86.5%

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

   if -31 < z < 3.5e22

   1. Initial program 96.3%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 95.2%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -31 \lor \neg \left(z \leq 3.5 \cdot 10^{+22}\right):\\ \;\;\;\;\frac{\left(t + y\right) \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.015 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{\left(t + y\right) \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -0.015) (not (<= z 1.0))) (/ (* (+ t y) x) z) (* y (/ x z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -0.015) || !(z <= 1.0)) {
		tmp = ((t + y) * x) / z;
	} else {
		tmp = y * (x / z);
	}
	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 <= (-0.015d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = ((t + y) * x) / z
    else
        tmp = y * (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -0.015) || !(z <= 1.0)) {
		tmp = ((t + y) * x) / z;
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -0.015) or not (z <= 1.0):
		tmp = ((t + y) * x) / z
	else:
		tmp = y * (x / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -0.015) || !(z <= 1.0))
		tmp = Float64(Float64(Float64(t + y) * x) / z);
	else
		tmp = Float64(y * Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -0.015) || ~((z <= 1.0)))
		tmp = ((t + y) * x) / z;
	else
		tmp = y * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -0.015], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(N[(N[(t + y), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -0.014999999999999999 or 1 < z

   1. Initial program 97.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 86.3

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

   5. Applied rewrites 86.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-+.f64 86.3

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

   8. Applied rewrites 86.3%

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

   if -0.014999999999999999 < z < 1

   1. Initial program 96.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 52.8

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

   5. Applied rewrites 52.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 69.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 73.3

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

   10. Applied rewrites 73.3%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.015 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{\left(t + y\right) \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -31:\\ \;\;\;\;\left(y + t\right) \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+22}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t + y\right) \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -31.0)
   (* (+ y t) (/ x z))
   (if (<= z 3.5e+22) (* x (- (/ y z) t)) (/ (* (+ t y) x) z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -31.0) {
		tmp = (y + t) * (x / z);
	} else if (z <= 3.5e+22) {
		tmp = x * ((y / z) - t);
	} else {
		tmp = ((t + y) * x) / z;
	}
	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 <= (-31.0d0)) then
        tmp = (y + t) * (x / z)
    else if (z <= 3.5d+22) then
        tmp = x * ((y / z) - t)
    else
        tmp = ((t + y) * x) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -31.0) {
		tmp = (y + t) * (x / z);
	} else if (z <= 3.5e+22) {
		tmp = x * ((y / z) - t);
	} else {
		tmp = ((t + y) * x) / z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -31.0:
		tmp = (y + t) * (x / z)
	elif z <= 3.5e+22:
		tmp = x * ((y / z) - t)
	else:
		tmp = ((t + y) * x) / z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -31.0)
		tmp = Float64(Float64(y + t) * Float64(x / z));
	elseif (z <= 3.5e+22)
		tmp = Float64(x * Float64(Float64(y / z) - t));
	else
		tmp = Float64(Float64(Float64(t + y) * x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -31.0)
		tmp = (y + t) * (x / z);
	elseif (z <= 3.5e+22)
		tmp = x * ((y / z) - t);
	else
		tmp = ((t + y) * x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -31.0], N[(N[(y + t), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.5e+22], N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t + y), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -31

   1. Initial program 97.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 87.6

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

   5. Applied rewrites 87.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-/.f64 87.8

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

   7. Applied rewrites 87.8%

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

   if -31 < z < 3.5e22

   1. Initial program 96.3%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 95.2%

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

   if 3.5e22 < z

   1. Initial program 96.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 85.4

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

   5. Applied rewrites 85.4%

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

   6. Taylor expanded in y around 0

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

      1. lower-+.f64 85.4

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

   8. Applied rewrites 85.4%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -31:\\ \;\;\;\;\left(y + t\right) \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+22}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t + y\right) \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.26 \cdot 10^{+16} \lor \neg \left(t \leq 1.9 \cdot 10^{+73}\right):\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1.26e+16) (not (<= t 1.9e+73))) (* x (/ t z)) (* y (/ x z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.26e+16) || !(t <= 1.9e+73)) {
		tmp = x * (t / z);
	} else {
		tmp = y * (x / z);
	}
	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 ((t <= (-1.26d+16)) .or. (.not. (t <= 1.9d+73))) then
        tmp = x * (t / z)
    else
        tmp = y * (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.26e+16) || !(t <= 1.9e+73)) {
		tmp = x * (t / z);
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -1.26e+16) or not (t <= 1.9e+73):
		tmp = x * (t / z)
	else:
		tmp = y * (x / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1.26e+16) || !(t <= 1.9e+73))
		tmp = Float64(x * Float64(t / z));
	else
		tmp = Float64(y * Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -1.26e+16) || ~((t <= 1.9e+73)))
		tmp = x * (t / z);
	else
		tmp = y * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -1.26e+16], N[Not[LessEqual[t, 1.9e+73]], $MachinePrecision]], N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.26e16 or 1.90000000000000011e73 < t

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lift--.f64 78.1

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

   5. Applied rewrites 78.1%

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

   6. Taylor expanded in z around inf

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

      1. lower-/.f64 55.0

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

   8. Applied rewrites 55.0%

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

   if -1.26e16 < t < 1.90000000000000011e73

   1. Initial program 94.3%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 79.1

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 81.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 84.1

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

   10. Applied rewrites 84.1%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.26 \cdot 10^{+16} \lor \neg \left(t \leq 1.9 \cdot 10^{+73}\right):\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.7 \cdot 10^{+139} \lor \neg \left(t \leq 1.05 \cdot 10^{+74}\right):\\ \;\;\;\;\frac{t \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -4.7e+139) (not (<= t 1.05e+74))) (/ (* t x) z) (* y (/ x z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -4.7e+139) || !(t <= 1.05e+74)) {
		tmp = (t * x) / z;
	} else {
		tmp = y * (x / z);
	}
	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 ((t <= (-4.7d+139)) .or. (.not. (t <= 1.05d+74))) then
        tmp = (t * x) / z
    else
        tmp = y * (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -4.7e+139) || !(t <= 1.05e+74)) {
		tmp = (t * x) / z;
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -4.7e+139) or not (t <= 1.05e+74):
		tmp = (t * x) / z
	else:
		tmp = y * (x / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -4.7e+139) || !(t <= 1.05e+74))
		tmp = Float64(Float64(t * x) / z);
	else
		tmp = Float64(y * Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -4.7e+139) || ~((t <= 1.05e+74)))
		tmp = (t * x) / z;
	else
		tmp = y * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -4.7e+139], N[Not[LessEqual[t, 1.05e+74]], $MachinePrecision]], N[(N[(t * x), $MachinePrecision] / z), $MachinePrecision], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.7000000000000001e139 or 1.0499999999999999e74 < t

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 53.4

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

   5. Applied rewrites 53.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 50.8%

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

   if -4.7000000000000001e139 < t < 1.0499999999999999e74

   1. Initial program 95.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 75.7

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 75.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 78.3

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

   10. Applied rewrites 78.3%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.7 \cdot 10^{+139} \lor \neg \left(t \leq 1.05 \cdot 10^{+74}\right):\\ \;\;\;\;\frac{t \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 42.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -31 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, z, -x\right) \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -31.0) (not (<= z 1.0)))
   (* t (/ x z))
   (* (fma (- x) z (- x)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -31.0) || !(z <= 1.0)) {
		tmp = t * (x / z);
	} else {
		tmp = fma(-x, z, -x) * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -31.0) || !(z <= 1.0))
		tmp = Float64(t * Float64(x / z));
	else
		tmp = Float64(fma(Float64(-x), z, Float64(-x)) * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -31.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(t * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(N[((-x) * z + (-x)), $MachinePrecision] * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -31 or 1 < z

   1. Initial program 96.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 86.8

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

   5. Applied rewrites 86.8%

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

   6. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 49.1

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

   8. Applied rewrites 49.1%

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

   if -31 < z < 1

   1. Initial program 96.2%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 81.3

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

   5. Applied rewrites 81.3%

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

   6. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lift--.f64 33.5

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

   8. Applied rewrites 33.5%

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

   9. Taylor expanded in z around 0

      \[\leadsto \left(-1 \cdot x + -1 \cdot \left(x \cdot z\right)\right) \cdot t \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(-1 \cdot \left(x \cdot z\right) + -1 \cdot x\right) \cdot t \]

       2. associate-*r* N/A

          \[\leadsto \left(\left(-1 \cdot x\right) \cdot z + -1 \cdot x\right) \cdot t \]

       3. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(-1 \cdot x, z, -1 \cdot x\right) \cdot t \]

       4. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), z, -1 \cdot x\right) \cdot t \]

       5. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(-x, z, -1 \cdot x\right) \cdot t \]

       6. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(-x, z, \mathsf{neg}\left(x\right)\right) \cdot t \]

       7. lift-neg.f64 32.5

          \[\leadsto \mathsf{fma}\left(-x, z, -x\right) \cdot t \]

   11. Applied rewrites 32.5%

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

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -31 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, z, -x\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 22.5% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \left(-t\right) \cdot x \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.6%

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

3. Taylor expanded in z around 0

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

   1. lower-/.f64 N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 63.0

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

5. Applied rewrites 63.0%

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

6. Taylor expanded in y around 0

   \[\leadsto -1 \cdot \color{blue}{\left(t \cdot x\right)} \]
7. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(-1 \cdot t\right) \cdot x \]

   2. lower-*.f64 N/A

      \[\leadsto \left(-1 \cdot t\right) \cdot x \]

   3. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(t\right)\right) \cdot x \]

   4. lift-neg.f64 22.3

      \[\leadsto \left(-t\right) \cdot x \]

8. Applied rewrites 22.3%

   \[\leadsto \left(-t\right) \cdot \color{blue}{x} \]
9. Add Preprocessing

Developer Target 1: 94.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ t_2 := x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \mathbf{if}\;t\_2 < -7.623226303312042 \cdot 10^{-196}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 < 1.4133944927702302 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z))))))
        (t_2 (* x (- (/ y z) (/ t (- 1.0 z))))))
   (if (< t_2 -7.623226303312042e-196)
     t_1
     (if (< t_2 1.4133944927702302e-211)
       (+ (/ (* y x) z) (- (/ (* t x) (- 1.0 z))))
       t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * ((y / z) - (t * (1.0 / (1.0 - z))));
	double t_2 = x * ((y / z) - (t / (1.0 - z)));
	double tmp;
	if (t_2 < -7.623226303312042e-196) {
		tmp = t_1;
	} else if (t_2 < 1.4133944927702302e-211) {
		tmp = ((y * x) / z) + -((t * x) / (1.0 - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x * ((y / z) - (t * (1.0d0 / (1.0d0 - z))))
    t_2 = x * ((y / z) - (t / (1.0d0 - z)))
    if (t_2 < (-7.623226303312042d-196)) then
        tmp = t_1
    else if (t_2 < 1.4133944927702302d-211) then
        tmp = ((y * x) / z) + -((t * x) / (1.0d0 - z))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * ((y / z) - (t * (1.0 / (1.0 - z))));
	double t_2 = x * ((y / z) - (t / (1.0 - z)));
	double tmp;
	if (t_2 < -7.623226303312042e-196) {
		tmp = t_1;
	} else if (t_2 < 1.4133944927702302e-211) {
		tmp = ((y * x) / z) + -((t * x) / (1.0 - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * ((y / z) - (t * (1.0 / (1.0 - z))))
	t_2 = x * ((y / z) - (t / (1.0 - z)))
	tmp = 0
	if t_2 < -7.623226303312042e-196:
		tmp = t_1
	elif t_2 < 1.4133944927702302e-211:
		tmp = ((y * x) / z) + -((t * x) / (1.0 - z))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(Float64(y / z) - Float64(t * Float64(1.0 / Float64(1.0 - z)))))
	t_2 = Float64(x * Float64(Float64(y / z) - Float64(t / Float64(1.0 - z))))
	tmp = 0.0
	if (t_2 < -7.623226303312042e-196)
		tmp = t_1;
	elseif (t_2 < 1.4133944927702302e-211)
		tmp = Float64(Float64(Float64(y * x) / z) + Float64(-Float64(Float64(t * x) / Float64(1.0 - z))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * ((y / z) - (t * (1.0 / (1.0 - z))));
	t_2 = x * ((y / z) - (t / (1.0 - z)));
	tmp = 0.0;
	if (t_2 < -7.623226303312042e-196)
		tmp = t_1;
	elseif (t_2 < 1.4133944927702302e-211)
		tmp = ((y * x) / z) + -((t * x) / (1.0 - z));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(N[(y / z), $MachinePrecision] - N[(t * N[(1.0 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$2, -7.623226303312042e-196], t$95$1, If[Less[t$95$2, 1.4133944927702302e-211], N[(N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision] + (-N[(N[(t * x), $MachinePrecision] / N[(1.0 - z), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], t$95$1]]]]

Reproduce

herbie shell --seed 2025037 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, C"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (* x (- (/ y z) (/ t (- 1 z)))) -3811613151656021/5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* x (- (/ y z) (* t (/ 1 (- 1 z))))) (if (< (* x (- (/ y z) (/ t (- 1 z)))) 7066972463851151/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (/ (* y x) z) (- (/ (* t x) (- 1 z)))) (* x (- (/ y z) (* t (/ 1 (- 1 z))))))))

  (* x (- (/ y z) (/ t (- 1.0 z)))))