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

Percentage Accurate: 94.8% → 98.2%

Time: 3.3s

Alternatives: 9

Speedup: 0.3×

• 20.6% 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: 94.8% 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: 98.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{z} \cdot y\\ t_2 := \frac{y}{z} - \frac{t}{1 - z}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+297}:\\ \;\;\;\;x \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ x z) y)) (t_2 (- (/ y z) (/ t (- 1.0 z)))))
   (if (<= t_2 (- INFINITY)) t_1 (if (<= t_2 4e+297) (* x t_2) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / z) * y;
	double t_2 = (y / z) - (t / (1.0 - z));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 4e+297) {
		tmp = x * t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x / z) * y;
	double t_2 = (y / z) - (t / (1.0 - z));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= 4e+297) {
		tmp = x * t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x / z) * y
	t_2 = (y / z) - (t / (1.0 - z))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= 4e+297:
		tmp = x * t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / z) * y)
	t_2 = Float64(Float64(y / z) - Float64(t / Float64(1.0 - z)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 4e+297)
		tmp = Float64(x * t_2);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x / z) * y;
	t_2 = (y / z) - (t / (1.0 - z));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= 4e+297)
		tmp = x * t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 4e+297], N[(x * t$95$2), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < -inf.0 or 4.0000000000000001e297 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z)))

   1. Initial program 71.3%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lower-/.f64 98.7

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in y around inf

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

      1. lift-/.f64 97.3

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

   7. Applied rewrites 97.3%

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

   if -inf.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < 4.0000000000000001e297

   1. Initial program 98.4%

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

Alternative 2: 97.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{-t}{y}, \frac{x}{1 - z}, \frac{x}{z}\right) \cdot y\\ t_2 := x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+293}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (fma (/ (- t) y) (/ x (- 1.0 z)) (/ x z)) y))
        (t_2 (* x (- (/ y z) (/ t (- 1.0 z))))))
   (if (<= t_2 (- INFINITY)) t_1 (if (<= t_2 5e+293) t_2 t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((-t / y), (x / (1.0 - z)), (x / z)) * y;
	double t_2 = x * ((y / z) - (t / (1.0 - z)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 5e+293) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(Float64(Float64(-t) / y), Float64(x / Float64(1.0 - z)), Float64(x / z)) * y)
	t_2 = Float64(x * Float64(Float64(y / z) - Float64(t / Float64(1.0 - z))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 5e+293)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[((-t) / y), $MachinePrecision] * N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(x / z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 5e+293], t$95$2, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z)))) < -inf.0 or 5.00000000000000033e293 < (*.f64 x (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))))

   1. Initial program 85.0%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lower-/.f64 94.7

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

   4. Applied rewrites 94.7%

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

   if -inf.0 < (*.f64 x (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z)))) < 5.00000000000000033e293

   1. Initial program 98.1%

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

Alternative 3: 93.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{t + y}{z}\\ \mathbf{if}\;z \leq -48:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ (+ t y) z))))
   (if (<= z -48.0) t_1 (if (<= z 1.1e-14) (* x (- (/ y z) t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * ((t + y) / z);
	double tmp;
	if (z <= -48.0) {
		tmp = t_1;
	} else if (z <= 1.1e-14) {
		tmp = x * ((y / z) - t);
	} 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 = x * ((t + y) / z)
    if (z <= (-48.0d0)) then
        tmp = t_1
    else if (z <= 1.1d-14) then
        tmp = x * ((y / z) - t)
    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 * ((t + y) / z);
	double tmp;
	if (z <= -48.0) {
		tmp = t_1;
	} else if (z <= 1.1e-14) {
		tmp = x * ((y / z) - t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * ((t + y) / z)
	tmp = 0
	if z <= -48.0:
		tmp = t_1
	elif z <= 1.1e-14:
		tmp = x * ((y / z) - t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(Float64(t + y) / z))
	tmp = 0.0
	if (z <= -48.0)
		tmp = t_1;
	elseif (z <= 1.1e-14)
		tmp = Float64(x * Float64(Float64(y / z) - t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * ((t + y) / z);
	tmp = 0.0;
	if (z <= -48.0)
		tmp = t_1;
	elseif (z <= 1.1e-14)
		tmp = x * ((y / z) - t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(N[(t + y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -48.0], t$95$1, If[LessEqual[z, 1.1e-14], N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -48 or 1.1e-14 < z

   1. Initial program 97.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 95.9

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

   4. Applied rewrites 95.9%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 95.9

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

   7. Applied rewrites 95.9%

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

   if -48 < z < 1.1e-14

   1. Initial program 92.2%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 91.7%

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

Alternative 4: 89.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(t + y\right) \cdot x}{z}\\ \mathbf{if}\;z \leq -48:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* (+ t y) x) z)))
   (if (<= z -48.0) t_1 (if (<= z 1.1e-14) (* x (- (/ y z) t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((t + y) * x) / z;
	double tmp;
	if (z <= -48.0) {
		tmp = t_1;
	} else if (z <= 1.1e-14) {
		tmp = x * ((y / z) - t);
	} 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 + y) * x) / z
    if (z <= (-48.0d0)) then
        tmp = t_1
    else if (z <= 1.1d-14) then
        tmp = x * ((y / z) - t)
    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 + y) * x) / z;
	double tmp;
	if (z <= -48.0) {
		tmp = t_1;
	} else if (z <= 1.1e-14) {
		tmp = x * ((y / z) - t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((t + y) * x) / z
	tmp = 0
	if z <= -48.0:
		tmp = t_1
	elif z <= 1.1e-14:
		tmp = x * ((y / z) - t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(t + y) * x) / z)
	tmp = 0.0
	if (z <= -48.0)
		tmp = t_1;
	elseif (z <= 1.1e-14)
		tmp = Float64(x * Float64(Float64(y / z) - t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((t + y) * x) / z;
	tmp = 0.0;
	if (z <= -48.0)
		tmp = t_1;
	elseif (z <= 1.1e-14)
		tmp = x * ((y / z) - t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(t + y), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[z, -48.0], t$95$1, If[LessEqual[z, 1.1e-14], N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -48 or 1.1e-14 < z

   1. Initial program 97.4%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{x \cdot \left(y - -1 \cdot t\right)}{z}} \]
   3. 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

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

   4. Applied rewrites 86.3%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 86.3

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

   7. Applied rewrites 86.3%

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

   if -48 < z < 1.1e-14

   1. Initial program 92.2%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 91.7%

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

Alternative 5: 74.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{y}{z}\\ \mathbf{if}\;z \leq -3.6 \cdot 10^{+196}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{+27}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;z \leq 4800:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+135}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ y z))))
   (if (<= z -3.6e+196)
     t_1
     (if (<= z -8.5e+27)
       (* x (/ t z))
       (if (<= z 4800.0)
         (* x (- (/ y z) t))
         (if (<= z 2.8e+135) (/ (* t x) z) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (y / z);
	double tmp;
	if (z <= -3.6e+196) {
		tmp = t_1;
	} else if (z <= -8.5e+27) {
		tmp = x * (t / z);
	} else if (z <= 4800.0) {
		tmp = x * ((y / z) - t);
	} else if (z <= 2.8e+135) {
		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 = x * (y / z)
    if (z <= (-3.6d+196)) then
        tmp = t_1
    else if (z <= (-8.5d+27)) then
        tmp = x * (t / z)
    else if (z <= 4800.0d0) then
        tmp = x * ((y / z) - t)
    else if (z <= 2.8d+135) 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 = x * (y / z);
	double tmp;
	if (z <= -3.6e+196) {
		tmp = t_1;
	} else if (z <= -8.5e+27) {
		tmp = x * (t / z);
	} else if (z <= 4800.0) {
		tmp = x * ((y / z) - t);
	} else if (z <= 2.8e+135) {
		tmp = (t * x) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (y / z)
	tmp = 0
	if z <= -3.6e+196:
		tmp = t_1
	elif z <= -8.5e+27:
		tmp = x * (t / z)
	elif z <= 4800.0:
		tmp = x * ((y / z) - t)
	elif z <= 2.8e+135:
		tmp = (t * x) / z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(y / z))
	tmp = 0.0
	if (z <= -3.6e+196)
		tmp = t_1;
	elseif (z <= -8.5e+27)
		tmp = Float64(x * Float64(t / z));
	elseif (z <= 4800.0)
		tmp = Float64(x * Float64(Float64(y / z) - t));
	elseif (z <= 2.8e+135)
		tmp = Float64(Float64(t * x) / z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * (y / z);
	tmp = 0.0;
	if (z <= -3.6e+196)
		tmp = t_1;
	elseif (z <= -8.5e+27)
		tmp = x * (t / z);
	elseif (z <= 4800.0)
		tmp = x * ((y / z) - t);
	elseif (z <= 2.8e+135)
		tmp = (t * x) / z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.6e+196], t$95$1, If[LessEqual[z, -8.5e+27], N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4800.0], N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.8e+135], N[(N[(t * x), $MachinePrecision] / z), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -3.60000000000000007e196 or 2.80000000000000002e135 < z

   1. Initial program 95.8%

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

   2. Taylor expanded in y around inf

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

      1. lift-/.f64 60.5

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

   4. Applied rewrites 60.5%

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

   if -3.60000000000000007e196 < z < -8.5e27

   1. Initial program 98.3%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 55.7%

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

   if -8.5e27 < z < 4800

   1. Initial program 92.7%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 90.1%

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

   if 4800 < z < 2.80000000000000002e135

   1. Initial program 98.9%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{x \cdot \left(y - -1 \cdot t\right)}{z}} \]
   3. 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 92.2

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

   4. Applied rewrites 92.2%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 49.0%

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

Alternative 6: 66.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ \mathbf{if}\;t \leq -1 \cdot 10^{+26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{+83}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ t z))))
   (if (<= t -1e+26) t_1 (if (<= t 1.55e+83) (* (/ x z) y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (t / z);
	double tmp;
	if (t <= -1e+26) {
		tmp = t_1;
	} else if (t <= 1.55e+83) {
		tmp = (x / z) * y;
	} 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 = x * (t / z)
    if (t <= (-1d+26)) then
        tmp = t_1
    else if (t <= 1.55d+83) then
        tmp = (x / z) * y
    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 * (t / z);
	double tmp;
	if (t <= -1e+26) {
		tmp = t_1;
	} else if (t <= 1.55e+83) {
		tmp = (x / z) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (t / z)
	tmp = 0
	if t <= -1e+26:
		tmp = t_1
	elif t <= 1.55e+83:
		tmp = (x / z) * y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(t / z))
	tmp = 0.0
	if (t <= -1e+26)
		tmp = t_1;
	elseif (t <= 1.55e+83)
		tmp = Float64(Float64(x / z) * y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * (t / z);
	tmp = 0.0;
	if (t <= -1e+26)
		tmp = t_1;
	elseif (t <= 1.55e+83)
		tmp = (x / z) * y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1e+26], t$95$1, If[LessEqual[t, 1.55e+83], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.00000000000000005e26 or 1.54999999999999996e83 < t

   1. Initial program 96.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 66.1

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

   4. Applied rewrites 66.1%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 50.7%

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

   if -1.00000000000000005e26 < t < 1.54999999999999996e83

   1. Initial program 93.9%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lower-/.f64 89.5

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

   4. Applied rewrites 89.5%

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

   5. Taylor expanded in y around inf

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

      1. lift-/.f64 77.5

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

   7. Applied rewrites 77.5%

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

Alternative 7: 63.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{+26}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{+83}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1e+26)
   (/ (* t x) z)
   (if (<= t 1.6e+83) (* (/ x z) y) (* (/ x z) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1e+26) {
		tmp = (t * x) / z;
	} else if (t <= 1.6e+83) {
		tmp = (x / z) * y;
	} else {
		tmp = (x / 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 (t <= (-1d+26)) then
        tmp = (t * x) / z
    else if (t <= 1.6d+83) then
        tmp = (x / z) * y
    else
        tmp = (x / z) * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1e+26) {
		tmp = (t * x) / z;
	} else if (t <= 1.6e+83) {
		tmp = (x / z) * y;
	} else {
		tmp = (x / z) * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1e+26:
		tmp = (t * x) / z
	elif t <= 1.6e+83:
		tmp = (x / z) * y
	else:
		tmp = (x / z) * t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1e+26)
		tmp = Float64(Float64(t * x) / z);
	elseif (t <= 1.6e+83)
		tmp = Float64(Float64(x / z) * y);
	else
		tmp = Float64(Float64(x / z) * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1e+26)
		tmp = (t * x) / z;
	elseif (t <= 1.6e+83)
		tmp = (x / z) * y;
	else
		tmp = (x / z) * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1e+26], N[(N[(t * x), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t, 1.6e+83], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.00000000000000005e26

   1. Initial program 96.4%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{x \cdot \left(y - -1 \cdot t\right)}{z}} \]
   3. 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 59.0

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

   4. Applied rewrites 59.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 42.7%

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

   if -1.00000000000000005e26 < t < 1.5999999999999999e83

   1. Initial program 93.9%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lower-/.f64 89.5

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

   4. Applied rewrites 89.5%

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

   5. Taylor expanded in y around inf

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

      1. lift-/.f64 77.5

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

   7. Applied rewrites 77.5%

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

   if 1.5999999999999999e83 < t

   1. Initial program 95.8%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{x \cdot \left(y - -1 \cdot t\right)}{z}} \]
   3. 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 58.8

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

   4. Applied rewrites 58.8%

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

   5. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 45.2

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

   7. Applied rewrites 45.2%

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

Alternative 8: 63.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{z} \cdot t\\ \mathbf{if}\;t \leq -1 \cdot 10^{+26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{+83}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ x z) t)))
   (if (<= t -1e+26) t_1 (if (<= t 1.6e+83) (* (/ x z) y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / z) * t;
	double tmp;
	if (t <= -1e+26) {
		tmp = t_1;
	} else if (t <= 1.6e+83) {
		tmp = (x / z) * y;
	} 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 = (x / z) * t
    if (t <= (-1d+26)) then
        tmp = t_1
    else if (t <= 1.6d+83) then
        tmp = (x / z) * y
    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 / z) * t;
	double tmp;
	if (t <= -1e+26) {
		tmp = t_1;
	} else if (t <= 1.6e+83) {
		tmp = (x / z) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x / z) * t
	tmp = 0
	if t <= -1e+26:
		tmp = t_1
	elif t <= 1.6e+83:
		tmp = (x / z) * y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / z) * t)
	tmp = 0.0
	if (t <= -1e+26)
		tmp = t_1;
	elseif (t <= 1.6e+83)
		tmp = Float64(Float64(x / z) * y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x / z) * t;
	tmp = 0.0;
	if (t <= -1e+26)
		tmp = t_1;
	elseif (t <= 1.6e+83)
		tmp = (x / z) * y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t, -1e+26], t$95$1, If[LessEqual[t, 1.6e+83], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.00000000000000005e26 or 1.5999999999999999e83 < t

   1. Initial program 96.1%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{x \cdot \left(y - -1 \cdot t\right)}{z}} \]
   3. 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 58.9

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

   4. Applied rewrites 58.9%

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

   5. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 43.7

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

   7. Applied rewrites 43.7%

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

   if -1.00000000000000005e26 < t < 1.5999999999999999e83

   1. Initial program 93.9%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lower-/.f64 89.5

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

   4. Applied rewrites 89.5%

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

   5. Taylor expanded in y around inf

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

      1. lift-/.f64 77.5

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

   7. Applied rewrites 77.5%

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

Alternative 9: 34.3% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.8%

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

2. Taylor expanded in z around inf

   \[\leadsto \color{blue}{\frac{x \cdot \left(y - -1 \cdot t\right)}{z}} \]
3. 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 71.6

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

4. Applied rewrites 71.6%

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

5. Taylor expanded in y around 0

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lift-/.f64 34.3

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

7. Applied rewrites 34.3%

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

Reproduce

herbie shell --seed 2025112 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, C"
  :precision binary64
  (* x (- (/ y z) (/ t (- 1.0 z)))))