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

Percentage Accurate: 94.4% → 98.5%

Time: 3.6s

Alternatives: 11

Speedup: 0.3×

• 21.0% 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: 94.4% 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: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{x}{z}\\ t_2 := \frac{y}{z} - \frac{t}{1 - z}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+308}:\\ \;\;\;\;x \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (/ x z))) (t_2 (- (/ y z) (/ t (- 1.0 z)))))
   (if (<= t_2 (- INFINITY)) t_1 (if (<= t_2 1e+308) (* x t_2) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * (x / z);
	double t_2 = (y / z) - (t / (1.0 - z));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 1e+308) {
		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 = y * (x / z);
	double t_2 = (y / z) - (t / (1.0 - z));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= 1e+308) {
		tmp = x * t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = y * (x / z)
	t_2 = (y / z) - (t / (1.0 - z))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= 1e+308:
		tmp = x * t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(x / z))
	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 <= 1e+308)
		tmp = Float64(x * t_2);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = y * (x / z);
	t_2 = (y / z) - (t / (1.0 - z));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= 1e+308)
		tmp = x * t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(x / z), $MachinePrecision]), $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, 1e+308], 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 1e308 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z)))

   1. Initial program 67.8%

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

   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 87.0

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

   4. Applied rewrites 87.0%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 99.6%

      \[\leadsto \frac{y \cdot x}{z} \]
   8. 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 99.6

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

   9. Applied rewrites 99.6%

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

   if -inf.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < 1e308

   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: 90.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 97.3%

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

   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 85.6

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

   4. Applied rewrites 85.6%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 85.6

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

   7. Applied rewrites 85.6%

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

   if -1 < z < 1

   1. Initial program 91.6%

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

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{-1 \cdot \left(t \cdot \left(x \cdot z\right)\right) + x \cdot y}{z}} \]
   3. 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 89.8

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

   4. Applied rewrites 89.8%

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

   if 1 < 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 96.6

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

   4. Applied rewrites 96.6%

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

Alternative 3: 75.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(t + y\right) \cdot x}{z}\\ \mathbf{if}\;z \leq -1.6 \cdot 10^{-36}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-167}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-114}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* (+ t y) x) z)))
   (if (<= z -1.6e-36)
     t_1
     (if (<= z 1.4e-167) (/ (* y x) z) (if (<= z 6.2e-114) (* x (- 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 <= -1.6e-36) {
		tmp = t_1;
	} else if (z <= 1.4e-167) {
		tmp = (y * x) / z;
	} else if (z <= 6.2e-114) {
		tmp = x * -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 <= (-1.6d-36)) then
        tmp = t_1
    else if (z <= 1.4d-167) then
        tmp = (y * x) / z
    else if (z <= 6.2d-114) then
        tmp = x * -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 <= -1.6e-36) {
		tmp = t_1;
	} else if (z <= 1.4e-167) {
		tmp = (y * x) / z;
	} else if (z <= 6.2e-114) {
		tmp = x * -t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((t + y) * x) / z
	tmp = 0
	if z <= -1.6e-36:
		tmp = t_1
	elif z <= 1.4e-167:
		tmp = (y * x) / z
	elif z <= 6.2e-114:
		tmp = x * -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 <= -1.6e-36)
		tmp = t_1;
	elseif (z <= 1.4e-167)
		tmp = Float64(Float64(y * x) / z);
	elseif (z <= 6.2e-114)
		tmp = Float64(x * Float64(-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 <= -1.6e-36)
		tmp = t_1;
	elseif (z <= 1.4e-167)
		tmp = (y * x) / z;
	elseif (z <= 6.2e-114)
		tmp = x * -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, -1.6e-36], t$95$1, If[LessEqual[z, 1.4e-167], N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 6.2e-114], N[(x * (-t)), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.60000000000000011e-36 or 6.2e-114 < z

   1. Initial program 97.2%

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

   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 80.4

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

   4. Applied rewrites 80.4%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 80.4

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

   7. Applied rewrites 80.4%

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

   if -1.60000000000000011e-36 < z < 1.39999999999999993e-167

   1. Initial program 90.0%

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

   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 57.6

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

   4. Applied rewrites 57.6%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 72.2%

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

   if 1.39999999999999993e-167 < z < 6.2e-114

   1. Initial program 90.4%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 90.4

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

   4. Applied rewrites 90.4%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 34.7

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

   7. Applied rewrites 34.7%

      \[\leadsto x \cdot \left(-t\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 91.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.8:\\ \;\;\;\;\frac{\left(t + y\right) \cdot x}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - \left(-t\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -0.8)
   (/ (* (+ t y) x) z)
   (if (<= z 1.0) (* x (- (/ y z) t)) (* x (/ (- y (- t)) z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -0.8) {
		tmp = ((t + y) * x) / z;
	} else if (z <= 1.0) {
		tmp = x * ((y / z) - t);
	} else {
		tmp = x * ((y - -t) / 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 <= (-0.8d0)) then
        tmp = ((t + y) * x) / z
    else if (z <= 1.0d0) then
        tmp = x * ((y / z) - t)
    else
        tmp = x * ((y - -t) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -0.8) {
		tmp = ((t + y) * x) / z;
	} else if (z <= 1.0) {
		tmp = x * ((y / z) - t);
	} else {
		tmp = x * ((y - -t) / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -0.8:
		tmp = ((t + y) * x) / z
	elif z <= 1.0:
		tmp = x * ((y / z) - t)
	else:
		tmp = x * ((y - -t) / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -0.8)
		tmp = Float64(Float64(Float64(t + y) * x) / z);
	elseif (z <= 1.0)
		tmp = Float64(x * Float64(Float64(y / z) - t));
	else
		tmp = Float64(x * Float64(Float64(y - Float64(-t)) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -0.8)
		tmp = ((t + y) * x) / z;
	elseif (z <= 1.0)
		tmp = x * ((y / z) - t);
	else
		tmp = x * ((y - -t) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -0.8], N[(N[(N[(t + y), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 1.0], N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y - (-t)), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.80000000000000004

   1. Initial program 97.3%

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

   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 85.6

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

   4. Applied rewrites 85.6%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 85.6

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

   7. Applied rewrites 85.6%

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

   if -0.80000000000000004 < z < 1

   1. Initial program 91.6%

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

   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 90.8%

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

   if 1 < 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 96.6

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

   4. Applied rewrites 96.6%

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

Alternative 5: 88.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.8:\\ \;\;\;\;\frac{\left(t + y\right) \cdot x}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - \left(-t\right)\right) \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -0.8)
   (/ (* (+ t y) x) z)
   (if (<= z 1.0) (* x (- (/ y z) t)) (* (- y (- t)) (/ x z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -0.8) {
		tmp = ((t + y) * x) / z;
	} else if (z <= 1.0) {
		tmp = x * ((y / z) - t);
	} else {
		tmp = (y - -t) * (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 <= (-0.8d0)) then
        tmp = ((t + y) * x) / z
    else if (z <= 1.0d0) then
        tmp = x * ((y / z) - t)
    else
        tmp = (y - -t) * (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 <= -0.8) {
		tmp = ((t + y) * x) / z;
	} else if (z <= 1.0) {
		tmp = x * ((y / z) - t);
	} else {
		tmp = (y - -t) * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -0.8:
		tmp = ((t + y) * x) / z
	elif z <= 1.0:
		tmp = x * ((y / z) - t)
	else:
		tmp = (y - -t) * (x / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -0.8)
		tmp = Float64(Float64(Float64(t + y) * x) / z);
	elseif (z <= 1.0)
		tmp = Float64(x * Float64(Float64(y / z) - t));
	else
		tmp = Float64(Float64(y - Float64(-t)) * Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -0.8)
		tmp = ((t + y) * x) / z;
	elseif (z <= 1.0)
		tmp = x * ((y / z) - t);
	else
		tmp = (y - -t) * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -0.8], N[(N[(N[(t + y), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 1.0], N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[(y - (-t)), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.80000000000000004

   1. Initial program 97.3%

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

   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 85.6

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

   4. Applied rewrites 85.6%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 85.6

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

   7. Applied rewrites 85.6%

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

   if -0.80000000000000004 < z < 1

   1. Initial program 91.6%

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

   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 90.8%

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

   if 1 < 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 85.8

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

   4. Applied rewrites 85.8%

      \[\leadsto \color{blue}{\frac{\left(y - \left(-t\right)\right) \cdot x}{z}} \]
   5. 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. lower-/.f64 86.7

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

   6. Applied rewrites 86.7%

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

Alternative 6: 88.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(t + y\right) \cdot x}{z}\\ \mathbf{if}\;z \leq -0.8:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* (+ t y) x) z)))
   (if (<= z -0.8) t_1 (if (<= z 1.0) (* 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 <= -0.8) {
		tmp = t_1;
	} else if (z <= 1.0) {
		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 <= (-0.8d0)) then
        tmp = t_1
    else if (z <= 1.0d0) 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 <= -0.8) {
		tmp = t_1;
	} else if (z <= 1.0) {
		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 <= -0.8:
		tmp = t_1
	elif z <= 1.0:
		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 <= -0.8)
		tmp = t_1;
	elseif (z <= 1.0)
		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 <= -0.8)
		tmp = t_1;
	elseif (z <= 1.0)
		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, -0.8], t$95$1, If[LessEqual[z, 1.0], N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.80000000000000004 or 1 < z

   1. Initial program 97.3%

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

   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 85.7

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

   4. Applied rewrites 85.7%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 85.7

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

   7. Applied rewrites 85.7%

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

   if -0.80000000000000004 < z < 1

   1. Initial program 91.6%

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

   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 90.8%

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

Alternative 7: 62.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.82 \cdot 10^{-183}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{-251}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.82e-183)
   (/ (* y x) z)
   (if (<= y 7.6e-251) (* x (- t)) (* y (/ x z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.82e-183) {
		tmp = (y * x) / z;
	} else if (y <= 7.6e-251) {
		tmp = x * -t;
	} 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 (y <= (-1.82d-183)) then
        tmp = (y * x) / z
    else if (y <= 7.6d-251) then
        tmp = x * -t
    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 (y <= -1.82e-183) {
		tmp = (y * x) / z;
	} else if (y <= 7.6e-251) {
		tmp = x * -t;
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -1.82e-183:
		tmp = (y * x) / z
	elif y <= 7.6e-251:
		tmp = x * -t
	else:
		tmp = y * (x / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.82e-183)
		tmp = Float64(Float64(y * x) / z);
	elseif (y <= 7.6e-251)
		tmp = Float64(x * Float64(-t));
	else
		tmp = Float64(y * Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -1.82e-183)
		tmp = (y * x) / z;
	elseif (y <= 7.6e-251)
		tmp = x * -t;
	else
		tmp = y * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.82e-183], N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[y, 7.6e-251], N[(x * (-t)), $MachinePrecision], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.82e-183

   1. Initial program 93.7%

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

   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 75.0

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

   4. Applied rewrites 75.0%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 67.7%

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

   if -1.82e-183 < y < 7.5999999999999994e-251

   1. Initial program 97.2%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 58.5

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

   4. Applied rewrites 58.5%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 44.3

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

   7. Applied rewrites 44.3%

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

   if 7.5999999999999994e-251 < y

   1. Initial program 94.2%

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

   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 72.6

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 64.2%

      \[\leadsto \frac{y \cdot x}{z} \]
   8. 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 64.6

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

   9. Applied rewrites 64.6%

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

Alternative 8: 64.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{+193}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{+235}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.5e+193)
   (* x (- t))
   (if (<= t 8.2e+235) (* y (/ x z)) (/ (* t x) z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.5e+193) {
		tmp = x * -t;
	} else if (t <= 8.2e+235) {
		tmp = y * (x / z);
	} else {
		tmp = (t * 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 <= (-1.5d+193)) then
        tmp = x * -t
    else if (t <= 8.2d+235) then
        tmp = y * (x / z)
    else
        tmp = (t * 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 <= -1.5e+193) {
		tmp = x * -t;
	} else if (t <= 8.2e+235) {
		tmp = y * (x / z);
	} else {
		tmp = (t * x) / z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.5e+193:
		tmp = x * -t
	elif t <= 8.2e+235:
		tmp = y * (x / z)
	else:
		tmp = (t * x) / z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.5e+193)
		tmp = Float64(x * Float64(-t));
	elseif (t <= 8.2e+235)
		tmp = Float64(y * Float64(x / z));
	else
		tmp = Float64(Float64(t * x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1.5e+193)
		tmp = x * -t;
	elseif (t <= 8.2e+235)
		tmp = y * (x / z);
	else
		tmp = (t * x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.5e+193], N[(x * (-t)), $MachinePrecision], If[LessEqual[t, 8.2e+235], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(N[(t * x), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.5e193

   1. Initial program 96.3%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 56.4

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

   4. Applied rewrites 56.4%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 45.4

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

   7. Applied rewrites 45.4%

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

   if -1.5e193 < t < 8.2000000000000003e235

   1. Initial program 94.2%

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

   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 74.4

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

   4. Applied rewrites 74.4%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 67.0%

      \[\leadsto \frac{y \cdot x}{z} \]
   8. 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 67.3

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

   9. Applied rewrites 67.3%

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

   if 8.2000000000000003e235 < t

   1. Initial program 95.2%

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

   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 55.9

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

   4. Applied rewrites 55.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 52.0%

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

Alternative 9: 64.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{+193}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{+235}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.5e+193)
   (* x (- t))
   (if (<= t 8.2e+235) (* y (/ x z)) (* t (/ x z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.5e+193) {
		tmp = x * -t;
	} else if (t <= 8.2e+235) {
		tmp = y * (x / z);
	} else {
		tmp = t * (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 <= (-1.5d+193)) then
        tmp = x * -t
    else if (t <= 8.2d+235) then
        tmp = y * (x / z)
    else
        tmp = t * (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 <= -1.5e+193) {
		tmp = x * -t;
	} else if (t <= 8.2e+235) {
		tmp = y * (x / z);
	} else {
		tmp = t * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.5e+193:
		tmp = x * -t
	elif t <= 8.2e+235:
		tmp = y * (x / z)
	else:
		tmp = t * (x / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.5e+193)
		tmp = Float64(x * Float64(-t));
	elseif (t <= 8.2e+235)
		tmp = Float64(y * Float64(x / z));
	else
		tmp = Float64(t * Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1.5e+193)
		tmp = x * -t;
	elseif (t <= 8.2e+235)
		tmp = y * (x / z);
	else
		tmp = t * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.5e+193], N[(x * (-t)), $MachinePrecision], If[LessEqual[t, 8.2e+235], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(t * N[(x / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.5e193

   1. Initial program 96.3%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 56.4

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

   4. Applied rewrites 56.4%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 45.4

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

   7. Applied rewrites 45.4%

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

   if -1.5e193 < t < 8.2000000000000003e235

   1. Initial program 94.2%

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

   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 74.4

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

   4. Applied rewrites 74.4%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 67.0%

      \[\leadsto \frac{y \cdot x}{z} \]
   8. 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 67.3

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

   9. Applied rewrites 67.3%

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

   if 8.2000000000000003e235 < t

   1. Initial program 95.2%

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

   2. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lift--.f64 72.7

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

   4. Applied rewrites 72.7%

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

   5. Taylor expanded in z around inf

      \[\leadsto \frac{t \cdot x}{\color{blue}{z}} \]
   6. 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. lower-/.f64 51.6

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

   7. Applied rewrites 51.6%

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

Alternative 10: 42.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -0.016:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1860:\\ \;\;\;\;x \cdot \left(\left(\left(-z\right) - 1\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* t (/ x z))))
   (if (<= z -0.016) t_1 (if (<= z 1860.0) (* x (* (- (- z) 1.0) t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = t * (x / z);
	double tmp;
	if (z <= -0.016) {
		tmp = t_1;
	} else if (z <= 1860.0) {
		tmp = x * ((-z - 1.0) * 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 * (x / z)
    if (z <= (-0.016d0)) then
        tmp = t_1
    else if (z <= 1860.0d0) then
        tmp = x * ((-z - 1.0d0) * 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 * (x / z);
	double tmp;
	if (z <= -0.016) {
		tmp = t_1;
	} else if (z <= 1860.0) {
		tmp = x * ((-z - 1.0) * t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = t * (x / z)
	tmp = 0
	if z <= -0.016:
		tmp = t_1
	elif z <= 1860.0:
		tmp = x * ((-z - 1.0) * t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(t * Float64(x / z))
	tmp = 0.0
	if (z <= -0.016)
		tmp = t_1;
	elseif (z <= 1860.0)
		tmp = Float64(x * Float64(Float64(Float64(-z) - 1.0) * t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = t * (x / z);
	tmp = 0.0;
	if (z <= -0.016)
		tmp = t_1;
	elseif (z <= 1860.0)
		tmp = x * ((-z - 1.0) * t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(t * N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -0.016], t$95$1, If[LessEqual[z, 1860.0], N[(x * N[(N[((-z) - 1.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.016 or 1860 < z

   1. Initial program 97.3%

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

   2. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lift--.f64 53.3

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

   4. Applied rewrites 53.3%

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

   5. Taylor expanded in z around inf

      \[\leadsto \frac{t \cdot x}{\color{blue}{z}} \]
   6. 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. lower-/.f64 52.7

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

   7. Applied rewrites 52.7%

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

   if -0.016 < z < 1860

   1. Initial program 91.6%

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

   2. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \frac{t}{1 - z}\right)} \]
   3. 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 33.8

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

   4. Applied rewrites 33.8%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lift-neg.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lift-neg.f64 33.2

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

   7. Applied rewrites 33.2%

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

   8. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-neg.f64 33.2

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

   10. Applied rewrites 33.2%

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

Alternative 11: 23.3% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.4%

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

2. Taylor expanded in z around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 64.6

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

4. Applied rewrites 64.6%

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

5. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

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

   2. lift-neg.f64 23.3

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

7. Applied rewrites 23.3%

   \[\leadsto x \cdot \left(-t\right) \]
8. Add Preprocessing

Developer Target 1: 94.8% 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 2025093 
(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)))))