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

Percentage Accurate: 94.7% → 98.3%

Time: 4.1s

Alternatives: 11

Speedup: 0.3×

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

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.7% 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: 98.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{z} - \frac{t}{1 - z}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+282}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-t, z \cdot x, y \cdot x\right)}{z}\\ \mathbf{elif}\;t\_1 \leq 10^{+303}:\\ \;\;\;\;x \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (/ y z) (/ t (- 1.0 z)))))
   (if (<= t_1 -1e+282)
     (/ (fma (- t) (* z x) (* y x)) z)
     (if (<= t_1 1e+303) (* x t_1) (* (/ x z) y)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y / z) - (t / (1.0 - z));
	double tmp;
	if (t_1 <= -1e+282) {
		tmp = fma(-t, (z * x), (y * x)) / z;
	} else if (t_1 <= 1e+303) {
		tmp = x * t_1;
	} else {
		tmp = (x / z) * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y / z) - Float64(t / Float64(1.0 - z)))
	tmp = 0.0
	if (t_1 <= -1e+282)
		tmp = Float64(fma(Float64(-t), Float64(z * x), Float64(y * x)) / z);
	elseif (t_1 <= 1e+303)
		tmp = Float64(x * t_1);
	else
		tmp = Float64(Float64(x / z) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+282], N[(N[((-t) * N[(z * x), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 1e+303], N[(x * t$95$1), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < -1.00000000000000003e282

   1. Initial program 76.5%

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

   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 96.9

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

   4. Applied rewrites 96.9%

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

   if -1.00000000000000003e282 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < 1e303

   1. Initial program 98.2%

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

   if 1e303 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z)))

   1. Initial program 70.9%

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

   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 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 98.3

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

   7. Applied rewrites 98.3%

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

Alternative 2: 98.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{z} \cdot y\\ t_2 := \frac{y}{z} - \frac{t}{1 - z}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+303}:\\ \;\;\;\;x \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 1e+303) (* 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 <= 1e+303) {
		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 <= 1e+303) {
		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 <= 1e+303:
		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 <= 1e+303)
		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 <= 1e+303)
		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, 1e+303], 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 1e303 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z)))

   1. Initial program 69.9%

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

   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 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in y around inf

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

      1. lift-/.f64 99.1

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

   7. Applied rewrites 99.1%

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

   if -inf.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < 1e303

   1. Initial program 98.2%

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

Alternative 3: 94.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{y - \left(-t\right)}{z}\\ \mathbf{if}\;z \leq -0.8:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 0.0128:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \mathsf{fma}\left(t, z, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ (- y (- t)) z))))
   (if (<= z -0.8) t_1 (if (<= z 0.0128) (* x (- (/ y z) (fma t z t))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * ((y - -t) / z);
	double tmp;
	if (z <= -0.8) {
		tmp = t_1;
	} else if (z <= 0.0128) {
		tmp = x * ((y / z) - fma(t, z, t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(Float64(y - Float64(-t)) / z))
	tmp = 0.0
	if (z <= -0.8)
		tmp = t_1;
	elseif (z <= 0.0128)
		tmp = Float64(x * Float64(Float64(y / z) - fma(t, z, t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(N[(y - (-t)), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -0.8], t$95$1, If[LessEqual[z, 0.0128], N[(x * N[(N[(y / z), $MachinePrecision] - N[(t * z + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.80000000000000004 or 0.0128000000000000006 < z

   1. Initial program 97.0%

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

   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 96.2

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

   4. Applied rewrites 96.2%

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

   if -0.80000000000000004 < z < 0.0128000000000000006

   1. Initial program 92.4%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 92.0

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

   4. Applied rewrites 92.0%

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

Alternative 4: 93.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{y - \left(-t\right)}{z}\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 0.0128:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ (- y (- t)) z))))
   (if (<= z -1.0) t_1 (if (<= z 0.0128) (* x (- (/ y z) t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * ((y - -t) / z);
	double tmp;
	if (z <= -1.0) {
		tmp = t_1;
	} else if (z <= 0.0128) {
		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 * ((y - -t) / z)
    if (z <= (-1.0d0)) then
        tmp = t_1
    else if (z <= 0.0128d0) 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 * ((y - -t) / z);
	double tmp;
	if (z <= -1.0) {
		tmp = t_1;
	} else if (z <= 0.0128) {
		tmp = x * ((y / z) - t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * ((y - -t) / z)
	tmp = 0
	if z <= -1.0:
		tmp = t_1
	elif z <= 0.0128:
		tmp = x * ((y / z) - t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(Float64(y - Float64(-t)) / z))
	tmp = 0.0
	if (z <= -1.0)
		tmp = t_1;
	elseif (z <= 0.0128)
		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 * ((y - -t) / z);
	tmp = 0.0;
	if (z <= -1.0)
		tmp = t_1;
	elseif (z <= 0.0128)
		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[(y - (-t)), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.0], t$95$1, If[LessEqual[z, 0.0128], N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 0.0128000000000000006 < z

   1. Initial program 97.0%

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

   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 96.2

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

   4. Applied rewrites 96.2%

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

   if -1 < z < 0.0128000000000000006

   1. Initial program 92.4%

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

   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 91.6%

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

Alternative 5: 89.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(y - \left(-t\right)\right) \cdot x}{z}\\ \mathbf{if}\;z \leq -0.02:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 0.0128:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* (- y (- t)) x) z)))
   (if (<= z -0.02) t_1 (if (<= z 0.0128) (* x (- (/ y z) t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((y - -t) * x) / z;
	double tmp;
	if (z <= -0.02) {
		tmp = t_1;
	} else if (z <= 0.0128) {
		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 = ((y - -t) * x) / z
    if (z <= (-0.02d0)) then
        tmp = t_1
    else if (z <= 0.0128d0) 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 = ((y - -t) * x) / z;
	double tmp;
	if (z <= -0.02) {
		tmp = t_1;
	} else if (z <= 0.0128) {
		tmp = x * ((y / z) - t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((y - -t) * x) / z
	tmp = 0
	if z <= -0.02:
		tmp = t_1
	elif z <= 0.0128:
		tmp = x * ((y / z) - t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(y - Float64(-t)) * x) / z)
	tmp = 0.0
	if (z <= -0.02)
		tmp = t_1;
	elseif (z <= 0.0128)
		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 = ((y - -t) * x) / z;
	tmp = 0.0;
	if (z <= -0.02)
		tmp = t_1;
	elseif (z <= 0.0128)
		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[(y - (-t)), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[z, -0.02], t$95$1, If[LessEqual[z, 0.0128], N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.0200000000000000004 or 0.0128000000000000006 < z

   1. Initial program 97.0%

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

   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 86.2

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

   4. Applied rewrites 86.2%

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

   if -0.0200000000000000004 < z < 0.0128000000000000006

   1. Initial program 92.3%

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

   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 6: 72.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-x}{1 - z} \cdot t\\ \mathbf{if}\;t \leq -9.5 \cdot 10^{+45}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+24}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ (- x) (- 1.0 z)) t)))
   (if (<= t -9.5e+45) t_1 (if (<= t 4.5e+24) (* (/ x z) y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (-x / (1.0 - z)) * t;
	double tmp;
	if (t <= -9.5e+45) {
		tmp = t_1;
	} else if (t <= 4.5e+24) {
		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 / (1.0d0 - z)) * t
    if (t <= (-9.5d+45)) then
        tmp = t_1
    else if (t <= 4.5d+24) 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 / (1.0 - z)) * t;
	double tmp;
	if (t <= -9.5e+45) {
		tmp = t_1;
	} else if (t <= 4.5e+24) {
		tmp = (x / z) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (-x / (1.0 - z)) * t
	tmp = 0
	if t <= -9.5e+45:
		tmp = t_1
	elif t <= 4.5e+24:
		tmp = (x / z) * y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(-x) / Float64(1.0 - z)) * t)
	tmp = 0.0
	if (t <= -9.5e+45)
		tmp = t_1;
	elseif (t <= 4.5e+24)
		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 / (1.0 - z)) * t;
	tmp = 0.0;
	if (t <= -9.5e+45)
		tmp = t_1;
	elseif (t <= 4.5e+24)
		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) / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t, -9.5e+45], t$95$1, If[LessEqual[t, 4.5e+24], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -9.4999999999999998e45 or 4.50000000000000019e24 < t

   1. Initial program 96.2%

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

   2. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 87.9

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

   4. Applied rewrites 87.9%

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

   5. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lift--.f64 63.1

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

   7. Applied rewrites 63.1%

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

   if -9.4999999999999998e45 < t < 4.50000000000000019e24

   1. Initial program 93.5%

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

   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 90.9

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

   4. Applied rewrites 90.9%

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

   5. Taylor expanded in y around inf

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

      1. lift-/.f64 79.9

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

   7. Applied rewrites 79.9%

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

Alternative 7: 72.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+160}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{+67}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{elif}\;z \leq 1.06 \cdot 10^{+141}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -3.8e+160)
   (* x (/ t z))
   (if (<= z 1.95e+67)
     (* x (- (/ y z) t))
     (if (<= z 1.06e+141) (* (/ x z) t) (* x (/ y z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.8e+160) {
		tmp = x * (t / z);
	} else if (z <= 1.95e+67) {
		tmp = x * ((y / z) - t);
	} else if (z <= 1.06e+141) {
		tmp = (x / z) * t;
	} else {
		tmp = x * (y / 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 <= (-3.8d+160)) then
        tmp = x * (t / z)
    else if (z <= 1.95d+67) then
        tmp = x * ((y / z) - t)
    else if (z <= 1.06d+141) then
        tmp = (x / z) * t
    else
        tmp = x * (y / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.8e+160) {
		tmp = x * (t / z);
	} else if (z <= 1.95e+67) {
		tmp = x * ((y / z) - t);
	} else if (z <= 1.06e+141) {
		tmp = (x / z) * t;
	} else {
		tmp = x * (y / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -3.8e+160:
		tmp = x * (t / z)
	elif z <= 1.95e+67:
		tmp = x * ((y / z) - t)
	elif z <= 1.06e+141:
		tmp = (x / z) * t
	else:
		tmp = x * (y / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -3.8e+160)
		tmp = Float64(x * Float64(t / z));
	elseif (z <= 1.95e+67)
		tmp = Float64(x * Float64(Float64(y / z) - t));
	elseif (z <= 1.06e+141)
		tmp = Float64(Float64(x / z) * t);
	else
		tmp = Float64(x * Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -3.8e+160)
		tmp = x * (t / z);
	elseif (z <= 1.95e+67)
		tmp = x * ((y / z) - t);
	elseif (z <= 1.06e+141)
		tmp = (x / z) * t;
	else
		tmp = x * (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -3.8e+160], N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.95e+67], N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.06e+141], N[(N[(x / z), $MachinePrecision] * t), $MachinePrecision], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -3.80000000000000012e160

   1. Initial program 94.6%

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

   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 58.5

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

   4. Applied rewrites 58.5%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 58.5

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

   7. Applied rewrites 58.5%

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

   if -3.80000000000000012e160 < z < 1.95000000000000003e67

   1. Initial program 94.2%

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

   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 80.3%

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

   if 1.95000000000000003e67 < z < 1.05999999999999997e141

   1. Initial program 99.2%

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

   2. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 84.6

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

   4. Applied rewrites 84.6%

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

   5. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lift--.f64 50.4

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

   7. Applied rewrites 50.4%

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

   8. Taylor expanded in z around inf

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

      1. lift-/.f64 50.4

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

   10. Applied rewrites 50.4%

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

   if 1.05999999999999997e141 < z

   1. Initial program 95.2%

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

   2. Taylor expanded in y around inf

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

      1. lift-/.f64 57.0

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

   4. Applied rewrites 57.0%

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

Alternative 8: 64.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{+159}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+24}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -5.5e+159)
   (* x (- t))
   (if (<= t 4.5e+24) (* (/ x z) y) (* x (/ t z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -5.5e+159) {
		tmp = x * -t;
	} else if (t <= 4.5e+24) {
		tmp = (x / z) * y;
	} else {
		tmp = x * (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 (t <= (-5.5d+159)) then
        tmp = x * -t
    else if (t <= 4.5d+24) then
        tmp = (x / z) * y
    else
        tmp = x * (t / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -5.5e+159) {
		tmp = x * -t;
	} else if (t <= 4.5e+24) {
		tmp = (x / z) * y;
	} else {
		tmp = x * (t / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -5.5e+159:
		tmp = x * -t
	elif t <= 4.5e+24:
		tmp = (x / z) * y
	else:
		tmp = x * (t / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -5.5e+159)
		tmp = Float64(x * Float64(-t));
	elseif (t <= 4.5e+24)
		tmp = Float64(Float64(x / z) * y);
	else
		tmp = Float64(x * Float64(t / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -5.5e+159)
		tmp = x * -t;
	elseif (t <= 4.5e+24)
		tmp = (x / z) * y;
	else
		tmp = x * (t / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -5.5e+159], N[(x * (-t)), $MachinePrecision], If[LessEqual[t, 4.5e+24], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision], N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.4999999999999998e159

   1. Initial program 95.8%

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

   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 80.5

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

   4. Applied rewrites 80.5%

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

   5. Taylor expanded in z 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 40.2

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

   7. Applied rewrites 40.2%

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

   if -5.4999999999999998e159 < t < 4.50000000000000019e24

   1. Initial program 93.8%

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

   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 87.9

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

   4. Applied rewrites 87.9%

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

   5. Taylor expanded in y around inf

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

      1. lift-/.f64 75.0

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

   7. Applied rewrites 75.0%

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

   if 4.50000000000000019e24 < t

   1. Initial program 96.5%

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

   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 69.2

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

   4. Applied rewrites 69.2%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 49.4

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

   7. Applied rewrites 49.4%

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

Alternative 9: 64.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{+159}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{+133}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -5.5e+159)
   (* x (- t))
   (if (<= t 3.5e+133) (* (/ x z) y) (* (/ x z) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -5.5e+159) {
		tmp = x * -t;
	} else if (t <= 3.5e+133) {
		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 <= (-5.5d+159)) then
        tmp = x * -t
    else if (t <= 3.5d+133) 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 <= -5.5e+159) {
		tmp = x * -t;
	} else if (t <= 3.5e+133) {
		tmp = (x / z) * y;
	} else {
		tmp = (x / z) * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -5.5e+159:
		tmp = x * -t
	elif t <= 3.5e+133:
		tmp = (x / z) * y
	else:
		tmp = (x / z) * t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -5.5e+159)
		tmp = Float64(x * Float64(-t));
	elseif (t <= 3.5e+133)
		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 <= -5.5e+159)
		tmp = x * -t;
	elseif (t <= 3.5e+133)
		tmp = (x / z) * y;
	else
		tmp = (x / z) * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -5.5e+159], N[(x * (-t)), $MachinePrecision], If[LessEqual[t, 3.5e+133], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.4999999999999998e159

   1. Initial program 95.8%

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

   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 80.5

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

   4. Applied rewrites 80.5%

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

   5. Taylor expanded in z 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 40.2

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

   7. Applied rewrites 40.2%

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

   if -5.4999999999999998e159 < t < 3.4999999999999998e133

   1. Initial program 94.3%

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

   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 86.4

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

   4. Applied rewrites 86.4%

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

   5. Taylor expanded in y around inf

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

      1. lift-/.f64 72.0

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

   7. Applied rewrites 72.0%

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

   if 3.4999999999999998e133 < t

   1. Initial program 95.8%

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

   2. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 86.0

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

   4. Applied rewrites 86.0%

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

   5. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lift--.f64 67.6

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

   7. Applied rewrites 67.6%

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

   8. Taylor expanded in z around inf

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

      1. lift-/.f64 45.1

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

   10. Applied rewrites 45.1%

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

Alternative 10: 42.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{z} \cdot t\\ \mathbf{if}\;z \leq -1 \cdot 10^{-39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 0.0128:\\ \;\;\;\;x \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ x z) t)))
   (if (<= z -1e-39) t_1 (if (<= z 0.0128) (* x (- t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / z) * t;
	double tmp;
	if (z <= -1e-39) {
		tmp = t_1;
	} else if (z <= 0.0128) {
		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 = (x / z) * t
    if (z <= (-1d-39)) then
        tmp = t_1
    else if (z <= 0.0128d0) 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 = (x / z) * t;
	double tmp;
	if (z <= -1e-39) {
		tmp = t_1;
	} else if (z <= 0.0128) {
		tmp = x * -t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x / z) * t
	tmp = 0
	if z <= -1e-39:
		tmp = t_1
	elif z <= 0.0128:
		tmp = x * -t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / z) * t)
	tmp = 0.0
	if (z <= -1e-39)
		tmp = t_1;
	elseif (z <= 0.0128)
		tmp = Float64(x * Float64(-t));
	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 (z <= -1e-39)
		tmp = t_1;
	elseif (z <= 0.0128)
		tmp = x * -t;
	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[z, -1e-39], t$95$1, If[LessEqual[z, 0.0128], N[(x * (-t)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.99999999999999929e-40 or 0.0128000000000000006 < z

   1. Initial program 97.1%

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

   2. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 82.9

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

   4. Applied rewrites 82.9%

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

   5. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lift--.f64 52.7

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

   7. Applied rewrites 52.7%

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

   8. Taylor expanded in z around inf

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

      1. lift-/.f64 50.0

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

   10. Applied rewrites 50.0%

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

   if -9.99999999999999929e-40 < z < 0.0128000000000000006

   1. Initial program 92.0%

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

   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 34.5

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

   4. Applied rewrites 34.5%

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

   5. Taylor expanded in z 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 34.2

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

   7. Applied rewrites 34.2%

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

Alternative 11: 23.6% accurate, 3.2× 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 94.7%

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

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 46.6

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

4. Applied rewrites 46.6%

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

5. Taylor expanded in z 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 23.6

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

7. Applied rewrites 23.6%

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

Reproduce

herbie shell --seed 2025119 
(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)))))