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

Percentage Accurate: 94.8% → 96.8%

Time: 3.4s

Alternatives: 11

Speedup: 0.3×

• 21.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+246}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{y}, \frac{x\_m}{1 - z}, \frac{x\_m}{z}\right) \cdot y\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+268}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-t, z, y\right) \cdot x\_m}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (* x_m (- (/ y z) (/ t (- 1.0 z))))))
   (*
    x_s
    (if (<= t_1 -1e+246)
      (* (fma (/ (- t) y) (/ x_m (- 1.0 z)) (/ x_m z)) y)
      (if (<= t_1 4e+268) t_1 (/ (* (fma (- t) z y) x_m) z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m * ((y / z) - (t / (1.0 - z)));
	double tmp;
	if (t_1 <= -1e+246) {
		tmp = fma((-t / y), (x_m / (1.0 - z)), (x_m / z)) * y;
	} else if (t_1 <= 4e+268) {
		tmp = t_1;
	} else {
		tmp = (fma(-t, z, y) * x_m) / z;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(x_m * Float64(Float64(y / z) - Float64(t / Float64(1.0 - z))))
	tmp = 0.0
	if (t_1 <= -1e+246)
		tmp = Float64(fma(Float64(Float64(-t) / y), Float64(x_m / Float64(1.0 - z)), Float64(x_m / z)) * y);
	elseif (t_1 <= 4e+268)
		tmp = t_1;
	else
		tmp = Float64(Float64(fma(Float64(-t), z, y) * x_m) / z);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m * N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$1, -1e+246], N[(N[(N[((-t) / y), $MachinePrecision] * N[(x$95$m / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$1, 4e+268], t$95$1, N[(N[(N[((-t) * z + y), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z)))) < -1.00000000000000007e246

   1. Initial program 94.8%

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

   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 80.6

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

   4. Applied rewrites 80.6%

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

   if -1.00000000000000007e246 < (*.f64 x (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z)))) < 3.9999999999999999e268

   1. Initial program 94.8%

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

   if 3.9999999999999999e268 < (*.f64 x (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))))

   1. Initial program 94.8%

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

   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 61.1

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

   4. Applied rewrites 61.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lift-neg.f64 N/A

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

      8. mul-1-neg N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-in N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lift-neg.f64 64.0

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

   6. Applied rewrites 64.0%

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

Alternative 2: 95.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(-t, z, y\right) \cdot x\_m}{z}\\ t_2 := \frac{y}{z} - \frac{t}{1 - z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+153}:\\ \;\;\;\;x\_m \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ (* (fma (- t) z y) x_m) z)) (t_2 (- (/ y z) (/ t (- 1.0 z)))))
   (*
    x_s
    (if (<= t_2 (- INFINITY)) t_1 (if (<= t_2 5e+153) (* x_m t_2) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (fma(-t, z, y) * x_m) / z;
	double t_2 = (y / z) - (t / (1.0 - z));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 5e+153) {
		tmp = x_m * t_2;
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(Float64(fma(Float64(-t), z, y) * x_m) / 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 <= 5e+153)
		tmp = Float64(x_m * t_2);
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[((-t) * z + y), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 5e+153], N[(x$95$m * t$95$2), $MachinePrecision], t$95$1]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < -inf.0 or 5.00000000000000018e153 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z)))

   1. Initial program 94.8%

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

   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 61.1

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

   4. Applied rewrites 61.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lift-neg.f64 N/A

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

      8. mul-1-neg N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-in N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lift-neg.f64 64.0

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

   6. Applied rewrites 64.0%

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

   if -inf.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < 5.00000000000000018e153

   1. Initial program 94.8%

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

Alternative 3: 91.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -0.95:\\ \;\;\;\;\frac{\left(t + y\right) \cdot x\_m}{z}\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-304}:\\ \;\;\;\;x\_m \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-t, z, y\right) \cdot x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{t + y}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -0.95)
    (/ (* (+ t y) x_m) z)
    (if (<= z 1.4e-304)
      (* x_m (- (/ y z) t))
      (if (<= z 2.6e-23)
        (/ (* (fma (- t) z y) x_m) z)
        (* x_m (/ (+ t y) z)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -0.95) {
		tmp = ((t + y) * x_m) / z;
	} else if (z <= 1.4e-304) {
		tmp = x_m * ((y / z) - t);
	} else if (z <= 2.6e-23) {
		tmp = (fma(-t, z, y) * x_m) / z;
	} else {
		tmp = x_m * ((t + y) / z);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -0.95)
		tmp = Float64(Float64(Float64(t + y) * x_m) / z);
	elseif (z <= 1.4e-304)
		tmp = Float64(x_m * Float64(Float64(y / z) - t));
	elseif (z <= 2.6e-23)
		tmp = Float64(Float64(fma(Float64(-t), z, y) * x_m) / z);
	else
		tmp = Float64(x_m * Float64(Float64(t + y) / z));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -0.95], N[(N[(N[(t + y), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 1.4e-304], N[(x$95$m * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.6e-23], N[(N[(N[((-t) * z + y), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(x$95$m * N[(N[(t + y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if z < -0.94999999999999996

   1. Initial program 94.8%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 71.4

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

   4. Applied rewrites 71.4%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 71.4

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

   7. Applied rewrites 71.4%

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

   if -0.94999999999999996 < z < 1.3999999999999999e-304

   1. Initial program 94.8%

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

   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 65.0%

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

   if 1.3999999999999999e-304 < z < 2.6e-23

   1. Initial program 94.8%

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

   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 61.1

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

   4. Applied rewrites 61.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lift-neg.f64 N/A

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

      8. mul-1-neg N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-in N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lift-neg.f64 64.0

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

   6. Applied rewrites 64.0%

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

   if 2.6e-23 < z

   1. Initial program 94.8%

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

   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 73.9

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

   4. Applied rewrites 73.9%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 73.9

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

   7. Applied rewrites 73.9%

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

Alternative 4: 90.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -0.95:\\ \;\;\;\;\frac{\left(t + y\right) \cdot x\_m}{z}\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-23}:\\ \;\;\;\;x\_m \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{t + y}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -0.95)
    (/ (* (+ t y) x_m) z)
    (if (<= z 2.6e-23) (* x_m (- (/ y z) t)) (* x_m (/ (+ t y) z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -0.95) {
		tmp = ((t + y) * x_m) / z;
	} else if (z <= 2.6e-23) {
		tmp = x_m * ((y / z) - t);
	} else {
		tmp = x_m * ((t + y) / z);
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-0.95d0)) then
        tmp = ((t + y) * x_m) / z
    else if (z <= 2.6d-23) then
        tmp = x_m * ((y / z) - t)
    else
        tmp = x_m * ((t + y) / z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -0.95) {
		tmp = ((t + y) * x_m) / z;
	} else if (z <= 2.6e-23) {
		tmp = x_m * ((y / z) - t);
	} else {
		tmp = x_m * ((t + y) / z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= -0.95:
		tmp = ((t + y) * x_m) / z
	elif z <= 2.6e-23:
		tmp = x_m * ((y / z) - t)
	else:
		tmp = x_m * ((t + y) / z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -0.95)
		tmp = Float64(Float64(Float64(t + y) * x_m) / z);
	elseif (z <= 2.6e-23)
		tmp = Float64(x_m * Float64(Float64(y / z) - t));
	else
		tmp = Float64(x_m * Float64(Float64(t + y) / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= -0.95)
		tmp = ((t + y) * x_m) / z;
	elseif (z <= 2.6e-23)
		tmp = x_m * ((y / z) - t);
	else
		tmp = x_m * ((t + y) / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -0.95], N[(N[(N[(t + y), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 2.6e-23], N[(x$95$m * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(t + y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -0.94999999999999996

   1. Initial program 94.8%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 71.4

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

   4. Applied rewrites 71.4%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 71.4

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

   7. Applied rewrites 71.4%

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

   if -0.94999999999999996 < z < 2.6e-23

   1. Initial program 94.8%

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

   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 65.0%

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

   if 2.6e-23 < z

   1. Initial program 94.8%

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

   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 73.9

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

   4. Applied rewrites 73.9%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 73.9

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

   7. Applied rewrites 73.9%

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

Alternative 5: 88.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -0.95:\\ \;\;\;\;\frac{\left(t + y\right) \cdot x\_m}{z}\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-23}:\\ \;\;\;\;x\_m \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t + y\right) \cdot \frac{x\_m}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -0.95)
    (/ (* (+ t y) x_m) z)
    (if (<= z 2.6e-23) (* x_m (- (/ y z) t)) (* (+ t y) (/ x_m z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -0.95) {
		tmp = ((t + y) * x_m) / z;
	} else if (z <= 2.6e-23) {
		tmp = x_m * ((y / z) - t);
	} else {
		tmp = (t + y) * (x_m / z);
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-0.95d0)) then
        tmp = ((t + y) * x_m) / z
    else if (z <= 2.6d-23) then
        tmp = x_m * ((y / z) - t)
    else
        tmp = (t + y) * (x_m / z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -0.95) {
		tmp = ((t + y) * x_m) / z;
	} else if (z <= 2.6e-23) {
		tmp = x_m * ((y / z) - t);
	} else {
		tmp = (t + y) * (x_m / z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= -0.95:
		tmp = ((t + y) * x_m) / z
	elif z <= 2.6e-23:
		tmp = x_m * ((y / z) - t)
	else:
		tmp = (t + y) * (x_m / z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -0.95)
		tmp = Float64(Float64(Float64(t + y) * x_m) / z);
	elseif (z <= 2.6e-23)
		tmp = Float64(x_m * Float64(Float64(y / z) - t));
	else
		tmp = Float64(Float64(t + y) * Float64(x_m / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= -0.95)
		tmp = ((t + y) * x_m) / z;
	elseif (z <= 2.6e-23)
		tmp = x_m * ((y / z) - t);
	else
		tmp = (t + y) * (x_m / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -0.95], N[(N[(N[(t + y), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 2.6e-23], N[(x$95$m * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[(t + y), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -0.94999999999999996

   1. Initial program 94.8%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 71.4

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

   4. Applied rewrites 71.4%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 71.4

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

   7. Applied rewrites 71.4%

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

   if -0.94999999999999996 < z < 2.6e-23

   1. Initial program 94.8%

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

   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 65.0%

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

   if 2.6e-23 < z

   1. Initial program 94.8%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 71.4

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

   4. Applied rewrites 71.4%

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

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

      4. lift-neg.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-neg.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-/.f64 71.6

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

   6. Applied rewrites 71.6%

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

   7. Taylor expanded in y around 0

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

      1. lower-+.f64 71.6

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

   9. Applied rewrites 71.6%

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

Alternative 6: 88.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \left(t + y\right) \cdot \frac{x\_m}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -0.98:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-23}:\\ \;\;\;\;x\_m \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (* (+ t y) (/ x_m z))))
   (*
    x_s
    (if (<= z -0.98) t_1 (if (<= z 2.6e-23) (* x_m (- (/ y z) t)) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (t + y) * (x_m / z);
	double tmp;
	if (z <= -0.98) {
		tmp = t_1;
	} else if (z <= 2.6e-23) {
		tmp = x_m * ((y / z) - t);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    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_m / z)
    if (z <= (-0.98d0)) then
        tmp = t_1
    else if (z <= 2.6d-23) then
        tmp = x_m * ((y / z) - t)
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (t + y) * (x_m / z);
	double tmp;
	if (z <= -0.98) {
		tmp = t_1;
	} else if (z <= 2.6e-23) {
		tmp = x_m * ((y / z) - t);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = (t + y) * (x_m / z)
	tmp = 0
	if z <= -0.98:
		tmp = t_1
	elif z <= 2.6e-23:
		tmp = x_m * ((y / z) - t)
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(Float64(t + y) * Float64(x_m / z))
	tmp = 0.0
	if (z <= -0.98)
		tmp = t_1;
	elseif (z <= 2.6e-23)
		tmp = Float64(x_m * Float64(Float64(y / z) - t));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = (t + y) * (x_m / z);
	tmp = 0.0;
	if (z <= -0.98)
		tmp = t_1;
	elseif (z <= 2.6e-23)
		tmp = x_m * ((y / z) - t);
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t + y), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -0.98], t$95$1, If[LessEqual[z, 2.6e-23], N[(x$95$m * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -0.97999999999999998 or 2.6e-23 < z

   1. Initial program 94.8%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 71.4

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

   4. Applied rewrites 71.4%

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

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

      4. lift-neg.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-neg.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-/.f64 71.6

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

   6. Applied rewrites 71.6%

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

   7. Taylor expanded in y around 0

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

      1. lower-+.f64 71.6

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

   9. Applied rewrites 71.6%

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

   if -0.97999999999999998 < z < 2.6e-23

   1. Initial program 94.8%

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

   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 65.0%

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

Alternative 7: 73.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := y \cdot \frac{x\_m}{z}\\ t_2 := \frac{y}{z} - \frac{t}{1 - z}\\ t_3 := x\_m \cdot \left(\frac{y}{z} - t\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-109}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{-154}:\\ \;\;\;\;t \cdot \frac{x\_m}{z}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (* y (/ x_m z)))
        (t_2 (- (/ y z) (/ t (- 1.0 z))))
        (t_3 (* x_m (- (/ y z) t))))
   (*
    x_s
    (if (<= t_2 (- INFINITY))
      t_1
      (if (<= t_2 -5e-109)
        t_3
        (if (<= t_2 4e-154) (* t (/ x_m z)) (if (<= t_2 5e+302) t_3 t_1)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = y * (x_m / z);
	double t_2 = (y / z) - (t / (1.0 - z));
	double t_3 = x_m * ((y / z) - t);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= -5e-109) {
		tmp = t_3;
	} else if (t_2 <= 4e-154) {
		tmp = t * (x_m / z);
	} else if (t_2 <= 5e+302) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = y * (x_m / z);
	double t_2 = (y / z) - (t / (1.0 - z));
	double t_3 = x_m * ((y / z) - t);
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= -5e-109) {
		tmp = t_3;
	} else if (t_2 <= 4e-154) {
		tmp = t * (x_m / z);
	} else if (t_2 <= 5e+302) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = y * (x_m / z)
	t_2 = (y / z) - (t / (1.0 - z))
	t_3 = x_m * ((y / z) - t)
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= -5e-109:
		tmp = t_3
	elif t_2 <= 4e-154:
		tmp = t * (x_m / z)
	elif t_2 <= 5e+302:
		tmp = t_3
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(y * Float64(x_m / z))
	t_2 = Float64(Float64(y / z) - Float64(t / Float64(1.0 - z)))
	t_3 = Float64(x_m * Float64(Float64(y / z) - t))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= -5e-109)
		tmp = t_3;
	elseif (t_2 <= 4e-154)
		tmp = Float64(t * Float64(x_m / z));
	elseif (t_2 <= 5e+302)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = y * (x_m / z);
	t_2 = (y / z) - (t / (1.0 - z));
	t_3 = x_m * ((y / z) - t);
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= -5e-109)
		tmp = t_3;
	elseif (t_2 <= 4e-154)
		tmp = t * (x_m / z);
	elseif (t_2 <= 5e+302)
		tmp = t_3;
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(x$95$m * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, -5e-109], t$95$3, If[LessEqual[t$95$2, 4e-154], N[(t * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+302], t$95$3, t$95$1]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < -inf.0 or 5e302 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z)))

   1. Initial program 94.8%

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

   2. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 61.1

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

   4. Applied rewrites 61.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 60.4

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

   6. Applied rewrites 60.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      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. lift-/.f64 61.3

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

   8. Applied rewrites 61.3%

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

   if -inf.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < -5.0000000000000002e-109 or 3.9999999999999999e-154 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < 5e302

   1. Initial program 94.8%

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

   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 65.0%

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

   if -5.0000000000000002e-109 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < 3.9999999999999999e-154

   1. Initial program 94.8%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 71.4

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

   4. Applied rewrites 71.4%

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

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

      4. lift-neg.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-neg.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-/.f64 71.6

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

   6. Applied rewrites 71.6%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 34.4%

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

Alternative 8: 65.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{+98}:\\ \;\;\;\;x\_m \cdot \frac{t}{z}\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{+209}:\\ \;\;\;\;y \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= t -3.8e+98)
    (* x_m (/ t z))
    (if (<= t 2.2e+209) (* y (/ x_m z)) (* (- t) x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= -3.8e+98) {
		tmp = x_m * (t / z);
	} else if (t <= 2.2e+209) {
		tmp = y * (x_m / z);
	} else {
		tmp = -t * x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-3.8d+98)) then
        tmp = x_m * (t / z)
    else if (t <= 2.2d+209) then
        tmp = y * (x_m / z)
    else
        tmp = -t * x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= -3.8e+98) {
		tmp = x_m * (t / z);
	} else if (t <= 2.2e+209) {
		tmp = y * (x_m / z);
	} else {
		tmp = -t * x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if t <= -3.8e+98:
		tmp = x_m * (t / z)
	elif t <= 2.2e+209:
		tmp = y * (x_m / z)
	else:
		tmp = -t * x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (t <= -3.8e+98)
		tmp = Float64(x_m * Float64(t / z));
	elseif (t <= 2.2e+209)
		tmp = Float64(y * Float64(x_m / z));
	else
		tmp = Float64(Float64(-t) * x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (t <= -3.8e+98)
		tmp = x_m * (t / z);
	elseif (t <= 2.2e+209)
		tmp = y * (x_m / z);
	else
		tmp = -t * x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[t, -3.8e+98], N[(x$95$m * N[(t / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.2e+209], N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[((-t) * x$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -3.7999999999999999e98

   1. Initial program 94.8%

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

   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 73.9

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

   4. Applied rewrites 73.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 35.6%

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

   if -3.7999999999999999e98 < t < 2.1999999999999999e209

   1. Initial program 94.8%

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

   2. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 61.1

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

   4. Applied rewrites 61.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 60.4

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

   6. Applied rewrites 60.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      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. lift-/.f64 61.3

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

   8. Applied rewrites 61.3%

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

   if 2.1999999999999999e209 < t

   1. Initial program 94.8%

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

   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 61.1

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

   4. Applied rewrites 61.1%

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

   5. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 22.7

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

   7. Applied rewrites 22.7%

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

Alternative 9: 63.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \left(-t\right) \cdot x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -9.2 \cdot 10^{+152}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{+209}:\\ \;\;\;\;y \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (* (- t) x_m)))
   (* x_s (if (<= t -9.2e+152) t_1 (if (<= t 2.2e+209) (* y (/ x_m z)) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = -t * x_m;
	double tmp;
	if (t <= -9.2e+152) {
		tmp = t_1;
	} else if (t <= 2.2e+209) {
		tmp = y * (x_m / z);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    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_m
    if (t <= (-9.2d+152)) then
        tmp = t_1
    else if (t <= 2.2d+209) then
        tmp = y * (x_m / z)
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = -t * x_m;
	double tmp;
	if (t <= -9.2e+152) {
		tmp = t_1;
	} else if (t <= 2.2e+209) {
		tmp = y * (x_m / z);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = -t * x_m
	tmp = 0
	if t <= -9.2e+152:
		tmp = t_1
	elif t <= 2.2e+209:
		tmp = y * (x_m / z)
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(Float64(-t) * x_m)
	tmp = 0.0
	if (t <= -9.2e+152)
		tmp = t_1;
	elseif (t <= 2.2e+209)
		tmp = Float64(y * Float64(x_m / z));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = -t * x_m;
	tmp = 0.0;
	if (t <= -9.2e+152)
		tmp = t_1;
	elseif (t <= 2.2e+209)
		tmp = y * (x_m / z);
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[((-t) * x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t, -9.2e+152], t$95$1, If[LessEqual[t, 2.2e+209], N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -9.1999999999999994e152 or 2.1999999999999999e209 < t

   1. Initial program 94.8%

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

   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 61.1

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

   4. Applied rewrites 61.1%

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

   5. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 22.7

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

   7. Applied rewrites 22.7%

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

   if -9.1999999999999994e152 < t < 2.1999999999999999e209

   1. Initial program 94.8%

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

   2. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 61.1

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

   4. Applied rewrites 61.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 60.4

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

   6. Applied rewrites 60.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      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. lift-/.f64 61.3

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

   8. Applied rewrites 61.3%

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

Alternative 10: 63.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \left(-t\right) \cdot x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{+209}:\\ \;\;\;\;x\_m \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (* (- t) x_m)))
   (* x_s (if (<= t -5e+104) t_1 (if (<= t 1.7e+209) (* x_m (/ y z)) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = -t * x_m;
	double tmp;
	if (t <= -5e+104) {
		tmp = t_1;
	} else if (t <= 1.7e+209) {
		tmp = x_m * (y / z);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    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_m
    if (t <= (-5d+104)) then
        tmp = t_1
    else if (t <= 1.7d+209) then
        tmp = x_m * (y / z)
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = -t * x_m;
	double tmp;
	if (t <= -5e+104) {
		tmp = t_1;
	} else if (t <= 1.7e+209) {
		tmp = x_m * (y / z);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = -t * x_m
	tmp = 0
	if t <= -5e+104:
		tmp = t_1
	elif t <= 1.7e+209:
		tmp = x_m * (y / z)
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(Float64(-t) * x_m)
	tmp = 0.0
	if (t <= -5e+104)
		tmp = t_1;
	elseif (t <= 1.7e+209)
		tmp = Float64(x_m * Float64(y / z));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = -t * x_m;
	tmp = 0.0;
	if (t <= -5e+104)
		tmp = t_1;
	elseif (t <= 1.7e+209)
		tmp = x_m * (y / z);
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[((-t) * x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t, -5e+104], t$95$1, If[LessEqual[t, 1.7e+209], N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.9999999999999997e104 or 1.6999999999999998e209 < t

   1. Initial program 94.8%

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

   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 61.1

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

   4. Applied rewrites 61.1%

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

   5. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 22.7

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

   7. Applied rewrites 22.7%

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

   if -4.9999999999999997e104 < t < 1.6999999999999998e209

   1. Initial program 94.8%

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

   2. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 61.1

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

   4. Applied rewrites 61.1%

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

Alternative 11: 22.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\left(-t\right) \cdot x\_m\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t) :precision binary64 (* x_s (* (- t) x_m)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * (-t * x_m);
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x_s * (-t * x_m)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * (-t * x_m);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	return x_s * (-t * x_m)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	return Float64(x_s * Float64(Float64(-t) * x_m))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z, t)
	tmp = x_s * (-t * x_m);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * N[((-t) * x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.8%

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

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 61.1

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

4. Applied rewrites 61.1%

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

5. Taylor expanded in y around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lift-neg.f64 22.7

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

7. Applied rewrites 22.7%

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

Reproduce

herbie shell --seed 2025142 
(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)))))