Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2

Percentage Accurate: 88.6% → 98.7%

Time: 3.6s

Alternatives: 7

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (1.0 / x) / (y * (1.0 + (z * 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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (1.0d0 / x) / (y * (1.0d0 + (z * z)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (1.0 / x) / (y * (1.0 + (z * 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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (1.0d0 / x) / (y * (1.0d0 + (z * z)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \cdot \left(1 + z \cdot z\right) \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{\mathsf{fma}\left(y\_m \cdot z, z, y\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{z}}{x\_m \cdot z}}{y\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* y_m (+ 1.0 (* z z))) 2e+307)
     (/ (/ 1.0 x_m) (fma (* y_m z) z y_m))
     (/ (/ (/ 1.0 z) (* x_m z)) y_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((y_m * (1.0 + (z * z))) <= 2e+307) {
		tmp = (1.0 / x_m) / fma((y_m * z), z, y_m);
	} else {
		tmp = ((1.0 / z) / (x_m * z)) / y_m;
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(y_m * Float64(1.0 + Float64(z * z))) <= 2e+307)
		tmp = Float64(Float64(1.0 / x_m) / fma(Float64(y_m * z), z, y_m));
	else
		tmp = Float64(Float64(Float64(1.0 / z) / Float64(x_m * z)) / y_m);
	end
	return Float64(y_s * 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]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(y$95$m * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+307], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(N[(y$95$m * z), $MachinePrecision] * z + y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / z), $MachinePrecision] / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 1.99999999999999997e307

   1. Initial program 88.6%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. *-commutative N/A

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

      5. pow2 N/A

         \[\leadsto \frac{\frac{1}{x}}{\left(1 + \color{blue}{{z}^{2}}\right) \cdot y} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{1}{x}}{\color{blue}{y \cdot \left(1 + {z}^{2}\right)}} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{1}{x}}{y \cdot \color{blue}{\left({z}^{2} + 1\right)}} \]

      8. distribute-rgt-in N/A

         \[\leadsto \frac{\frac{1}{x}}{\color{blue}{{z}^{2} \cdot y + 1 \cdot y}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\frac{1}{x}}{\color{blue}{y \cdot {z}^{2}} + 1 \cdot y} \]

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. *-lft-identity N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 90.7

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

   3. Applied rewrites 90.7%

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

   if 1.99999999999999997e307 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 88.6%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. pow2 N/A

          \[\leadsto \frac{\frac{\frac{1}{x}}{1 + \color{blue}{{z}^{2}}}}{y} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{\frac{1}{x}}{\color{blue}{{z}^{2} + 1}}}{y} \]

      13. pow2 N/A

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

      14. lower-fma.f64 92.7

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

   3. Applied rewrites 92.7%

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

   4. Taylor expanded in z around inf

      \[\leadsto \frac{\color{blue}{\frac{1}{x \cdot {z}^{2}}}}{y} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{x \cdot {z}^{2}}}}{y} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{1}{{z}^{2} \cdot \color{blue}{x}}}{y} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{{z}^{2} \cdot \color{blue}{x}}}{y} \]

      4. pow2 N/A

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

      5. lift-*.f64 52.4

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

   6. Applied rewrites 52.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 58.4

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

   8. Applied rewrites 58.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 58.6

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

   10. Applied rewrites 58.6%

       \[\leadsto \frac{\frac{\frac{1}{z}}{\color{blue}{x \cdot z}}}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 94.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 3.4 \cdot 10^{+72}:\\ \;\;\;\;\frac{1}{\left(y\_m \cdot x\_m\right) \cdot \mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z \cdot \left(z \cdot x\_m\right)}}{y\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z 3.4e+72)
     (/ 1.0 (* (* y_m x_m) (fma z z 1.0)))
     (/ (/ 1.0 (* z (* z x_m))) y_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 3.4e+72) {
		tmp = 1.0 / ((y_m * x_m) * fma(z, z, 1.0));
	} else {
		tmp = (1.0 / (z * (z * x_m))) / y_m;
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (z <= 3.4e+72)
		tmp = Float64(1.0 / Float64(Float64(y_m * x_m) * fma(z, z, 1.0)));
	else
		tmp = Float64(Float64(1.0 / Float64(z * Float64(z * x_m))) / y_m);
	end
	return Float64(y_s * 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]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z, 3.4e+72], N[(1.0 / N[(N[(y$95$m * x$95$m), $MachinePrecision] * N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(z * N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.3999999999999998e72

   1. Initial program 88.6%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. pow2 N/A

          \[\leadsto \frac{\frac{\frac{1}{x}}{1 + \color{blue}{{z}^{2}}}}{y} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{\frac{1}{x}}{\color{blue}{{z}^{2} + 1}}}{y} \]

      13. pow2 N/A

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

      14. lower-fma.f64 92.7

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

   3. Applied rewrites 92.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-/l/ N/A

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

      6. pow2 N/A

         \[\leadsto \frac{\frac{1}{x}}{\left(\color{blue}{{z}^{2}} + 1\right) \cdot y} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\left(1 + {z}^{2}\right)} \cdot y} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{1}{x}}{\color{blue}{y \cdot \left(1 + {z}^{2}\right)}} \]

      9. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot \left(1 + {z}^{2}\right)\right)}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot \left(1 + {z}^{2}\right)\right)}} \]

      11. associate-*r* N/A

          \[\leadsto \frac{1}{\color{blue}{\left(x \cdot y\right) \cdot \left(1 + {z}^{2}\right)}} \]

      12. +-commutative N/A

          \[\leadsto \frac{1}{\left(x \cdot y\right) \cdot \color{blue}{\left({z}^{2} + 1\right)}} \]

      13. pow2 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lift-fma.f64 90.6

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

   5. Applied rewrites 90.6%

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

   if 3.3999999999999998e72 < z

   1. Initial program 88.6%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. pow2 N/A

          \[\leadsto \frac{\frac{\frac{1}{x}}{1 + \color{blue}{{z}^{2}}}}{y} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{\frac{1}{x}}{\color{blue}{{z}^{2} + 1}}}{y} \]

      13. pow2 N/A

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

      14. lower-fma.f64 92.7

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

   3. Applied rewrites 92.7%

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

   4. Taylor expanded in z around inf

      \[\leadsto \frac{\color{blue}{\frac{1}{x \cdot {z}^{2}}}}{y} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{x \cdot {z}^{2}}}}{y} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{1}{{z}^{2} \cdot \color{blue}{x}}}{y} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{{z}^{2} \cdot \color{blue}{x}}}{y} \]

      4. pow2 N/A

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

      5. lift-*.f64 52.4

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

   6. Applied rewrites 52.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 58.4

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

   8. Applied rewrites 58.4%

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

Alternative 3: 92.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 0.88:\\ \;\;\;\;\frac{\frac{1 - z \cdot z}{x\_m}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z \cdot \left(z \cdot x\_m\right)}}{y\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z 0.88)
     (/ (/ (- 1.0 (* z z)) x_m) y_m)
     (/ (/ 1.0 (* z (* z x_m))) y_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 0.88) {
		tmp = ((1.0 - (z * z)) / x_m) / y_m;
	} else {
		tmp = (1.0 / (z * (z * x_m))) / y_m;
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
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(y_s, x_s, x_m, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 0.88d0) then
        tmp = ((1.0d0 - (z * z)) / x_m) / y_m
    else
        tmp = (1.0d0 / (z * (z * x_m))) / y_m
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 0.88) {
		tmp = ((1.0 - (z * z)) / x_m) / y_m;
	} else {
		tmp = (1.0 / (z * (z * x_m))) / y_m;
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if z <= 0.88:
		tmp = ((1.0 - (z * z)) / x_m) / y_m
	else:
		tmp = (1.0 / (z * (z * x_m))) / y_m
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (z <= 0.88)
		tmp = Float64(Float64(Float64(1.0 - Float64(z * z)) / x_m) / y_m);
	else
		tmp = Float64(Float64(1.0 / Float64(z * Float64(z * x_m))) / y_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (z <= 0.88)
		tmp = ((1.0 - (z * z)) / x_m) / y_m;
	else
		tmp = (1.0 / (z * (z * x_m))) / y_m;
	end
	tmp_2 = y_s * (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]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z, 0.88], N[(N[(N[(1.0 - N[(z * z), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(1.0 / N[(z * N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 0.880000000000000004

   1. Initial program 88.6%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. pow2 N/A

          \[\leadsto \frac{\frac{\frac{1}{x}}{1 + \color{blue}{{z}^{2}}}}{y} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{\frac{1}{x}}{\color{blue}{{z}^{2} + 1}}}{y} \]

      13. pow2 N/A

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

      14. lower-fma.f64 92.7

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

   3. Applied rewrites 92.7%

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

   4. Taylor expanded in z around 0

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{{z}^{2}}{x} + \frac{1}{x}}}{y} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{1}{x} + \color{blue}{-1 \cdot \frac{{z}^{2}}{x}}}{y} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\frac{1}{x} + \left(\mathsf{neg}\left(\frac{{z}^{2}}{x}\right)\right)}{y} \]

      3. sub-flip-reverse N/A

         \[\leadsto \frac{\frac{1}{x} - \color{blue}{\frac{{z}^{2}}{x}}}{y} \]

      4. sub-div N/A

         \[\leadsto \frac{\frac{1 - {z}^{2}}{\color{blue}{x}}}{y} \]

      5. pow2 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 51.0

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

   6. Applied rewrites 51.0%

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

   if 0.880000000000000004 < z

   1. Initial program 88.6%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. pow2 N/A

          \[\leadsto \frac{\frac{\frac{1}{x}}{1 + \color{blue}{{z}^{2}}}}{y} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{\frac{1}{x}}{\color{blue}{{z}^{2} + 1}}}{y} \]

      13. pow2 N/A

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

      14. lower-fma.f64 92.7

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

   3. Applied rewrites 92.7%

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

   4. Taylor expanded in z around inf

      \[\leadsto \frac{\color{blue}{\frac{1}{x \cdot {z}^{2}}}}{y} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{x \cdot {z}^{2}}}}{y} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{1}{{z}^{2} \cdot \color{blue}{x}}}{y} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{{z}^{2} \cdot \color{blue}{x}}}{y} \]

      4. pow2 N/A

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

      5. lift-*.f64 52.4

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

   6. Applied rewrites 52.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 58.4

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

   8. Applied rewrites 58.4%

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

Alternative 4: 92.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := 1 + z \cdot z\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2:\\ \;\;\;\;\frac{\frac{1 - z \cdot z}{x\_m}}{y\_m}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+295}:\\ \;\;\;\;\frac{1}{\left(z \cdot z\right) \cdot \left(x\_m \cdot y\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m \cdot \left(\left(y\_m \cdot z\right) \cdot z\right)}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* z z))))
   (*
    y_s
    (*
     x_s
     (if (<= t_0 2.0)
       (/ (/ (- 1.0 (* z z)) x_m) y_m)
       (if (<= t_0 4e+295)
         (/ 1.0 (* (* z z) (* x_m y_m)))
         (/ 1.0 (* x_m (* (* y_m z) z)))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = 1.0 + (z * z);
	double tmp;
	if (t_0 <= 2.0) {
		tmp = ((1.0 - (z * z)) / x_m) / y_m;
	} else if (t_0 <= 4e+295) {
		tmp = 1.0 / ((z * z) * (x_m * y_m));
	} else {
		tmp = 1.0 / (x_m * ((y_m * z) * z));
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
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(y_s, x_s, x_m, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (z * z)
    if (t_0 <= 2.0d0) then
        tmp = ((1.0d0 - (z * z)) / x_m) / y_m
    else if (t_0 <= 4d+295) then
        tmp = 1.0d0 / ((z * z) * (x_m * y_m))
    else
        tmp = 1.0d0 / (x_m * ((y_m * z) * z))
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = 1.0 + (z * z);
	double tmp;
	if (t_0 <= 2.0) {
		tmp = ((1.0 - (z * z)) / x_m) / y_m;
	} else if (t_0 <= 4e+295) {
		tmp = 1.0 / ((z * z) * (x_m * y_m));
	} else {
		tmp = 1.0 / (x_m * ((y_m * z) * z));
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	t_0 = 1.0 + (z * z)
	tmp = 0
	if t_0 <= 2.0:
		tmp = ((1.0 - (z * z)) / x_m) / y_m
	elif t_0 <= 4e+295:
		tmp = 1.0 / ((z * z) * (x_m * y_m))
	else:
		tmp = 1.0 / (x_m * ((y_m * z) * z))
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(1.0 + Float64(z * z))
	tmp = 0.0
	if (t_0 <= 2.0)
		tmp = Float64(Float64(Float64(1.0 - Float64(z * z)) / x_m) / y_m);
	elseif (t_0 <= 4e+295)
		tmp = Float64(1.0 / Float64(Float64(z * z) * Float64(x_m * y_m)));
	else
		tmp = Float64(1.0 / Float64(x_m * Float64(Float64(y_m * z) * z)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	t_0 = 1.0 + (z * z);
	tmp = 0.0;
	if (t_0 <= 2.0)
		tmp = ((1.0 - (z * z)) / x_m) / y_m;
	elseif (t_0 <= 4e+295)
		tmp = 1.0 / ((z * z) * (x_m * y_m));
	else
		tmp = 1.0 / (x_m * ((y_m * z) * z));
	end
	tmp_2 = y_s * (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]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$0, 2.0], N[(N[(N[(1.0 - N[(z * z), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision], If[LessEqual[t$95$0, 4e+295], N[(1.0 / N[(N[(z * z), $MachinePrecision] * N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x$95$m * N[(N[(y$95$m * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 #s(literal 1 binary64) (*.f64 z z)) < 2

   1. Initial program 88.6%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. pow2 N/A

          \[\leadsto \frac{\frac{\frac{1}{x}}{1 + \color{blue}{{z}^{2}}}}{y} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{\frac{1}{x}}{\color{blue}{{z}^{2} + 1}}}{y} \]

      13. pow2 N/A

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

      14. lower-fma.f64 92.7

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

   3. Applied rewrites 92.7%

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

   4. Taylor expanded in z around 0

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{{z}^{2}}{x} + \frac{1}{x}}}{y} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{1}{x} + \color{blue}{-1 \cdot \frac{{z}^{2}}{x}}}{y} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\frac{1}{x} + \left(\mathsf{neg}\left(\frac{{z}^{2}}{x}\right)\right)}{y} \]

      3. sub-flip-reverse N/A

         \[\leadsto \frac{\frac{1}{x} - \color{blue}{\frac{{z}^{2}}{x}}}{y} \]

      4. sub-div N/A

         \[\leadsto \frac{\frac{1 - {z}^{2}}{\color{blue}{x}}}{y} \]

      5. pow2 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 51.0

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

   6. Applied rewrites 51.0%

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

   if 2 < (+.f64 #s(literal 1 binary64) (*.f64 z z)) < 3.9999999999999999e295

   1. Initial program 88.6%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot {z}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(y \cdot {z}^{2}\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{\left(y \cdot {z}^{2}\right) \cdot \color{blue}{x}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{\left(y \cdot {z}^{2}\right) \cdot \color{blue}{x}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\left({z}^{2} \cdot y\right) \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\left({z}^{2} \cdot y\right) \cdot x} \]

      6. pow2 N/A

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

      7. lift-*.f64 48.4

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

   4. Applied rewrites 48.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

         \[\leadsto \frac{1}{\left({z}^{2} \cdot y\right) \cdot x} \]

      5. associate-*l* N/A

         \[\leadsto \frac{1}{{z}^{2} \cdot \color{blue}{\left(y \cdot x\right)}} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{{z}^{2} \cdot \left(x \cdot \color{blue}{y}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{{z}^{2} \cdot \color{blue}{\left(x \cdot y\right)}} \]

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 52.4

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

   6. Applied rewrites 52.4%

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

   if 3.9999999999999999e295 < (+.f64 #s(literal 1 binary64) (*.f64 z z))

   1. Initial program 88.6%

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

   2. Taylor expanded in z around inf

      \[\leadsto \frac{\frac{1}{x}}{\color{blue}{y \cdot {z}^{2}}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\frac{1}{x}}{{z}^{2} \cdot \color{blue}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{x}}{{z}^{2} \cdot \color{blue}{y}} \]

      3. pow2 N/A

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

      4. lift-*.f64 48.4

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

   4. Applied rewrites 48.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 48.4

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 50.5

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

   6. Applied rewrites 50.5%

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

Alternative 5: 74.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := 1 + z \cdot z\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2:\\ \;\;\;\;\frac{1 - z \cdot z}{y\_m \cdot x\_m}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+295}:\\ \;\;\;\;\frac{1}{\left(z \cdot z\right) \cdot \left(x\_m \cdot y\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m \cdot \left(\left(y\_m \cdot z\right) \cdot z\right)}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* z z))))
   (*
    y_s
    (*
     x_s
     (if (<= t_0 2.0)
       (/ (- 1.0 (* z z)) (* y_m x_m))
       (if (<= t_0 4e+295)
         (/ 1.0 (* (* z z) (* x_m y_m)))
         (/ 1.0 (* x_m (* (* y_m z) z)))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = 1.0 + (z * z);
	double tmp;
	if (t_0 <= 2.0) {
		tmp = (1.0 - (z * z)) / (y_m * x_m);
	} else if (t_0 <= 4e+295) {
		tmp = 1.0 / ((z * z) * (x_m * y_m));
	} else {
		tmp = 1.0 / (x_m * ((y_m * z) * z));
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
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(y_s, x_s, x_m, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (z * z)
    if (t_0 <= 2.0d0) then
        tmp = (1.0d0 - (z * z)) / (y_m * x_m)
    else if (t_0 <= 4d+295) then
        tmp = 1.0d0 / ((z * z) * (x_m * y_m))
    else
        tmp = 1.0d0 / (x_m * ((y_m * z) * z))
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = 1.0 + (z * z);
	double tmp;
	if (t_0 <= 2.0) {
		tmp = (1.0 - (z * z)) / (y_m * x_m);
	} else if (t_0 <= 4e+295) {
		tmp = 1.0 / ((z * z) * (x_m * y_m));
	} else {
		tmp = 1.0 / (x_m * ((y_m * z) * z));
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	t_0 = 1.0 + (z * z)
	tmp = 0
	if t_0 <= 2.0:
		tmp = (1.0 - (z * z)) / (y_m * x_m)
	elif t_0 <= 4e+295:
		tmp = 1.0 / ((z * z) * (x_m * y_m))
	else:
		tmp = 1.0 / (x_m * ((y_m * z) * z))
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(1.0 + Float64(z * z))
	tmp = 0.0
	if (t_0 <= 2.0)
		tmp = Float64(Float64(1.0 - Float64(z * z)) / Float64(y_m * x_m));
	elseif (t_0 <= 4e+295)
		tmp = Float64(1.0 / Float64(Float64(z * z) * Float64(x_m * y_m)));
	else
		tmp = Float64(1.0 / Float64(x_m * Float64(Float64(y_m * z) * z)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	t_0 = 1.0 + (z * z);
	tmp = 0.0;
	if (t_0 <= 2.0)
		tmp = (1.0 - (z * z)) / (y_m * x_m);
	elseif (t_0 <= 4e+295)
		tmp = 1.0 / ((z * z) * (x_m * y_m));
	else
		tmp = 1.0 / (x_m * ((y_m * z) * z));
	end
	tmp_2 = y_s * (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]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$0, 2.0], N[(N[(1.0 - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+295], N[(1.0 / N[(N[(z * z), $MachinePrecision] * N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x$95$m * N[(N[(y$95$m * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 #s(literal 1 binary64) (*.f64 z z)) < 2

   1. Initial program 88.6%

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

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{{z}^{2}}{x \cdot y} + \frac{1}{x \cdot y}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{x \cdot y} + \color{blue}{-1 \cdot \frac{{z}^{2}}{x \cdot y}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{1}{x \cdot y} + \left(\mathsf{neg}\left(\frac{{z}^{2}}{x \cdot y}\right)\right) \]

      3. sub-flip-reverse N/A

         \[\leadsto \frac{1}{x \cdot y} - \color{blue}{\frac{{z}^{2}}{x \cdot y}} \]

      4. sub-div N/A

         \[\leadsto \frac{1 - {z}^{2}}{\color{blue}{x \cdot y}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{1 - {z}^{2}}{\color{blue}{x \cdot y}} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{1 - {z}^{2}}{\color{blue}{x} \cdot y} \]

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 52.9

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

   4. Applied rewrites 52.9%

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

   if 2 < (+.f64 #s(literal 1 binary64) (*.f64 z z)) < 3.9999999999999999e295

   1. Initial program 88.6%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot {z}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(y \cdot {z}^{2}\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{\left(y \cdot {z}^{2}\right) \cdot \color{blue}{x}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{\left(y \cdot {z}^{2}\right) \cdot \color{blue}{x}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\left({z}^{2} \cdot y\right) \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\left({z}^{2} \cdot y\right) \cdot x} \]

      6. pow2 N/A

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

      7. lift-*.f64 48.4

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

   4. Applied rewrites 48.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

         \[\leadsto \frac{1}{\left({z}^{2} \cdot y\right) \cdot x} \]

      5. associate-*l* N/A

         \[\leadsto \frac{1}{{z}^{2} \cdot \color{blue}{\left(y \cdot x\right)}} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{{z}^{2} \cdot \left(x \cdot \color{blue}{y}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{{z}^{2} \cdot \color{blue}{\left(x \cdot y\right)}} \]

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 52.4

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

   6. Applied rewrites 52.4%

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

   if 3.9999999999999999e295 < (+.f64 #s(literal 1 binary64) (*.f64 z z))

   1. Initial program 88.6%

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

   2. Taylor expanded in z around inf

      \[\leadsto \frac{\frac{1}{x}}{\color{blue}{y \cdot {z}^{2}}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\frac{1}{x}}{{z}^{2} \cdot \color{blue}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{x}}{{z}^{2} \cdot \color{blue}{y}} \]

      3. pow2 N/A

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

      4. lift-*.f64 48.4

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

   4. Applied rewrites 48.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 48.4

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 50.5

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

   6. Applied rewrites 50.5%

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

Alternative 6: 71.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 0.85:\\ \;\;\;\;\frac{1 - z \cdot z}{y\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(z \cdot z\right) \cdot \left(x\_m \cdot y\_m\right)}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z 0.85)
     (/ (- 1.0 (* z z)) (* y_m x_m))
     (/ 1.0 (* (* z z) (* x_m y_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 0.85) {
		tmp = (1.0 - (z * z)) / (y_m * x_m);
	} else {
		tmp = 1.0 / ((z * z) * (x_m * y_m));
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
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(y_s, x_s, x_m, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 0.85d0) then
        tmp = (1.0d0 - (z * z)) / (y_m * x_m)
    else
        tmp = 1.0d0 / ((z * z) * (x_m * y_m))
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 0.85) {
		tmp = (1.0 - (z * z)) / (y_m * x_m);
	} else {
		tmp = 1.0 / ((z * z) * (x_m * y_m));
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if z <= 0.85:
		tmp = (1.0 - (z * z)) / (y_m * x_m)
	else:
		tmp = 1.0 / ((z * z) * (x_m * y_m))
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (z <= 0.85)
		tmp = Float64(Float64(1.0 - Float64(z * z)) / Float64(y_m * x_m));
	else
		tmp = Float64(1.0 / Float64(Float64(z * z) * Float64(x_m * y_m)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (z <= 0.85)
		tmp = (1.0 - (z * z)) / (y_m * x_m);
	else
		tmp = 1.0 / ((z * z) * (x_m * y_m));
	end
	tmp_2 = y_s * (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]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z, 0.85], N[(N[(1.0 - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(z * z), $MachinePrecision] * N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 0.849999999999999978

   1. Initial program 88.6%

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

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{{z}^{2}}{x \cdot y} + \frac{1}{x \cdot y}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{x \cdot y} + \color{blue}{-1 \cdot \frac{{z}^{2}}{x \cdot y}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{1}{x \cdot y} + \left(\mathsf{neg}\left(\frac{{z}^{2}}{x \cdot y}\right)\right) \]

      3. sub-flip-reverse N/A

         \[\leadsto \frac{1}{x \cdot y} - \color{blue}{\frac{{z}^{2}}{x \cdot y}} \]

      4. sub-div N/A

         \[\leadsto \frac{1 - {z}^{2}}{\color{blue}{x \cdot y}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{1 - {z}^{2}}{\color{blue}{x \cdot y}} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{1 - {z}^{2}}{\color{blue}{x} \cdot y} \]

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 52.9

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

   4. Applied rewrites 52.9%

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

   if 0.849999999999999978 < z

   1. Initial program 88.6%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot {z}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(y \cdot {z}^{2}\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{\left(y \cdot {z}^{2}\right) \cdot \color{blue}{x}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{\left(y \cdot {z}^{2}\right) \cdot \color{blue}{x}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\left({z}^{2} \cdot y\right) \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\left({z}^{2} \cdot y\right) \cdot x} \]

      6. pow2 N/A

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

      7. lift-*.f64 48.4

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

   4. Applied rewrites 48.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

         \[\leadsto \frac{1}{\left({z}^{2} \cdot y\right) \cdot x} \]

      5. associate-*l* N/A

         \[\leadsto \frac{1}{{z}^{2} \cdot \color{blue}{\left(y \cdot x\right)}} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{{z}^{2} \cdot \left(x \cdot \color{blue}{y}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{{z}^{2} \cdot \color{blue}{\left(x \cdot y\right)}} \]

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 52.4

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

   6. Applied rewrites 52.4%

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

Alternative 7: 58.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \frac{1}{y\_m \cdot x\_m}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (/ 1.0 (* y_m x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * (1.0 / (y_m * x_m)));
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
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(y_s, x_s, x_m, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = y_s * (x_s * (1.0d0 / (y_m * x_m)))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * (1.0 / (y_m * x_m)));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	return y_s * (x_s * (1.0 / (y_m * x_m)))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	return Float64(y_s * Float64(x_s * Float64(1.0 / Float64(y_m * x_m))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp = code(y_s, x_s, x_m, y_m, z)
	tmp = y_s * (x_s * (1.0 / (y_m * 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]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * N[(1.0 / N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.6%

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

2. Taylor expanded in z around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 58.5

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

4. Applied rewrites 58.5%

   \[\leadsto \color{blue}{\frac{1}{y \cdot x}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025134 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64
  (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))