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

Percentage Accurate: 88.6% → 98.7%

Time: 2.9s

Alternatives: 8

Speedup: 1.0×

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, 0.8× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 140000000:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(z\_m, z\_m, 1\right) \cdot y\_m}}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z\_m \cdot \left(z\_m \cdot x\_m\right)}}{y\_m}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z_m 140000000.0)
     (/ (/ 1.0 (* (fma z_m z_m 1.0) y_m)) x_m)
     (/ (/ 1.0 (* z_m (* z_m x_m))) y_m)))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 140000000.0) {
		tmp = (1.0 / (fma(z_m, z_m, 1.0) * y_m)) / x_m;
	} else {
		tmp = (1.0 / (z_m * (z_m * x_m))) / y_m;
	}
	return y_s * (x_s * tmp);
}

Julia

z_m = abs(z)
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_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (z_m <= 140000000.0)
		tmp = Float64(Float64(1.0 / Float64(fma(z_m, z_m, 1.0) * y_m)) / x_m);
	else
		tmp = Float64(Float64(1.0 / Float64(z_m * Float64(z_m * x_m))) / y_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 140000000.0], N[(N[(1.0 / N[(N[(z$95$m * z$95$m + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision], N[(N[(1.0 / N[(z$95$m * N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.4e8

   1. Initial program 99.6%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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. associate-/l/ N/A

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

      7. *-commutative N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. +-commutative N/A

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

      15. pow2 N/A

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

      16. lower-fma.f64 99.7

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

   3. Applied rewrites 99.7%

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

   if 1.4e8 < z

   1. Initial program 77.2%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 85.8

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

   3. Applied rewrites 85.8%

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

   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 85.8

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

   6. Applied rewrites 85.8%

      \[\leadsto \frac{\color{blue}{\frac{1}{\left(z \cdot z\right) \cdot x}}}{y} \]
   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 97.7

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

   8. Applied rewrites 97.7%

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

Alternative 2: 98.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \cdot \left(1 + z\_m \cdot z\_m\right) \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z\_m \cdot y\_m, z\_m, y\_m\right) \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z\_m \cdot \left(z\_m \cdot x\_m\right)}}{y\_m}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* y_m (+ 1.0 (* z_m z_m))) 2e+305)
     (/ 1.0 (* (fma (* z_m y_m) z_m y_m) x_m))
     (/ (/ 1.0 (* z_m (* z_m x_m))) y_m)))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((y_m * (1.0 + (z_m * z_m))) <= 2e+305) {
		tmp = 1.0 / (fma((z_m * y_m), z_m, y_m) * x_m);
	} else {
		tmp = (1.0 / (z_m * (z_m * x_m))) / y_m;
	}
	return y_s * (x_s * tmp);
}

Julia

z_m = abs(z)
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_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(y_m * Float64(1.0 + Float64(z_m * z_m))) <= 2e+305)
		tmp = Float64(1.0 / Float64(fma(Float64(z_m * y_m), z_m, y_m) * x_m));
	else
		tmp = Float64(Float64(1.0 / Float64(z_m * Float64(z_m * x_m))) / y_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(y$95$m * N[(1.0 + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+305], N[(1.0 / N[(N[(N[(z$95$m * y$95$m), $MachinePrecision] * z$95$m + y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(z$95$m * N[(z$95$m * x$95$m), $MachinePrecision]), $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))) < 1.9999999999999999e305

   1. Initial program 99.6%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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. pow2 N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 99.6

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

   3. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. inv-pow N/A

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

      3. sqr-pow N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 N/A

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

      8. metadata-eval 99.2

         \[\leadsto \frac{{x}^{-0.5} \cdot {x}^{\color{blue}{-0.5}}}{\mathsf{fma}\left(y \cdot z, z, y\right)} \]

   5. Applied rewrites 99.2%

      \[\leadsto \frac{\color{blue}{{x}^{-0.5} \cdot {x}^{-0.5}}}{\mathsf{fma}\left(y \cdot z, z, y\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow-prod-up N/A

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

      6. metadata-eval N/A

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

      7. inv-pow N/A

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

      8. lift-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. associate-*l* N/A

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

      11. pow2 N/A

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

      12. +-commutative N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. +-commutative N/A

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

      17. pow2 N/A

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

      18. associate-*l* N/A

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

      19. lower-*.f64 N/A

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

      20. lift-fma.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 99.1

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

   7. Applied rewrites 99.1%

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

   if 1.9999999999999999e305 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 69.3%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 81.0

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

   3. Applied rewrites 81.0%

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

   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 79.5

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

   6. Applied rewrites 79.5%

      \[\leadsto \frac{\color{blue}{\frac{1}{\left(z \cdot z\right) \cdot x}}}{y} \]
   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 95.6

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

   8. Applied rewrites 95.6%

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

Alternative 3: 97.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 51000000000:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(z\_m, z\_m, 1\right) \cdot y\_m\right) \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z\_m \cdot \left(z\_m \cdot x\_m\right)}}{y\_m}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z_m 51000000000.0)
     (/ 1.0 (* (* (fma z_m z_m 1.0) y_m) x_m))
     (/ (/ 1.0 (* z_m (* z_m x_m))) y_m)))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 51000000000.0) {
		tmp = 1.0 / ((fma(z_m, z_m, 1.0) * y_m) * x_m);
	} else {
		tmp = (1.0 / (z_m * (z_m * x_m))) / y_m;
	}
	return y_s * (x_s * tmp);
}

Julia

z_m = abs(z)
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_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (z_m <= 51000000000.0)
		tmp = Float64(1.0 / Float64(Float64(fma(z_m, z_m, 1.0) * y_m) * x_m));
	else
		tmp = Float64(Float64(1.0 / Float64(z_m * Float64(z_m * x_m))) / y_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 51000000000.0], N[(1.0 / N[(N[(N[(z$95$m * z$95$m + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(z$95$m * N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 5.1e10

   1. Initial program 99.6%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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. pow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-fma.f64 99.1

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

   3. Applied rewrites 99.1%

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

   if 5.1e10 < z

   1. Initial program 77.0%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 85.7

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

   3. Applied rewrites 85.7%

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

   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 85.7

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

   6. Applied rewrites 85.7%

      \[\leadsto \frac{\color{blue}{\frac{1}{\left(z \cdot z\right) \cdot x}}}{y} \]
   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 97.7

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

   8. Applied rewrites 97.7%

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

Alternative 4: 97.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 0.86:\\ \;\;\;\;\frac{1 - z\_m \cdot z\_m}{y\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z\_m \cdot \left(z\_m \cdot x\_m\right)}}{y\_m}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z_m 0.86)
     (/ (- 1.0 (* z_m z_m)) (* y_m x_m))
     (/ (/ 1.0 (* z_m (* z_m x_m))) y_m)))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 0.86) {
		tmp = (1.0 - (z_m * z_m)) / (y_m * x_m);
	} else {
		tmp = (1.0 / (z_m * (z_m * x_m))) / y_m;
	}
	return y_s * (x_s * tmp);
}

Fortran

z_m =     private
x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
NOTE: x_m, y_m, and z_m 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_m)
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_m
    real(8) :: tmp
    if (z_m <= 0.86d0) then
        tmp = (1.0d0 - (z_m * z_m)) / (y_m * x_m)
    else
        tmp = (1.0d0 / (z_m * (z_m * x_m))) / y_m
    end if
    code = y_s * (x_s * tmp)
end function

Java

z_m = Math.abs(z);
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_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 0.86) {
		tmp = (1.0 - (z_m * z_m)) / (y_m * x_m);
	} else {
		tmp = (1.0 / (z_m * (z_m * x_m))) / y_m;
	}
	return y_s * (x_s * tmp);
}

Python

z_m = math.fabs(z)
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_m] = sort([x_m, y_m, z_m])
def code(y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if z_m <= 0.86:
		tmp = (1.0 - (z_m * z_m)) / (y_m * x_m)
	else:
		tmp = (1.0 / (z_m * (z_m * x_m))) / y_m
	return y_s * (x_s * tmp)

Julia

z_m = abs(z)
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_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (z_m <= 0.86)
		tmp = Float64(Float64(1.0 - Float64(z_m * z_m)) / Float64(y_m * x_m));
	else
		tmp = Float64(Float64(1.0 / Float64(z_m * Float64(z_m * x_m))) / y_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

z_m = abs(z);
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_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (z_m <= 0.86)
		tmp = (1.0 - (z_m * z_m)) / (y_m * x_m);
	else
		tmp = (1.0 / (z_m * (z_m * x_m))) / y_m;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 0.86], N[(N[(1.0 - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(z$95$m * N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 0.859999999999999987

   1. Initial program 99.6%

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

   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. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. mul-1-neg N/A

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

      5. pow2 N/A

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

      6. distribute-lft-neg-in N/A

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. pow2 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 98.7

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

   4. Applied rewrites 98.7%

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

   if 0.859999999999999987 < z

   1. Initial program 77.7%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 86.2

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

   3. Applied rewrites 86.2%

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

   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 85.4

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

   6. Applied rewrites 85.4%

      \[\leadsto \frac{\color{blue}{\frac{1}{\left(z \cdot z\right) \cdot x}}}{y} \]
   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 96.9

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

   8. Applied rewrites 96.9%

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

Alternative 5: 96.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 0.86:\\ \;\;\;\;\frac{1 - z\_m \cdot z\_m}{y\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y\_m \cdot z\_m}}{x\_m \cdot z\_m}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z_m 0.86)
     (/ (- 1.0 (* z_m z_m)) (* y_m x_m))
     (/ (/ 1.0 (* y_m z_m)) (* x_m z_m))))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 0.86) {
		tmp = (1.0 - (z_m * z_m)) / (y_m * x_m);
	} else {
		tmp = (1.0 / (y_m * z_m)) / (x_m * z_m);
	}
	return y_s * (x_s * tmp);
}

Fortran

z_m =     private
x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
NOTE: x_m, y_m, and z_m 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_m)
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_m
    real(8) :: tmp
    if (z_m <= 0.86d0) then
        tmp = (1.0d0 - (z_m * z_m)) / (y_m * x_m)
    else
        tmp = (1.0d0 / (y_m * z_m)) / (x_m * z_m)
    end if
    code = y_s * (x_s * tmp)
end function

Java

z_m = Math.abs(z);
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_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 0.86) {
		tmp = (1.0 - (z_m * z_m)) / (y_m * x_m);
	} else {
		tmp = (1.0 / (y_m * z_m)) / (x_m * z_m);
	}
	return y_s * (x_s * tmp);
}

Python

z_m = math.fabs(z)
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_m] = sort([x_m, y_m, z_m])
def code(y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if z_m <= 0.86:
		tmp = (1.0 - (z_m * z_m)) / (y_m * x_m)
	else:
		tmp = (1.0 / (y_m * z_m)) / (x_m * z_m)
	return y_s * (x_s * tmp)

Julia

z_m = abs(z)
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_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (z_m <= 0.86)
		tmp = Float64(Float64(1.0 - Float64(z_m * z_m)) / Float64(y_m * x_m));
	else
		tmp = Float64(Float64(1.0 / Float64(y_m * z_m)) / Float64(x_m * z_m));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

z_m = abs(z);
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_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (z_m <= 0.86)
		tmp = (1.0 - (z_m * z_m)) / (y_m * x_m);
	else
		tmp = (1.0 / (y_m * z_m)) / (x_m * z_m);
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 0.86], N[(N[(1.0 - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(y$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 0.859999999999999987

   1. Initial program 99.6%

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

   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. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. mul-1-neg N/A

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

      5. pow2 N/A

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

      6. distribute-lft-neg-in N/A

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. pow2 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 98.7

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

   4. Applied rewrites 98.7%

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

   if 0.859999999999999987 < z

   1. Initial program 77.7%

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

   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 76.8

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

   4. Applied rewrites 76.8%

      \[\leadsto \color{blue}{\frac{1}{\left(\left(z \cdot z\right) \cdot y\right) \cdot x}} \]
   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. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 93.9

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

   6. Applied rewrites 93.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 94.4

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 94.4

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

   8. Applied rewrites 94.4%

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

Alternative 6: 96.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 0.86:\\ \;\;\;\;\frac{1 - z\_m \cdot z\_m}{y\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(y\_m \cdot z\_m\right) \cdot \left(z\_m \cdot x\_m\right)}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z_m 0.86)
     (/ (- 1.0 (* z_m z_m)) (* y_m x_m))
     (/ 1.0 (* (* y_m z_m) (* z_m x_m)))))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 0.86) {
		tmp = (1.0 - (z_m * z_m)) / (y_m * x_m);
	} else {
		tmp = 1.0 / ((y_m * z_m) * (z_m * x_m));
	}
	return y_s * (x_s * tmp);
}

Fortran

z_m =     private
x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
NOTE: x_m, y_m, and z_m 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_m)
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_m
    real(8) :: tmp
    if (z_m <= 0.86d0) then
        tmp = (1.0d0 - (z_m * z_m)) / (y_m * x_m)
    else
        tmp = 1.0d0 / ((y_m * z_m) * (z_m * x_m))
    end if
    code = y_s * (x_s * tmp)
end function

Java

z_m = Math.abs(z);
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_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 0.86) {
		tmp = (1.0 - (z_m * z_m)) / (y_m * x_m);
	} else {
		tmp = 1.0 / ((y_m * z_m) * (z_m * x_m));
	}
	return y_s * (x_s * tmp);
}

Python

z_m = math.fabs(z)
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_m] = sort([x_m, y_m, z_m])
def code(y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if z_m <= 0.86:
		tmp = (1.0 - (z_m * z_m)) / (y_m * x_m)
	else:
		tmp = 1.0 / ((y_m * z_m) * (z_m * x_m))
	return y_s * (x_s * tmp)

Julia

z_m = abs(z)
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_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (z_m <= 0.86)
		tmp = Float64(Float64(1.0 - Float64(z_m * z_m)) / Float64(y_m * x_m));
	else
		tmp = Float64(1.0 / Float64(Float64(y_m * z_m) * Float64(z_m * x_m)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

z_m = abs(z);
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_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (z_m <= 0.86)
		tmp = (1.0 - (z_m * z_m)) / (y_m * x_m);
	else
		tmp = 1.0 / ((y_m * z_m) * (z_m * x_m));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 0.86], N[(N[(1.0 - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(y$95$m * z$95$m), $MachinePrecision] * N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 0.859999999999999987

   1. Initial program 99.6%

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

   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. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. mul-1-neg N/A

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

      5. pow2 N/A

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

      6. distribute-lft-neg-in N/A

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. pow2 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 98.7

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

   4. Applied rewrites 98.7%

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

   if 0.859999999999999987 < z

   1. Initial program 77.7%

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

   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 76.8

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

   4. Applied rewrites 76.8%

      \[\leadsto \color{blue}{\frac{1}{\left(\left(z \cdot z\right) \cdot y\right) \cdot x}} \]
   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. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 93.9

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

   6. Applied rewrites 93.9%

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

Alternative 7: 95.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 0.86:\\ \;\;\;\;\frac{1 - z\_m \cdot z\_m}{y\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x\_m \cdot \left(y\_m \cdot z\_m\right)\right) \cdot z\_m}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z_m 0.86)
     (/ (- 1.0 (* z_m z_m)) (* y_m x_m))
     (/ 1.0 (* (* x_m (* y_m z_m)) z_m))))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 0.86) {
		tmp = (1.0 - (z_m * z_m)) / (y_m * x_m);
	} else {
		tmp = 1.0 / ((x_m * (y_m * z_m)) * z_m);
	}
	return y_s * (x_s * tmp);
}

Fortran

z_m =     private
x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
NOTE: x_m, y_m, and z_m 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_m)
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_m
    real(8) :: tmp
    if (z_m <= 0.86d0) then
        tmp = (1.0d0 - (z_m * z_m)) / (y_m * x_m)
    else
        tmp = 1.0d0 / ((x_m * (y_m * z_m)) * z_m)
    end if
    code = y_s * (x_s * tmp)
end function

Java

z_m = Math.abs(z);
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_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 0.86) {
		tmp = (1.0 - (z_m * z_m)) / (y_m * x_m);
	} else {
		tmp = 1.0 / ((x_m * (y_m * z_m)) * z_m);
	}
	return y_s * (x_s * tmp);
}

Python

z_m = math.fabs(z)
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_m] = sort([x_m, y_m, z_m])
def code(y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if z_m <= 0.86:
		tmp = (1.0 - (z_m * z_m)) / (y_m * x_m)
	else:
		tmp = 1.0 / ((x_m * (y_m * z_m)) * z_m)
	return y_s * (x_s * tmp)

Julia

z_m = abs(z)
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_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (z_m <= 0.86)
		tmp = Float64(Float64(1.0 - Float64(z_m * z_m)) / Float64(y_m * x_m));
	else
		tmp = Float64(1.0 / Float64(Float64(x_m * Float64(y_m * z_m)) * z_m));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

z_m = abs(z);
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_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (z_m <= 0.86)
		tmp = (1.0 - (z_m * z_m)) / (y_m * x_m);
	else
		tmp = 1.0 / ((x_m * (y_m * z_m)) * z_m);
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 0.86], N[(N[(1.0 - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(x$95$m * N[(y$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 0.859999999999999987

   1. Initial program 99.6%

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

   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. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. mul-1-neg N/A

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

      5. pow2 N/A

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

      6. distribute-lft-neg-in N/A

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. pow2 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 98.7

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

   4. Applied rewrites 98.7%

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

   if 0.859999999999999987 < z

   1. Initial program 77.7%

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

   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 76.8

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

   4. Applied rewrites 76.8%

      \[\leadsto \color{blue}{\frac{1}{\left(\left(z \cdot z\right) \cdot y\right) \cdot x}} \]
   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. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 92.4

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

   6. Applied rewrites 92.4%

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

Alternative 8: 57.5% accurate, 2.2× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (* y_s (* x_s (/ 1.0 (* y_m x_m)))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	return y_s * (x_s * (1.0 / (y_m * x_m)));
}

Fortran

z_m =     private
x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
NOTE: x_m, y_m, and z_m 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_m)
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_m
    code = y_s * (x_s * (1.0d0 / (y_m * x_m)))
end function

Java

z_m = Math.abs(z);
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_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	return y_s * (x_s * (1.0 / (y_m * x_m)));
}

Python

z_m = math.fabs(z)
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_m] = sort([x_m, y_m, z_m])
def code(y_s, x_s, x_m, y_m, z_m):
	return y_s * (x_s * (1.0 / (y_m * x_m)))

Julia

z_m = abs(z)
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_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	return Float64(y_s * Float64(x_s * Float64(1.0 / Float64(y_m * x_m))))
end

MATLAB

z_m = abs(z);
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_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp = code(y_s, x_s, x_m, y_m, z_m)
	tmp = y_s * (x_s * (1.0 / (y_m * x_m)));
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := 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 57.5

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

4. Applied rewrites 57.5%

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

Reproduce

herbie shell --seed 2025115 
(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)))))