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

Percentage Accurate: 90.7% → 96.6%

Time: 3.5s

Alternatives: 11

Speedup: 0.8×

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: 90.7% 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: 96.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(z \cdot y, z, y\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(y \cdot x\right)}^{-1}}{\mathsf{fma}\left(z, z, 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 2e+14)
   (/ (/ 1.0 (fma (* z y) z y)) x)
   (/ (pow (* y x) -1.0) (fma z z 1.0))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e+14) {
		tmp = (1.0 / fma((z * y), z, y)) / x;
	} else {
		tmp = pow((y * x), -1.0) / fma(z, z, 1.0);
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 2e+14)
		tmp = Float64(Float64(1.0 / fma(Float64(z * y), z, y)) / x);
	else
		tmp = Float64((Float64(y * x) ^ -1.0) / fma(z, z, 1.0));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 2e+14], N[(N[(1.0 / N[(N[(z * y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[Power[N[(y * x), $MachinePrecision], -1.0], $MachinePrecision] / N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2e14

   1. Initial program 89.3%

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

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

      3. lift-*.f64 N/A

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

      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 96.1

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

   3. Applied rewrites 96.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 96.1

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

   5. Applied rewrites 96.1%

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

   if 2e14 < y

   1. Initial program 92.7%

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

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. inv-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. +-commutative N/A

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

      15. pow2 N/A

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

      16. lower-fma.f64 97.4

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

   3. Applied rewrites 97.4%

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

Alternative 2: 94.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 5 \cdot 10^{+303}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(y \cdot z, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(y \cdot x\right)}^{-1} \cdot {z}^{-2}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* y (+ 1.0 (* z z))) 5e+303)
   (/ (/ 1.0 x) (fma (* y z) z y))
   (* (pow (* y x) -1.0) (pow z -2.0))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((y * (1.0 + (z * z))) <= 5e+303) {
		tmp = (1.0 / x) / fma((y * z), z, y);
	} else {
		tmp = pow((y * x), -1.0) * pow(z, -2.0);
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(y * Float64(1.0 + Float64(z * z))) <= 5e+303)
		tmp = Float64(Float64(1.0 / x) / fma(Float64(y * z), z, y));
	else
		tmp = Float64((Float64(y * x) ^ -1.0) * (z ^ -2.0));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+303], N[(N[(1.0 / x), $MachinePrecision] / N[(N[(y * z), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(y * x), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[z, -2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 4.9999999999999997e303

   1. Initial program 97.2%

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

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

      3. lift-*.f64 N/A

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

      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 99.2

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

   3. Applied rewrites 99.2%

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

   if 4.9999999999999997e303 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 71.0%

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

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

      5. lift-*.f64 N/A

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

      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. inv-pow N/A

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

      9. associate-*r* N/A

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

      10. pow2 N/A

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

      11. unpow-prod-down N/A

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

      12. inv-pow N/A

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

      13. lower-*.f64 N/A

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

      14. inv-pow N/A

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

      15. lower-pow.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. pow2 N/A

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

      20. +-commutative N/A

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

      21. pow2 N/A

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

      22. lower-fma.f64 79.1

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

   3. Applied rewrites 79.1%

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

   4. Taylor expanded in z around inf

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

      1. pow-flip N/A

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

      2. lower-pow.f64 N/A

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

      3. metadata-eval 78.8

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

   6. Applied rewrites 78.8%

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

Alternative 3: 92.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 5 \cdot 10^{+303}:\\ \;\;\;\;\frac{\frac{{y}^{-1}}{\mathsf{fma}\left(z, z, 1\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;{\left(y \cdot x\right)}^{-1} \cdot {z}^{-2}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* y (+ 1.0 (* z z))) 5e+303)
   (/ (/ (pow y -1.0) (fma z z 1.0)) x)
   (* (pow (* y x) -1.0) (pow z -2.0))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((y * (1.0 + (z * z))) <= 5e+303) {
		tmp = (pow(y, -1.0) / fma(z, z, 1.0)) / x;
	} else {
		tmp = pow((y * x), -1.0) * pow(z, -2.0);
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(y * Float64(1.0 + Float64(z * z))) <= 5e+303)
		tmp = Float64(Float64((y ^ -1.0) / fma(z, z, 1.0)) / x);
	else
		tmp = Float64((Float64(y * x) ^ -1.0) * (z ^ -2.0));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+303], N[(N[(N[Power[y, -1.0], $MachinePrecision] / N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[Power[N[(y * x), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[z, -2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 4.9999999999999997e303

   1. Initial program 97.2%

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

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

      3. lift-*.f64 N/A

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

      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 99.2

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

   3. Applied rewrites 99.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in y around -inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 97.2

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

   8. Applied rewrites 97.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. distribute-frac-neg N/A

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

      5. lift--.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

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

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

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

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

      12. metadata-eval N/A

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

      13. distribute-neg-in N/A

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

      14. +-commutative N/A

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

      15. frac-2neg N/A

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

      16. lower-/.f64 N/A

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

      17. inv-pow N/A

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

      18. lower-pow.f64 N/A

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

      19. +-commutative N/A

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

      20. pow2 N/A

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

      21. lower-fma.f64 97.2

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

   10. Applied rewrites 97.2%

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

   if 4.9999999999999997e303 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 71.0%

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

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

      5. lift-*.f64 N/A

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

      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. inv-pow N/A

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

      9. associate-*r* N/A

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

      10. pow2 N/A

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

      11. unpow-prod-down N/A

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

      12. inv-pow N/A

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

      13. lower-*.f64 N/A

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

      14. inv-pow N/A

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

      15. lower-pow.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. pow2 N/A

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

      20. +-commutative N/A

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

      21. pow2 N/A

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

      22. lower-fma.f64 79.1

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

   3. Applied rewrites 79.1%

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

   4. Taylor expanded in z around inf

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

      1. pow-flip N/A

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

      2. lower-pow.f64 N/A

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

      3. metadata-eval 78.8

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

   6. Applied rewrites 78.8%

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

Alternative 4: 71.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 0.000205:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, z, 1\right)}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{-1}}{z \cdot z}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z 0.000205)
   (/ (fma (- z) z 1.0) (* y x))
   (/ (/ (pow x -1.0) (* z z)) y)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.000205) {
		tmp = fma(-z, z, 1.0) / (y * x);
	} else {
		tmp = (pow(x, -1.0) / (z * z)) / y;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (z <= 0.000205)
		tmp = Float64(fma(Float64(-z), z, 1.0) / Float64(y * x));
	else
		tmp = Float64(Float64((x ^ -1.0) / Float64(z * z)) / y);
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[z, 0.000205], N[(N[((-z) * z + 1.0), $MachinePrecision] / N[(y * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[x, -1.0], $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.05e-4

   1. Initial program 93.5%

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

   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. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. mul-1-neg N/A

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

      5. pow2 N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 68.7

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

   4. Applied rewrites 68.7%

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

   if 2.05e-4 < z

   1. Initial program 82.3%

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

   2. Taylor expanded in z around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 80.6

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

   4. Applied rewrites 80.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 81.6

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

      10. pow2 81.6

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

      11. +-commutative 81.6

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

      12. pow2 81.6

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

   6. Applied rewrites 81.6%

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

Alternative 5: 71.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 0.000205:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, z, 1\right)}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\left(z \cdot z\right) \cdot x\right)}^{-1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z 0.000205)
   (/ (fma (- z) z 1.0) (* y x))
   (/ (pow (* (* z z) x) -1.0) y)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.000205) {
		tmp = fma(-z, z, 1.0) / (y * x);
	} else {
		tmp = pow(((z * z) * x), -1.0) / y;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (z <= 0.000205)
		tmp = Float64(fma(Float64(-z), z, 1.0) / Float64(y * x));
	else
		tmp = Float64((Float64(Float64(z * z) * x) ^ -1.0) / y);
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[z, 0.000205], N[(N[((-z) * z + 1.0), $MachinePrecision] / N[(y * x), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(z * z), $MachinePrecision] * x), $MachinePrecision], -1.0], $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.05e-4

   1. Initial program 93.5%

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

   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. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. mul-1-neg N/A

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

      5. pow2 N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 68.7

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

   4. Applied rewrites 68.7%

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

   if 2.05e-4 < z

   1. Initial program 82.3%

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

   2. Taylor expanded in z around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 80.6

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

   4. Applied rewrites 80.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 81.6

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

      10. pow2 81.6

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

      11. +-commutative 81.6

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

      12. pow2 81.6

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

   6. Applied rewrites 81.6%

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

   7. Taylor expanded in z around inf

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 81.2

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

   9. Applied rewrites 81.2%

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

Alternative 6: 71.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 0.000205:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, z, 1\right)}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;{\left(y \cdot x\right)}^{-1} \cdot {z}^{-2}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z 0.000205)
   (/ (fma (- z) z 1.0) (* y x))
   (* (pow (* y x) -1.0) (pow z -2.0))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.000205) {
		tmp = fma(-z, z, 1.0) / (y * x);
	} else {
		tmp = pow((y * x), -1.0) * pow(z, -2.0);
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (z <= 0.000205)
		tmp = Float64(fma(Float64(-z), z, 1.0) / Float64(y * x));
	else
		tmp = Float64((Float64(y * x) ^ -1.0) * (z ^ -2.0));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[z, 0.000205], N[(N[((-z) * z + 1.0), $MachinePrecision] / N[(y * x), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(y * x), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[z, -2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.05e-4

   1. Initial program 93.5%

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

   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. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. mul-1-neg N/A

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

      5. pow2 N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 68.7

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

   4. Applied rewrites 68.7%

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

   if 2.05e-4 < z

   1. Initial program 82.3%

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

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

      5. lift-*.f64 N/A

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

      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. inv-pow N/A

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

      9. associate-*r* N/A

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

      10. pow2 N/A

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

      11. unpow-prod-down N/A

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

      12. inv-pow N/A

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

      13. lower-*.f64 N/A

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

      14. inv-pow N/A

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

      15. lower-pow.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. pow2 N/A

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

      20. +-commutative N/A

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

      21. pow2 N/A

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

      22. lower-fma.f64 82.7

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

   3. Applied rewrites 82.7%

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

   4. Taylor expanded in z around inf

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

      1. pow-flip N/A

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

      2. lower-pow.f64 N/A

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

      3. metadata-eval 81.5

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

   6. Applied rewrites 81.5%

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

Alternative 7: 71.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 0.000205:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, z, 1\right)}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(z \cdot z\right) \cdot y\right) \cdot x\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z 0.000205)
   (/ (fma (- z) z 1.0) (* y x))
   (pow (* (* (* z z) y) x) -1.0)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.000205) {
		tmp = fma(-z, z, 1.0) / (y * x);
	} else {
		tmp = pow((((z * z) * y) * x), -1.0);
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (z <= 0.000205)
		tmp = Float64(fma(Float64(-z), z, 1.0) / Float64(y * x));
	else
		tmp = Float64(Float64(Float64(z * z) * y) * x) ^ -1.0;
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[z, 0.000205], N[(N[((-z) * z + 1.0), $MachinePrecision] / N[(y * x), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision] * x), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.05e-4

   1. Initial program 93.5%

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

   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. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. mul-1-neg N/A

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

      5. pow2 N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 68.7

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

   4. Applied rewrites 68.7%

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

   if 2.05e-4 < z

   1. Initial program 82.3%

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

   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. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 80.4

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

   4. Applied rewrites 80.4%

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

Alternative 8: 63.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;1 + z \cdot z \leq 1.00000002:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, z, 1\right)}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(y \cdot y\right)}^{-0.5}}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (+ 1.0 (* z z)) 1.00000002)
   (/ (fma (- z) z 1.0) (* y x))
   (/ (pow (* y y) -0.5) x)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((1.0 + (z * z)) <= 1.00000002) {
		tmp = fma(-z, z, 1.0) / (y * x);
	} else {
		tmp = pow((y * y), -0.5) / x;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(1.0 + Float64(z * z)) <= 1.00000002)
		tmp = Float64(fma(Float64(-z), z, 1.0) / Float64(y * x));
	else
		tmp = Float64((Float64(y * y) ^ -0.5) / x);
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision], 1.00000002], N[(N[((-z) * z + 1.0), $MachinePrecision] / N[(y * x), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(y * y), $MachinePrecision], -0.5], $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 #s(literal 1 binary64) (*.f64 z z)) < 1.0000000200000001

   1. Initial program 99.7%

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

   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. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. mul-1-neg N/A

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

      5. pow2 N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.2

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

   4. Applied rewrites 99.2%

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

   if 1.0000000200000001 < (+.f64 #s(literal 1 binary64) (*.f64 z z))

   1. Initial program 81.8%

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

   2. Taylor expanded in z around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. *-commutative N/A

         \[\leadsto {\left(y \cdot x\right)}^{-1} \]

      4. lower-*.f64 20.5

         \[\leadsto {\left(y \cdot x\right)}^{-1} \]

   4. Applied rewrites 20.5%

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

      1. lift-*.f64 N/A

         \[\leadsto {\left(y \cdot x\right)}^{-1} \]

      2. lift-pow.f64 N/A

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

      3. unpow-1 N/A

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

      4. associate-/r* N/A

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

      5. unpow-1 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 19.9

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

   6. Applied rewrites 19.9%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-down N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval 27.9

         \[\leadsto \frac{{\left(y \cdot y\right)}^{-0.5}}{x} \]

   8. Applied rewrites 27.9%

      \[\leadsto \frac{{\left(y \cdot y\right)}^{-0.5}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 63.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;1 + z \cdot z \leq 1.00000002:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(y \cdot y\right)}^{-0.5}}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (+ 1.0 (* z z)) 1.00000002) (/ (/ 1.0 x) y) (/ (pow (* y y) -0.5) x)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((1.0 + (z * z)) <= 1.00000002) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = pow((y * y), -0.5) / x;
	}
	return tmp;
}

Fortran

NOTE: x, y, 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(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((1.0d0 + (z * z)) <= 1.00000002d0) then
        tmp = (1.0d0 / x) / y
    else
        tmp = ((y * y) ** (-0.5d0)) / x
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if ((1.0 + (z * z)) <= 1.00000002) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = Math.pow((y * y), -0.5) / x;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if (1.0 + (z * z)) <= 1.00000002:
		tmp = (1.0 / x) / y
	else:
		tmp = math.pow((y * y), -0.5) / x
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(1.0 + Float64(z * z)) <= 1.00000002)
		tmp = Float64(Float64(1.0 / x) / y);
	else
		tmp = Float64((Float64(y * y) ^ -0.5) / x);
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((1.0 + (z * z)) <= 1.00000002)
		tmp = (1.0 / x) / y;
	else
		tmp = ((y * y) ^ -0.5) / x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision], 1.00000002], N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision], N[(N[Power[N[(y * y), $MachinePrecision], -0.5], $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 #s(literal 1 binary64) (*.f64 z z)) < 1.0000000200000001

   1. Initial program 99.7%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 99.5%

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

   if 1.0000000200000001 < (+.f64 #s(literal 1 binary64) (*.f64 z z))

   1. Initial program 81.8%

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

   2. Taylor expanded in z around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. *-commutative N/A

         \[\leadsto {\left(y \cdot x\right)}^{-1} \]

      4. lower-*.f64 20.5

         \[\leadsto {\left(y \cdot x\right)}^{-1} \]

   4. Applied rewrites 20.5%

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

      1. lift-*.f64 N/A

         \[\leadsto {\left(y \cdot x\right)}^{-1} \]

      2. lift-pow.f64 N/A

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

      3. unpow-1 N/A

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

      4. associate-/r* N/A

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

      5. unpow-1 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 19.9

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

   6. Applied rewrites 19.9%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-down N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval 27.9

         \[\leadsto \frac{{\left(y \cdot y\right)}^{-0.5}}{x} \]

   8. Applied rewrites 27.9%

      \[\leadsto \frac{{\left(y \cdot y\right)}^{-0.5}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 59.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ {\left(y \cdot x\right)}^{-1} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (pow (* y x) -1.0))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return pow((y * x), -1.0);
}

Fortran

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

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	return Math.pow((y * x), -1.0);
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return math.pow((y * x), -1.0)

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(y * x) ^ -1.0
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = (y * x) ^ -1.0;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[Power[N[(y * x), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 90.7%

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

2. Taylor expanded in z around 0

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

   1. inv-pow N/A

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

   2. lower-pow.f64 N/A

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

   3. *-commutative N/A

      \[\leadsto {\left(y \cdot x\right)}^{-1} \]

   4. lower-*.f64 59.6

      \[\leadsto {\left(y \cdot x\right)}^{-1} \]

4. Applied rewrites 59.6%

   \[\leadsto \color{blue}{{\left(y \cdot x\right)}^{-1}} \]
5. Add Preprocessing

Alternative 11: 59.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \frac{{y}^{-1}}{x} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (/ (pow y -1.0) x))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return pow(y, -1.0) / x;
}

Fortran

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

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	return Math.pow(y, -1.0) / x;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return math.pow(y, -1.0) / x

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64((y ^ -1.0) / x)
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = (y ^ -1.0) / x;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(N[Power[y, -1.0], $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 90.7%

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

2. Taylor expanded in z around 0

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

   1. inv-pow N/A

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

   2. lower-pow.f64 N/A

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

   3. *-commutative N/A

      \[\leadsto {\left(y \cdot x\right)}^{-1} \]

   4. lower-*.f64 59.6

      \[\leadsto {\left(y \cdot x\right)}^{-1} \]

4. Applied rewrites 59.6%

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

   1. lift-*.f64 N/A

      \[\leadsto {\left(y \cdot x\right)}^{-1} \]

   2. lift-pow.f64 N/A

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

   3. unpow-1 N/A

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

   4. associate-/r* N/A

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

   5. unpow-1 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-pow.f64 59.5

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

6. Applied rewrites 59.5%

   \[\leadsto \frac{{y}^{-1}}{\color{blue}{x}} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025101 
(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)))))