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

Percentage Accurate: 91.0% → 97.7%

Time: 3.4s

Alternatives: 12

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.0% 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: 97.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.0%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
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. 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 90.6

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

3. Applied rewrites 90.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. *-commutative N/A

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

   7. pow2 N/A

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

   8. distribute-rgt-in N/A

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

   9. *-lft-identity N/A

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

   10. *-commutative N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

   13. pow2 N/A

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

   14. associate-*l* N/A

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

   15. associate-*l* N/A

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

   16. lower-fma.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. lower-*.f64 97.7

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

5. Applied rewrites 97.7%

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

Alternative 2: 92.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.08 \cdot 10^{+142}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{\mathsf{fma}\left(z, z, 1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(x \cdot y\right) \cdot z\right) \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.08e+142)
   (/ (/ (/ 1.0 x) (fma z z 1.0)) y)
   (/ 1.0 (* (* (* x y) z) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.08e+142) {
		tmp = ((1.0 / x) / fma(z, z, 1.0)) / y;
	} else {
		tmp = 1.0 / (((x * y) * z) * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.08e+142)
		tmp = Float64(Float64(Float64(1.0 / x) / fma(z, z, 1.0)) / y);
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(x * y) * z) * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.08e+142], N[(N[(N[(1.0 / x), $MachinePrecision] / N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(1.0 / N[(N[(N[(x * y), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.08e142

   1. Initial program 93.4%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 93.3

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

   3. Applied rewrites 93.3%

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

   if 1.08e142 < z

   1. Initial program 74.1%

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

   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 74.1

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

   4. Applied rewrites 74.1%

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

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 73.3

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

      10. pow2 73.3

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

      11. +-commutative 73.3

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

      12. pow2 73.3

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

   6. Applied rewrites 73.3%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 88.2

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

   9. Applied rewrites 88.2%

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

   10. Taylor expanded in z around inf

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

   12. Applied rewrites 88.2%

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

Alternative 3: 92.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \leq 0:\\ \;\;\;\;\frac{1}{\left(\left(x \cdot y\right) \cdot z\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(z, z, 1\right) \cdot y}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))) 0.0)
   (/ 1.0 (* (* (* x y) z) z))
   (/ (/ 1.0 (* (fma z z 1.0) y)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((1.0 / x) / (y * (1.0 + (z * z)))) <= 0.0) {
		tmp = 1.0 / (((x * y) * z) * z);
	} else {
		tmp = (1.0 / (fma(z, z, 1.0) * y)) / x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(1.0 / x) / Float64(y * Float64(1.0 + Float64(z * z)))) <= 0.0)
		tmp = Float64(1.0 / Float64(Float64(Float64(x * y) * z) * z));
	else
		tmp = Float64(Float64(1.0 / Float64(fma(z, z, 1.0) * y)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(1.0 / x), $MachinePrecision] / N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(1.0 / N[(N[(N[(x * y), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(z * z + 1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 1 binary64) x) (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))) < 0.0

   1. Initial program 87.4%

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

   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 54.2

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

   4. Applied rewrites 54.2%

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

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 58.0

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

      10. pow2 58.0

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

      11. +-commutative 58.0

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

      12. pow2 58.0

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

   6. Applied rewrites 58.0%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 75.0

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

   9. Applied rewrites 75.0%

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

   10. Taylor expanded in z around inf

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

   12. Applied rewrites 68.2%

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

   if 0.0 < (/.f64 (/.f64 #s(literal 1 binary64) x) (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))))

   1. Initial program 99.5%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 99.6

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

   3. Applied rewrites 99.6%

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

Alternative 4: 77.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \leq 0:\\ \;\;\;\;\frac{1}{\left(\left(x \cdot y\right) \cdot z\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(y \cdot z, z, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))) 0.0)
   (/ 1.0 (* (* (* x y) z) z))
   (/ (/ 1.0 x) (fma (* y z) z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((1.0 / x) / (y * (1.0 + (z * z)))) <= 0.0) {
		tmp = 1.0 / (((x * y) * z) * z);
	} else {
		tmp = (1.0 / x) / fma((y * z), z, y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(1.0 / x) / Float64(y * Float64(1.0 + Float64(z * z)))) <= 0.0)
		tmp = Float64(1.0 / Float64(Float64(Float64(x * y) * z) * z));
	else
		tmp = Float64(Float64(1.0 / x) / fma(Float64(y * z), z, y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(1.0 / x), $MachinePrecision] / N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(1.0 / N[(N[(N[(x * y), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / N[(N[(y * z), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 1 binary64) x) (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))) < 0.0

   1. Initial program 87.4%

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

   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 54.2

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

   4. Applied rewrites 54.2%

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

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 58.0

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

      10. pow2 58.0

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

      11. +-commutative 58.0

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

      12. pow2 58.0

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

   6. Applied rewrites 58.0%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 75.0

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

   9. Applied rewrites 75.0%

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

   10. Taylor expanded in z around inf

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

   12. Applied rewrites 68.2%

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

   if 0.0 < (/.f64 (/.f64 #s(literal 1 binary64) x) (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))))

   1. Initial program 99.5%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 99.5

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

   3. Applied rewrites 99.5%

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

Alternative 5: 77.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \leq 10^{-317}:\\ \;\;\;\;\frac{1}{\left(\left(x \cdot y\right) \cdot z\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(z, z, 1\right) \cdot y\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))) 1e-317)
   (/ 1.0 (* (* (* x y) z) z))
   (/ 1.0 (* (* (fma z z 1.0) y) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((1.0 / x) / (y * (1.0 + (z * z)))) <= 1e-317) {
		tmp = 1.0 / (((x * y) * z) * z);
	} else {
		tmp = 1.0 / ((fma(z, z, 1.0) * y) * x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(1.0 / x) / Float64(y * Float64(1.0 + Float64(z * z)))) <= 1e-317)
		tmp = Float64(1.0 / Float64(Float64(Float64(x * y) * z) * z));
	else
		tmp = Float64(1.0 / Float64(Float64(fma(z, z, 1.0) * y) * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(1.0 / x), $MachinePrecision] / N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-317], N[(1.0 / N[(N[(N[(x * y), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(z * z + 1.0), $MachinePrecision] * y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 1 binary64) x) (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))) < 1.00000023e-317

   1. Initial program 87.4%

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

   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 54.1

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

   4. Applied rewrites 54.1%

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

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 58.0

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

      10. pow2 58.0

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

      11. +-commutative 58.0

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

      12. pow2 58.0

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

   6. Applied rewrites 58.0%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 75.0

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

   9. Applied rewrites 75.0%

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

   10. Taylor expanded in z around inf

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

   12. Applied rewrites 68.3%

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

   if 1.00000023e-317 < (/.f64 (/.f64 #s(literal 1 binary64) x) (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))))

   1. Initial program 99.5%

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

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

Alternative 6: 77.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{elif}\;z \leq 1.92 \cdot 10^{+143}:\\ \;\;\;\;\frac{\frac{1}{\left(z \cdot z\right) \cdot x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(x \cdot y\right) \cdot z\right) \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.0)
   (/ (/ 1.0 x) y)
   (if (<= z 1.92e+143)
     (/ (/ 1.0 (* (* z z) x)) y)
     (/ 1.0 (* (* (* x y) z) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / x) / y;
	} else if (z <= 1.92e+143) {
		tmp = (1.0 / ((z * z) * x)) / y;
	} else {
		tmp = 1.0 / (((x * y) * z) * z);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 1.0d0) then
        tmp = (1.0d0 / x) / y
    else if (z <= 1.92d+143) then
        tmp = (1.0d0 / ((z * z) * x)) / y
    else
        tmp = 1.0d0 / (((x * y) * z) * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / x) / y;
	} else if (z <= 1.92e+143) {
		tmp = (1.0 / ((z * z) * x)) / y;
	} else {
		tmp = 1.0 / (((x * y) * z) * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.0:
		tmp = (1.0 / x) / y
	elif z <= 1.92e+143:
		tmp = (1.0 / ((z * z) * x)) / y
	else:
		tmp = 1.0 / (((x * y) * z) * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.0)
		tmp = Float64(Float64(1.0 / x) / y);
	elseif (z <= 1.92e+143)
		tmp = Float64(Float64(1.0 / Float64(Float64(z * z) * x)) / y);
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(x * y) * z) * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.0)
		tmp = (1.0 / x) / y;
	elseif (z <= 1.92e+143)
		tmp = (1.0 / ((z * z) * x)) / y;
	else
		tmp = 1.0 / (((x * y) * z) * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.0], N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[z, 1.92e+143], N[(N[(1.0 / N[(N[(z * z), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(1.0 / N[(N[(N[(x * y), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 1

   1. Initial program 93.8%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 71.3%

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

   if 1 < z < 1.9199999999999999e143

   1. Initial program 91.2%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 91.7

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

   3. Applied rewrites 91.7%

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

   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 89.4

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

   6. Applied rewrites 89.4%

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

   if 1.9199999999999999e143 < z

   1. Initial program 74.0%

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

   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 74.0

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

   4. Applied rewrites 74.0%

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

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 73.3

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

      10. pow2 73.3

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

      11. +-commutative 73.3

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

      12. pow2 73.3

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

   6. Applied rewrites 73.3%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 88.3

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

   9. Applied rewrites 88.3%

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

   10. Taylor expanded in z around inf

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

   12. Applied rewrites 88.3%

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

Alternative 7: 75.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 10^{+141}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(z, z, 1\right) \cdot x\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(x \cdot y\right) \cdot z\right) \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1e+141)
   (/ 1.0 (* (* (fma z z 1.0) x) y))
   (/ 1.0 (* (* (* x y) z) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1e+141) {
		tmp = 1.0 / ((fma(z, z, 1.0) * x) * y);
	} else {
		tmp = 1.0 / (((x * y) * z) * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1e+141)
		tmp = Float64(1.0 / Float64(Float64(fma(z, z, 1.0) * x) * y));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(x * y) * z) * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1e+141], N[(1.0 / N[(N[(N[(z * z + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(x * y), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.00000000000000002e141

   1. Initial program 93.4%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 93.1

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

   3. Applied rewrites 93.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-fma.f64 92.8

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

   5. Applied rewrites 92.8%

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

   if 1.00000000000000002e141 < z

   1. Initial program 74.3%

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

   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 74.3

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

   4. Applied rewrites 74.3%

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

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 73.5

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

      10. pow2 73.5

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

      11. +-commutative 73.5

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

      12. pow2 73.5

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

   6. Applied rewrites 73.5%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 88.3

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

   9. Applied rewrites 88.3%

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

   10. Taylor expanded in z around inf

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

   12. Applied rewrites 88.3%

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

Alternative 8: 75.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{elif}\;z \leq 10^{+141}:\\ \;\;\;\;\frac{1}{\left(\left(z \cdot z\right) \cdot x\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(x \cdot y\right) \cdot z\right) \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.0)
   (/ (/ 1.0 x) y)
   (if (<= z 1e+141) (/ 1.0 (* (* (* z z) x) y)) (/ 1.0 (* (* (* x y) z) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / x) / y;
	} else if (z <= 1e+141) {
		tmp = 1.0 / (((z * z) * x) * y);
	} else {
		tmp = 1.0 / (((x * y) * z) * z);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 1.0d0) then
        tmp = (1.0d0 / x) / y
    else if (z <= 1d+141) then
        tmp = 1.0d0 / (((z * z) * x) * y)
    else
        tmp = 1.0d0 / (((x * y) * z) * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / x) / y;
	} else if (z <= 1e+141) {
		tmp = 1.0 / (((z * z) * x) * y);
	} else {
		tmp = 1.0 / (((x * y) * z) * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.0:
		tmp = (1.0 / x) / y
	elif z <= 1e+141:
		tmp = 1.0 / (((z * z) * x) * y)
	else:
		tmp = 1.0 / (((x * y) * z) * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.0)
		tmp = Float64(Float64(1.0 / x) / y);
	elseif (z <= 1e+141)
		tmp = Float64(1.0 / Float64(Float64(Float64(z * z) * x) * y));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(x * y) * z) * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.0)
		tmp = (1.0 / x) / y;
	elseif (z <= 1e+141)
		tmp = 1.0 / (((z * z) * x) * y);
	else
		tmp = 1.0 / (((x * y) * z) * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.0], N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[z, 1e+141], N[(1.0 / N[(N[(N[(z * z), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(x * y), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 1

   1. Initial program 93.8%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 71.3%

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

   if 1 < z < 1.00000000000000002e141

   1. Initial program 91.1%

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

   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 89.4

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

   4. Applied rewrites 89.4%

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

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 91.9

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

      10. pow2 91.9

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

      11. +-commutative 91.9

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

      12. pow2 91.9

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

   6. Applied rewrites 91.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 89.0

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

   8. Applied rewrites 89.0%

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

   if 1.00000000000000002e141 < z

   1. Initial program 74.3%

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

   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 74.3

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

   4. Applied rewrites 74.3%

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

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 73.5

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

      10. pow2 73.5

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

      11. +-commutative 73.5

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

      12. pow2 73.5

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

   6. Applied rewrites 73.5%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 88.3

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

   9. Applied rewrites 88.3%

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

   10. Taylor expanded in z around inf

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

   12. Applied rewrites 88.3%

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

Alternative 9: 75.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(x \cdot y\right) \cdot z\right) \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.0) (/ (/ 1.0 x) y) (/ 1.0 (* (* (* x y) z) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / (((x * y) * z) * z);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / (((x * y) * z) * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.0:
		tmp = (1.0 / x) / y
	else:
		tmp = 1.0 / (((x * y) * z) * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.0)
		tmp = Float64(Float64(1.0 / x) / y);
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(x * y) * z) * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.0)
		tmp = (1.0 / x) / y;
	else
		tmp = 1.0 / (((x * y) * z) * z);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 93.8%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 71.3%

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

   if 1 < z

   1. Initial program 82.4%

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

   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 81.5

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

   4. Applied rewrites 81.5%

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

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 82.3

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

      10. pow2 82.3

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

      11. +-commutative 82.3

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

      12. pow2 82.3

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

   6. Applied rewrites 82.3%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 90.9

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

   9. Applied rewrites 90.9%

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

   10. Taylor expanded in z around inf

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

   12. Applied rewrites 90.0%

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

Alternative 10: 75.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(y \cdot z\right) \cdot z\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.0) (/ (/ 1.0 x) y) (/ 1.0 (* (* (* y z) z) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / (((y * z) * z) * x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / (((y * z) * z) * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.0:
		tmp = (1.0 / x) / y
	else:
		tmp = 1.0 / (((y * z) * z) * x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.0)
		tmp = Float64(Float64(1.0 / x) / y);
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(y * z) * z) * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.0)
		tmp = (1.0 / x) / y;
	else
		tmp = 1.0 / (((y * z) * z) * x);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 93.8%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 71.3%

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

   if 1 < z

   1. Initial program 82.4%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 82.2

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

   3. Applied rewrites 82.2%

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

   4. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot {z}^{2}\right)}} \]
   5. 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. pow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 88.8

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

   6. Applied rewrites 88.8%

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

Alternative 11: 73.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(z \cdot z\right) \cdot y\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.0) (/ (/ 1.0 x) y) (/ 1.0 (* (* (* z z) y) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / (((z * z) * y) * x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / (((z * z) * y) * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.0:
		tmp = (1.0 / x) / y
	else:
		tmp = 1.0 / (((z * z) * y) * x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.0)
		tmp = Float64(Float64(1.0 / x) / y);
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(z * z) * y) * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.0)
		tmp = (1.0 / x) / y;
	else
		tmp = 1.0 / (((z * z) * y) * x);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 93.8%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 71.3%

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

   if 1 < z

   1. Initial program 82.4%

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

   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 81.5

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

   4. Applied rewrites 81.5%

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

Alternative 12: 58.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{y \cdot x} \end{array} \]

FPCore

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

C

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

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 / (y * x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.0%

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

2. Taylor expanded in z around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 58.2

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

4. Applied rewrites 58.2%

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

Reproduce

herbie shell --seed 2025113 
(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)))))