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

Percentage Accurate: 90.3% → 97.4%

Time: 6.7s

Alternatives: 9

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.3% 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: 97.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.5%

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

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 91.2

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lower-fma.f64 91.2

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

4. Applied rewrites 91.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. distribute-lft-in N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f64 N/A

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

   13. distribute-rgt-in N/A

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

   14. *-rgt-identity N/A

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

   15. distribute-rgt-in N/A

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

   16. +-commutative N/A

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

   17. lift-fma.f64 N/A

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

   18. lift-fma.f64 N/A

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

   19. distribute-lft-in N/A

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

   20. associate-*r* N/A

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

   21. lower-fma.f64 N/A

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

6. Applied rewrites 97.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lower-*.f64 98.5

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 98.5

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

8. Applied rewrites 98.5%

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

Alternative 2: 77.2% accurate, 0.5× speedup

Math

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

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 y) z 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 * y), z, 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(1.0 / Float64(fma(Float64(z * y), z, 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[(1.0 / N[(N[(N[(z * y), $MachinePrecision] * z + 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)))) < 0.0

   1. Initial program 87.8%

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

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 87.9

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 87.9

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

   4. Applied rewrites 87.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-in N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-in N/A

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

      14. *-rgt-identity N/A

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

      15. distribute-rgt-in N/A

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

      16. +-commutative N/A

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

      17. lift-fma.f64 N/A

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

      18. lift-fma.f64 N/A

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

      19. distribute-lft-in N/A

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

      20. *-commutative N/A

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

      21. associate-*l* N/A

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

      22. lower-fma.f64 N/A

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

   6. Applied rewrites 97.0%

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

   7. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. *-commutative N/A

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

      7. distribute-lft1-in N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

      18. *-commutative N/A

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

      19. associate-*r* N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 N/A

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

   9. Applied rewrites 72.9%

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

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. lift-*.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/r* N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 98.3

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

   6. Applied rewrites 98.3%

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

Alternative 3: 77.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \leq 2 \cdot 10^{-319}:\\ \;\;\;\;\frac{1}{\left(\left(x \cdot y\right) \cdot z\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))) 2e-319)
   (/ 1.0 (* (* (* x y) z) z))
   (/ 1.0 (* (* y (fma z z 1.0)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((1.0 / x) / (y * (1.0 + (z * z)))) <= 2e-319) {
		tmp = 1.0 / (((x * y) * z) * z);
	} else {
		tmp = 1.0 / ((y * fma(z, z, 1.0)) * 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)))) <= 2e-319)
		tmp = Float64(1.0 / Float64(Float64(Float64(x * y) * z) * z));
	else
		tmp = Float64(1.0 / Float64(Float64(y * fma(z, z, 1.0)) * 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], 2e-319], N[(1.0 / N[(N[(N[(x * y), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(y * N[(z * z + 1.0), $MachinePrecision]), $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)))) < 1.99998e-319

   1. Initial program 87.9%

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

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 87.9

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 87.9

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

   4. Applied rewrites 87.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-in N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-in N/A

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

      14. *-rgt-identity N/A

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

      15. distribute-rgt-in N/A

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

      16. +-commutative N/A

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

      17. lift-fma.f64 N/A

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

      18. lift-fma.f64 N/A

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

      19. distribute-lft-in N/A

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

      20. *-commutative N/A

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

      21. associate-*l* N/A

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

      22. lower-fma.f64 N/A

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

   6. Applied rewrites 96.9%

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

   7. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. *-commutative N/A

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

      7. distribute-lft1-in N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

      18. *-commutative N/A

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

      19. associate-*r* N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 N/A

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

   9. Applied rewrites 72.9%

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

   if 1.99998e-319 < (/.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. Add Preprocessing
   3. 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. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 98.5

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 98.5

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

   4. Applied rewrites 98.5%

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

Alternative 4: 96.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.6 \cdot 10^{-100}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z \cdot y, z, y\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, z \cdot \left(y \cdot x\right), y \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 8.6e-100)
   (/ 1.0 (* (fma (* z y) z y) x))
   (/ 1.0 (fma z (* z (* y x)) (* y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.6e-100) {
		tmp = 1.0 / (fma((z * y), z, y) * x);
	} else {
		tmp = 1.0 / fma(z, (z * (y * x)), (y * x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 8.6e-100)
		tmp = Float64(1.0 / Float64(fma(Float64(z * y), z, y) * x));
	else
		tmp = Float64(1.0 / fma(z, Float64(z * Float64(y * x)), Float64(y * x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 8.6e-100], N[(1.0 / N[(N[(N[(z * y), $MachinePrecision] * z + y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(z * N[(z * N[(y * x), $MachinePrecision]), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.59999999999999997e-100

   1. Initial program 91.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 96.2

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

   4. Applied rewrites 96.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. lift-*.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/r* N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 91.5

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

   6. Applied rewrites 95.6%

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

   if 8.59999999999999997e-100 < y

   1. Initial program 90.8%

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

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 90.6

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 90.6

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

   4. Applied rewrites 90.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-in N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-in N/A

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

      14. *-rgt-identity N/A

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

      15. distribute-rgt-in N/A

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

      16. +-commutative N/A

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

      17. lift-fma.f64 N/A

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

      18. lift-fma.f64 N/A

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

      19. distribute-lft-in N/A

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

   6. Applied rewrites 98.7%

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

Alternative 5: 73.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.88:\\ \;\;\;\;\frac{1 - z \cdot z}{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(x \cdot y\right) \cdot z\right) \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 0.88) (/ (- 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 <= 0.88) {
		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 <= 0.88d0) 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 <= 0.88) {
		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 <= 0.88:
		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 <= 0.88)
		tmp = Float64(Float64(1.0 - Float64(z * z)) / Float64(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 <= 0.88)
		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, 0.88], N[(N[(1.0 - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(x * 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 < 0.880000000000000004

   1. Initial program 95.5%

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

   3. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 36.8

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

   5. Applied rewrites 36.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      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. lower-*.f64 42.3

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

   7. Applied rewrites 42.3%

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

      1. lift-/.f64 N/A

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

      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

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 42.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 42.3

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

   9. Applied rewrites 42.3%

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

   10. Taylor expanded in z around 0

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

       1. mul-1-neg N/A

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

       2. unpow2 N/A

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

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

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

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

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

       5. unpow2 N/A

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

       6. lower--.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 72.3

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

   12. Applied rewrites 72.3%

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

   if 0.880000000000000004 < z

   1. Initial program 79.3%

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

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 79.0

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 79.0

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

   4. Applied rewrites 79.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-in N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-in N/A

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

      14. *-rgt-identity N/A

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

      15. distribute-rgt-in N/A

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

      16. +-commutative N/A

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

      17. lift-fma.f64 N/A

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

      18. lift-fma.f64 N/A

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

      19. distribute-lft-in N/A

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

      20. *-commutative N/A

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

      21. associate-*l* N/A

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

      22. lower-fma.f64 N/A

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

   6. Applied rewrites 93.9%

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

   7. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. *-commutative N/A

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

      7. distribute-lft1-in N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

      18. *-commutative N/A

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

      19. associate-*r* N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 N/A

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

   9. Applied rewrites 92.1%

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

Alternative 6: 71.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.88:\\ \;\;\;\;\frac{1 - z \cdot z}{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(z \cdot z\right) \cdot y\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 0.88) (/ (- 1.0 (* z z)) (* x y)) (/ 1.0 (* (* (* z z) y) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.88) {
		tmp = (1.0 - (z * z)) / (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 <= 0.88d0) then
        tmp = (1.0d0 - (z * z)) / (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 <= 0.88) {
		tmp = (1.0 - (z * z)) / (x * y);
	} else {
		tmp = 1.0 / (((z * z) * y) * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 0.88:
		tmp = (1.0 - (z * z)) / (x * y)
	else:
		tmp = 1.0 / (((z * z) * y) * x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 0.88)
		tmp = Float64(Float64(1.0 - Float64(z * z)) / Float64(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 <= 0.88)
		tmp = (1.0 - (z * z)) / (x * y);
	else
		tmp = 1.0 / (((z * z) * y) * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 0.88], N[(N[(1.0 - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(x * y), $MachinePrecision]), $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 < 0.880000000000000004

   1. Initial program 95.5%

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

   3. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 36.8

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

   5. Applied rewrites 36.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      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. lower-*.f64 42.3

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

   7. Applied rewrites 42.3%

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

      1. lift-/.f64 N/A

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

      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

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 42.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 42.3

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

   9. Applied rewrites 42.3%

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

   10. Taylor expanded in z around 0

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

       1. mul-1-neg N/A

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

       2. unpow2 N/A

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

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

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

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

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

       5. unpow2 N/A

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

       6. lower--.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 72.3

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

   12. Applied rewrites 72.3%

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

   if 0.880000000000000004 < z

   1. Initial program 79.3%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot {z}^{2}\right)}} \]
   4. 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. unpow2 N/A

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

      7. lower-*.f64 79.0

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

   5. Applied rewrites 79.0%

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

Alternative 7: 97.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.5%

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

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 91.2

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lower-fma.f64 91.2

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

4. Applied rewrites 91.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. distribute-lft-in N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f64 N/A

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

   13. distribute-rgt-in N/A

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

   14. *-rgt-identity N/A

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

   15. distribute-rgt-in N/A

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

   16. +-commutative N/A

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

   17. lift-fma.f64 N/A

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

   18. lift-fma.f64 N/A

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

   19. distribute-lft-in N/A

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

   20. *-commutative N/A

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

   21. associate-*l* N/A

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

   22. lower-fma.f64 N/A

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

6. Applied rewrites 97.4%

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

Alternative 8: 97.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.5%

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

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 91.2

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lower-fma.f64 91.2

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

4. Applied rewrites 91.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. distribute-lft-in N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f64 N/A

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

   13. distribute-rgt-in N/A

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

   14. *-rgt-identity N/A

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

   15. distribute-rgt-in N/A

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

   16. +-commutative N/A

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

   17. lift-fma.f64 N/A

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

   18. lift-fma.f64 N/A

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

   19. distribute-lft-in N/A

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

   20. associate-*r* N/A

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

   21. lower-fma.f64 N/A

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

6. Applied rewrites 97.0%

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

Alternative 9: 58.7% accurate, 2.1× 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 91.5%

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

3. Taylor expanded in z around 0

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

5. Applied rewrites 59.9%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 60.0

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

7. Applied rewrites 60.0%

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

Developer Target 1: 91.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + z \cdot z\\ t_1 := y \cdot t\_0\\ t_2 := \frac{\frac{1}{y}}{t\_0 \cdot x}\\ \mathbf{if}\;t\_1 < -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{t\_0 \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* z z))) (t_1 (* y t_0)) (t_2 (/ (/ 1.0 y) (* t_0 x))))
   (if (< t_1 (- INFINITY))
     t_2
     (if (< t_1 8.680743250567252e+305) (/ (/ 1.0 x) (* t_0 y)) t_2))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 + (z * z);
	double t_1 = y * t_0;
	double t_2 = (1.0 / y) / (t_0 * x);
	double tmp;
	if (t_1 < -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 < 8.680743250567252e+305) {
		tmp = (1.0 / x) / (t_0 * y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 + (z * z);
	double t_1 = y * t_0;
	double t_2 = (1.0 / y) / (t_0 * x);
	double tmp;
	if (t_1 < -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if (t_1 < 8.680743250567252e+305) {
		tmp = (1.0 / x) / (t_0 * y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 + (z * z)
	t_1 = y * t_0
	t_2 = (1.0 / y) / (t_0 * x)
	tmp = 0
	if t_1 < -math.inf:
		tmp = t_2
	elif t_1 < 8.680743250567252e+305:
		tmp = (1.0 / x) / (t_0 * y)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 + Float64(z * z))
	t_1 = Float64(y * t_0)
	t_2 = Float64(Float64(1.0 / y) / Float64(t_0 * x))
	tmp = 0.0
	if (t_1 < Float64(-Inf))
		tmp = t_2;
	elseif (t_1 < 8.680743250567252e+305)
		tmp = Float64(Float64(1.0 / x) / Float64(t_0 * y));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 + (z * z);
	t_1 = y * t_0;
	t_2 = (1.0 / y) / (t_0 * x);
	tmp = 0.0;
	if (t_1 < -Inf)
		tmp = t_2;
	elseif (t_1 < 8.680743250567252e+305)
		tmp = (1.0 / x) / (t_0 * y);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 / y), $MachinePrecision] / N[(t$95$0 * x), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$1, (-Infinity)], t$95$2, If[Less[t$95$1, 8.680743250567252e+305], N[(N[(1.0 / x), $MachinePrecision] / N[(t$95$0 * y), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Reproduce

herbie shell --seed 2025026 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (* y (+ 1 (* z z))) -inf.0) (/ (/ 1 y) (* (+ 1 (* z z)) x)) (if (< (* y (+ 1 (* z z))) 868074325056725200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (/ 1 x) (* (+ 1 (* z z)) y)) (/ (/ 1 y) (* (+ 1 (* z z)) x)))))

  (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))