Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A

Percentage Accurate: 70.1% → 99.9%

Time: 3.6s

Alternatives: 10

Speedup: 1.0×

• 41.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))

C

double code(double x, double y, double z) {
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
}

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 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
}

Python

def code(x, y, z):
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 70.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))

C

double code(double x, double y, double z) {
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
}

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 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
}

Python

def code(x, y, z):
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \mathsf{fma}\left(0.5, y, \left(\left(z\_m + x\right) \cdot \frac{x - z\_m}{y}\right) \cdot 0.5\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (fma 0.5 y (* (* (+ z_m x) (/ (- x z_m) y)) 0.5)))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	return fma(0.5, y, (((z_m + x) * ((x - z_m) / y)) * 0.5));
}

Julia

z_m = abs(z)
function code(x, y, z_m)
	return fma(0.5, y, Float64(Float64(Float64(z_m + x) * Float64(Float64(x - z_m) / y)) * 0.5))
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := N[(0.5 * y + N[(N[(N[(z$95$m + x), $MachinePrecision] * N[(N[(x - z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.1%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. pow2 N/A

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

   9. div-sub N/A

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

   10. lower--.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. pow2 N/A

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

   13. pow2 N/A

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

   14. +-commutative N/A

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

   15. pow2 N/A

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

   16. lower-fma.f64 N/A

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

   17. pow2 N/A

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. count-2-rev N/A

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

   21. lower-+.f64 N/A

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

3. Applied rewrites 68.1%

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

4. Taylor expanded in x around 0

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

   1. associate--l+ N/A

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

   2. associate-*r/ N/A

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

   3. associate-*r/ N/A

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

   4. div-sub N/A

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

   5. distribute-lft-out-- N/A

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

   6. pow2 N/A

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

   7. pow2 N/A

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

   8. difference-of-squares-rev N/A

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

   9. associate-*r/ N/A

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

   10. lower-fma.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

6. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, y, \left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot 0.5\right)} \]
7. Add Preprocessing

Alternative 2: 66.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2} \leq 0:\\ \;\;\;\;0.5 \cdot y - z\_m \cdot \frac{z\_m}{y + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, y\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0)) 0.0)
   (- (* 0.5 y) (* z_m (/ z_m (+ y y))))
   (* (fma (/ x y) x y) 0.5)))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double tmp;
	if (((((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0)) <= 0.0) {
		tmp = (0.5 * y) - (z_m * (z_m / (y + y)));
	} else {
		tmp = fma((x / y), x, y) * 0.5;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0)) <= 0.0)
		tmp = Float64(Float64(0.5 * y) - Float64(z_m * Float64(z_m / Float64(y + y))));
	else
		tmp = Float64(fma(Float64(x / y), x, y) * 0.5);
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := If[LessEqual[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(0.5 * y), $MachinePrecision] - N[(z$95$m * N[(z$95$m / N[(y + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] * x + y), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 77.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. pow2 N/A

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

      9. div-sub N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. +-commutative N/A

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

      15. pow2 N/A

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

      16. lower-fma.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. count-2-rev N/A

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

      21. lower-+.f64 N/A

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

   3. Applied rewrites 75.1%

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

   4. Taylor expanded in x around 0

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

      1. lift-*.f64 67.2

         \[\leadsto 0.5 \cdot \color{blue}{y} - z \cdot \frac{z}{y + y} \]

   6. Applied rewrites 67.2%

      \[\leadsto \color{blue}{0.5 \cdot y} - z \cdot \frac{z}{y + y} \]

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

   1. Initial program 64.3%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 62.4%

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

   5. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. pow2 N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{y}, x, y\right) \cdot \frac{1}{2} \]

      8. lift-/.f64 65.3

         \[\leadsto \mathsf{fma}\left(\frac{x}{y}, x, y\right) \cdot 0.5 \]

   7. Applied rewrites 65.3%

      \[\leadsto \mathsf{fma}\left(\frac{x}{y}, x, y\right) \cdot \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 66.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2} \leq 0:\\ \;\;\;\;\left(\left(z\_m + x\right) \cdot \frac{x - z\_m}{y}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, y\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0)) 0.0)
   (* (* (+ z_m x) (/ (- x z_m) y)) 0.5)
   (* (fma (/ x y) x y) 0.5)))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double tmp;
	if (((((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0)) <= 0.0) {
		tmp = ((z_m + x) * ((x - z_m) / y)) * 0.5;
	} else {
		tmp = fma((x / y), x, y) * 0.5;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0)) <= 0.0)
		tmp = Float64(Float64(Float64(z_m + x) * Float64(Float64(x - z_m) / y)) * 0.5);
	else
		tmp = Float64(fma(Float64(x / y), x, y) * 0.5);
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := If[LessEqual[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(z$95$m + x), $MachinePrecision] * N[(N[(x - z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] * x + y), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 77.3%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 80.0

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

   4. Applied rewrites 80.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 67.4

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

   7. Applied rewrites 67.4%

      \[\leadsto \left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot \color{blue}{0.5} \]

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

   1. Initial program 64.3%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 62.4%

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

   5. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. pow2 N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{y}, x, y\right) \cdot \frac{1}{2} \]

      8. lift-/.f64 65.3

         \[\leadsto \mathsf{fma}\left(\frac{x}{y}, x, y\right) \cdot 0.5 \]

   7. Applied rewrites 65.3%

      \[\leadsto \mathsf{fma}\left(\frac{x}{y}, x, y\right) \cdot \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 53.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2} \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\left(\frac{z\_m}{y} \cdot -0.5\right) \cdot z\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} \cdot x + y\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0)) -2e-11)
   (* (* (/ z_m y) -0.5) z_m)
   (* (+ (* (/ x y) x) y) 0.5)))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double tmp;
	if (((((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0)) <= -2e-11) {
		tmp = ((z_m / y) * -0.5) * z_m;
	} else {
		tmp = (((x / y) * x) + y) * 0.5;
	}
	return tmp;
}

Fortran

z_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (((((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0d0)) <= (-2d-11)) then
        tmp = ((z_m / y) * (-0.5d0)) * z_m
    else
        tmp = (((x / y) * x) + y) * 0.5d0
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	double tmp;
	if (((((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0)) <= -2e-11) {
		tmp = ((z_m / y) * -0.5) * z_m;
	} else {
		tmp = (((x / y) * x) + y) * 0.5;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	tmp = 0
	if ((((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0)) <= -2e-11:
		tmp = ((z_m / y) * -0.5) * z_m
	else:
		tmp = (((x / y) * x) + y) * 0.5
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0)) <= -2e-11)
		tmp = Float64(Float64(Float64(z_m / y) * -0.5) * z_m);
	else
		tmp = Float64(Float64(Float64(Float64(x / y) * x) + y) * 0.5);
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m)
	tmp = 0.0;
	if (((((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0)) <= -2e-11)
		tmp = ((z_m / y) * -0.5) * z_m;
	else
		tmp = (((x / y) * x) + y) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := If[LessEqual[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], -2e-11], N[(N[(N[(z$95$m / y), $MachinePrecision] * -0.5), $MachinePrecision] * z$95$m), $MachinePrecision], N[(N[(N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] + y), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -1.99999999999999988e-11

   1. Initial program 78.2%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 72.8%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 35.3

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

   7. Applied rewrites 35.3%

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

   if -1.99999999999999988e-11 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 64.6%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 60.5%

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

   5. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. pow2 N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{y}, x, y\right) \cdot \frac{1}{2} \]

      8. lift-/.f64 66.1

         \[\leadsto \mathsf{fma}\left(\frac{x}{y}, x, y\right) \cdot 0.5 \]

   7. Applied rewrites 66.1%

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

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{y}, x, y\right) \cdot \frac{1}{2} \]

      2. lift-fma.f64 N/A

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

      3. associate-*l/ N/A

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

      4. pow2 N/A

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

      5. lower-+.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 66.1

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

   9. Applied rewrites 66.1%

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

Alternative 5: 53.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2} \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\left(\frac{z\_m}{y} \cdot -0.5\right) \cdot z\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, y\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0)) -2e-11)
   (* (* (/ z_m y) -0.5) z_m)
   (* (fma (/ x y) x y) 0.5)))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double tmp;
	if (((((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0)) <= -2e-11) {
		tmp = ((z_m / y) * -0.5) * z_m;
	} else {
		tmp = fma((x / y), x, y) * 0.5;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0)) <= -2e-11)
		tmp = Float64(Float64(Float64(z_m / y) * -0.5) * z_m);
	else
		tmp = Float64(fma(Float64(x / y), x, y) * 0.5);
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := If[LessEqual[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], -2e-11], N[(N[(N[(z$95$m / y), $MachinePrecision] * -0.5), $MachinePrecision] * z$95$m), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] * x + y), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -1.99999999999999988e-11

   1. Initial program 78.2%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 72.8%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 35.3

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

   7. Applied rewrites 35.3%

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

   if -1.99999999999999988e-11 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 64.6%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 60.5%

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

   5. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. pow2 N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{y}, x, y\right) \cdot \frac{1}{2} \]

      8. lift-/.f64 66.1

         \[\leadsto \mathsf{fma}\left(\frac{x}{y}, x, y\right) \cdot 0.5 \]

   7. Applied rewrites 66.1%

      \[\leadsto \mathsf{fma}\left(\frac{x}{y}, x, y\right) \cdot \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 42.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\left(\frac{z\_m}{y} \cdot -0.5\right) \cdot z\_m\\ \mathbf{elif}\;t\_0 \leq 10^{+143}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{x \cdot x}{y + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0))))
   (if (<= t_0 0.0)
     (* (* (/ z_m y) -0.5) z_m)
     (if (<= t_0 1e+143)
       (* 0.5 y)
       (if (<= t_0 INFINITY) (/ (* x x) (+ y y)) (* 0.5 y))))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = ((z_m / y) * -0.5) * z_m;
	} else if (t_0 <= 1e+143) {
		tmp = 0.5 * y;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (x * x) / (y + y);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	double t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = ((z_m / y) * -0.5) * z_m;
	} else if (t_0 <= 1e+143) {
		tmp = 0.5 * y;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (x * x) / (y + y);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0)
	tmp = 0
	if t_0 <= 0.0:
		tmp = ((z_m / y) * -0.5) * z_m
	elif t_0 <= 1e+143:
		tmp = 0.5 * y
	elif t_0 <= math.inf:
		tmp = (x * x) / (y + y)
	else:
		tmp = 0.5 * y
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(z_m / y) * -0.5) * z_m);
	elseif (t_0 <= 1e+143)
		tmp = Float64(0.5 * y);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(x * x) / Float64(y + y));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m)
	t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = ((z_m / y) * -0.5) * z_m;
	elseif (t_0 <= 1e+143)
		tmp = 0.5 * y;
	elseif (t_0 <= Inf)
		tmp = (x * x) / (y + y);
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(N[(z$95$m / y), $MachinePrecision] * -0.5), $MachinePrecision] * z$95$m), $MachinePrecision], If[LessEqual[t$95$0, 1e+143], N[(0.5 * y), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(x * x), $MachinePrecision] / N[(y + y), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 77.3%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 69.3%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 34.3

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

   7. Applied rewrites 34.3%

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

   if 0.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 1e143 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 51.1%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 42.6

         \[\leadsto 0.5 \cdot \color{blue}{y} \]

   4. Applied rewrites 42.6%

      \[\leadsto \color{blue}{0.5 \cdot y} \]

   if 1e143 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 72.9%

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

   2. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 37.2

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

   4. Applied rewrites 37.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f64 37.2

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

   6. Applied rewrites 37.2%

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

Alternative 7: 38.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\left(\frac{z\_m}{y} \cdot -0.5\right) \cdot z\_m\\ \mathbf{elif}\;t\_0 \leq 10^{+143}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} \cdot x\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0))))
   (if (<= t_0 0.0)
     (* (* (/ z_m y) -0.5) z_m)
     (if (<= t_0 1e+143) (* 0.5 y) (* (* (/ x y) x) 0.5)))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = ((z_m / y) * -0.5) * z_m;
	} else if (t_0 <= 1e+143) {
		tmp = 0.5 * y;
	} else {
		tmp = ((x / y) * x) * 0.5;
	}
	return tmp;
}

Fortran

z_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0d0)
    if (t_0 <= 0.0d0) then
        tmp = ((z_m / y) * (-0.5d0)) * z_m
    else if (t_0 <= 1d+143) then
        tmp = 0.5d0 * y
    else
        tmp = ((x / y) * x) * 0.5d0
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	double t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = ((z_m / y) * -0.5) * z_m;
	} else if (t_0 <= 1e+143) {
		tmp = 0.5 * y;
	} else {
		tmp = ((x / y) * x) * 0.5;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0)
	tmp = 0
	if t_0 <= 0.0:
		tmp = ((z_m / y) * -0.5) * z_m
	elif t_0 <= 1e+143:
		tmp = 0.5 * y
	else:
		tmp = ((x / y) * x) * 0.5
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(z_m / y) * -0.5) * z_m);
	elseif (t_0 <= 1e+143)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(Float64(Float64(x / y) * x) * 0.5);
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m)
	t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = ((z_m / y) * -0.5) * z_m;
	elseif (t_0 <= 1e+143)
		tmp = 0.5 * y;
	else
		tmp = ((x / y) * x) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(N[(z$95$m / y), $MachinePrecision] * -0.5), $MachinePrecision] * z$95$m), $MachinePrecision], If[LessEqual[t$95$0, 1e+143], N[(0.5 * y), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 77.3%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 69.3%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 34.3

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

   7. Applied rewrites 34.3%

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

   if 0.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 1e143

   1. Initial program 99.4%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 59.0

         \[\leadsto 0.5 \cdot \color{blue}{y} \]

   4. Applied rewrites 59.0%

      \[\leadsto \color{blue}{0.5 \cdot y} \]

   if 1e143 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 55.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. pow2 N/A

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

      9. div-sub N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. +-commutative N/A

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

      15. pow2 N/A

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

      16. lower-fma.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. count-2-rev N/A

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

      21. lower-+.f64 N/A

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

   3. Applied rewrites 53.1%

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

   4. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. div-sub N/A

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

      5. distribute-lft-out-- N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. difference-of-squares-rev N/A

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

      9. associate-*r/ N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   6. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lower-/.f64 96.2

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

   8. Applied rewrites 96.2%

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

   9. Taylor expanded in x around inf

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

       1. pow2 N/A

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

       2. pow2 N/A

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

       3. +-commutative N/A

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

       4. pow2 N/A

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

       5. pow2 N/A

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

       6. count-2-rev N/A

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

       7. *-commutative N/A

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

       8. associate-*r/ N/A

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

       9. pow2 N/A

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

       10. count-2-rev N/A

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

       11. *-commutative N/A

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

       12. div-sub N/A

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

       13. pow2 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 N/A

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

       16. pow2 N/A

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

       17. associate-*l/ N/A

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

       18. lower-*.f64 N/A

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

       19. lift-/.f64 37.0

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

   11. Applied rewrites 37.0%

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

Alternative 8: 37.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-11}:\\ \;\;\;\;-0.5 \cdot \frac{z\_m \cdot z\_m}{y}\\ \mathbf{elif}\;t\_0 \leq 10^{+143}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{x \cdot x}{y + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0))))
   (if (<= t_0 -2e-11)
     (* -0.5 (/ (* z_m z_m) y))
     (if (<= t_0 1e+143)
       (* 0.5 y)
       (if (<= t_0 INFINITY) (/ (* x x) (+ y y)) (* 0.5 y))))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_0 <= -2e-11) {
		tmp = -0.5 * ((z_m * z_m) / y);
	} else if (t_0 <= 1e+143) {
		tmp = 0.5 * y;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (x * x) / (y + y);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	double t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_0 <= -2e-11) {
		tmp = -0.5 * ((z_m * z_m) / y);
	} else if (t_0 <= 1e+143) {
		tmp = 0.5 * y;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (x * x) / (y + y);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0)
	tmp = 0
	if t_0 <= -2e-11:
		tmp = -0.5 * ((z_m * z_m) / y)
	elif t_0 <= 1e+143:
		tmp = 0.5 * y
	elif t_0 <= math.inf:
		tmp = (x * x) / (y + y)
	else:
		tmp = 0.5 * y
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= -2e-11)
		tmp = Float64(-0.5 * Float64(Float64(z_m * z_m) / y));
	elseif (t_0 <= 1e+143)
		tmp = Float64(0.5 * y);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(x * x) / Float64(y + y));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m)
	t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	tmp = 0.0;
	if (t_0 <= -2e-11)
		tmp = -0.5 * ((z_m * z_m) / y);
	elseif (t_0 <= 1e+143)
		tmp = 0.5 * y;
	elseif (t_0 <= Inf)
		tmp = (x * x) / (y + y);
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-11], N[(-0.5 * N[(N[(z$95$m * z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+143], N[(0.5 * y), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(x * x), $MachinePrecision] / N[(y + y), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -1.99999999999999988e-11

   1. Initial program 78.2%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 33.7

         \[\leadsto -0.5 \cdot \frac{z \cdot z}{y} \]

   4. Applied rewrites 33.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{z \cdot z}{y}} \]

   if -1.99999999999999988e-11 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 1e143 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 54.0%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 43.9

         \[\leadsto 0.5 \cdot \color{blue}{y} \]

   4. Applied rewrites 43.9%

      \[\leadsto \color{blue}{0.5 \cdot y} \]

   if 1e143 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 72.9%

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

   2. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 37.2

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

   4. Applied rewrites 37.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f64 37.2

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

   6. Applied rewrites 37.2%

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

Alternative 9: 37.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{+57}:\\ \;\;\;\;\frac{x \cdot x}{y + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= y 1.85e+57) (/ (* x x) (+ y y)) (* 0.5 y)))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double tmp;
	if (y <= 1.85e+57) {
		tmp = (x * x) / (y + y);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

z_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y <= 1.85d+57) then
        tmp = (x * x) / (y + y)
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	double tmp;
	if (y <= 1.85e+57) {
		tmp = (x * x) / (y + y);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	tmp = 0
	if y <= 1.85e+57:
		tmp = (x * x) / (y + y)
	else:
		tmp = 0.5 * y
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m)
	tmp = 0.0
	if (y <= 1.85e+57)
		tmp = Float64(Float64(x * x) / Float64(y + y));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m)
	tmp = 0.0;
	if (y <= 1.85e+57)
		tmp = (x * x) / (y + y);
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := If[LessEqual[y, 1.85e+57], N[(N[(x * x), $MachinePrecision] / N[(y + y), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.85000000000000003e57

   1. Initial program 78.0%

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

   2. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 36.4

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

   4. Applied rewrites 36.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f64 36.4

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

   6. Applied rewrites 36.4%

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

   if 1.85000000000000003e57 < y

   1. Initial program 39.2%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 64.3

         \[\leadsto 0.5 \cdot \color{blue}{y} \]

   4. Applied rewrites 64.3%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 33.8% accurate, 5.4× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ 0.5 \cdot y \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m) :precision binary64 (* 0.5 y))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	return 0.5 * y;
}

Fortran

z_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    code = 0.5d0 * y
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	return 0.5 * y;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	return 0.5 * y

Julia

z_m = abs(z)
function code(x, y, z_m)
	return Float64(0.5 * y)
end

MATLAB

z_m = abs(z);
function tmp = code(x, y, z_m)
	tmp = 0.5 * y;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := N[(0.5 * y), $MachinePrecision]

Derivation

1. Initial program 70.1%

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

2. Taylor expanded in y around inf

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

   1. lower-*.f64 33.8

      \[\leadsto 0.5 \cdot \color{blue}{y} \]

4. Applied rewrites 33.8%

   \[\leadsto \color{blue}{0.5 \cdot y} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025115 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64
  (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))