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

Percentage Accurate: 69.4% → 99.9%

Time: 4.8s

Alternatives: 10

Speedup: 1.0×

• 41.8% 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: 69.4% 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

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

FPCore

(FPCore (x y z)
 :precision binary64
 (fma (+ z x) (* (/ (- x z) y) 0.5) (* 0.5 y)))

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[(z + x), $MachinePrecision] * N[(N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision] * 0.5), $MachinePrecision] + N[(0.5 * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.4%

   \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
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 67.6%

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

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-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. +-commutative N/A

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

   8. associate-*l* N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift--.f64 N/A

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

   14. lift-*.f64 99.9

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

8. Applied rewrites 99.9%

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

Alternative 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (fma 0.5 y (* (* (+ z x) (/ (- x z) y)) 0.5)))

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(0.5 * y + N[(N[(N[(z + x), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.4%

   \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
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 67.6%

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

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 3: 74.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - z}{y}\\ \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -5 \cdot 10^{-129}:\\ \;\;\;\;\left(\left(z + x\right) \cdot t\_0\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, y, \left(x \cdot t\_0\right) \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x z) y)))
   (if (<= (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)) -5e-129)
     (* (* (+ z x) t_0) 0.5)
     (fma 0.5 y (* (* x t_0) 0.5)))))

C

double code(double x, double y, double z) {
	double t_0 = (x - z) / y;
	double tmp;
	if (((((x * x) + (y * y)) - (z * z)) / (y * 2.0)) <= -5e-129) {
		tmp = ((z + x) * t_0) * 0.5;
	} else {
		tmp = fma(0.5, y, ((x * t_0) * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - z) / y)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0)) <= -5e-129)
		tmp = Float64(Float64(Float64(z + x) * t_0) * 0.5);
	else
		tmp = fma(0.5, y, Float64(Float64(x * t_0) * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], -5e-129], N[(N[(N[(z + x), $MachinePrecision] * t$95$0), $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * y + N[(N[(x * t$95$0), $MachinePrecision] * 0.5), $MachinePrecision]), $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))) < -5.00000000000000027e-129

   1. Initial program 79.0%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right)} \]
   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 83.2

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

   4. Applied rewrites 83.2%

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

   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 66.8

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

   7. Applied rewrites 66.8%

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

   if -5.00000000000000027e-129 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 61.8%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 61.8%

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

   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. Taylor expanded in x around inf

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

   9. Applied rewrites 72.8%

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

Alternative 4: 73.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot 0.5\\ t_1 := \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-129}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;0.5 \cdot \left(\frac{x \cdot x}{y} + y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* (+ z x) (/ (- x z) y)) 0.5))
        (t_1 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))))
   (if (<= t_1 -5e-129)
     t_0
     (if (<= t_1 INFINITY) (* 0.5 (+ (/ (* x x) y) y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((z + x) * ((x - z) / y)) * 0.5;
	double t_1 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_1 <= -5e-129) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 0.5 * (((x * x) / y) + y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = ((z + x) * ((x - z) / y)) * 0.5;
	double t_1 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_1 <= -5e-129) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 0.5 * (((x * x) / y) + y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((z + x) * ((x - z) / y)) * 0.5
	t_1 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0)
	tmp = 0
	if t_1 <= -5e-129:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = 0.5 * (((x * x) / y) + y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(z + x) * Float64(Float64(x - z) / y)) * 0.5)
	t_1 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_1 <= -5e-129)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(0.5 * Float64(Float64(Float64(x * x) / y) + y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((z + x) * ((x - z) / y)) * 0.5;
	t_1 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	tmp = 0.0;
	if (t_1 <= -5e-129)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = 0.5 * (((x * x) / y) + y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(z + x), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-129], t$95$0, If[LessEqual[t$95$1, Infinity], N[(0.5 * N[(N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

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))) < -5.00000000000000027e-129 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 63.0%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right)} \]
   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 74.9

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

   4. Applied rewrites 74.9%

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

   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 68.1

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

   7. Applied rewrites 68.1%

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

   if -5.00000000000000027e-129 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.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 74.8%

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

   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. Taylor expanded in z around 0

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 63.4

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

   9. Applied rewrites 63.4%

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

Alternative 5: 70.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-129}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;0.5 \cdot \left(\frac{x \cdot x}{y} + y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{x - z}{y}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))))
   (if (<= t_0 -5e-129)
     (* 0.5 (- y (* z (/ z y))))
     (if (<= t_0 INFINITY)
       (* 0.5 (+ (/ (* x x) y) y))
       (* (* x (/ (- x z) y)) 0.5)))))

C

double code(double x, double y, double z) {
	double t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_0 <= -5e-129) {
		tmp = 0.5 * (y - (z * (z / y)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 0.5 * (((x * x) / y) + y);
	} else {
		tmp = (x * ((x - z) / y)) * 0.5;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_0 <= -5e-129) {
		tmp = 0.5 * (y - (z * (z / y)));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = 0.5 * (((x * x) / y) + y);
	} else {
		tmp = (x * ((x - z) / y)) * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0)
	tmp = 0
	if t_0 <= -5e-129:
		tmp = 0.5 * (y - (z * (z / y)))
	elif t_0 <= math.inf:
		tmp = 0.5 * (((x * x) / y) + y)
	else:
		tmp = (x * ((x - z) / y)) * 0.5
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= -5e-129)
		tmp = Float64(0.5 * Float64(y - Float64(z * Float64(z / y))));
	elseif (t_0 <= Inf)
		tmp = Float64(0.5 * Float64(Float64(Float64(x * x) / y) + y));
	else
		tmp = Float64(Float64(x * Float64(Float64(x - z) / y)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	tmp = 0.0;
	if (t_0 <= -5e-129)
		tmp = 0.5 * (y - (z * (z / y)));
	elseif (t_0 <= Inf)
		tmp = 0.5 * (((x * x) / y) + y);
	else
		tmp = (x * ((x - z) / y)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-129], N[(0.5 * N[(y - N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(0.5 * N[(N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $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))) < -5.00000000000000027e-129

   1. Initial program 79.0%

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

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

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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 65.2

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

   6. Applied rewrites 65.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 66.3

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

   8. Applied rewrites 66.3%

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

   if -5.00000000000000027e-129 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.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 74.8%

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

   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. Taylor expanded in z around 0

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 63.4

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

   9. Applied rewrites 63.4%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right)} \]
   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 42.3

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

   4. Applied rewrites 42.3%

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

   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 73.3

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

   7. Applied rewrites 73.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 40.9%

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

Alternative 6: 66.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{+53}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{x - z}{y}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 7e+53) (* 0.5 (- y (* z (/ z y)))) (* (* x (/ (- x z) y)) 0.5)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 7e+53) {
		tmp = 0.5 * (y - (z * (z / y)));
	} else {
		tmp = (x * ((x - z) / y)) * 0.5;
	}
	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 (x <= 7d+53) then
        tmp = 0.5d0 * (y - (z * (z / y)))
    else
        tmp = (x * ((x - z) / y)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 7e+53) {
		tmp = 0.5 * (y - (z * (z / y)));
	} else {
		tmp = (x * ((x - z) / y)) * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 7e+53:
		tmp = 0.5 * (y - (z * (z / y)))
	else:
		tmp = (x * ((x - z) / y)) * 0.5
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 7e+53)
		tmp = Float64(0.5 * Float64(y - Float64(z * Float64(z / y))));
	else
		tmp = Float64(Float64(x * Float64(Float64(x - z) / y)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 7e+53)
		tmp = 0.5 * (y - (z * (z / y)));
	else
		tmp = (x * ((x - z) / y)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 7e+53], N[(0.5 * N[(y - N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.00000000000000038e53

   1. Initial program 70.7%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 70.2%

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

   4. Taylor expanded in x around 0

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

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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 69.3

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

   6. Applied rewrites 69.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 75.3

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

   8. Applied rewrites 75.3%

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

   if 7.00000000000000038e53 < x

   1. Initial program 64.4%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right)} \]
   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 77.2

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

   4. Applied rewrites 77.2%

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

   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 83.7

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

   7. Applied rewrites 83.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 72.0%

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

Alternative 7: 62.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{+121}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{y + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2.8e+121) (* 0.5 (- y (* z (/ z y)))) (/ (* x x) (+ y y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.8e+121) {
		tmp = 0.5 * (y - (z * (z / y)));
	} else {
		tmp = (x * x) / (y + y);
	}
	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 (x <= 2.8d+121) then
        tmp = 0.5d0 * (y - (z * (z / y)))
    else
        tmp = (x * x) / (y + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.8e+121) {
		tmp = 0.5 * (y - (z * (z / y)));
	} else {
		tmp = (x * x) / (y + y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 2.8e+121:
		tmp = 0.5 * (y - (z * (z / y)))
	else:
		tmp = (x * x) / (y + y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2.8e+121)
		tmp = Float64(0.5 * Float64(y - Float64(z * Float64(z / y))));
	else
		tmp = Float64(Float64(x * x) / Float64(y + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 2.8e+121)
		tmp = 0.5 * (y - (z * (z / y)));
	else
		tmp = (x * x) / (y + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2.8e+121], N[(0.5 * N[(y - N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] / N[(y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.80000000000000006e121

   1. Initial program 71.2%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 70.0%

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

   4. Taylor expanded in x around 0

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

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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 68.2

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

   6. Applied rewrites 68.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 74.2

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

   8. Applied rewrites 74.2%

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

   if 2.80000000000000006e121 < x

   1. Initial program 59.6%

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

   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 66.6

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

   4. Applied rewrites 66.6%

      \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot 2} \]
   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 66.6

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

   6. Applied rewrites 66.6%

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

Alternative 8: 42.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-129}:\\ \;\;\;\;-0.5 \cdot \frac{z \cdot z}{y}\\ \mathbf{elif}\;t\_0 \leq 10^{+151}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{y + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))))
   (if (<= t_0 -5e-129)
     (* -0.5 (/ (* z z) y))
     (if (<= t_0 1e+151) (* 0.5 y) (/ (* x x) (+ y y))))))

C

double code(double x, double y, double z) {
	double t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_0 <= -5e-129) {
		tmp = -0.5 * ((z * z) / y);
	} else if (t_0 <= 1e+151) {
		tmp = 0.5 * y;
	} else {
		tmp = (x * x) / (y + y);
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0d0)
    if (t_0 <= (-5d-129)) then
        tmp = (-0.5d0) * ((z * z) / y)
    else if (t_0 <= 1d+151) then
        tmp = 0.5d0 * y
    else
        tmp = (x * x) / (y + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_0 <= -5e-129) {
		tmp = -0.5 * ((z * z) / y);
	} else if (t_0 <= 1e+151) {
		tmp = 0.5 * y;
	} else {
		tmp = (x * x) / (y + y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0)
	tmp = 0
	if t_0 <= -5e-129:
		tmp = -0.5 * ((z * z) / y)
	elif t_0 <= 1e+151:
		tmp = 0.5 * y
	else:
		tmp = (x * x) / (y + y)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= -5e-129)
		tmp = Float64(-0.5 * Float64(Float64(z * z) / y));
	elseif (t_0 <= 1e+151)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(Float64(x * x) / Float64(y + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	tmp = 0.0;
	if (t_0 <= -5e-129)
		tmp = -0.5 * ((z * z) / y);
	elseif (t_0 <= 1e+151)
		tmp = 0.5 * y;
	else
		tmp = (x * x) / (y + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-129], N[(-0.5 * N[(N[(z * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+151], N[(0.5 * y), $MachinePrecision], N[(N[(x * x), $MachinePrecision] / N[(y + y), $MachinePrecision]), $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))) < -5.00000000000000027e-129

   1. Initial program 79.0%

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

   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 31.6

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

   4. Applied rewrites 31.6%

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

   if -5.00000000000000027e-129 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 1.00000000000000002e151

   1. Initial program 90.1%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 58.9

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

   4. Applied rewrites 58.9%

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

   if 1.00000000000000002e151 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 53.3%

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

   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 35.3

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

   4. Applied rewrites 35.3%

      \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot 2} \]
   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 35.3

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

   6. Applied rewrites 35.3%

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

Alternative 9: 36.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+48}:\\ \;\;\;\;\frac{x \cdot x}{y + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4e+48) (/ (* x x) (+ y y)) (* 0.5 y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4e+48) {
		tmp = (x * x) / (y + y);
	} else {
		tmp = 0.5 * y;
	}
	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 (y <= 4d+48) then
        tmp = (x * x) / (y + y)
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4e+48) {
		tmp = (x * x) / (y + y);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4e+48:
		tmp = (x * x) / (y + y)
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4e+48)
		tmp = Float64(Float64(x * x) / Float64(y + y));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4e+48)
		tmp = (x * x) / (y + y);
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4e+48], 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 < 4.00000000000000018e48

   1. Initial program 77.2%

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

   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 36.8

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

   4. Applied rewrites 36.8%

      \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot 2} \]
   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 36.8

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

   6. Applied rewrites 36.8%

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

   if 4.00000000000000018e48 < y

   1. Initial program 40.3%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 63.5

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

   4. Applied rewrites 63.5%

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

Alternative 10: 34.5% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* 0.5 y))

C

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

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 = 0.5d0 * y
end function

Java

public static double code(double x, double y, double z) {
	return 0.5 * y;
}

Python

def code(x, y, z):
	return 0.5 * y

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(0.5 * y), $MachinePrecision]

Derivation

1. Initial program 69.4%

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

2. Taylor expanded in y around inf

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

   1. lower-*.f64 34.5

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

4. Applied rewrites 34.5%

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

Reproduce

herbie shell --seed 2025120 
(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)))