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

Percentage Accurate: 68.5% → 99.9%

Time: 3.2s

Alternatives: 9

Speedup: 0.9×

• 58.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 68.5% accurate, 1.0× speedup

Math

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

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, 0.9× speedup

Math

\[y \cdot 0.5 + \frac{x - z}{y} \cdot \frac{z + x}{2} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (+ (* y 0.5) (* (/ (- x z) y) (/ (+ z x) 2.0))))

C

double code(double x, double y, double z) {
	return (y * 0.5) + (((x - z) / y) * ((z + x) / 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 = (y * 0.5d0) + (((x - z) / y) * ((z + x) / 2.0d0))
end function

Java

public static double code(double x, double y, double z) {
	return (y * 0.5) + (((x - z) / y) * ((z + x) / 2.0));
}

Python

def code(x, y, z):
	return (y * 0.5) + (((x - z) / y) * ((z + x) / 2.0))

Julia

function code(x, y, z)
	return Float64(Float64(y * 0.5) + Float64(Float64(Float64(x - z) / y) * Float64(Float64(z + x) / 2.0)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (y * 0.5) + (((x - z) / y) * ((z + x) / 2.0));
end

Wolfram

code[x_, y_, z_] := N[(N[(y * 0.5), $MachinePrecision] + N[(N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision] * N[(N[(z + x), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.5%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. div-sub N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. div-add N/A

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

   7. associate--l+ N/A

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

   8. lower-+.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. count-2-rev N/A

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

   16. lower-+.f64 N/A

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

   17. lower--.f64 N/A

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

3. Applied rewrites 90.9%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*r/ N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. associate-*l/ N/A

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

   9. lift-*.f64 N/A

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

   10. sub-div N/A

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

   11. sub-negate N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

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

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

   15. lift-+.f64 N/A

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

   16. lift--.f64 N/A

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

   17. *-commutative N/A

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

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

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

   19. lift--.f64 N/A

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

   20. sub-negate-rev N/A

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

   21. lift-+.f64 N/A

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

   22. count-2 N/A

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

   23. *-commutative N/A

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

5. Applied rewrites 99.9%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 99.9%

   \[\leadsto y \cdot \color{blue}{0.5} + \frac{x - z}{y} \cdot \frac{z + x}{2} \]
9. Add Preprocessing

Alternative 2: 96.2% accurate, 0.2× speedup

Math

\[\mathsf{copysign}\left(1, y\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(\left|x\right| \cdot \left|x\right| + \left|y\right| \cdot \left|y\right|\right) - \left|z\right| \cdot \left|z\right|}{\left|y\right| \cdot 2} \leq 0:\\ \;\;\;\;\left(\left|z\right| + \left|x\right|\right) \cdot \left(\frac{\left|z\right| - \left|x\right|}{\left|y\right|} \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left|y\right| \cdot 0.5 + \frac{\left|x\right| - \left|z\right|}{\left|y\right|} \cdot \left(0.5 \cdot \left|x\right|\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (*
 (copysign 1.0 y)
 (if (<=
      (/
       (-
        (+ (* (fabs x) (fabs x)) (* (fabs y) (fabs y)))
        (* (fabs z) (fabs z)))
       (* (fabs y) 2.0))
      0.0)
   (*
    (+ (fabs z) (fabs x))
    (* (/ (- (fabs z) (fabs x)) (fabs y)) -0.5))
   (+
    (* (fabs y) 0.5)
    (* (/ (- (fabs x) (fabs z)) (fabs y)) (* 0.5 (fabs x)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((((fabs(x) * fabs(x)) + (fabs(y) * fabs(y))) - (fabs(z) * fabs(z))) / (fabs(y) * 2.0)) <= 0.0) {
		tmp = (fabs(z) + fabs(x)) * (((fabs(z) - fabs(x)) / fabs(y)) * -0.5);
	} else {
		tmp = (fabs(y) * 0.5) + (((fabs(x) - fabs(z)) / fabs(y)) * (0.5 * fabs(x)));
	}
	return copysign(1.0, y) * tmp;
}

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (((((Math.abs(x) * Math.abs(x)) + (Math.abs(y) * Math.abs(y))) - (Math.abs(z) * Math.abs(z))) / (Math.abs(y) * 2.0)) <= 0.0) {
		tmp = (Math.abs(z) + Math.abs(x)) * (((Math.abs(z) - Math.abs(x)) / Math.abs(y)) * -0.5);
	} else {
		tmp = (Math.abs(y) * 0.5) + (((Math.abs(x) - Math.abs(z)) / Math.abs(y)) * (0.5 * Math.abs(x)));
	}
	return Math.copySign(1.0, y) * tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((((math.fabs(x) * math.fabs(x)) + (math.fabs(y) * math.fabs(y))) - (math.fabs(z) * math.fabs(z))) / (math.fabs(y) * 2.0)) <= 0.0:
		tmp = (math.fabs(z) + math.fabs(x)) * (((math.fabs(z) - math.fabs(x)) / math.fabs(y)) * -0.5)
	else:
		tmp = (math.fabs(y) * 0.5) + (((math.fabs(x) - math.fabs(z)) / math.fabs(y)) * (0.5 * math.fabs(x)))
	return math.copysign(1.0, y) * tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(abs(x) * abs(x)) + Float64(abs(y) * abs(y))) - Float64(abs(z) * abs(z))) / Float64(abs(y) * 2.0)) <= 0.0)
		tmp = Float64(Float64(abs(z) + abs(x)) * Float64(Float64(Float64(abs(z) - abs(x)) / abs(y)) * -0.5));
	else
		tmp = Float64(Float64(abs(y) * 0.5) + Float64(Float64(Float64(abs(x) - abs(z)) / abs(y)) * Float64(0.5 * abs(x))));
	end
	return Float64(copysign(1.0, y) * tmp)
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((((abs(x) * abs(x)) + (abs(y) * abs(y))) - (abs(z) * abs(z))) / (abs(y) * 2.0)) <= 0.0)
		tmp = (abs(z) + abs(x)) * (((abs(z) - abs(x)) / abs(y)) * -0.5);
	else
		tmp = (abs(y) * 0.5) + (((abs(x) - abs(z)) / abs(y)) * (0.5 * abs(x)));
	end
	tmp_2 = (sign(y) * abs(1.0)) * tmp;
end

Wolfram

code[x_, y_, z_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[Abs[y], $MachinePrecision] * N[Abs[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Abs[z], $MachinePrecision] * N[Abs[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[y], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[Abs[z], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Abs[z], $MachinePrecision] - N[Abs[x], $MachinePrecision]), $MachinePrecision] / N[Abs[y], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[y], $MachinePrecision] * 0.5), $MachinePrecision] + N[(N[(N[(N[Abs[x], $MachinePrecision] - N[Abs[z], $MachinePrecision]), $MachinePrecision] / N[Abs[y], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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))) < 0.0

   1. Initial program 68.5%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-+.f64 N/A

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

      4. associate--r+ N/A

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

      5. sub-negate N/A

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

      6. lower--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 73.7%

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f64 73.7%

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

   3. Applied rewrites 73.7%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower--.f64 61.6%

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

   6. Applied rewrites 61.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 67.0%

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

   8. Applied rewrites 67.0%

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

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

   1. Initial program 68.5%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. div-add N/A

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

      7. associate--l+ N/A

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

      8. lower-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f64 N/A

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

      17. lower--.f64 N/A

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

   3. Applied rewrites 90.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lift-*.f64 N/A

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

      10. sub-div N/A

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

      11. sub-negate N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

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

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

      15. lift-+.f64 N/A

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

      16. lift--.f64 N/A

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

      17. *-commutative N/A

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

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

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

      19. lift--.f64 N/A

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

      20. sub-negate-rev N/A

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

      21. lift-+.f64 N/A

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

      22. count-2 N/A

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

      23. *-commutative N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 99.9%

      \[\leadsto y \cdot \color{blue}{0.5} + \frac{x - z}{y} \cdot \frac{z + x}{2} \]

   9. Taylor expanded in x around inf

      \[\leadsto y \cdot 0.5 + \frac{x - z}{y} \cdot \color{blue}{\left(\frac{1}{2} \cdot x\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 73.9%

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

   11. Applied rewrites 73.9%

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

Alternative 3: 93.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2.1 \cdot 10^{-137}:\\ \;\;\;\;y \cdot 0.5 - \frac{z}{y + y} \cdot z\\ \mathbf{elif}\;\left|x\right| \leq 1.05 \cdot 10^{+151}:\\ \;\;\;\;\left(y + \frac{\left(\left|x\right| + z\right) \cdot \left(\left|x\right| - z\right)}{y}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(z + \left|x\right|\right) \cdot \left(\frac{z - \left|x\right|}{y} \cdot -0.5\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (if (<= (fabs x) 2.1e-137)
  (- (* y 0.5) (* (/ z (+ y y)) z))
  (if (<= (fabs x) 1.05e+151)
    (* (+ y (/ (* (+ (fabs x) z) (- (fabs x) z)) y)) 0.5)
    (* (+ z (fabs x)) (* (/ (- z (fabs x)) y) -0.5)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (fabs(x) <= 2.1e-137) {
		tmp = (y * 0.5) - ((z / (y + y)) * z);
	} else if (fabs(x) <= 1.05e+151) {
		tmp = (y + (((fabs(x) + z) * (fabs(x) - z)) / y)) * 0.5;
	} else {
		tmp = (z + fabs(x)) * (((z - fabs(x)) / 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 (abs(x) <= 2.1d-137) then
        tmp = (y * 0.5d0) - ((z / (y + y)) * z)
    else if (abs(x) <= 1.05d+151) then
        tmp = (y + (((abs(x) + z) * (abs(x) - z)) / y)) * 0.5d0
    else
        tmp = (z + abs(x)) * (((z - abs(x)) / y) * (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (Math.abs(x) <= 2.1e-137) {
		tmp = (y * 0.5) - ((z / (y + y)) * z);
	} else if (Math.abs(x) <= 1.05e+151) {
		tmp = (y + (((Math.abs(x) + z) * (Math.abs(x) - z)) / y)) * 0.5;
	} else {
		tmp = (z + Math.abs(x)) * (((z - Math.abs(x)) / y) * -0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if math.fabs(x) <= 2.1e-137:
		tmp = (y * 0.5) - ((z / (y + y)) * z)
	elif math.fabs(x) <= 1.05e+151:
		tmp = (y + (((math.fabs(x) + z) * (math.fabs(x) - z)) / y)) * 0.5
	else:
		tmp = (z + math.fabs(x)) * (((z - math.fabs(x)) / y) * -0.5)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (abs(x) <= 2.1e-137)
		tmp = Float64(Float64(y * 0.5) - Float64(Float64(z / Float64(y + y)) * z));
	elseif (abs(x) <= 1.05e+151)
		tmp = Float64(Float64(y + Float64(Float64(Float64(abs(x) + z) * Float64(abs(x) - z)) / y)) * 0.5);
	else
		tmp = Float64(Float64(z + abs(x)) * Float64(Float64(Float64(z - abs(x)) / y) * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (abs(x) <= 2.1e-137)
		tmp = (y * 0.5) - ((z / (y + y)) * z);
	elseif (abs(x) <= 1.05e+151)
		tmp = (y + (((abs(x) + z) * (abs(x) - z)) / y)) * 0.5;
	else
		tmp = (z + abs(x)) * (((z - abs(x)) / y) * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Abs[x], $MachinePrecision], 2.1e-137], N[(N[(y * 0.5), $MachinePrecision] - N[(N[(z / N[(y + y), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[x], $MachinePrecision], 1.05e+151], N[(N[(y + N[(N[(N[(N[Abs[x], $MachinePrecision] + z), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(z + N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(z - N[Abs[x], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.0999999999999999e-137

   1. Initial program 68.5%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. mult-flip N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. add-to-fraction-rev N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. lift-*.f64 N/A

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

      17. associate-/l* N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

   3. Applied rewrites 84.2%

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

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y} \cdot 0.5 - \frac{z}{y + y} \cdot z \]
   5. Step-by-step derivation

   6. Applied rewrites 66.2%

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

   if 2.0999999999999999e-137 < x < 1.05e151

   1. Initial program 68.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. mult-flip N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 88.6%

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

   if 1.05e151 < x

   1. Initial program 68.5%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-+.f64 N/A

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

      4. associate--r+ N/A

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

      5. sub-negate N/A

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

      6. lower--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 73.7%

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f64 73.7%

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

   3. Applied rewrites 73.7%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower--.f64 61.6%

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

   6. Applied rewrites 61.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 67.0%

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

   8. Applied rewrites 67.0%

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

Alternative 4: 85.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 3.65 \cdot 10^{+61}:\\ \;\;\;\;y \cdot 0.5 - \frac{z}{y + y} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(z + \left|x\right|\right) \cdot \left(\frac{z - \left|x\right|}{y} \cdot -0.5\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (if (<= (fabs x) 3.65e+61)
  (- (* y 0.5) (* (/ z (+ y y)) z))
  (* (+ z (fabs x)) (* (/ (- z (fabs x)) y) -0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if (fabs(x) <= 3.65e+61) {
		tmp = (y * 0.5) - ((z / (y + y)) * z);
	} else {
		tmp = (z + fabs(x)) * (((z - fabs(x)) / 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 (abs(x) <= 3.65d+61) then
        tmp = (y * 0.5d0) - ((z / (y + y)) * z)
    else
        tmp = (z + abs(x)) * (((z - abs(x)) / y) * (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (Math.abs(x) <= 3.65e+61) {
		tmp = (y * 0.5) - ((z / (y + y)) * z);
	} else {
		tmp = (z + Math.abs(x)) * (((z - Math.abs(x)) / y) * -0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if math.fabs(x) <= 3.65e+61:
		tmp = (y * 0.5) - ((z / (y + y)) * z)
	else:
		tmp = (z + math.fabs(x)) * (((z - math.fabs(x)) / y) * -0.5)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (abs(x) <= 3.65e+61)
		tmp = Float64(Float64(y * 0.5) - Float64(Float64(z / Float64(y + y)) * z));
	else
		tmp = Float64(Float64(z + abs(x)) * Float64(Float64(Float64(z - abs(x)) / y) * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (abs(x) <= 3.65e+61)
		tmp = (y * 0.5) - ((z / (y + y)) * z);
	else
		tmp = (z + abs(x)) * (((z - abs(x)) / y) * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Abs[x], $MachinePrecision], 3.65e+61], N[(N[(y * 0.5), $MachinePrecision] - N[(N[(z / N[(y + y), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(N[(z + N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(z - N[Abs[x], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.6500000000000001e61

   1. Initial program 68.5%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. mult-flip N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. add-to-fraction-rev N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. lift-*.f64 N/A

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

      17. associate-/l* N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

   3. Applied rewrites 84.2%

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

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y} \cdot 0.5 - \frac{z}{y + y} \cdot z \]
   5. Step-by-step derivation

   6. Applied rewrites 66.2%

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

   if 3.6500000000000001e61 < x

   1. Initial program 68.5%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-+.f64 N/A

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

      4. associate--r+ N/A

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

      5. sub-negate N/A

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

      6. lower--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 73.7%

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f64 73.7%

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

   3. Applied rewrites 73.7%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower--.f64 61.6%

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

   6. Applied rewrites 61.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 67.0%

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

   8. Applied rewrites 67.0%

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

Alternative 5: 80.8% accurate, 0.3× speedup

Math

\[\mathsf{copysign}\left(1, y\right) \cdot \begin{array}{l} \mathbf{if}\;\left|y\right| \leq 1.1 \cdot 10^{+116}:\\ \;\;\;\;\left(z + x\right) \cdot \left(\frac{z - x}{\left|y\right|} \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left|y\right|\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (*
 (copysign 1.0 y)
 (if (<= (fabs y) 1.1e+116)
   (* (+ z x) (* (/ (- z x) (fabs y)) -0.5))
   (* 0.5 (fabs y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (fabs(y) <= 1.1e+116) {
		tmp = (z + x) * (((z - x) / fabs(y)) * -0.5);
	} else {
		tmp = 0.5 * fabs(y);
	}
	return copysign(1.0, y) * tmp;
}

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (Math.abs(y) <= 1.1e+116) {
		tmp = (z + x) * (((z - x) / Math.abs(y)) * -0.5);
	} else {
		tmp = 0.5 * Math.abs(y);
	}
	return Math.copySign(1.0, y) * tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if math.fabs(y) <= 1.1e+116:
		tmp = (z + x) * (((z - x) / math.fabs(y)) * -0.5)
	else:
		tmp = 0.5 * math.fabs(y)
	return math.copysign(1.0, y) * tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (abs(y) <= 1.1e+116)
		tmp = Float64(Float64(z + x) * Float64(Float64(Float64(z - x) / abs(y)) * -0.5));
	else
		tmp = Float64(0.5 * abs(y));
	end
	return Float64(copysign(1.0, y) * tmp)
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (abs(y) <= 1.1e+116)
		tmp = (z + x) * (((z - x) / abs(y)) * -0.5);
	else
		tmp = 0.5 * abs(y);
	end
	tmp_2 = (sign(y) * abs(1.0)) * tmp;
end

Wolfram

code[x_, y_, z_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[y], $MachinePrecision], 1.1e+116], N[(N[(z + x), $MachinePrecision] * N[(N[(N[(z - x), $MachinePrecision] / N[Abs[y], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Abs[y], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.0999999999999999e116

   1. Initial program 68.5%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-+.f64 N/A

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

      4. associate--r+ N/A

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

      5. sub-negate N/A

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

      6. lower--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 73.7%

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f64 73.7%

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

   3. Applied rewrites 73.7%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower--.f64 61.6%

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

   6. Applied rewrites 61.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 67.0%

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

   8. Applied rewrites 67.0%

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

   if 1.0999999999999999e116 < y

   1. Initial program 68.5%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 34.4%

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

   4. Applied rewrites 34.4%

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

Alternative 6: 78.9% accurate, 0.3× speedup

Math

\[\mathsf{copysign}\left(1, y\right) \cdot \begin{array}{l} \mathbf{if}\;\left|y\right| \leq 1.02 \cdot 10^{+104}:\\ \;\;\;\;-0.5 \cdot \frac{\left(x + z\right) \cdot \left(z - x\right)}{\left|y\right|}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left|y\right|\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (*
 (copysign 1.0 y)
 (if (<= (fabs y) 1.02e+104)
   (* -0.5 (/ (* (+ x z) (- z x)) (fabs y)))
   (* 0.5 (fabs y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (fabs(y) <= 1.02e+104) {
		tmp = -0.5 * (((x + z) * (z - x)) / fabs(y));
	} else {
		tmp = 0.5 * fabs(y);
	}
	return copysign(1.0, y) * tmp;
}

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (Math.abs(y) <= 1.02e+104) {
		tmp = -0.5 * (((x + z) * (z - x)) / Math.abs(y));
	} else {
		tmp = 0.5 * Math.abs(y);
	}
	return Math.copySign(1.0, y) * tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if math.fabs(y) <= 1.02e+104:
		tmp = -0.5 * (((x + z) * (z - x)) / math.fabs(y))
	else:
		tmp = 0.5 * math.fabs(y)
	return math.copysign(1.0, y) * tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (abs(y) <= 1.02e+104)
		tmp = Float64(-0.5 * Float64(Float64(Float64(x + z) * Float64(z - x)) / abs(y)));
	else
		tmp = Float64(0.5 * abs(y));
	end
	return Float64(copysign(1.0, y) * tmp)
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (abs(y) <= 1.02e+104)
		tmp = -0.5 * (((x + z) * (z - x)) / abs(y));
	else
		tmp = 0.5 * abs(y);
	end
	tmp_2 = (sign(y) * abs(1.0)) * tmp;
end

Wolfram

code[x_, y_, z_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[y], $MachinePrecision], 1.02e+104], N[(-0.5 * N[(N[(N[(x + z), $MachinePrecision] * N[(z - x), $MachinePrecision]), $MachinePrecision] / N[Abs[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Abs[y], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.02e104

   1. Initial program 68.5%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-+.f64 N/A

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

      4. associate--r+ N/A

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

      5. sub-negate N/A

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

      6. lower--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 73.7%

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f64 73.7%

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

   3. Applied rewrites 73.7%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower--.f64 61.6%

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

   6. Applied rewrites 61.6%

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

   if 1.02e104 < y

   1. Initial program 68.5%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 34.4%

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

   4. Applied rewrites 34.4%

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

Alternative 7: 67.3% accurate, 0.2× speedup

Math

\[\mathsf{copysign}\left(1, y\right) \cdot \begin{array}{l} \mathbf{if}\;\left|y\right| \leq 1.15 \cdot 10^{-77}:\\ \;\;\;\;-0.5 \cdot \left(\left(\left|z\right| - \left|x\right|\right) \cdot \frac{\left|z\right|}{\left|y\right|}\right)\\ \mathbf{elif}\;\left|y\right| \leq 1.35 \cdot 10^{+152}:\\ \;\;\;\;\frac{\left(\left|y\right| + \left|z\right|\right) \cdot \left(\left|y\right| - \left|z\right|\right)}{\left|y\right| + \left|y\right|}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left|y\right|\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (*
 (copysign 1.0 y)
 (if (<= (fabs y) 1.15e-77)
   (* -0.5 (* (- (fabs z) (fabs x)) (/ (fabs z) (fabs y))))
   (if (<= (fabs y) 1.35e+152)
     (/
      (* (+ (fabs y) (fabs z)) (- (fabs y) (fabs z)))
      (+ (fabs y) (fabs y)))
     (* 0.5 (fabs y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (fabs(y) <= 1.15e-77) {
		tmp = -0.5 * ((fabs(z) - fabs(x)) * (fabs(z) / fabs(y)));
	} else if (fabs(y) <= 1.35e+152) {
		tmp = ((fabs(y) + fabs(z)) * (fabs(y) - fabs(z))) / (fabs(y) + fabs(y));
	} else {
		tmp = 0.5 * fabs(y);
	}
	return copysign(1.0, y) * tmp;
}

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (Math.abs(y) <= 1.15e-77) {
		tmp = -0.5 * ((Math.abs(z) - Math.abs(x)) * (Math.abs(z) / Math.abs(y)));
	} else if (Math.abs(y) <= 1.35e+152) {
		tmp = ((Math.abs(y) + Math.abs(z)) * (Math.abs(y) - Math.abs(z))) / (Math.abs(y) + Math.abs(y));
	} else {
		tmp = 0.5 * Math.abs(y);
	}
	return Math.copySign(1.0, y) * tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if math.fabs(y) <= 1.15e-77:
		tmp = -0.5 * ((math.fabs(z) - math.fabs(x)) * (math.fabs(z) / math.fabs(y)))
	elif math.fabs(y) <= 1.35e+152:
		tmp = ((math.fabs(y) + math.fabs(z)) * (math.fabs(y) - math.fabs(z))) / (math.fabs(y) + math.fabs(y))
	else:
		tmp = 0.5 * math.fabs(y)
	return math.copysign(1.0, y) * tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (abs(y) <= 1.15e-77)
		tmp = Float64(-0.5 * Float64(Float64(abs(z) - abs(x)) * Float64(abs(z) / abs(y))));
	elseif (abs(y) <= 1.35e+152)
		tmp = Float64(Float64(Float64(abs(y) + abs(z)) * Float64(abs(y) - abs(z))) / Float64(abs(y) + abs(y)));
	else
		tmp = Float64(0.5 * abs(y));
	end
	return Float64(copysign(1.0, y) * tmp)
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (abs(y) <= 1.15e-77)
		tmp = -0.5 * ((abs(z) - abs(x)) * (abs(z) / abs(y)));
	elseif (abs(y) <= 1.35e+152)
		tmp = ((abs(y) + abs(z)) * (abs(y) - abs(z))) / (abs(y) + abs(y));
	else
		tmp = 0.5 * abs(y);
	end
	tmp_2 = (sign(y) * abs(1.0)) * tmp;
end

Wolfram

code[x_, y_, z_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[y], $MachinePrecision], 1.15e-77], N[(-0.5 * N[(N[(N[Abs[z], $MachinePrecision] - N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[z], $MachinePrecision] / N[Abs[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[y], $MachinePrecision], 1.35e+152], N[(N[(N[(N[Abs[y], $MachinePrecision] + N[Abs[z], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[y], $MachinePrecision] - N[Abs[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[y], $MachinePrecision] + N[Abs[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Abs[y], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < 1.15e-77

   1. Initial program 68.5%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-+.f64 N/A

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

      4. associate--r+ N/A

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

      5. sub-negate N/A

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

      6. lower--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 73.7%

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f64 73.7%

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

   3. Applied rewrites 73.7%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower--.f64 61.6%

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

   6. Applied rewrites 61.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 66.9%

          \[\leadsto -0.5 \cdot \left(\left(z - x\right) \cdot \left(\left(z + x\right) \cdot \frac{1}{\color{blue}{y}}\right)\right) \]

   8. Applied rewrites 66.9%

      \[\leadsto -0.5 \cdot \left(\left(z - x\right) \cdot \color{blue}{\left(\left(z + x\right) \cdot \frac{1}{y}\right)}\right) \]

   9. Taylor expanded in x around 0

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

       1. lower-/.f64 37.3%

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

   11. Applied rewrites 37.3%

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

   if 1.15e-77 < y < 1.3500000000000001e152

   1. Initial program 68.5%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-+.f64 N/A

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

      4. associate--r+ N/A

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

      5. sub-negate N/A

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

      6. lower--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 73.7%

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f64 73.7%

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

   3. Applied rewrites 73.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate--r- N/A

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

      9. lower-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 70.0%

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

   5. Applied rewrites 70.0%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 46.2%

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

   8. Applied rewrites 46.2%

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

   if 1.3500000000000001e152 < y

   1. Initial program 68.5%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 34.4%

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

   4. Applied rewrites 34.4%

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

Alternative 8: 64.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := -0.5 \cdot \left(\left(\left|z\right| - \left|x\right|\right) \cdot \frac{\left|z\right|}{\left|y\right|}\right)\\ t_1 := \frac{\left(\left|x\right| \cdot \left|x\right| + \left|y\right| \cdot \left|y\right|\right) - \left|z\right| \cdot \left|z\right|}{\left|y\right| \cdot 2}\\ \mathsf{copysign}\left(1, y\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;0.5 \cdot \left|y\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (* -0.5 (* (- (fabs z) (fabs x)) (/ (fabs z) (fabs y)))))
       (t_1
        (/
         (-
          (+ (* (fabs x) (fabs x)) (* (fabs y) (fabs y)))
          (* (fabs z) (fabs z)))
         (* (fabs y) 2.0))))
  (*
   (copysign 1.0 y)
   (if (<= t_1 0.0) t_0 (if (<= t_1 INFINITY) (* 0.5 (fabs y)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = -0.5 * ((fabs(z) - fabs(x)) * (fabs(z) / fabs(y)));
	double t_1 = (((fabs(x) * fabs(x)) + (fabs(y) * fabs(y))) - (fabs(z) * fabs(z))) / (fabs(y) * 2.0);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 0.5 * fabs(y);
	} else {
		tmp = t_0;
	}
	return copysign(1.0, y) * tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = -0.5 * ((Math.abs(z) - Math.abs(x)) * (Math.abs(z) / Math.abs(y)));
	double t_1 = (((Math.abs(x) * Math.abs(x)) + (Math.abs(y) * Math.abs(y))) - (Math.abs(z) * Math.abs(z))) / (Math.abs(y) * 2.0);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 0.5 * Math.abs(y);
	} else {
		tmp = t_0;
	}
	return Math.copySign(1.0, y) * tmp;
}

Python

def code(x, y, z):
	t_0 = -0.5 * ((math.fabs(z) - math.fabs(x)) * (math.fabs(z) / math.fabs(y)))
	t_1 = (((math.fabs(x) * math.fabs(x)) + (math.fabs(y) * math.fabs(y))) - (math.fabs(z) * math.fabs(z))) / (math.fabs(y) * 2.0)
	tmp = 0
	if t_1 <= 0.0:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = 0.5 * math.fabs(y)
	else:
		tmp = t_0
	return math.copysign(1.0, y) * tmp

Julia

function code(x, y, z)
	t_0 = Float64(-0.5 * Float64(Float64(abs(z) - abs(x)) * Float64(abs(z) / abs(y))))
	t_1 = Float64(Float64(Float64(Float64(abs(x) * abs(x)) + Float64(abs(y) * abs(y))) - Float64(abs(z) * abs(z))) / Float64(abs(y) * 2.0))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(0.5 * abs(y));
	else
		tmp = t_0;
	end
	return Float64(copysign(1.0, y) * tmp)
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -0.5 * ((abs(z) - abs(x)) * (abs(z) / abs(y)));
	t_1 = (((abs(x) * abs(x)) + (abs(y) * abs(y))) - (abs(z) * abs(z))) / (abs(y) * 2.0);
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = 0.5 * abs(y);
	else
		tmp = t_0;
	end
	tmp_2 = (sign(y) * abs(1.0)) * tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(-0.5 * N[(N[(N[Abs[z], $MachinePrecision] - N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[z], $MachinePrecision] / N[Abs[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[Abs[y], $MachinePrecision] * N[Abs[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Abs[z], $MachinePrecision] * N[Abs[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[y], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$1, 0.0], t$95$0, If[LessEqual[t$95$1, Infinity], N[(0.5 * N[Abs[y], $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 68.5%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-+.f64 N/A

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

      4. associate--r+ N/A

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

      5. sub-negate N/A

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

      6. lower--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 73.7%

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f64 73.7%

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

   3. Applied rewrites 73.7%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower--.f64 61.6%

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

   6. Applied rewrites 61.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 66.9%

          \[\leadsto -0.5 \cdot \left(\left(z - x\right) \cdot \left(\left(z + x\right) \cdot \frac{1}{\color{blue}{y}}\right)\right) \]

   8. Applied rewrites 66.9%

      \[\leadsto -0.5 \cdot \left(\left(z - x\right) \cdot \color{blue}{\left(\left(z + x\right) \cdot \frac{1}{y}\right)}\right) \]

   9. Taylor expanded in x around 0

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

       1. lower-/.f64 37.3%

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

   11. Applied rewrites 37.3%

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

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

   1. Initial program 68.5%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 34.4%

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

   4. Applied rewrites 34.4%

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

Alternative 9: 34.4% accurate, 6.3× speedup

Math

\[0.5 \cdot y \]

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 68.5%

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

2. Taylor expanded in y around inf

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

   1. lower-*.f64 34.4%

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

4. Applied rewrites 34.4%

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

Reproduce

herbie shell --seed 2025258 
(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)))