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

Percentage Accurate: 69.3% → 99.9%

Time: 10.0s

Alternatives: 9

Speedup: 1.2×

• 41.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (((x * x) + (y * y)) - (z * z)) / (y * 2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (((x * x) + (y * y)) - (z * z)) / (y * 2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.2× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z)
 :precision binary64
 (/ (+ y (* (+ z x_m) (/ (- x_m z) y))) 2.0))

C

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

Fortran

x_m = abs(x)
real(8) function code(x_m, y, z)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (y + ((z + x_m) * ((x_m - z) / y))) / 2.0d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y, double z) {
	return (y + ((z + x_m) * ((x_m - z) / y))) / 2.0;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z):
	return (y + ((z + x_m) * ((x_m - z) / y))) / 2.0

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_] := N[(N[(y + N[(N[(z + x$95$m), $MachinePrecision] * N[(N[(x$95$m - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 68.4%

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

   1. associate-/r* N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y}\right), \color{blue}{2}\right) \]

   3. associate--l+ N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{x \cdot x + \left(y \cdot y - z \cdot z\right)}{y}\right), 2\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(y \cdot y - z \cdot z\right) + x \cdot x}{y}\right), 2\right) \]

   5. associate-+l- N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot y - \left(z \cdot z - x \cdot x\right)}{y}\right), 2\right) \]

   6. div-sub N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot y}{y} - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

   7. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{y}{y} - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{y}{y} \cdot y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

   9. *-inverses N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

   10. *-lft-identity N/A

       \[\leadsto \mathsf{/.f64}\left(\left(y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(\frac{z \cdot z - x \cdot x}{y}\right)\right), 2\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

   13. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

   15. *-lowering-*.f64 85.6%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

3. Simplified 85.6%

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

   1. difference-of-squares N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(\frac{\left(z + x\right) \cdot \left(z - x\right)}{y}\right)\right), 2\right) \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(\left(z + x\right) \cdot \frac{z - x}{y}\right)\right), 2\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{*.f64}\left(\left(z + x\right), \left(\frac{z - x}{y}\right)\right)\right), 2\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(z, x\right), \left(\frac{z - x}{y}\right)\right)\right), 2\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(z, x\right), \mathsf{/.f64}\left(\left(z - x\right), y\right)\right)\right), 2\right) \]

   6. --lowering--.f64 99.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(z, x\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(z, x\right), y\right)\right)\right), 2\right) \]

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 87.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \cdot x\_m \leq 2 \cdot 10^{-105}:\\ \;\;\;\;\frac{y - z \cdot \frac{z}{y}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x\_m \cdot \frac{x\_m - z}{y}}{2}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z)
 :precision binary64
 (if (<= (* x_m x_m) 2e-105)
   (/ (- y (* z (/ z y))) 2.0)
   (/ (+ y (* x_m (/ (- x_m z) y))) 2.0)))

C

x_m = fabs(x);
double code(double x_m, double y, double z) {
	double tmp;
	if ((x_m * x_m) <= 2e-105) {
		tmp = (y - (z * (z / y))) / 2.0;
	} else {
		tmp = (y + (x_m * ((x_m - z) / y))) / 2.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y, z)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x_m * x_m) <= 2d-105) then
        tmp = (y - (z * (z / y))) / 2.0d0
    else
        tmp = (y + (x_m * ((x_m - z) / y))) / 2.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y, double z) {
	double tmp;
	if ((x_m * x_m) <= 2e-105) {
		tmp = (y - (z * (z / y))) / 2.0;
	} else {
		tmp = (y + (x_m * ((x_m - z) / y))) / 2.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z):
	tmp = 0
	if (x_m * x_m) <= 2e-105:
		tmp = (y - (z * (z / y))) / 2.0
	else:
		tmp = (y + (x_m * ((x_m - z) / y))) / 2.0
	return tmp

Julia

x_m = abs(x)
function code(x_m, y, z)
	tmp = 0.0
	if (Float64(x_m * x_m) <= 2e-105)
		tmp = Float64(Float64(y - Float64(z * Float64(z / y))) / 2.0);
	else
		tmp = Float64(Float64(y + Float64(x_m * Float64(Float64(x_m - z) / y))) / 2.0);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y, z)
	tmp = 0.0;
	if ((x_m * x_m) <= 2e-105)
		tmp = (y - (z * (z / y))) / 2.0;
	else
		tmp = (y + (x_m * ((x_m - z) / y))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_] := If[LessEqual[N[(x$95$m * x$95$m), $MachinePrecision], 2e-105], N[(N[(y - N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(y + N[(x$95$m * N[(N[(x$95$m - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.99999999999999993e-105

   1. Initial program 68.6%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y}\right), \color{blue}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x \cdot x + \left(y \cdot y - z \cdot z\right)}{y}\right), 2\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(y \cdot y - z \cdot z\right) + x \cdot x}{y}\right), 2\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot y - \left(z \cdot z - x \cdot x\right)}{y}\right), 2\right) \]

      6. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot y}{y} - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{y}{y} - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y}{y} \cdot y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(\frac{z \cdot z - x \cdot x}{y}\right)\right), 2\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      15. *-lowering-*.f64 92.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

   3. Simplified 92.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(y - \frac{{z}^{2}}{y}\right)}, 2\right) \]
   6. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(\frac{{z}^{2}}{y}\right)\right), 2\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left({z}^{2}\right), y\right)\right), 2\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z\right), y\right)\right), 2\right) \]

      4. *-lowering-*.f64 90.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), y\right)\right), 2\right) \]

   7. Simplified 90.8%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(z \cdot \frac{z}{y}\right)\right), 2\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(\frac{z}{y} \cdot z\right)\right), 2\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{*.f64}\left(\left(\frac{z}{y}\right), z\right)\right), 2\right) \]

      4. /-lowering-/.f64 95.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, y\right), z\right)\right), 2\right) \]

   9. Applied egg-rr 95.7%

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

   if 1.99999999999999993e-105 < (*.f64 x x)

   1. Initial program 68.2%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y}\right), \color{blue}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x \cdot x + \left(y \cdot y - z \cdot z\right)}{y}\right), 2\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(y \cdot y - z \cdot z\right) + x \cdot x}{y}\right), 2\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot y - \left(z \cdot z - x \cdot x\right)}{y}\right), 2\right) \]

      6. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot y}{y} - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{y}{y} - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y}{y} \cdot y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(\frac{z \cdot z - x \cdot x}{y}\right)\right), 2\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      15. *-lowering-*.f64 79.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

   3. Simplified 79.9%

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

      1. difference-of-squares N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(\frac{\left(z + x\right) \cdot \left(z - x\right)}{y}\right)\right), 2\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(\left(z + x\right) \cdot \frac{z - x}{y}\right)\right), 2\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{*.f64}\left(\left(z + x\right), \left(\frac{z - x}{y}\right)\right)\right), 2\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(z, x\right), \left(\frac{z - x}{y}\right)\right)\right), 2\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(z, x\right), \mathsf{/.f64}\left(\left(z - x\right), y\right)\right)\right), 2\right) \]

      6. --lowering--.f64 99.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(z, x\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(z, x\right), y\right)\right)\right), 2\right) \]

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in z around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(z, x\right), y\right)\right)\right), 2\right) \]
   8. Step-by-step derivation

   9. Simplified 80.1%

      \[\leadsto \frac{y - \color{blue}{x} \cdot \frac{z - x}{y}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-105}:\\ \;\;\;\;\frac{y - z \cdot \frac{z}{y}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x \cdot \frac{x - z}{y}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;y \leq 5.4 \cdot 10^{+44}:\\ \;\;\;\;\frac{\left(z + x\_m\right) \cdot \frac{x\_m - z}{y}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z \cdot \frac{z}{y}}{2}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z)
 :precision binary64
 (if (<= y 5.4e+44)
   (/ (* (+ z x_m) (/ (- x_m z) y)) 2.0)
   (/ (- y (* z (/ z y))) 2.0)))

C

x_m = fabs(x);
double code(double x_m, double y, double z) {
	double tmp;
	if (y <= 5.4e+44) {
		tmp = ((z + x_m) * ((x_m - z) / y)) / 2.0;
	} else {
		tmp = (y - (z * (z / y))) / 2.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y, z)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 5.4d+44) then
        tmp = ((z + x_m) * ((x_m - z) / y)) / 2.0d0
    else
        tmp = (y - (z * (z / y))) / 2.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y, double z) {
	double tmp;
	if (y <= 5.4e+44) {
		tmp = ((z + x_m) * ((x_m - z) / y)) / 2.0;
	} else {
		tmp = (y - (z * (z / y))) / 2.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z):
	tmp = 0
	if y <= 5.4e+44:
		tmp = ((z + x_m) * ((x_m - z) / y)) / 2.0
	else:
		tmp = (y - (z * (z / y))) / 2.0
	return tmp

Julia

x_m = abs(x)
function code(x_m, y, z)
	tmp = 0.0
	if (y <= 5.4e+44)
		tmp = Float64(Float64(Float64(z + x_m) * Float64(Float64(x_m - z) / y)) / 2.0);
	else
		tmp = Float64(Float64(y - Float64(z * Float64(z / y))) / 2.0);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y, z)
	tmp = 0.0;
	if (y <= 5.4e+44)
		tmp = ((z + x_m) * ((x_m - z) / y)) / 2.0;
	else
		tmp = (y - (z * (z / y))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_] := If[LessEqual[y, 5.4e+44], N[(N[(N[(z + x$95$m), $MachinePrecision] * N[(N[(x$95$m - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(y - N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.4e44

   1. Initial program 73.8%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y}\right), \color{blue}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x \cdot x + \left(y \cdot y - z \cdot z\right)}{y}\right), 2\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(y \cdot y - z \cdot z\right) + x \cdot x}{y}\right), 2\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot y - \left(z \cdot z - x \cdot x\right)}{y}\right), 2\right) \]

      6. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot y}{y} - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{y}{y} - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y}{y} \cdot y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(\frac{z \cdot z - x \cdot x}{y}\right)\right), 2\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      15. *-lowering-*.f64 87.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

   3. Simplified 87.1%

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

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{{x}^{2} - {z}^{2}}{y}\right)}, 2\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x \cdot x - {z}^{2}}{y}\right), 2\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x \cdot x - z \cdot z}{y}\right), 2\right) \]

      3. difference-of-squares N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{y}\right), 2\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(x + z\right) \cdot \frac{x - z}{y}\right), 2\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(x + z\right), \left(\frac{x - z}{y}\right)\right), 2\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, z\right), \left(\frac{x - z}{y}\right)\right), 2\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, z\right), \mathsf{/.f64}\left(\left(x - z\right), y\right)\right), 2\right) \]

      8. --lowering--.f64 75.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, z\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, z\right), y\right)\right), 2\right) \]

   7. Simplified 75.9%

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

   if 5.4e44 < y

   1. Initial program 50.2%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y}\right), \color{blue}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x \cdot x + \left(y \cdot y - z \cdot z\right)}{y}\right), 2\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(y \cdot y - z \cdot z\right) + x \cdot x}{y}\right), 2\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot y - \left(z \cdot z - x \cdot x\right)}{y}\right), 2\right) \]

      6. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot y}{y} - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{y}{y} - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y}{y} \cdot y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(\frac{z \cdot z - x \cdot x}{y}\right)\right), 2\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      15. *-lowering-*.f64 80.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

   3. Simplified 80.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(y - \frac{{z}^{2}}{y}\right)}, 2\right) \]
   6. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(\frac{{z}^{2}}{y}\right)\right), 2\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left({z}^{2}\right), y\right)\right), 2\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z\right), y\right)\right), 2\right) \]

      4. *-lowering-*.f64 78.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), y\right)\right), 2\right) \]

   7. Simplified 78.9%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(z \cdot \frac{z}{y}\right)\right), 2\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(\frac{z}{y} \cdot z\right)\right), 2\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{*.f64}\left(\left(\frac{z}{y}\right), z\right)\right), 2\right) \]

      4. /-lowering-/.f64 88.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, y\right), z\right)\right), 2\right) \]

   9. Applied egg-rr 88.8%

      \[\leadsto \frac{y - \color{blue}{\frac{z}{y} \cdot z}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.4 \cdot 10^{+44}:\\ \;\;\;\;\frac{\left(z + x\right) \cdot \frac{x - z}{y}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z \cdot \frac{z}{y}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{+44}:\\ \;\;\;\;\left(z + x\_m\right) \cdot \left(\left(x\_m - z\right) \cdot \frac{0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z \cdot \frac{z}{y}}{2}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z)
 :precision binary64
 (if (<= y 7.5e+44)
   (* (+ z x_m) (* (- x_m z) (/ 0.5 y)))
   (/ (- y (* z (/ z y))) 2.0)))

C

x_m = fabs(x);
double code(double x_m, double y, double z) {
	double tmp;
	if (y <= 7.5e+44) {
		tmp = (z + x_m) * ((x_m - z) * (0.5 / y));
	} else {
		tmp = (y - (z * (z / y))) / 2.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y, z)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 7.5d+44) then
        tmp = (z + x_m) * ((x_m - z) * (0.5d0 / y))
    else
        tmp = (y - (z * (z / y))) / 2.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y, double z) {
	double tmp;
	if (y <= 7.5e+44) {
		tmp = (z + x_m) * ((x_m - z) * (0.5 / y));
	} else {
		tmp = (y - (z * (z / y))) / 2.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z):
	tmp = 0
	if y <= 7.5e+44:
		tmp = (z + x_m) * ((x_m - z) * (0.5 / y))
	else:
		tmp = (y - (z * (z / y))) / 2.0
	return tmp

Julia

x_m = abs(x)
function code(x_m, y, z)
	tmp = 0.0
	if (y <= 7.5e+44)
		tmp = Float64(Float64(z + x_m) * Float64(Float64(x_m - z) * Float64(0.5 / y)));
	else
		tmp = Float64(Float64(y - Float64(z * Float64(z / y))) / 2.0);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y, z)
	tmp = 0.0;
	if (y <= 7.5e+44)
		tmp = (z + x_m) * ((x_m - z) * (0.5 / y));
	else
		tmp = (y - (z * (z / y))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_] := If[LessEqual[y, 7.5e+44], N[(N[(z + x$95$m), $MachinePrecision] * N[(N[(x$95$m - z), $MachinePrecision] * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y - N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.50000000000000027e44

   1. Initial program 73.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left({x}^{2} - {z}^{2}\right)}, \mathsf{*.f64}\left(y, 2\right)\right) \]
   4. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left({x}^{2}\right), \left({z}^{2}\right)\right), \mathsf{*.f64}\left(\color{blue}{y}, 2\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot x\right), \left({z}^{2}\right)\right), \mathsf{*.f64}\left(y, 2\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({z}^{2}\right)\right), \mathsf{*.f64}\left(y, 2\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(z \cdot z\right)\right), \mathsf{*.f64}\left(y, 2\right)\right) \]

      5. *-lowering-*.f64 66.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(z, z\right)\right), \mathsf{*.f64}\left(y, 2\right)\right) \]

   5. Simplified 66.6%

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

      1. div-inv N/A

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

      2. difference-of-squares N/A

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

      3. +-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(z + x\right), \color{blue}{\left(\left(x - z\right) \cdot \frac{1}{y \cdot 2}\right)}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(z, x\right), \left(\color{blue}{\left(x - z\right)} \cdot \frac{1}{y \cdot 2}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(z, x\right), \mathsf{*.f64}\left(\left(x - z\right), \color{blue}{\left(\frac{1}{y \cdot 2}\right)}\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(z, x\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, z\right), \left(\frac{\color{blue}{1}}{y \cdot 2}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(z, x\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, z\right), \left(\frac{1}{2 \cdot \color{blue}{y}}\right)\right)\right) \]

      10. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(z, x\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, z\right), \left(\frac{\frac{1}{2}}{\color{blue}{y}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(z, x\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, z\right), \mathsf{/.f64}\left(\left(\frac{1}{2}\right), \color{blue}{y}\right)\right)\right) \]

      12. metadata-eval 75.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(z, x\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, z\right), \mathsf{/.f64}\left(\frac{1}{2}, y\right)\right)\right) \]

   7. Applied egg-rr 75.9%

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

   if 7.50000000000000027e44 < y

   1. Initial program 50.2%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y}\right), \color{blue}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x \cdot x + \left(y \cdot y - z \cdot z\right)}{y}\right), 2\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(y \cdot y - z \cdot z\right) + x \cdot x}{y}\right), 2\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot y - \left(z \cdot z - x \cdot x\right)}{y}\right), 2\right) \]

      6. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot y}{y} - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{y}{y} - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y}{y} \cdot y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(\frac{z \cdot z - x \cdot x}{y}\right)\right), 2\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      15. *-lowering-*.f64 80.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

   3. Simplified 80.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(y - \frac{{z}^{2}}{y}\right)}, 2\right) \]
   6. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(\frac{{z}^{2}}{y}\right)\right), 2\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left({z}^{2}\right), y\right)\right), 2\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z\right), y\right)\right), 2\right) \]

      4. *-lowering-*.f64 78.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), y\right)\right), 2\right) \]

   7. Simplified 78.9%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(z \cdot \frac{z}{y}\right)\right), 2\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(\frac{z}{y} \cdot z\right)\right), 2\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{*.f64}\left(\left(\frac{z}{y}\right), z\right)\right), 2\right) \]

      4. /-lowering-/.f64 88.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, y\right), z\right)\right), 2\right) \]

   9. Applied egg-rr 88.8%

      \[\leadsto \frac{y - \color{blue}{\frac{z}{y} \cdot z}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{+44}:\\ \;\;\;\;\left(z + x\right) \cdot \left(\left(x - z\right) \cdot \frac{0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z \cdot \frac{z}{y}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+17}:\\ \;\;\;\;\frac{y + x\_m \cdot \frac{x\_m}{y}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z \cdot \frac{z}{y}}{2}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z)
 :precision binary64
 (if (<= (* z z) 2e+17)
   (/ (+ y (* x_m (/ x_m y))) 2.0)
   (/ (- y (* z (/ z y))) 2.0)))

C

x_m = fabs(x);
double code(double x_m, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+17) {
		tmp = (y + (x_m * (x_m / y))) / 2.0;
	} else {
		tmp = (y - (z * (z / y))) / 2.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y, z)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 2d+17) then
        tmp = (y + (x_m * (x_m / y))) / 2.0d0
    else
        tmp = (y - (z * (z / y))) / 2.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+17) {
		tmp = (y + (x_m * (x_m / y))) / 2.0;
	} else {
		tmp = (y - (z * (z / y))) / 2.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z):
	tmp = 0
	if (z * z) <= 2e+17:
		tmp = (y + (x_m * (x_m / y))) / 2.0
	else:
		tmp = (y - (z * (z / y))) / 2.0
	return tmp

Julia

x_m = abs(x)
function code(x_m, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+17)
		tmp = Float64(Float64(y + Float64(x_m * Float64(x_m / y))) / 2.0);
	else
		tmp = Float64(Float64(y - Float64(z * Float64(z / y))) / 2.0);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e+17)
		tmp = (y + (x_m * (x_m / y))) / 2.0;
	else
		tmp = (y - (z * (z / y))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+17], N[(N[(y + N[(x$95$m * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(y - N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2e17

   1. Initial program 68.9%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y}\right), \color{blue}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x \cdot x + \left(y \cdot y - z \cdot z\right)}{y}\right), 2\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(y \cdot y - z \cdot z\right) + x \cdot x}{y}\right), 2\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot y - \left(z \cdot z - x \cdot x\right)}{y}\right), 2\right) \]

      6. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot y}{y} - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{y}{y} - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y}{y} \cdot y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(\frac{z \cdot z - x \cdot x}{y}\right)\right), 2\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      15. *-lowering-*.f64 90.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

   3. Simplified 90.0%

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

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(y + \frac{{x}^{2}}{y}\right)}, 2\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(y, \left(\frac{{x}^{2}}{y}\right)\right), 2\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(y, \mathsf{/.f64}\left(\left({x}^{2}\right), y\right)\right), 2\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(y, \mathsf{/.f64}\left(\left(x \cdot x\right), y\right)\right), 2\right) \]

      4. *-lowering-*.f64 82.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), y\right)\right), 2\right) \]

   7. Simplified 82.5%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(y, \left(x \cdot \frac{x}{y}\right)\right), 2\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(y, \left(\frac{x}{y} \cdot x\right)\right), 2\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(y, \mathsf{*.f64}\left(\left(\frac{x}{y}\right), x\right)\right), 2\right) \]

      4. /-lowering-/.f64 90.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(y, \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), x\right)\right), 2\right) \]

   9. Applied egg-rr 90.5%

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

   if 2e17 < (*.f64 z z)

   1. Initial program 67.8%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y}\right), \color{blue}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x \cdot x + \left(y \cdot y - z \cdot z\right)}{y}\right), 2\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(y \cdot y - z \cdot z\right) + x \cdot x}{y}\right), 2\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot y - \left(z \cdot z - x \cdot x\right)}{y}\right), 2\right) \]

      6. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot y}{y} - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{y}{y} - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y}{y} \cdot y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(\frac{z \cdot z - x \cdot x}{y}\right)\right), 2\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      15. *-lowering-*.f64 80.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

   3. Simplified 80.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(y - \frac{{z}^{2}}{y}\right)}, 2\right) \]
   6. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(\frac{{z}^{2}}{y}\right)\right), 2\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left({z}^{2}\right), y\right)\right), 2\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z\right), y\right)\right), 2\right) \]

      4. *-lowering-*.f64 80.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), y\right)\right), 2\right) \]

   7. Simplified 80.6%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(z \cdot \frac{z}{y}\right)\right), 2\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(\frac{z}{y} \cdot z\right)\right), 2\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{*.f64}\left(\left(\frac{z}{y}\right), z\right)\right), 2\right) \]

      4. /-lowering-/.f64 87.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, y\right), z\right)\right), 2\right) \]

   9. Applied egg-rr 87.6%

      \[\leadsto \frac{y - \color{blue}{\frac{z}{y} \cdot z}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+17}:\\ \;\;\;\;\frac{y + x \cdot \frac{x}{y}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z \cdot \frac{z}{y}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+293}:\\ \;\;\;\;\frac{y + x\_m \cdot \frac{x\_m}{y}}{2}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{z \cdot -0.5}{y}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z)
 :precision binary64
 (if (<= (* z z) 1e+293)
   (/ (+ y (* x_m (/ x_m y))) 2.0)
   (* z (/ (* z -0.5) y))))

C

x_m = fabs(x);
double code(double x_m, double y, double z) {
	double tmp;
	if ((z * z) <= 1e+293) {
		tmp = (y + (x_m * (x_m / y))) / 2.0;
	} else {
		tmp = z * ((z * -0.5) / y);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y, z)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 1d+293) then
        tmp = (y + (x_m * (x_m / y))) / 2.0d0
    else
        tmp = z * ((z * (-0.5d0)) / y)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y, double z) {
	double tmp;
	if ((z * z) <= 1e+293) {
		tmp = (y + (x_m * (x_m / y))) / 2.0;
	} else {
		tmp = z * ((z * -0.5) / y);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z):
	tmp = 0
	if (z * z) <= 1e+293:
		tmp = (y + (x_m * (x_m / y))) / 2.0
	else:
		tmp = z * ((z * -0.5) / y)
	return tmp

Julia

x_m = abs(x)
function code(x_m, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e+293)
		tmp = Float64(Float64(y + Float64(x_m * Float64(x_m / y))) / 2.0);
	else
		tmp = Float64(z * Float64(Float64(z * -0.5) / y));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y, z)
	tmp = 0.0;
	if ((z * z) <= 1e+293)
		tmp = (y + (x_m * (x_m / y))) / 2.0;
	else
		tmp = z * ((z * -0.5) / y);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+293], N[(N[(y + N[(x$95$m * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(z * N[(N[(z * -0.5), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 9.9999999999999992e292

   1. Initial program 70.3%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y}\right), \color{blue}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x \cdot x + \left(y \cdot y - z \cdot z\right)}{y}\right), 2\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(y \cdot y - z \cdot z\right) + x \cdot x}{y}\right), 2\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot y - \left(z \cdot z - x \cdot x\right)}{y}\right), 2\right) \]

      6. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot y}{y} - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{y}{y} - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y}{y} \cdot y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(\frac{z \cdot z - x \cdot x}{y}\right)\right), 2\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      15. *-lowering-*.f64 92.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

   3. Simplified 92.3%

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

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(y + \frac{{x}^{2}}{y}\right)}, 2\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(y, \left(\frac{{x}^{2}}{y}\right)\right), 2\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(y, \mathsf{/.f64}\left(\left({x}^{2}\right), y\right)\right), 2\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(y, \mathsf{/.f64}\left(\left(x \cdot x\right), y\right)\right), 2\right) \]

      4. *-lowering-*.f64 73.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), y\right)\right), 2\right) \]

   7. Simplified 73.9%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(y, \left(x \cdot \frac{x}{y}\right)\right), 2\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(y, \left(\frac{x}{y} \cdot x\right)\right), 2\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(y, \mathsf{*.f64}\left(\left(\frac{x}{y}\right), x\right)\right), 2\right) \]

      4. /-lowering-/.f64 80.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(y, \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), x\right)\right), 2\right) \]

   9. Applied egg-rr 80.1%

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

   if 9.9999999999999992e292 < (*.f64 z z)

   1. Initial program 63.3%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y}\right), \color{blue}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x \cdot x + \left(y \cdot y - z \cdot z\right)}{y}\right), 2\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(y \cdot y - z \cdot z\right) + x \cdot x}{y}\right), 2\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot y - \left(z \cdot z - x \cdot x\right)}{y}\right), 2\right) \]

      6. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot y}{y} - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{y}{y} - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y}{y} \cdot y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(\frac{z \cdot z - x \cdot x}{y}\right)\right), 2\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      15. *-lowering-*.f64 67.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

   3. Simplified 67.6%

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

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(\mathsf{neg}\left({z}^{2}\right)\right)\right), \color{blue}{y}\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(-1 \cdot {z}^{2}\right)\right), y\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot -1\right) \cdot {z}^{2}\right), y\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot {z}^{2}\right), y\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left({z}^{2} \cdot \frac{-1}{2}\right), y\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({z}^{2}\right), \frac{-1}{2}\right), y\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(z \cdot z\right), \frac{-1}{2}\right), y\right) \]

      12. *-lowering-*.f64 73.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \frac{-1}{2}\right), y\right) \]

   7. Simplified 73.8%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{z \cdot \frac{-1}{2}}{y}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\left(z \cdot \frac{-1}{2}\right), \color{blue}{y}\right)\right) \]

      5. *-lowering-*.f64 78.2%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \frac{-1}{2}\right), y\right)\right) \]

   9. Applied egg-rr 78.2%

      \[\leadsto \color{blue}{z \cdot \frac{z \cdot -0.5}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+293}:\\ \;\;\;\;\frac{y + x \cdot \frac{x}{y}}{2}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{z \cdot -0.5}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 44.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{+44}:\\ \;\;\;\;\frac{z}{\frac{y}{z \cdot -0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{2}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z)
 :precision binary64
 (if (<= y 1.2e+44) (/ z (/ y (* z -0.5))) (/ y 2.0)))

C

x_m = fabs(x);
double code(double x_m, double y, double z) {
	double tmp;
	if (y <= 1.2e+44) {
		tmp = z / (y / (z * -0.5));
	} else {
		tmp = y / 2.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y, z)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.2d+44) then
        tmp = z / (y / (z * (-0.5d0)))
    else
        tmp = y / 2.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y, double z) {
	double tmp;
	if (y <= 1.2e+44) {
		tmp = z / (y / (z * -0.5));
	} else {
		tmp = y / 2.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z):
	tmp = 0
	if y <= 1.2e+44:
		tmp = z / (y / (z * -0.5))
	else:
		tmp = y / 2.0
	return tmp

Julia

x_m = abs(x)
function code(x_m, y, z)
	tmp = 0.0
	if (y <= 1.2e+44)
		tmp = Float64(z / Float64(y / Float64(z * -0.5)));
	else
		tmp = Float64(y / 2.0);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y, z)
	tmp = 0.0;
	if (y <= 1.2e+44)
		tmp = z / (y / (z * -0.5));
	else
		tmp = y / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_] := If[LessEqual[y, 1.2e+44], N[(z / N[(y / N[(z * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.20000000000000007e44

   1. Initial program 74.2%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y}\right), \color{blue}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x \cdot x + \left(y \cdot y - z \cdot z\right)}{y}\right), 2\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(y \cdot y - z \cdot z\right) + x \cdot x}{y}\right), 2\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot y - \left(z \cdot z - x \cdot x\right)}{y}\right), 2\right) \]

      6. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot y}{y} - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{y}{y} - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y}{y} \cdot y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(\frac{z \cdot z - x \cdot x}{y}\right)\right), 2\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      15. *-lowering-*.f64 87.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

   3. Simplified 87.6%

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

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(\mathsf{neg}\left({z}^{2}\right)\right)\right), \color{blue}{y}\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(-1 \cdot {z}^{2}\right)\right), y\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot -1\right) \cdot {z}^{2}\right), y\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot {z}^{2}\right), y\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left({z}^{2} \cdot \frac{-1}{2}\right), y\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({z}^{2}\right), \frac{-1}{2}\right), y\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(z \cdot z\right), \frac{-1}{2}\right), y\right) \]

      12. *-lowering-*.f64 42.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \frac{-1}{2}\right), y\right) \]

   7. Simplified 42.7%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{z \cdot \frac{-1}{2}}{y}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\left(z \cdot \frac{-1}{2}\right), \color{blue}{y}\right)\right) \]

      5. *-lowering-*.f64 44.3%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \frac{-1}{2}\right), y\right)\right) \]

   9. Applied egg-rr 44.3%

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

       1. clear-num N/A

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

       2. un-div-inv N/A

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

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(z, \color{blue}{\left(\frac{y}{z \cdot \frac{-1}{2}}\right)}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(z, \mathsf{/.f64}\left(y, \color{blue}{\left(z \cdot \frac{-1}{2}\right)}\right)\right) \]

       5. *-lowering-*.f64 44.3%

          \[\leadsto \mathsf{/.f64}\left(z, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(z, \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

   11. Applied egg-rr 44.3%

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

   if 1.20000000000000007e44 < y

   1. Initial program 49.4%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y}\right), \color{blue}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x \cdot x + \left(y \cdot y - z \cdot z\right)}{y}\right), 2\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(y \cdot y - z \cdot z\right) + x \cdot x}{y}\right), 2\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot y - \left(z \cdot z - x \cdot x\right)}{y}\right), 2\right) \]

      6. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot y}{y} - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{y}{y} - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y}{y} \cdot y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(\frac{z \cdot z - x \cdot x}{y}\right)\right), 2\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      15. *-lowering-*.f64 79.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

   3. Simplified 79.0%

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

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{y}, 2\right) \]
   6. Step-by-step derivation

   7. Simplified 78.3%

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

Alternative 8: 44.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{+44}:\\ \;\;\;\;z \cdot \frac{z \cdot -0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{2}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z)
 :precision binary64
 (if (<= y 2.1e+44) (* z (/ (* z -0.5) y)) (/ y 2.0)))

C

x_m = fabs(x);
double code(double x_m, double y, double z) {
	double tmp;
	if (y <= 2.1e+44) {
		tmp = z * ((z * -0.5) / y);
	} else {
		tmp = y / 2.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y, z)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2.1d+44) then
        tmp = z * ((z * (-0.5d0)) / y)
    else
        tmp = y / 2.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y, double z) {
	double tmp;
	if (y <= 2.1e+44) {
		tmp = z * ((z * -0.5) / y);
	} else {
		tmp = y / 2.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z):
	tmp = 0
	if y <= 2.1e+44:
		tmp = z * ((z * -0.5) / y)
	else:
		tmp = y / 2.0
	return tmp

Julia

x_m = abs(x)
function code(x_m, y, z)
	tmp = 0.0
	if (y <= 2.1e+44)
		tmp = Float64(z * Float64(Float64(z * -0.5) / y));
	else
		tmp = Float64(y / 2.0);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y, z)
	tmp = 0.0;
	if (y <= 2.1e+44)
		tmp = z * ((z * -0.5) / y);
	else
		tmp = y / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_] := If[LessEqual[y, 2.1e+44], N[(z * N[(N[(z * -0.5), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(y / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.09999999999999987e44

   1. Initial program 74.2%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y}\right), \color{blue}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x \cdot x + \left(y \cdot y - z \cdot z\right)}{y}\right), 2\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(y \cdot y - z \cdot z\right) + x \cdot x}{y}\right), 2\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot y - \left(z \cdot z - x \cdot x\right)}{y}\right), 2\right) \]

      6. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot y}{y} - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{y}{y} - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y}{y} \cdot y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(\frac{z \cdot z - x \cdot x}{y}\right)\right), 2\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      15. *-lowering-*.f64 87.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

   3. Simplified 87.6%

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

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(\mathsf{neg}\left({z}^{2}\right)\right)\right), \color{blue}{y}\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(-1 \cdot {z}^{2}\right)\right), y\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot -1\right) \cdot {z}^{2}\right), y\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot {z}^{2}\right), y\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left({z}^{2} \cdot \frac{-1}{2}\right), y\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({z}^{2}\right), \frac{-1}{2}\right), y\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(z \cdot z\right), \frac{-1}{2}\right), y\right) \]

      12. *-lowering-*.f64 42.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \frac{-1}{2}\right), y\right) \]

   7. Simplified 42.7%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{z \cdot \frac{-1}{2}}{y}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\left(z \cdot \frac{-1}{2}\right), \color{blue}{y}\right)\right) \]

      5. *-lowering-*.f64 44.3%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \frac{-1}{2}\right), y\right)\right) \]

   9. Applied egg-rr 44.3%

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

   if 2.09999999999999987e44 < y

   1. Initial program 49.4%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y}\right), \color{blue}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x \cdot x + \left(y \cdot y - z \cdot z\right)}{y}\right), 2\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(y \cdot y - z \cdot z\right) + x \cdot x}{y}\right), 2\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot y - \left(z \cdot z - x \cdot x\right)}{y}\right), 2\right) \]

      6. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot y}{y} - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{y}{y} - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y}{y} \cdot y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(\frac{z \cdot z - x \cdot x}{y}\right)\right), 2\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      15. *-lowering-*.f64 79.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

   3. Simplified 79.0%

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

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{y}, 2\right) \]
   6. Step-by-step derivation

   7. Simplified 78.3%

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

Alternative 9: 34.8% accurate, 5.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{y}{2} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z) :precision binary64 (/ y 2.0))

C

x_m = fabs(x);
double code(double x_m, double y, double z) {
	return y / 2.0;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y, z)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = y / 2.0d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y, double z) {
	return y / 2.0;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z):
	return y / 2.0

Julia

x_m = abs(x)
function code(x_m, y, z)
	return Float64(y / 2.0)
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, y, z)
	tmp = y / 2.0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_] := N[(y / 2.0), $MachinePrecision]

Derivation

1. Initial program 68.4%

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

   1. associate-/r* N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y}\right), \color{blue}{2}\right) \]

   3. associate--l+ N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{x \cdot x + \left(y \cdot y - z \cdot z\right)}{y}\right), 2\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(y \cdot y - z \cdot z\right) + x \cdot x}{y}\right), 2\right) \]

   5. associate-+l- N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot y - \left(z \cdot z - x \cdot x\right)}{y}\right), 2\right) \]

   6. div-sub N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot y}{y} - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

   7. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{y}{y} - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{y}{y} \cdot y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

   9. *-inverses N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

   10. *-lft-identity N/A

       \[\leadsto \mathsf{/.f64}\left(\left(y - \frac{z \cdot z - x \cdot x}{y}\right), 2\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(\frac{z \cdot z - x \cdot x}{y}\right)\right), 2\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

   13. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

   15. *-lowering-*.f64 85.6%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

3. Simplified 85.6%

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

5. Taylor expanded in y around inf

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{y}, 2\right) \]
6. Step-by-step derivation

7. Simplified 37.9%

   \[\leadsto \frac{\color{blue}{y}}{2} \]
8. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (y * 0.5d0) - (((0.5d0 / y) * (z + x)) * (z - x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024158 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* y 1/2) (* (* (/ 1/2 y) (+ z x)) (- z x))))

  (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))