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

Percentage Accurate: 69.5% → 99.9%

Time: 11.6s

Alternatives: 11

Speedup: 1.2×

• 41.5% 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: 69.5% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.8%

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

   1. clear-num N/A

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

   2. associate-/l* N/A

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

   3. associate-/r* N/A

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

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

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

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

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

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

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

   7. associate--l+ N/A

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

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

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

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

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

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

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

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

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

   12. *-lowering-*.f64 70.7%

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

4. Applied egg-rr 70.7%

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

5. Taylor expanded in x around 0

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

   1. distribute-lft-out N/A

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

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

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

   3. div-sub N/A

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

   4. sub-neg N/A

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

   5. associate-+l+ N/A

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

   6. unpow2 N/A

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

   7. associate-/l* N/A

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

   8. *-inverses N/A

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

   9. *-rgt-identity N/A

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

   10. +-commutative N/A

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

   11. sub-neg N/A

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

   12. div-sub N/A

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

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

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

   14. unpow2 N/A

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

   15. unpow2 N/A

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

   16. difference-of-squares N/A

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

   17. associate-/l* N/A

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

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

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

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

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

   20. /-lowering-/.f64 N/A

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

   21. --lowering--.f64 99.9%

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

7. Simplified 99.9%

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

Alternative 2: 90.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 10^{-117}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x\_m}{\frac{y}{x\_m}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + z\_m \cdot \frac{x\_m - z\_m}{y}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
(FPCore (x_m y z_m)
 :precision binary64
 (if (<= (* z_m z_m) 1e-117)
   (* 0.5 (+ y (/ x_m (/ y x_m))))
   (* 0.5 (+ y (* z_m (/ (- x_m z_m) y))))))

C

x_m = fabs(x);
z_m = fabs(z);
double code(double x_m, double y, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 1e-117) {
		tmp = 0.5 * (y + (x_m / (y / x_m)));
	} else {
		tmp = 0.5 * (y + (z_m * ((x_m - z_m) / y)));
	}
	return tmp;
}

Fortran

x_m = abs(x)
z_m = abs(z)
real(8) function code(x_m, y, z_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if ((z_m * z_m) <= 1d-117) then
        tmp = 0.5d0 * (y + (x_m / (y / x_m)))
    else
        tmp = 0.5d0 * (y + (z_m * ((x_m - z_m) / y)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
public static double code(double x_m, double y, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 1e-117) {
		tmp = 0.5 * (y + (x_m / (y / x_m)));
	} else {
		tmp = 0.5 * (y + (z_m * ((x_m - z_m) / y)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
def code(x_m, y, z_m):
	tmp = 0
	if (z_m * z_m) <= 1e-117:
		tmp = 0.5 * (y + (x_m / (y / x_m)))
	else:
		tmp = 0.5 * (y + (z_m * ((x_m - z_m) / y)))
	return tmp

Julia

x_m = abs(x)
z_m = abs(z)
function code(x_m, y, z_m)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 1e-117)
		tmp = Float64(0.5 * Float64(y + Float64(x_m / Float64(y / x_m))));
	else
		tmp = Float64(0.5 * Float64(y + Float64(z_m * Float64(Float64(x_m - z_m) / y))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
z_m = abs(z);
function tmp_2 = code(x_m, y, z_m)
	tmp = 0.0;
	if ((z_m * z_m) <= 1e-117)
		tmp = 0.5 * (y + (x_m / (y / x_m)));
	else
		tmp = 0.5 * (y + (z_m * ((x_m - z_m) / y)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 1e-117], N[(0.5 * N[(y + N[(x$95$m / N[(y / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y + N[(z$95$m * N[(N[(x$95$m - z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.00000000000000003e-117

   1. Initial program 76.3%

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

   3. Taylor expanded in z around 0

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

      1. *-lft-identity N/A

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

      2. *-inverses N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-lft-in N/A

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

      7. +-commutative N/A

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

      8. associate-*l/ N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. *-inverses N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. *-commutative N/A

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

      16. associate-/r/ N/A

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

      17. unpow2 N/A

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

      18. associate-/l* N/A

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

      19. *-inverses N/A

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

      20. *-rgt-identity N/A

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

   5. Simplified 91.9%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 99.9%

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

   9. Applied egg-rr 99.9%

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

   if 1.00000000000000003e-117 < (*.f64 z z)

   1. Initial program 67.1%

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

      1. clear-num N/A

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

      2. associate-/l* N/A

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

      3. associate-/r* N/A

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

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

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

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

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

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

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

      7. associate--l+ N/A

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

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

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

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

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

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

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

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

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

      12. *-lowering-*.f64 67.0%

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

   4. Applied egg-rr 67.0%

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

   5. Taylor expanded in x around 0

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

      1. distribute-lft-out N/A

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

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

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. associate-+l+ N/A

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

      6. unpow2 N/A

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

      7. associate-/l* N/A

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

      8. *-inverses N/A

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

      9. *-rgt-identity N/A

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

      10. +-commutative N/A

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

      11. sub-neg N/A

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

      12. div-sub N/A

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

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

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

      14. unpow2 N/A

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

      15. unpow2 N/A

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

      16. difference-of-squares N/A

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

      17. associate-/l* N/A

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

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

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

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

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

      20. /-lowering-/.f64 N/A

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

      21. --lowering--.f64 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 78.3%

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

Alternative 3: 43.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-53}:\\ \;\;\;\;\frac{x\_m}{y} \cdot \left(0.5 \cdot x\_m\right)\\ \mathbf{elif}\;y \leq 0.00018:\\ \;\;\;\;\frac{z\_m}{\frac{\frac{y}{z\_m}}{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
(FPCore (x_m y z_m)
 :precision binary64
 (if (<= y 4e-53)
   (* (/ x_m y) (* 0.5 x_m))
   (if (<= y 0.00018) (/ z_m (/ (/ y z_m) -0.5)) (* 0.5 y))))

C

x_m = fabs(x);
z_m = fabs(z);
double code(double x_m, double y, double z_m) {
	double tmp;
	if (y <= 4e-53) {
		tmp = (x_m / y) * (0.5 * x_m);
	} else if (y <= 0.00018) {
		tmp = z_m / ((y / z_m) / -0.5);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

x_m = abs(x)
z_m = abs(z)
real(8) function code(x_m, y, z_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y <= 4d-53) then
        tmp = (x_m / y) * (0.5d0 * x_m)
    else if (y <= 0.00018d0) then
        tmp = z_m / ((y / z_m) / (-0.5d0))
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
public static double code(double x_m, double y, double z_m) {
	double tmp;
	if (y <= 4e-53) {
		tmp = (x_m / y) * (0.5 * x_m);
	} else if (y <= 0.00018) {
		tmp = z_m / ((y / z_m) / -0.5);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
def code(x_m, y, z_m):
	tmp = 0
	if y <= 4e-53:
		tmp = (x_m / y) * (0.5 * x_m)
	elif y <= 0.00018:
		tmp = z_m / ((y / z_m) / -0.5)
	else:
		tmp = 0.5 * y
	return tmp

Julia

x_m = abs(x)
z_m = abs(z)
function code(x_m, y, z_m)
	tmp = 0.0
	if (y <= 4e-53)
		tmp = Float64(Float64(x_m / y) * Float64(0.5 * x_m));
	elseif (y <= 0.00018)
		tmp = Float64(z_m / Float64(Float64(y / z_m) / -0.5));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

x_m = abs(x);
z_m = abs(z);
function tmp_2 = code(x_m, y, z_m)
	tmp = 0.0;
	if (y <= 4e-53)
		tmp = (x_m / y) * (0.5 * x_m);
	elseif (y <= 0.00018)
		tmp = z_m / ((y / z_m) / -0.5);
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := If[LessEqual[y, 4e-53], N[(N[(x$95$m / y), $MachinePrecision] * N[(0.5 * x$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00018], N[(z$95$m / N[(N[(y / z$95$m), $MachinePrecision] / -0.5), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.00000000000000012e-53

   1. Initial program 76.6%

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

      1. clear-num N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-/r* N/A

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

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

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

      6. metadata-eval N/A

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

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

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

      8. associate--l+ N/A

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

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

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

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

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

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

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

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

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

      13. *-lowering-*.f64 76.6%

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

   4. Applied egg-rr 76.6%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{y}{{x}^{2}}\right)}\right) \]
   6. Step-by-step derivation

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 43.0%

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

   7. Simplified 43.0%

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

      1. metadata-eval N/A

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

      2. associate-*l/ N/A

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

      3. associate-/r* N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. *-lowering-*.f64 46.0%

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

   9. Applied egg-rr 46.0%

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

   if 4.00000000000000012e-53 < y < 1.80000000000000011e-4

   1. Initial program 92.2%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{{z}^{2}}{y}} \]
   4. 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 46.7%

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

   5. Simplified 46.7%

      \[\leadsto \color{blue}{\frac{\left(z \cdot z\right) \cdot -0.5}{y}} \]
   6. 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 46.7%

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

   7. Applied egg-rr 46.7%

      \[\leadsto \color{blue}{z \cdot \frac{z \cdot -0.5}{y}} \]
   8. 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. associate-/r* N/A

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

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

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

      6. /-lowering-/.f64 46.7%

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

   9. Applied egg-rr 46.7%

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

   if 1.80000000000000011e-4 < y

   1. Initial program 49.3%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 56.4%

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

   5. Simplified 56.4%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-53}:\\ \;\;\;\;\frac{x}{y} \cdot \left(0.5 \cdot x\right)\\ \mathbf{elif}\;y \leq 0.00018:\\ \;\;\;\;\frac{z}{\frac{\frac{y}{z}}{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 43.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{-53}:\\ \;\;\;\;\frac{x\_m}{y} \cdot \left(0.5 \cdot x\_m\right)\\ \mathbf{elif}\;y \leq 0.00023:\\ \;\;\;\;z\_m \cdot \frac{z\_m \cdot -0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
(FPCore (x_m y z_m)
 :precision binary64
 (if (<= y 4.2e-53)
   (* (/ x_m y) (* 0.5 x_m))
   (if (<= y 0.00023) (* z_m (/ (* z_m -0.5) y)) (* 0.5 y))))

C

x_m = fabs(x);
z_m = fabs(z);
double code(double x_m, double y, double z_m) {
	double tmp;
	if (y <= 4.2e-53) {
		tmp = (x_m / y) * (0.5 * x_m);
	} else if (y <= 0.00023) {
		tmp = z_m * ((z_m * -0.5) / y);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

x_m = abs(x)
z_m = abs(z)
real(8) function code(x_m, y, z_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y <= 4.2d-53) then
        tmp = (x_m / y) * (0.5d0 * x_m)
    else if (y <= 0.00023d0) then
        tmp = z_m * ((z_m * (-0.5d0)) / y)
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
public static double code(double x_m, double y, double z_m) {
	double tmp;
	if (y <= 4.2e-53) {
		tmp = (x_m / y) * (0.5 * x_m);
	} else if (y <= 0.00023) {
		tmp = z_m * ((z_m * -0.5) / y);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
def code(x_m, y, z_m):
	tmp = 0
	if y <= 4.2e-53:
		tmp = (x_m / y) * (0.5 * x_m)
	elif y <= 0.00023:
		tmp = z_m * ((z_m * -0.5) / y)
	else:
		tmp = 0.5 * y
	return tmp

Julia

x_m = abs(x)
z_m = abs(z)
function code(x_m, y, z_m)
	tmp = 0.0
	if (y <= 4.2e-53)
		tmp = Float64(Float64(x_m / y) * Float64(0.5 * x_m));
	elseif (y <= 0.00023)
		tmp = Float64(z_m * Float64(Float64(z_m * -0.5) / y));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

x_m = abs(x);
z_m = abs(z);
function tmp_2 = code(x_m, y, z_m)
	tmp = 0.0;
	if (y <= 4.2e-53)
		tmp = (x_m / y) * (0.5 * x_m);
	elseif (y <= 0.00023)
		tmp = z_m * ((z_m * -0.5) / y);
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := If[LessEqual[y, 4.2e-53], N[(N[(x$95$m / y), $MachinePrecision] * N[(0.5 * x$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00023], N[(z$95$m * N[(N[(z$95$m * -0.5), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.19999999999999955e-53

   1. Initial program 76.6%

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

      1. clear-num N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-/r* N/A

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

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

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

      6. metadata-eval N/A

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

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

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

      8. associate--l+ N/A

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

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

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

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

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

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

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

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

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

      13. *-lowering-*.f64 76.6%

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

   4. Applied egg-rr 76.6%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{y}{{x}^{2}}\right)}\right) \]
   6. Step-by-step derivation

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 43.0%

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

   7. Simplified 43.0%

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

      1. metadata-eval N/A

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

      2. associate-*l/ N/A

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

      3. associate-/r* N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. *-lowering-*.f64 46.0%

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

   9. Applied egg-rr 46.0%

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

   if 4.19999999999999955e-53 < y < 2.3000000000000001e-4

   1. Initial program 92.2%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{{z}^{2}}{y}} \]
   4. 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 46.7%

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

   5. Simplified 46.7%

      \[\leadsto \color{blue}{\frac{\left(z \cdot z\right) \cdot -0.5}{y}} \]
   6. 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 46.7%

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

   7. Applied egg-rr 46.7%

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

   if 2.3000000000000001e-4 < y

   1. Initial program 49.3%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 56.4%

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

   5. Simplified 56.4%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{-53}:\\ \;\;\;\;\frac{x}{y} \cdot \left(0.5 \cdot x\right)\\ \mathbf{elif}\;y \leq 0.00023:\\ \;\;\;\;z \cdot \frac{z \cdot -0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 43.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-53}:\\ \;\;\;\;x\_m \cdot \frac{0.5}{\frac{y}{x\_m}}\\ \mathbf{elif}\;y \leq 0.00058:\\ \;\;\;\;z\_m \cdot \frac{z\_m \cdot -0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
(FPCore (x_m y z_m)
 :precision binary64
 (if (<= y 3.8e-53)
   (* x_m (/ 0.5 (/ y x_m)))
   (if (<= y 0.00058) (* z_m (/ (* z_m -0.5) y)) (* 0.5 y))))

C

x_m = fabs(x);
z_m = fabs(z);
double code(double x_m, double y, double z_m) {
	double tmp;
	if (y <= 3.8e-53) {
		tmp = x_m * (0.5 / (y / x_m));
	} else if (y <= 0.00058) {
		tmp = z_m * ((z_m * -0.5) / y);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

x_m = abs(x)
z_m = abs(z)
real(8) function code(x_m, y, z_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y <= 3.8d-53) then
        tmp = x_m * (0.5d0 / (y / x_m))
    else if (y <= 0.00058d0) then
        tmp = z_m * ((z_m * (-0.5d0)) / y)
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
public static double code(double x_m, double y, double z_m) {
	double tmp;
	if (y <= 3.8e-53) {
		tmp = x_m * (0.5 / (y / x_m));
	} else if (y <= 0.00058) {
		tmp = z_m * ((z_m * -0.5) / y);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
def code(x_m, y, z_m):
	tmp = 0
	if y <= 3.8e-53:
		tmp = x_m * (0.5 / (y / x_m))
	elif y <= 0.00058:
		tmp = z_m * ((z_m * -0.5) / y)
	else:
		tmp = 0.5 * y
	return tmp

Julia

x_m = abs(x)
z_m = abs(z)
function code(x_m, y, z_m)
	tmp = 0.0
	if (y <= 3.8e-53)
		tmp = Float64(x_m * Float64(0.5 / Float64(y / x_m)));
	elseif (y <= 0.00058)
		tmp = Float64(z_m * Float64(Float64(z_m * -0.5) / y));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

x_m = abs(x);
z_m = abs(z);
function tmp_2 = code(x_m, y, z_m)
	tmp = 0.0;
	if (y <= 3.8e-53)
		tmp = x_m * (0.5 / (y / x_m));
	elseif (y <= 0.00058)
		tmp = z_m * ((z_m * -0.5) / y);
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := If[LessEqual[y, 3.8e-53], N[(x$95$m * N[(0.5 / N[(y / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00058], N[(z$95$m * N[(N[(z$95$m * -0.5), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.7999999999999998e-53

   1. Initial program 76.6%

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

      1. clear-num N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-/r* N/A

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

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

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

      6. metadata-eval N/A

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

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

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

      8. associate--l+ N/A

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

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

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

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

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

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

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

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

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

      13. *-lowering-*.f64 76.6%

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

   4. Applied egg-rr 76.6%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{y}{{x}^{2}}\right)}\right) \]
   6. Step-by-step derivation

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 43.0%

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

   7. Simplified 43.0%

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

      1. associate-/r* N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

      5. /-lowering-/.f64 46.0%

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

   9. Applied egg-rr 46.0%

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

   if 3.7999999999999998e-53 < y < 5.8e-4

   1. Initial program 92.2%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{{z}^{2}}{y}} \]
   4. 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 46.7%

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

   5. Simplified 46.7%

      \[\leadsto \color{blue}{\frac{\left(z \cdot z\right) \cdot -0.5}{y}} \]
   6. 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 46.7%

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

   7. Applied egg-rr 46.7%

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

   if 5.8e-4 < y

   1. Initial program 49.3%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 56.4%

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

   5. Simplified 56.4%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-53}:\\ \;\;\;\;x \cdot \frac{0.5}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq 0.00058:\\ \;\;\;\;z \cdot \frac{z \cdot -0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 42.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{-53}:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot \frac{0.5}{y}\\ \mathbf{elif}\;y \leq 0.000125:\\ \;\;\;\;z\_m \cdot \frac{z\_m \cdot -0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
(FPCore (x_m y z_m)
 :precision binary64
 (if (<= y 1e-53)
   (* (* x_m x_m) (/ 0.5 y))
   (if (<= y 0.000125) (* z_m (/ (* z_m -0.5) y)) (* 0.5 y))))

C

x_m = fabs(x);
z_m = fabs(z);
double code(double x_m, double y, double z_m) {
	double tmp;
	if (y <= 1e-53) {
		tmp = (x_m * x_m) * (0.5 / y);
	} else if (y <= 0.000125) {
		tmp = z_m * ((z_m * -0.5) / y);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

x_m = abs(x)
z_m = abs(z)
real(8) function code(x_m, y, z_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y <= 1d-53) then
        tmp = (x_m * x_m) * (0.5d0 / y)
    else if (y <= 0.000125d0) then
        tmp = z_m * ((z_m * (-0.5d0)) / y)
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
public static double code(double x_m, double y, double z_m) {
	double tmp;
	if (y <= 1e-53) {
		tmp = (x_m * x_m) * (0.5 / y);
	} else if (y <= 0.000125) {
		tmp = z_m * ((z_m * -0.5) / y);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
def code(x_m, y, z_m):
	tmp = 0
	if y <= 1e-53:
		tmp = (x_m * x_m) * (0.5 / y)
	elif y <= 0.000125:
		tmp = z_m * ((z_m * -0.5) / y)
	else:
		tmp = 0.5 * y
	return tmp

Julia

x_m = abs(x)
z_m = abs(z)
function code(x_m, y, z_m)
	tmp = 0.0
	if (y <= 1e-53)
		tmp = Float64(Float64(x_m * x_m) * Float64(0.5 / y));
	elseif (y <= 0.000125)
		tmp = Float64(z_m * Float64(Float64(z_m * -0.5) / y));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

x_m = abs(x);
z_m = abs(z);
function tmp_2 = code(x_m, y, z_m)
	tmp = 0.0;
	if (y <= 1e-53)
		tmp = (x_m * x_m) * (0.5 / y);
	elseif (y <= 0.000125)
		tmp = z_m * ((z_m * -0.5) / y);
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := If[LessEqual[y, 1e-53], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.5 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.000125], N[(z$95$m * N[(N[(z$95$m * -0.5), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.00000000000000003e-53

   1. Initial program 76.6%

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

      1. clear-num N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-/r* N/A

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

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

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

      6. metadata-eval N/A

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

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

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

      8. associate--l+ N/A

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

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

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

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

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

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

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

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

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

      13. *-lowering-*.f64 76.6%

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

   4. Applied egg-rr 76.6%

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

   5. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. /-lowering-/.f64 62.8%

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

   7. Simplified 62.8%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({x}^{2}\right)}, \mathsf{/.f64}\left(\frac{1}{2}, y\right)\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

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

      2. *-lowering-*.f64 43.0%

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

   10. Simplified 43.0%

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

   if 1.00000000000000003e-53 < y < 1.25e-4

   1. Initial program 92.2%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{{z}^{2}}{y}} \]
   4. 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 46.7%

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

   5. Simplified 46.7%

      \[\leadsto \color{blue}{\frac{\left(z \cdot z\right) \cdot -0.5}{y}} \]
   6. 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 46.7%

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

   7. Applied egg-rr 46.7%

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

   if 1.25e-4 < y

   1. Initial program 49.3%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 56.4%

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

   5. Simplified 56.4%

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

Alternative 7: 81.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \cdot x\_m \leq 10^{-69}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z\_m \cdot z\_m}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + x\_m \cdot \frac{x\_m}{y}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
(FPCore (x_m y z_m)
 :precision binary64
 (if (<= (* x_m x_m) 1e-69)
   (* 0.5 (- y (/ (* z_m z_m) y)))
   (* 0.5 (+ y (* x_m (/ x_m y))))))

C

x_m = fabs(x);
z_m = fabs(z);
double code(double x_m, double y, double z_m) {
	double tmp;
	if ((x_m * x_m) <= 1e-69) {
		tmp = 0.5 * (y - ((z_m * z_m) / y));
	} else {
		tmp = 0.5 * (y + (x_m * (x_m / y)));
	}
	return tmp;
}

Fortran

x_m = abs(x)
z_m = abs(z)
real(8) function code(x_m, y, z_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if ((x_m * x_m) <= 1d-69) then
        tmp = 0.5d0 * (y - ((z_m * z_m) / y))
    else
        tmp = 0.5d0 * (y + (x_m * (x_m / y)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
public static double code(double x_m, double y, double z_m) {
	double tmp;
	if ((x_m * x_m) <= 1e-69) {
		tmp = 0.5 * (y - ((z_m * z_m) / y));
	} else {
		tmp = 0.5 * (y + (x_m * (x_m / y)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
def code(x_m, y, z_m):
	tmp = 0
	if (x_m * x_m) <= 1e-69:
		tmp = 0.5 * (y - ((z_m * z_m) / y))
	else:
		tmp = 0.5 * (y + (x_m * (x_m / y)))
	return tmp

Julia

x_m = abs(x)
z_m = abs(z)
function code(x_m, y, z_m)
	tmp = 0.0
	if (Float64(x_m * x_m) <= 1e-69)
		tmp = Float64(0.5 * Float64(y - Float64(Float64(z_m * z_m) / y)));
	else
		tmp = Float64(0.5 * Float64(y + Float64(x_m * Float64(x_m / y))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
z_m = abs(z);
function tmp_2 = code(x_m, y, z_m)
	tmp = 0.0;
	if ((x_m * x_m) <= 1e-69)
		tmp = 0.5 * (y - ((z_m * z_m) / y));
	else
		tmp = 0.5 * (y + (x_m * (x_m / y)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := If[LessEqual[N[(x$95$m * x$95$m), $MachinePrecision], 1e-69], N[(0.5 * N[(y - N[(N[(z$95$m * z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y + N[(x$95$m * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 9.9999999999999996e-70

   1. Initial program 75.6%

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

   3. Taylor expanded in x around 0

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. *-rgt-identity N/A

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

      8. +-commutative N/A

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

      9. sub-neg N/A

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

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

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

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

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 86.1%

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

   5. Simplified 86.1%

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

   if 9.9999999999999996e-70 < (*.f64 x x)

   1. Initial program 67.2%

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

   3. Taylor expanded in z around 0

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

      1. *-lft-identity N/A

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

      2. *-inverses N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-lft-in N/A

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

      7. +-commutative N/A

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

      8. associate-*l/ N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. *-inverses N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. *-commutative N/A

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

      16. associate-/r/ N/A

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

      17. unpow2 N/A

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

      18. associate-/l* N/A

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

      19. *-inverses N/A

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

      20. *-rgt-identity N/A

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

   5. Simplified 76.8%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 84.9%

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

   7. Applied egg-rr 84.9%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-69}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z \cdot z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 1.6 \cdot 10^{+162}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x\_m}{\frac{y}{x\_m}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z\_m}{\frac{\frac{y}{z\_m}}{-0.5}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
(FPCore (x_m y z_m)
 :precision binary64
 (if (<= z_m 1.6e+162)
   (* 0.5 (+ y (/ x_m (/ y x_m))))
   (/ z_m (/ (/ y z_m) -0.5))))

C

x_m = fabs(x);
z_m = fabs(z);
double code(double x_m, double y, double z_m) {
	double tmp;
	if (z_m <= 1.6e+162) {
		tmp = 0.5 * (y + (x_m / (y / x_m)));
	} else {
		tmp = z_m / ((y / z_m) / -0.5);
	}
	return tmp;
}

Fortran

x_m = abs(x)
z_m = abs(z)
real(8) function code(x_m, y, z_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 1.6d+162) then
        tmp = 0.5d0 * (y + (x_m / (y / x_m)))
    else
        tmp = z_m / ((y / z_m) / (-0.5d0))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
public static double code(double x_m, double y, double z_m) {
	double tmp;
	if (z_m <= 1.6e+162) {
		tmp = 0.5 * (y + (x_m / (y / x_m)));
	} else {
		tmp = z_m / ((y / z_m) / -0.5);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
def code(x_m, y, z_m):
	tmp = 0
	if z_m <= 1.6e+162:
		tmp = 0.5 * (y + (x_m / (y / x_m)))
	else:
		tmp = z_m / ((y / z_m) / -0.5)
	return tmp

Julia

x_m = abs(x)
z_m = abs(z)
function code(x_m, y, z_m)
	tmp = 0.0
	if (z_m <= 1.6e+162)
		tmp = Float64(0.5 * Float64(y + Float64(x_m / Float64(y / x_m))));
	else
		tmp = Float64(z_m / Float64(Float64(y / z_m) / -0.5));
	end
	return tmp
end

MATLAB

x_m = abs(x);
z_m = abs(z);
function tmp_2 = code(x_m, y, z_m)
	tmp = 0.0;
	if (z_m <= 1.6e+162)
		tmp = 0.5 * (y + (x_m / (y / x_m)));
	else
		tmp = z_m / ((y / z_m) / -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := If[LessEqual[z$95$m, 1.6e+162], N[(0.5 * N[(y + N[(x$95$m / N[(y / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z$95$m / N[(N[(y / z$95$m), $MachinePrecision] / -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.6000000000000001e162

   1. Initial program 73.5%

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

   3. Taylor expanded in z around 0

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

      1. *-lft-identity N/A

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

      2. *-inverses N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-lft-in N/A

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

      7. +-commutative N/A

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

      8. associate-*l/ N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. *-inverses N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. *-commutative N/A

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

      16. associate-/r/ N/A

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

      17. unpow2 N/A

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

      18. associate-/l* N/A

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

      19. *-inverses N/A

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

      20. *-rgt-identity N/A

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

   5. Simplified 74.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 80.3%

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

   7. Applied egg-rr 80.3%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 80.3%

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

   9. Applied egg-rr 80.3%

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

   if 1.6000000000000001e162 < z

   1. Initial program 49.7%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{{z}^{2}}{y}} \]
   4. 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 67.4%

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

   5. Simplified 67.4%

      \[\leadsto \color{blue}{\frac{\left(z \cdot z\right) \cdot -0.5}{y}} \]
   6. 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 89.6%

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

   7. Applied egg-rr 89.6%

      \[\leadsto \color{blue}{z \cdot \frac{z \cdot -0.5}{y}} \]
   8. 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. associate-/r* N/A

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

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

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

      6. /-lowering-/.f64 89.7%

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

   9. Applied egg-rr 89.7%

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

Alternative 9: 80.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 3.5 \cdot 10^{+162}:\\ \;\;\;\;0.5 \cdot \left(y + x\_m \cdot \frac{x\_m}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z\_m}{\frac{\frac{y}{z\_m}}{-0.5}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
(FPCore (x_m y z_m)
 :precision binary64
 (if (<= z_m 3.5e+162)
   (* 0.5 (+ y (* x_m (/ x_m y))))
   (/ z_m (/ (/ y z_m) -0.5))))

C

x_m = fabs(x);
z_m = fabs(z);
double code(double x_m, double y, double z_m) {
	double tmp;
	if (z_m <= 3.5e+162) {
		tmp = 0.5 * (y + (x_m * (x_m / y)));
	} else {
		tmp = z_m / ((y / z_m) / -0.5);
	}
	return tmp;
}

Fortran

x_m = abs(x)
z_m = abs(z)
real(8) function code(x_m, y, z_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 3.5d+162) then
        tmp = 0.5d0 * (y + (x_m * (x_m / y)))
    else
        tmp = z_m / ((y / z_m) / (-0.5d0))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
public static double code(double x_m, double y, double z_m) {
	double tmp;
	if (z_m <= 3.5e+162) {
		tmp = 0.5 * (y + (x_m * (x_m / y)));
	} else {
		tmp = z_m / ((y / z_m) / -0.5);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
def code(x_m, y, z_m):
	tmp = 0
	if z_m <= 3.5e+162:
		tmp = 0.5 * (y + (x_m * (x_m / y)))
	else:
		tmp = z_m / ((y / z_m) / -0.5)
	return tmp

Julia

x_m = abs(x)
z_m = abs(z)
function code(x_m, y, z_m)
	tmp = 0.0
	if (z_m <= 3.5e+162)
		tmp = Float64(0.5 * Float64(y + Float64(x_m * Float64(x_m / y))));
	else
		tmp = Float64(z_m / Float64(Float64(y / z_m) / -0.5));
	end
	return tmp
end

MATLAB

x_m = abs(x);
z_m = abs(z);
function tmp_2 = code(x_m, y, z_m)
	tmp = 0.0;
	if (z_m <= 3.5e+162)
		tmp = 0.5 * (y + (x_m * (x_m / y)));
	else
		tmp = z_m / ((y / z_m) / -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := If[LessEqual[z$95$m, 3.5e+162], N[(0.5 * N[(y + N[(x$95$m * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z$95$m / N[(N[(y / z$95$m), $MachinePrecision] / -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.50000000000000018e162

   1. Initial program 73.5%

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

   3. Taylor expanded in z around 0

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

      1. *-lft-identity N/A

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

      2. *-inverses N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-lft-in N/A

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

      7. +-commutative N/A

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

      8. associate-*l/ N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. *-inverses N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. *-commutative N/A

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

      16. associate-/r/ N/A

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

      17. unpow2 N/A

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

      18. associate-/l* N/A

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

      19. *-inverses N/A

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

      20. *-rgt-identity N/A

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

   5. Simplified 74.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 80.3%

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

   7. Applied egg-rr 80.3%

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

   if 3.50000000000000018e162 < z

   1. Initial program 49.7%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{{z}^{2}}{y}} \]
   4. 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 67.4%

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

   5. Simplified 67.4%

      \[\leadsto \color{blue}{\frac{\left(z \cdot z\right) \cdot -0.5}{y}} \]
   6. 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 89.6%

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

   7. Applied egg-rr 89.6%

      \[\leadsto \color{blue}{z \cdot \frac{z \cdot -0.5}{y}} \]
   8. 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. associate-/r* N/A

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

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

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

      6. /-lowering-/.f64 89.7%

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

   9. Applied egg-rr 89.7%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.5 \cdot 10^{+162}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{\frac{y}{z}}{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 2.4 \cdot 10^{+125}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\_m \cdot \frac{z\_m \cdot -0.5}{y}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
(FPCore (x_m y z_m)
 :precision binary64
 (if (<= z_m 2.4e+125) (* 0.5 y) (* z_m (/ (* z_m -0.5) y))))

C

x_m = fabs(x);
z_m = fabs(z);
double code(double x_m, double y, double z_m) {
	double tmp;
	if (z_m <= 2.4e+125) {
		tmp = 0.5 * y;
	} else {
		tmp = z_m * ((z_m * -0.5) / y);
	}
	return tmp;
}

Fortran

x_m = abs(x)
z_m = abs(z)
real(8) function code(x_m, y, z_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 2.4d+125) then
        tmp = 0.5d0 * y
    else
        tmp = z_m * ((z_m * (-0.5d0)) / y)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
public static double code(double x_m, double y, double z_m) {
	double tmp;
	if (z_m <= 2.4e+125) {
		tmp = 0.5 * y;
	} else {
		tmp = z_m * ((z_m * -0.5) / y);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
def code(x_m, y, z_m):
	tmp = 0
	if z_m <= 2.4e+125:
		tmp = 0.5 * y
	else:
		tmp = z_m * ((z_m * -0.5) / y)
	return tmp

Julia

x_m = abs(x)
z_m = abs(z)
function code(x_m, y, z_m)
	tmp = 0.0
	if (z_m <= 2.4e+125)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(z_m * Float64(Float64(z_m * -0.5) / y));
	end
	return tmp
end

MATLAB

x_m = abs(x);
z_m = abs(z);
function tmp_2 = code(x_m, y, z_m)
	tmp = 0.0;
	if (z_m <= 2.4e+125)
		tmp = 0.5 * y;
	else
		tmp = z_m * ((z_m * -0.5) / y);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := If[LessEqual[z$95$m, 2.4e+125], N[(0.5 * y), $MachinePrecision], N[(z$95$m * N[(N[(z$95$m * -0.5), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.4e125

   1. Initial program 73.3%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 39.5%

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

   5. Simplified 39.5%

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

   if 2.4e125 < z

   1. Initial program 54.4%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{{z}^{2}}{y}} \]
   4. 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 60.7%

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

   5. Simplified 60.7%

      \[\leadsto \color{blue}{\frac{\left(z \cdot z\right) \cdot -0.5}{y}} \]
   6. 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 82.3%

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

   7. Applied egg-rr 82.3%

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

Alternative 11: 34.8% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

x_m = abs(x)
z_m = abs(z)
real(8) function code(x_m, y, z_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    code = 0.5d0 * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.8%

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

3. Taylor expanded in y around inf

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

   1. *-lowering-*.f64 35.2%

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

5. Simplified 35.2%

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

Developer Target 1: 99.8% 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 2024192 
(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)))