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

Percentage Accurate: 68.6% → 92.3%

Time: 9.9s

Alternatives: 9

Speedup: 0.8×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.6% 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: 92.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right) + 0.5 \cdot \left(x \cdot \frac{x}{y}\right)\\ \mathbf{if}\;x \leq 6 \cdot 10^{+88}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+139}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y, \mathsf{fma}\left(x, x, z \cdot \left(-z\right)\right)\right)}{y \cdot 2}\\ \mathbf{elif}\;x \leq 8.4 \cdot 10^{+214}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (* 0.5 (- y (/ z (/ y z)))) (* 0.5 (* x (/ x y))))))
   (if (<= x 6e+88)
     t_0
     (if (<= x 3.5e+139)
       (/ (fma y y (fma x x (* z (- z)))) (* y 2.0))
       (if (<= x 8.4e+214) t_0 (* (/ x y) (/ x 2.0)))))))

C

x = abs(x);
double code(double x, double y, double z) {
	double t_0 = (0.5 * (y - (z / (y / z)))) + (0.5 * (x * (x / y)));
	double tmp;
	if (x <= 6e+88) {
		tmp = t_0;
	} else if (x <= 3.5e+139) {
		tmp = fma(y, y, fma(x, x, (z * -z))) / (y * 2.0);
	} else if (x <= 8.4e+214) {
		tmp = t_0;
	} else {
		tmp = (x / y) * (x / 2.0);
	}
	return tmp;
}

Julia

x = abs(x)
function code(x, y, z)
	t_0 = Float64(Float64(0.5 * Float64(y - Float64(z / Float64(y / z)))) + Float64(0.5 * Float64(x * Float64(x / y))))
	tmp = 0.0
	if (x <= 6e+88)
		tmp = t_0;
	elseif (x <= 3.5e+139)
		tmp = Float64(fma(y, y, fma(x, x, Float64(z * Float64(-z)))) / Float64(y * 2.0));
	elseif (x <= 8.4e+214)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / y) * Float64(x / 2.0));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_, z_] := Block[{t$95$0 = N[(N[(0.5 * N[(y - N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 6e+88], t$95$0, If[LessEqual[x, 3.5e+139], N[(N[(y * y + N[(x * x + N[(z * (-z)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.4e+214], t$95$0, N[(N[(x / y), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 6.00000000000000011e88 or 3.49999999999999978e139 < x < 8.4000000000000003e214

   1. Initial program 72.0%

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

   2. Taylor expanded in x around inf 82.8%

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

      1. unpow2 82.8%

         \[\leadsto 0.5 \cdot \left(y - \frac{{z}^{2}}{y}\right) + 0.5 \cdot \frac{\color{blue}{x \cdot x}}{y} \]

      2. *-un-lft-identity 82.8%

         \[\leadsto 0.5 \cdot \left(y - \frac{{z}^{2}}{y}\right) + 0.5 \cdot \frac{x \cdot x}{\color{blue}{1 \cdot y}} \]

      3. times-frac 88.7%

         \[\leadsto 0.5 \cdot \left(y - \frac{{z}^{2}}{y}\right) + 0.5 \cdot \color{blue}{\left(\frac{x}{1} \cdot \frac{x}{y}\right)} \]

   4. Applied egg-rr 88.7%

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

      1. pow2 88.7%

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

      2. *-un-lft-identity 88.7%

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

      3. times-frac 95.0%

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

   6. Applied egg-rr 95.0%

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

      1. /-rgt-identity 95.0%

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

      2. clear-num 95.0%

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

      3. un-div-inv 95.0%

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

   8. Applied egg-rr 95.0%

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

   if 6.00000000000000011e88 < x < 3.49999999999999978e139

   1. Initial program 86.4%

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

      1. sqr-neg 86.4%

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

      2. sqr-neg 86.4%

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

      3. +-commutative 86.4%

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

      4. associate--l+ 86.4%

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

      5. fma-def 93.6%

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

      6. fma-neg 93.6%

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

      7. distribute-rgt-neg-in 93.6%

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

   3. Simplified 93.6%

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

   if 8.4000000000000003e214 < x

   1. Initial program 63.1%

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

   2. Taylor expanded in x around inf 87.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{y}} \]
   3. Step-by-step derivation

      1. associate-*r/ 87.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot {x}^{2}}{y}} \]

      2. associate-/l* 87.0%

         \[\leadsto \color{blue}{\frac{0.5}{\frac{y}{{x}^{2}}}} \]

   4. Simplified 87.0%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{y}{{x}^{2}}}} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 87.0%

         \[\leadsto \frac{0.5}{\frac{\color{blue}{1 \cdot y}}{{x}^{2}}} \]

      2. unpow2 87.0%

         \[\leadsto \frac{0.5}{\frac{1 \cdot y}{\color{blue}{x \cdot x}}} \]

      3. times-frac 95.2%

         \[\leadsto \frac{0.5}{\color{blue}{\frac{1}{x} \cdot \frac{y}{x}}} \]

   6. Applied egg-rr 95.2%

      \[\leadsto \frac{0.5}{\color{blue}{\frac{1}{x} \cdot \frac{y}{x}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 95.2%

         \[\leadsto \frac{0.5}{\color{blue}{\frac{1 \cdot \frac{y}{x}}{x}}} \]

      2. *-lft-identity 95.2%

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

   8. Simplified 95.2%

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

      1. metadata-eval 95.2%

         \[\leadsto \frac{\color{blue}{1 \cdot 0.5}}{\frac{\frac{y}{x}}{x}} \]

      2. associate-*l/ 95.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{y}{x}}{x}} \cdot 0.5} \]

      3. associate-/l/ 87.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{y}{x \cdot x}}} \cdot 0.5 \]

      4. unpow2 87.0%

         \[\leadsto \frac{1}{\frac{y}{\color{blue}{{x}^{2}}}} \cdot 0.5 \]

      5. clear-num 87.0%

         \[\leadsto \color{blue}{\frac{{x}^{2}}{y}} \cdot 0.5 \]

      6. metadata-eval 87.0%

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

      7. metadata-eval 87.0%

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

      8. metadata-eval 87.0%

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

      9. sqrt-pow2 87.0%

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

      10. div-inv 87.0%

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

      11. associate-/r* 87.0%

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

      12. unpow2 87.0%

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

      13. times-frac 95.0%

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

      14. sqrt-pow2 95.2%

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

      15. metadata-eval 95.2%

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

      16. metadata-eval 95.2%

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

   10. Applied egg-rr 95.2%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{+88}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right) + 0.5 \cdot \left(x \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+139}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y, \mathsf{fma}\left(x, x, z \cdot \left(-z\right)\right)\right)}{y \cdot 2}\\ \mathbf{elif}\;x \leq 8.4 \cdot 10^{+214}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right) + 0.5 \cdot \left(x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \end{array} \]

Alternative 2: 94.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(y, \mathsf{hypot}\left(x, z\right)\right)\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+246}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right) + 0.5 \cdot \left(x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(t_0 \cdot \frac{0.5}{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (hypot y (hypot x z))))
   (if (<= (* x x) 5e+246)
     (+ (* 0.5 (- y (/ z (/ y z)))) (* 0.5 (* x (/ x y))))
     (* t_0 (* t_0 (/ 0.5 y))))))

C

x = abs(x);
double code(double x, double y, double z) {
	double t_0 = hypot(y, hypot(x, z));
	double tmp;
	if ((x * x) <= 5e+246) {
		tmp = (0.5 * (y - (z / (y / z)))) + (0.5 * (x * (x / y)));
	} else {
		tmp = t_0 * (t_0 * (0.5 / y));
	}
	return tmp;
}

Java

x = Math.abs(x);
public static double code(double x, double y, double z) {
	double t_0 = Math.hypot(y, Math.hypot(x, z));
	double tmp;
	if ((x * x) <= 5e+246) {
		tmp = (0.5 * (y - (z / (y / z)))) + (0.5 * (x * (x / y)));
	} else {
		tmp = t_0 * (t_0 * (0.5 / y));
	}
	return tmp;
}

Python

x = abs(x)
def code(x, y, z):
	t_0 = math.hypot(y, math.hypot(x, z))
	tmp = 0
	if (x * x) <= 5e+246:
		tmp = (0.5 * (y - (z / (y / z)))) + (0.5 * (x * (x / y)))
	else:
		tmp = t_0 * (t_0 * (0.5 / y))
	return tmp

Julia

x = abs(x)
function code(x, y, z)
	t_0 = hypot(y, hypot(x, z))
	tmp = 0.0
	if (Float64(x * x) <= 5e+246)
		tmp = Float64(Float64(0.5 * Float64(y - Float64(z / Float64(y / z)))) + Float64(0.5 * Float64(x * Float64(x / y))));
	else
		tmp = Float64(t_0 * Float64(t_0 * Float64(0.5 / y)));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, y, z)
	t_0 = hypot(y, hypot(x, z));
	tmp = 0.0;
	if ((x * x) <= 5e+246)
		tmp = (0.5 * (y - (z / (y / z)))) + (0.5 * (x * (x / y)));
	else
		tmp = t_0 * (t_0 * (0.5 / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_, z_] := Block[{t$95$0 = N[Sqrt[y ^ 2 + N[Sqrt[x ^ 2 + z ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 5e+246], N[(N[(0.5 * N[(y - N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(t$95$0 * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.99999999999999976e246

   1. Initial program 75.7%

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

   2. Taylor expanded in x around inf 89.0%

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

      1. unpow2 89.0%

         \[\leadsto 0.5 \cdot \left(y - \frac{{z}^{2}}{y}\right) + 0.5 \cdot \frac{\color{blue}{x \cdot x}}{y} \]

      2. *-un-lft-identity 89.0%

         \[\leadsto 0.5 \cdot \left(y - \frac{{z}^{2}}{y}\right) + 0.5 \cdot \frac{x \cdot x}{\color{blue}{1 \cdot y}} \]

      3. times-frac 90.6%

         \[\leadsto 0.5 \cdot \left(y - \frac{{z}^{2}}{y}\right) + 0.5 \cdot \color{blue}{\left(\frac{x}{1} \cdot \frac{x}{y}\right)} \]

   4. Applied egg-rr 90.6%

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

      1. pow2 90.6%

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

      2. *-un-lft-identity 90.6%

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

      3. times-frac 96.6%

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

   6. Applied egg-rr 96.6%

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

      1. /-rgt-identity 96.6%

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

      2. clear-num 96.6%

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

      3. un-div-inv 96.7%

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

   8. Applied egg-rr 96.7%

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

   if 4.99999999999999976e246 < (*.f64 x x)

   1. Initial program 63.5%

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

      1. div-inv 63.5%

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

      2. add-sqr-sqrt 60.9%

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

      3. associate-*l* 60.9%

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

   3. Applied egg-rr 92.1%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+246}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right) + 0.5 \cdot \left(x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(y, \mathsf{hypot}\left(x, z\right)\right) \cdot \left(\mathsf{hypot}\left(y, \mathsf{hypot}\left(x, z\right)\right) \cdot \frac{0.5}{y}\right)\\ \end{array} \]

Alternative 3: 52.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := \frac{z \cdot -0.5}{\frac{y}{z}}\\ \mathbf{if}\;y \leq -9 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;y \leq -5.4 \cdot 10^{-116}:\\ \;\;\;\;\frac{x \cdot 0.5}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{-224}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-254}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-70}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;y \leq 3300000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* z -0.5) (/ y z))))
   (if (<= y -9e+101)
     (* 0.5 y)
     (if (<= y -5.4e-116)
       (/ (* x 0.5) (/ y x))
       (if (<= y -2.7e-224)
         t_0
         (if (<= y 4.2e-254)
           (* (/ x y) (/ x 2.0))
           (if (<= y 1.3e-70)
             t_0
             (if (<= y 3.8e-8)
               (* x (* x (/ 0.5 y)))
               (if (<= y 3300000000.0) t_0 (* 0.5 y))))))))))

C

x = abs(x);
double code(double x, double y, double z) {
	double t_0 = (z * -0.5) / (y / z);
	double tmp;
	if (y <= -9e+101) {
		tmp = 0.5 * y;
	} else if (y <= -5.4e-116) {
		tmp = (x * 0.5) / (y / x);
	} else if (y <= -2.7e-224) {
		tmp = t_0;
	} else if (y <= 4.2e-254) {
		tmp = (x / y) * (x / 2.0);
	} else if (y <= 1.3e-70) {
		tmp = t_0;
	} else if (y <= 3.8e-8) {
		tmp = x * (x * (0.5 / y));
	} else if (y <= 3300000000.0) {
		tmp = t_0;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (z * (-0.5d0)) / (y / z)
    if (y <= (-9d+101)) then
        tmp = 0.5d0 * y
    else if (y <= (-5.4d-116)) then
        tmp = (x * 0.5d0) / (y / x)
    else if (y <= (-2.7d-224)) then
        tmp = t_0
    else if (y <= 4.2d-254) then
        tmp = (x / y) * (x / 2.0d0)
    else if (y <= 1.3d-70) then
        tmp = t_0
    else if (y <= 3.8d-8) then
        tmp = x * (x * (0.5d0 / y))
    else if (y <= 3300000000.0d0) then
        tmp = t_0
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x, double y, double z) {
	double t_0 = (z * -0.5) / (y / z);
	double tmp;
	if (y <= -9e+101) {
		tmp = 0.5 * y;
	} else if (y <= -5.4e-116) {
		tmp = (x * 0.5) / (y / x);
	} else if (y <= -2.7e-224) {
		tmp = t_0;
	} else if (y <= 4.2e-254) {
		tmp = (x / y) * (x / 2.0);
	} else if (y <= 1.3e-70) {
		tmp = t_0;
	} else if (y <= 3.8e-8) {
		tmp = x * (x * (0.5 / y));
	} else if (y <= 3300000000.0) {
		tmp = t_0;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

x = abs(x)
def code(x, y, z):
	t_0 = (z * -0.5) / (y / z)
	tmp = 0
	if y <= -9e+101:
		tmp = 0.5 * y
	elif y <= -5.4e-116:
		tmp = (x * 0.5) / (y / x)
	elif y <= -2.7e-224:
		tmp = t_0
	elif y <= 4.2e-254:
		tmp = (x / y) * (x / 2.0)
	elif y <= 1.3e-70:
		tmp = t_0
	elif y <= 3.8e-8:
		tmp = x * (x * (0.5 / y))
	elif y <= 3300000000.0:
		tmp = t_0
	else:
		tmp = 0.5 * y
	return tmp

Julia

x = abs(x)
function code(x, y, z)
	t_0 = Float64(Float64(z * -0.5) / Float64(y / z))
	tmp = 0.0
	if (y <= -9e+101)
		tmp = Float64(0.5 * y);
	elseif (y <= -5.4e-116)
		tmp = Float64(Float64(x * 0.5) / Float64(y / x));
	elseif (y <= -2.7e-224)
		tmp = t_0;
	elseif (y <= 4.2e-254)
		tmp = Float64(Float64(x / y) * Float64(x / 2.0));
	elseif (y <= 1.3e-70)
		tmp = t_0;
	elseif (y <= 3.8e-8)
		tmp = Float64(x * Float64(x * Float64(0.5 / y)));
	elseif (y <= 3300000000.0)
		tmp = t_0;
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, y, z)
	t_0 = (z * -0.5) / (y / z);
	tmp = 0.0;
	if (y <= -9e+101)
		tmp = 0.5 * y;
	elseif (y <= -5.4e-116)
		tmp = (x * 0.5) / (y / x);
	elseif (y <= -2.7e-224)
		tmp = t_0;
	elseif (y <= 4.2e-254)
		tmp = (x / y) * (x / 2.0);
	elseif (y <= 1.3e-70)
		tmp = t_0;
	elseif (y <= 3.8e-8)
		tmp = x * (x * (0.5 / y));
	elseif (y <= 3300000000.0)
		tmp = t_0;
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * -0.5), $MachinePrecision] / N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -9e+101], N[(0.5 * y), $MachinePrecision], If[LessEqual[y, -5.4e-116], N[(N[(x * 0.5), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2.7e-224], t$95$0, If[LessEqual[y, 4.2e-254], N[(N[(x / y), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3e-70], t$95$0, If[LessEqual[y, 3.8e-8], N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3300000000.0], t$95$0, N[(0.5 * y), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -9.0000000000000004e101 or 3.3e9 < y

   1. Initial program 35.0%

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

   2. Taylor expanded in y around inf 70.2%

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

   if -9.0000000000000004e101 < y < -5.4e-116

   1. Initial program 93.3%

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

   2. Taylor expanded in x around inf 57.8%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{y}} \]
   3. Step-by-step derivation

      1. associate-*r/ 57.8%

         \[\leadsto \color{blue}{\frac{0.5 \cdot {x}^{2}}{y}} \]

      2. associate-/l* 57.8%

         \[\leadsto \color{blue}{\frac{0.5}{\frac{y}{{x}^{2}}}} \]

   4. Simplified 57.8%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{y}{{x}^{2}}}} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 57.8%

         \[\leadsto \frac{0.5}{\frac{\color{blue}{1 \cdot y}}{{x}^{2}}} \]

      2. unpow2 57.8%

         \[\leadsto \frac{0.5}{\frac{1 \cdot y}{\color{blue}{x \cdot x}}} \]

      3. times-frac 57.8%

         \[\leadsto \frac{0.5}{\color{blue}{\frac{1}{x} \cdot \frac{y}{x}}} \]

   6. Applied egg-rr 57.8%

      \[\leadsto \frac{0.5}{\color{blue}{\frac{1}{x} \cdot \frac{y}{x}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 57.8%

         \[\leadsto \frac{0.5}{\color{blue}{\frac{1 \cdot \frac{y}{x}}{x}}} \]

      2. *-lft-identity 57.8%

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

   8. Simplified 57.8%

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

      1. associate-/l/ 57.8%

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

      2. unpow2 57.8%

         \[\leadsto \frac{0.5}{\frac{y}{\color{blue}{{x}^{2}}}} \]

      3. associate-/r/ 57.8%

         \[\leadsto \color{blue}{\frac{0.5}{y} \cdot {x}^{2}} \]

      4. metadata-eval 57.8%

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

      5. metadata-eval 57.8%

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

      6. metadata-eval 57.8%

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

      7. sqrt-pow2 57.5%

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

      8. associate-/r* 57.5%

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

      9. *-commutative 57.5%

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

      10. *-commutative 57.5%

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

      11. unpow2 57.5%

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

      12. associate-*l* 57.6%

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

      13. *-commutative 57.6%

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

      14. associate-/r* 57.6%

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

      15. sqrt-pow2 57.9%

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

      16. metadata-eval 57.9%

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

      17. metadata-eval 57.9%

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

      18. metadata-eval 57.9%

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

   10. Applied egg-rr 57.9%

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

       1. *-commutative 57.9%

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

       2. *-commutative 57.9%

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

       3. associate-*l/ 57.8%

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

       4. associate-*l/ 57.8%

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

       5. associate-*r/ 57.8%

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

       6. clear-num 57.8%

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

       7. un-div-inv 57.9%

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

   12. Applied egg-rr 57.9%

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

   if -5.4e-116 < y < -2.69999999999999998e-224 or 4.19999999999999993e-254 < y < 1.30000000000000001e-70 or 3.80000000000000028e-8 < y < 3.3e9

   1. Initial program 91.2%

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

   2. Taylor expanded in z around inf 59.5%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{{z}^{2}}{y}} \]
   3. Step-by-step derivation

      1. *-commutative 59.5%

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

   4. Simplified 59.5%

      \[\leadsto \color{blue}{\frac{{z}^{2}}{y} \cdot -0.5} \]
   5. Step-by-step derivation

      1. pow2 88.4%

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

      2. *-un-lft-identity 88.4%

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

      3. times-frac 88.4%

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

   6. Applied egg-rr 59.5%

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

      1. associate-/r/ 59.5%

         \[\leadsto \color{blue}{\frac{z}{\frac{1}{\frac{z}{y}}}} \cdot -0.5 \]

      2. clear-num 59.5%

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

      3. associate-*l/ 59.5%

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

   8. Applied egg-rr 59.5%

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

   if -2.69999999999999998e-224 < y < 4.19999999999999993e-254

   1. Initial program 87.7%

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

   2. Taylor expanded in x around inf 75.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{y}} \]
   3. Step-by-step derivation

      1. associate-*r/ 77.4%

         \[\leadsto \color{blue}{\frac{0.5 \cdot {x}^{2}}{y}} \]

      2. associate-/l* 77.3%

         \[\leadsto \color{blue}{\frac{0.5}{\frac{y}{{x}^{2}}}} \]

   4. Simplified 77.3%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{y}{{x}^{2}}}} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 77.3%

         \[\leadsto \frac{0.5}{\frac{\color{blue}{1 \cdot y}}{{x}^{2}}} \]

      2. unpow2 77.3%

         \[\leadsto \frac{0.5}{\frac{1 \cdot y}{\color{blue}{x \cdot x}}} \]

      3. times-frac 80.1%

         \[\leadsto \frac{0.5}{\color{blue}{\frac{1}{x} \cdot \frac{y}{x}}} \]

   6. Applied egg-rr 80.1%

      \[\leadsto \frac{0.5}{\color{blue}{\frac{1}{x} \cdot \frac{y}{x}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 80.2%

         \[\leadsto \frac{0.5}{\color{blue}{\frac{1 \cdot \frac{y}{x}}{x}}} \]

      2. *-lft-identity 80.2%

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

   8. Simplified 80.2%

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

      1. metadata-eval 80.2%

         \[\leadsto \frac{\color{blue}{1 \cdot 0.5}}{\frac{\frac{y}{x}}{x}} \]

      2. associate-*l/ 77.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{y}{x}}{x}} \cdot 0.5} \]

      3. associate-/l/ 74.9%

         \[\leadsto \frac{1}{\color{blue}{\frac{y}{x \cdot x}}} \cdot 0.5 \]

      4. unpow2 74.9%

         \[\leadsto \frac{1}{\frac{y}{\color{blue}{{x}^{2}}}} \cdot 0.5 \]

      5. clear-num 75.0%

         \[\leadsto \color{blue}{\frac{{x}^{2}}{y}} \cdot 0.5 \]

      6. metadata-eval 75.0%

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

      7. metadata-eval 75.0%

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

      8. metadata-eval 75.0%

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

      9. sqrt-pow2 74.7%

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

      10. div-inv 74.7%

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

      11. associate-/r* 77.1%

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

      12. unpow2 77.1%

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

      13. times-frac 80.0%

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

      14. sqrt-pow2 80.3%

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

      15. metadata-eval 80.3%

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

      16. metadata-eval 80.3%

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

   10. Applied egg-rr 80.3%

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

   if 1.30000000000000001e-70 < y < 3.80000000000000028e-8

   1. Initial program 91.4%

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

   2. Taylor expanded in x around inf 51.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{y}} \]
   3. Step-by-step derivation

      1. associate-*r/ 51.9%

         \[\leadsto \color{blue}{\frac{0.5 \cdot {x}^{2}}{y}} \]

      2. associate-/l* 52.2%

         \[\leadsto \color{blue}{\frac{0.5}{\frac{y}{{x}^{2}}}} \]

   4. Simplified 52.2%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{y}{{x}^{2}}}} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 52.2%

         \[\leadsto \frac{0.5}{\frac{\color{blue}{1 \cdot y}}{{x}^{2}}} \]

      2. unpow2 52.2%

         \[\leadsto \frac{0.5}{\frac{1 \cdot y}{\color{blue}{x \cdot x}}} \]

      3. times-frac 52.0%

         \[\leadsto \frac{0.5}{\color{blue}{\frac{1}{x} \cdot \frac{y}{x}}} \]

   6. Applied egg-rr 52.0%

      \[\leadsto \frac{0.5}{\color{blue}{\frac{1}{x} \cdot \frac{y}{x}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 52.0%

         \[\leadsto \frac{0.5}{\color{blue}{\frac{1 \cdot \frac{y}{x}}{x}}} \]

      2. *-lft-identity 52.0%

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

   8. Simplified 52.0%

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

      1. associate-/l/ 52.2%

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

      2. unpow2 52.2%

         \[\leadsto \frac{0.5}{\frac{y}{\color{blue}{{x}^{2}}}} \]

      3. associate-/r/ 51.9%

         \[\leadsto \color{blue}{\frac{0.5}{y} \cdot {x}^{2}} \]

      4. metadata-eval 51.9%

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

      5. metadata-eval 51.9%

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

      6. metadata-eval 51.9%

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

      7. sqrt-pow2 51.6%

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

      8. associate-/r* 51.6%

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

      9. *-commutative 51.6%

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

      10. *-commutative 51.6%

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

      11. unpow2 51.6%

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

      12. associate-*l* 51.6%

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

      13. *-commutative 51.6%

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

      14. associate-/r* 51.6%

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

      15. sqrt-pow2 52.2%

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

      16. metadata-eval 52.2%

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

      17. metadata-eval 52.2%

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

      18. metadata-eval 52.2%

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

   10. Applied egg-rr 52.2%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \frac{0.5}{y}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;y \leq -5.4 \cdot 10^{-116}:\\ \;\;\;\;\frac{x \cdot 0.5}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{-224}:\\ \;\;\;\;\frac{z \cdot -0.5}{\frac{y}{z}}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-254}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-70}:\\ \;\;\;\;\frac{z \cdot -0.5}{\frac{y}{z}}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;y \leq 3300000000:\\ \;\;\;\;\frac{z \cdot -0.5}{\frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]

Alternative 4: 52.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := \frac{z \cdot -0.5}{\frac{y}{z}}\\ \mathbf{if}\;y \leq -8.6 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;y \leq -1.7 \cdot 10^{-114}:\\ \;\;\;\;\frac{x \cdot 0.5}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq -5.4 \cdot 10^{-226}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-255}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{-67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;y \leq 30000000000:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* z -0.5) (/ y z))))
   (if (<= y -8.6e+101)
     (* 0.5 y)
     (if (<= y -1.7e-114)
       (/ (* x 0.5) (/ y x))
       (if (<= y -5.4e-226)
         t_0
         (if (<= y 4e-255)
           (* (/ x y) (/ x 2.0))
           (if (<= y 1.06e-67)
             t_0
             (if (<= y 5.5e-8)
               (* x (* x (/ 0.5 y)))
               (if (<= y 30000000000.0)
                 (* -0.5 (* z (/ z y)))
                 (* 0.5 y))))))))))

C

x = abs(x);
double code(double x, double y, double z) {
	double t_0 = (z * -0.5) / (y / z);
	double tmp;
	if (y <= -8.6e+101) {
		tmp = 0.5 * y;
	} else if (y <= -1.7e-114) {
		tmp = (x * 0.5) / (y / x);
	} else if (y <= -5.4e-226) {
		tmp = t_0;
	} else if (y <= 4e-255) {
		tmp = (x / y) * (x / 2.0);
	} else if (y <= 1.06e-67) {
		tmp = t_0;
	} else if (y <= 5.5e-8) {
		tmp = x * (x * (0.5 / y));
	} else if (y <= 30000000000.0) {
		tmp = -0.5 * (z * (z / y));
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (z * (-0.5d0)) / (y / z)
    if (y <= (-8.6d+101)) then
        tmp = 0.5d0 * y
    else if (y <= (-1.7d-114)) then
        tmp = (x * 0.5d0) / (y / x)
    else if (y <= (-5.4d-226)) then
        tmp = t_0
    else if (y <= 4d-255) then
        tmp = (x / y) * (x / 2.0d0)
    else if (y <= 1.06d-67) then
        tmp = t_0
    else if (y <= 5.5d-8) then
        tmp = x * (x * (0.5d0 / y))
    else if (y <= 30000000000.0d0) then
        tmp = (-0.5d0) * (z * (z / y))
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x, double y, double z) {
	double t_0 = (z * -0.5) / (y / z);
	double tmp;
	if (y <= -8.6e+101) {
		tmp = 0.5 * y;
	} else if (y <= -1.7e-114) {
		tmp = (x * 0.5) / (y / x);
	} else if (y <= -5.4e-226) {
		tmp = t_0;
	} else if (y <= 4e-255) {
		tmp = (x / y) * (x / 2.0);
	} else if (y <= 1.06e-67) {
		tmp = t_0;
	} else if (y <= 5.5e-8) {
		tmp = x * (x * (0.5 / y));
	} else if (y <= 30000000000.0) {
		tmp = -0.5 * (z * (z / y));
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

x = abs(x)
def code(x, y, z):
	t_0 = (z * -0.5) / (y / z)
	tmp = 0
	if y <= -8.6e+101:
		tmp = 0.5 * y
	elif y <= -1.7e-114:
		tmp = (x * 0.5) / (y / x)
	elif y <= -5.4e-226:
		tmp = t_0
	elif y <= 4e-255:
		tmp = (x / y) * (x / 2.0)
	elif y <= 1.06e-67:
		tmp = t_0
	elif y <= 5.5e-8:
		tmp = x * (x * (0.5 / y))
	elif y <= 30000000000.0:
		tmp = -0.5 * (z * (z / y))
	else:
		tmp = 0.5 * y
	return tmp

Julia

x = abs(x)
function code(x, y, z)
	t_0 = Float64(Float64(z * -0.5) / Float64(y / z))
	tmp = 0.0
	if (y <= -8.6e+101)
		tmp = Float64(0.5 * y);
	elseif (y <= -1.7e-114)
		tmp = Float64(Float64(x * 0.5) / Float64(y / x));
	elseif (y <= -5.4e-226)
		tmp = t_0;
	elseif (y <= 4e-255)
		tmp = Float64(Float64(x / y) * Float64(x / 2.0));
	elseif (y <= 1.06e-67)
		tmp = t_0;
	elseif (y <= 5.5e-8)
		tmp = Float64(x * Float64(x * Float64(0.5 / y)));
	elseif (y <= 30000000000.0)
		tmp = Float64(-0.5 * Float64(z * Float64(z / y)));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, y, z)
	t_0 = (z * -0.5) / (y / z);
	tmp = 0.0;
	if (y <= -8.6e+101)
		tmp = 0.5 * y;
	elseif (y <= -1.7e-114)
		tmp = (x * 0.5) / (y / x);
	elseif (y <= -5.4e-226)
		tmp = t_0;
	elseif (y <= 4e-255)
		tmp = (x / y) * (x / 2.0);
	elseif (y <= 1.06e-67)
		tmp = t_0;
	elseif (y <= 5.5e-8)
		tmp = x * (x * (0.5 / y));
	elseif (y <= 30000000000.0)
		tmp = -0.5 * (z * (z / y));
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * -0.5), $MachinePrecision] / N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -8.6e+101], N[(0.5 * y), $MachinePrecision], If[LessEqual[y, -1.7e-114], N[(N[(x * 0.5), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -5.4e-226], t$95$0, If[LessEqual[y, 4e-255], N[(N[(x / y), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.06e-67], t$95$0, If[LessEqual[y, 5.5e-8], N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 30000000000.0], N[(-0.5 * N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if y < -8.6000000000000002e101 or 3e10 < y

   1. Initial program 35.0%

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

   2. Taylor expanded in y around inf 70.2%

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

   if -8.6000000000000002e101 < y < -1.69999999999999991e-114

   1. Initial program 93.3%

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

   2. Taylor expanded in x around inf 57.8%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{y}} \]
   3. Step-by-step derivation

      1. associate-*r/ 57.8%

         \[\leadsto \color{blue}{\frac{0.5 \cdot {x}^{2}}{y}} \]

      2. associate-/l* 57.8%

         \[\leadsto \color{blue}{\frac{0.5}{\frac{y}{{x}^{2}}}} \]

   4. Simplified 57.8%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{y}{{x}^{2}}}} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 57.8%

         \[\leadsto \frac{0.5}{\frac{\color{blue}{1 \cdot y}}{{x}^{2}}} \]

      2. unpow2 57.8%

         \[\leadsto \frac{0.5}{\frac{1 \cdot y}{\color{blue}{x \cdot x}}} \]

      3. times-frac 57.8%

         \[\leadsto \frac{0.5}{\color{blue}{\frac{1}{x} \cdot \frac{y}{x}}} \]

   6. Applied egg-rr 57.8%

      \[\leadsto \frac{0.5}{\color{blue}{\frac{1}{x} \cdot \frac{y}{x}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 57.8%

         \[\leadsto \frac{0.5}{\color{blue}{\frac{1 \cdot \frac{y}{x}}{x}}} \]

      2. *-lft-identity 57.8%

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

   8. Simplified 57.8%

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

      1. associate-/l/ 57.8%

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

      2. unpow2 57.8%

         \[\leadsto \frac{0.5}{\frac{y}{\color{blue}{{x}^{2}}}} \]

      3. associate-/r/ 57.8%

         \[\leadsto \color{blue}{\frac{0.5}{y} \cdot {x}^{2}} \]

      4. metadata-eval 57.8%

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

      5. metadata-eval 57.8%

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

      6. metadata-eval 57.8%

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

      7. sqrt-pow2 57.5%

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

      8. associate-/r* 57.5%

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

      9. *-commutative 57.5%

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

      10. *-commutative 57.5%

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

      11. unpow2 57.5%

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

      12. associate-*l* 57.6%

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

      13. *-commutative 57.6%

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

      14. associate-/r* 57.6%

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

      15. sqrt-pow2 57.9%

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

      16. metadata-eval 57.9%

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

      17. metadata-eval 57.9%

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

      18. metadata-eval 57.9%

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

   10. Applied egg-rr 57.9%

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

       1. *-commutative 57.9%

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

       2. *-commutative 57.9%

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

       3. associate-*l/ 57.8%

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

       4. associate-*l/ 57.8%

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

       5. associate-*r/ 57.8%

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

       6. clear-num 57.8%

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

       7. un-div-inv 57.9%

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

   12. Applied egg-rr 57.9%

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

   if -1.69999999999999991e-114 < y < -5.40000000000000029e-226 or 4e-255 < y < 1.06e-67

   1. Initial program 90.7%

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

   2. Taylor expanded in z around inf 58.6%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{{z}^{2}}{y}} \]
   3. Step-by-step derivation

      1. *-commutative 58.6%

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

   4. Simplified 58.6%

      \[\leadsto \color{blue}{\frac{{z}^{2}}{y} \cdot -0.5} \]
   5. Step-by-step derivation

      1. pow2 87.7%

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

      2. *-un-lft-identity 87.7%

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

      3. times-frac 87.8%

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

   6. Applied egg-rr 58.6%

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

      1. associate-/r/ 58.7%

         \[\leadsto \color{blue}{\frac{z}{\frac{1}{\frac{z}{y}}}} \cdot -0.5 \]

      2. clear-num 58.7%

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

      3. associate-*l/ 58.7%

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

   8. Applied egg-rr 58.7%

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

   if -5.40000000000000029e-226 < y < 4e-255

   1. Initial program 87.7%

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

   2. Taylor expanded in x around inf 75.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{y}} \]
   3. Step-by-step derivation

      1. associate-*r/ 77.4%

         \[\leadsto \color{blue}{\frac{0.5 \cdot {x}^{2}}{y}} \]

      2. associate-/l* 77.3%

         \[\leadsto \color{blue}{\frac{0.5}{\frac{y}{{x}^{2}}}} \]

   4. Simplified 77.3%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{y}{{x}^{2}}}} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 77.3%

         \[\leadsto \frac{0.5}{\frac{\color{blue}{1 \cdot y}}{{x}^{2}}} \]

      2. unpow2 77.3%

         \[\leadsto \frac{0.5}{\frac{1 \cdot y}{\color{blue}{x \cdot x}}} \]

      3. times-frac 80.1%

         \[\leadsto \frac{0.5}{\color{blue}{\frac{1}{x} \cdot \frac{y}{x}}} \]

   6. Applied egg-rr 80.1%

      \[\leadsto \frac{0.5}{\color{blue}{\frac{1}{x} \cdot \frac{y}{x}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 80.2%

         \[\leadsto \frac{0.5}{\color{blue}{\frac{1 \cdot \frac{y}{x}}{x}}} \]

      2. *-lft-identity 80.2%

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

   8. Simplified 80.2%

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

      1. metadata-eval 80.2%

         \[\leadsto \frac{\color{blue}{1 \cdot 0.5}}{\frac{\frac{y}{x}}{x}} \]

      2. associate-*l/ 77.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{y}{x}}{x}} \cdot 0.5} \]

      3. associate-/l/ 74.9%

         \[\leadsto \frac{1}{\color{blue}{\frac{y}{x \cdot x}}} \cdot 0.5 \]

      4. unpow2 74.9%

         \[\leadsto \frac{1}{\frac{y}{\color{blue}{{x}^{2}}}} \cdot 0.5 \]

      5. clear-num 75.0%

         \[\leadsto \color{blue}{\frac{{x}^{2}}{y}} \cdot 0.5 \]

      6. metadata-eval 75.0%

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

      7. metadata-eval 75.0%

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

      8. metadata-eval 75.0%

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

      9. sqrt-pow2 74.7%

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

      10. div-inv 74.7%

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

      11. associate-/r* 77.1%

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

      12. unpow2 77.1%

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

      13. times-frac 80.0%

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

      14. sqrt-pow2 80.3%

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

      15. metadata-eval 80.3%

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

      16. metadata-eval 80.3%

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

   10. Applied egg-rr 80.3%

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

   if 1.06e-67 < y < 5.5000000000000003e-8

   1. Initial program 91.4%

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

   2. Taylor expanded in x around inf 51.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{y}} \]
   3. Step-by-step derivation

      1. associate-*r/ 51.9%

         \[\leadsto \color{blue}{\frac{0.5 \cdot {x}^{2}}{y}} \]

      2. associate-/l* 52.2%

         \[\leadsto \color{blue}{\frac{0.5}{\frac{y}{{x}^{2}}}} \]

   4. Simplified 52.2%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{y}{{x}^{2}}}} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 52.2%

         \[\leadsto \frac{0.5}{\frac{\color{blue}{1 \cdot y}}{{x}^{2}}} \]

      2. unpow2 52.2%

         \[\leadsto \frac{0.5}{\frac{1 \cdot y}{\color{blue}{x \cdot x}}} \]

      3. times-frac 52.0%

         \[\leadsto \frac{0.5}{\color{blue}{\frac{1}{x} \cdot \frac{y}{x}}} \]

   6. Applied egg-rr 52.0%

      \[\leadsto \frac{0.5}{\color{blue}{\frac{1}{x} \cdot \frac{y}{x}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 52.0%

         \[\leadsto \frac{0.5}{\color{blue}{\frac{1 \cdot \frac{y}{x}}{x}}} \]

      2. *-lft-identity 52.0%

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

   8. Simplified 52.0%

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

      1. associate-/l/ 52.2%

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

      2. unpow2 52.2%

         \[\leadsto \frac{0.5}{\frac{y}{\color{blue}{{x}^{2}}}} \]

      3. associate-/r/ 51.9%

         \[\leadsto \color{blue}{\frac{0.5}{y} \cdot {x}^{2}} \]

      4. metadata-eval 51.9%

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

      5. metadata-eval 51.9%

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

      6. metadata-eval 51.9%

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

      7. sqrt-pow2 51.6%

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

      8. associate-/r* 51.6%

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

      9. *-commutative 51.6%

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

      10. *-commutative 51.6%

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

      11. unpow2 51.6%

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

      12. associate-*l* 51.6%

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

      13. *-commutative 51.6%

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

      14. associate-/r* 51.6%

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

      15. sqrt-pow2 52.2%

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

      16. metadata-eval 52.2%

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

      17. metadata-eval 52.2%

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

      18. metadata-eval 52.2%

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

   10. Applied egg-rr 52.2%

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

   if 5.5000000000000003e-8 < y < 3e10

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 75.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{{z}^{2}}{y}} \]
   3. Step-by-step derivation

      1. *-commutative 75.3%

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

   4. Simplified 75.3%

      \[\leadsto \color{blue}{\frac{{z}^{2}}{y} \cdot -0.5} \]
   5. Step-by-step derivation

      1. pow2 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. times-frac 100.0%

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

   6. Applied egg-rr 75.3%

      \[\leadsto \color{blue}{\left(\frac{z}{1} \cdot \frac{z}{y}\right)} \cdot -0.5 \]
3. Recombined 6 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.6 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;y \leq -1.7 \cdot 10^{-114}:\\ \;\;\;\;\frac{x \cdot 0.5}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq -5.4 \cdot 10^{-226}:\\ \;\;\;\;\frac{z \cdot -0.5}{\frac{y}{z}}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-255}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{-67}:\\ \;\;\;\;\frac{z \cdot -0.5}{\frac{y}{z}}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;y \leq 30000000000:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]

Alternative 5: 85.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+149} \lor \neg \left(y \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.3e+149) (not (<= y 1.35e+154)))
   (* 0.5 y)
   (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))))

C

x = abs(x);
double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.3e+149) || !(y <= 1.35e+154)) {
		tmp = 0.5 * y;
	} else {
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-1.3d+149)) .or. (.not. (y <= 1.35d+154))) then
        tmp = 0.5d0 * y
    else
        tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0d0)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.3e+149) || !(y <= 1.35e+154)) {
		tmp = 0.5 * y;
	} else {
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	}
	return tmp;
}

Python

x = abs(x)
def code(x, y, z):
	tmp = 0
	if (y <= -1.3e+149) or not (y <= 1.35e+154):
		tmp = 0.5 * y
	else:
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0)
	return tmp

Julia

x = abs(x)
function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.3e+149) || !(y <= 1.35e+154))
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.3e+149) || ~((y <= 1.35e+154)))
		tmp = 0.5 * y;
	else
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_, z_] := If[Or[LessEqual[y, -1.3e+149], N[Not[LessEqual[y, 1.35e+154]], $MachinePrecision]], N[(0.5 * y), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.29999999999999989e149 or 1.35000000000000003e154 < y

   1. Initial program 10.9%

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

   2. Taylor expanded in y around inf 78.9%

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

   if -1.29999999999999989e149 < y < 1.35000000000000003e154

   1. Initial program 90.4%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+149} \lor \neg \left(y \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \end{array} \]

Alternative 6: 93.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{+214}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right) + 0.5 \cdot \left(x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= x 6.5e+214)
   (+ (* 0.5 (- y (/ z (/ y z)))) (* 0.5 (* x (/ x y))))
   (* (/ x y) (/ x 2.0))))

C

x = abs(x);
double code(double x, double y, double z) {
	double tmp;
	if (x <= 6.5e+214) {
		tmp = (0.5 * (y - (z / (y / z)))) + (0.5 * (x * (x / y)));
	} else {
		tmp = (x / y) * (x / 2.0);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 6.5d+214) then
        tmp = (0.5d0 * (y - (z / (y / z)))) + (0.5d0 * (x * (x / y)))
    else
        tmp = (x / y) * (x / 2.0d0)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 6.5e+214) {
		tmp = (0.5 * (y - (z / (y / z)))) + (0.5 * (x * (x / y)));
	} else {
		tmp = (x / y) * (x / 2.0);
	}
	return tmp;
}

Python

x = abs(x)
def code(x, y, z):
	tmp = 0
	if x <= 6.5e+214:
		tmp = (0.5 * (y - (z / (y / z)))) + (0.5 * (x * (x / y)))
	else:
		tmp = (x / y) * (x / 2.0)
	return tmp

Julia

x = abs(x)
function code(x, y, z)
	tmp = 0.0
	if (x <= 6.5e+214)
		tmp = Float64(Float64(0.5 * Float64(y - Float64(z / Float64(y / z)))) + Float64(0.5 * Float64(x * Float64(x / y))));
	else
		tmp = Float64(Float64(x / y) * Float64(x / 2.0));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 6.5e+214)
		tmp = (0.5 * (y - (z / (y / z)))) + (0.5 * (x * (x / y)));
	else
		tmp = (x / y) * (x / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[x, 6.5e+214], N[(N[(0.5 * N[(y - N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.5000000000000001e214

   1. Initial program 72.9%

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

   2. Taylor expanded in x around inf 82.6%

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

      1. unpow2 82.6%

         \[\leadsto 0.5 \cdot \left(y - \frac{{z}^{2}}{y}\right) + 0.5 \cdot \frac{\color{blue}{x \cdot x}}{y} \]

      2. *-un-lft-identity 82.6%

         \[\leadsto 0.5 \cdot \left(y - \frac{{z}^{2}}{y}\right) + 0.5 \cdot \frac{x \cdot x}{\color{blue}{1 \cdot y}} \]

      3. times-frac 88.1%

         \[\leadsto 0.5 \cdot \left(y - \frac{{z}^{2}}{y}\right) + 0.5 \cdot \color{blue}{\left(\frac{x}{1} \cdot \frac{x}{y}\right)} \]

   4. Applied egg-rr 88.1%

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

      1. pow2 88.1%

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

      2. *-un-lft-identity 88.1%

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

      3. times-frac 94.0%

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

   6. Applied egg-rr 94.0%

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

      1. /-rgt-identity 94.0%

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

      2. clear-num 94.0%

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

      3. un-div-inv 94.0%

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

   8. Applied egg-rr 94.0%

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

   if 6.5000000000000001e214 < x

   1. Initial program 63.1%

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

   2. Taylor expanded in x around inf 87.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{y}} \]
   3. Step-by-step derivation

      1. associate-*r/ 87.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot {x}^{2}}{y}} \]

      2. associate-/l* 87.0%

         \[\leadsto \color{blue}{\frac{0.5}{\frac{y}{{x}^{2}}}} \]

   4. Simplified 87.0%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{y}{{x}^{2}}}} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 87.0%

         \[\leadsto \frac{0.5}{\frac{\color{blue}{1 \cdot y}}{{x}^{2}}} \]

      2. unpow2 87.0%

         \[\leadsto \frac{0.5}{\frac{1 \cdot y}{\color{blue}{x \cdot x}}} \]

      3. times-frac 95.2%

         \[\leadsto \frac{0.5}{\color{blue}{\frac{1}{x} \cdot \frac{y}{x}}} \]

   6. Applied egg-rr 95.2%

      \[\leadsto \frac{0.5}{\color{blue}{\frac{1}{x} \cdot \frac{y}{x}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 95.2%

         \[\leadsto \frac{0.5}{\color{blue}{\frac{1 \cdot \frac{y}{x}}{x}}} \]

      2. *-lft-identity 95.2%

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

   8. Simplified 95.2%

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

      1. metadata-eval 95.2%

         \[\leadsto \frac{\color{blue}{1 \cdot 0.5}}{\frac{\frac{y}{x}}{x}} \]

      2. associate-*l/ 95.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{y}{x}}{x}} \cdot 0.5} \]

      3. associate-/l/ 87.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{y}{x \cdot x}}} \cdot 0.5 \]

      4. unpow2 87.0%

         \[\leadsto \frac{1}{\frac{y}{\color{blue}{{x}^{2}}}} \cdot 0.5 \]

      5. clear-num 87.0%

         \[\leadsto \color{blue}{\frac{{x}^{2}}{y}} \cdot 0.5 \]

      6. metadata-eval 87.0%

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

      7. metadata-eval 87.0%

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

      8. metadata-eval 87.0%

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

      9. sqrt-pow2 87.0%

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

      10. div-inv 87.0%

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

      11. associate-/r* 87.0%

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

      12. unpow2 87.0%

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

      13. times-frac 95.0%

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

      14. sqrt-pow2 95.2%

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

      15. metadata-eval 95.2%

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

      16. metadata-eval 95.2%

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

   10. Applied egg-rr 95.2%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{+214}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right) + 0.5 \cdot \left(x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \end{array} \]

Alternative 7: 52.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+101} \lor \neg \left(y \leq 2150000000000\right):\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.05e+101) (not (<= y 2150000000000.0)))
   (* 0.5 y)
   (* x (* x (/ 0.5 y)))))

C

x = abs(x);
double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.05e+101) || !(y <= 2150000000000.0)) {
		tmp = 0.5 * y;
	} else {
		tmp = x * (x * (0.5 / y));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-1.05d+101)) .or. (.not. (y <= 2150000000000.0d0))) then
        tmp = 0.5d0 * y
    else
        tmp = x * (x * (0.5d0 / y))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.05e+101) || !(y <= 2150000000000.0)) {
		tmp = 0.5 * y;
	} else {
		tmp = x * (x * (0.5 / y));
	}
	return tmp;
}

Python

x = abs(x)
def code(x, y, z):
	tmp = 0
	if (y <= -1.05e+101) or not (y <= 2150000000000.0):
		tmp = 0.5 * y
	else:
		tmp = x * (x * (0.5 / y))
	return tmp

Julia

x = abs(x)
function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.05e+101) || !(y <= 2150000000000.0))
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(x * Float64(x * Float64(0.5 / y)));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.05e+101) || ~((y <= 2150000000000.0)))
		tmp = 0.5 * y;
	else
		tmp = x * (x * (0.5 / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_, z_] := If[Or[LessEqual[y, -1.05e+101], N[Not[LessEqual[y, 2150000000000.0]], $MachinePrecision]], N[(0.5 * y), $MachinePrecision], N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.05e101 or 2.15e12 < y

   1. Initial program 35.8%

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

   2. Taylor expanded in y around inf 70.5%

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

   if -1.05e101 < y < 2.15e12

   1. Initial program 91.1%

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

   2. Taylor expanded in x around inf 49.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{y}} \]
   3. Step-by-step derivation

      1. associate-*r/ 49.6%

         \[\leadsto \color{blue}{\frac{0.5 \cdot {x}^{2}}{y}} \]

      2. associate-/l* 49.6%

         \[\leadsto \color{blue}{\frac{0.5}{\frac{y}{{x}^{2}}}} \]

   4. Simplified 49.6%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{y}{{x}^{2}}}} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 49.6%

         \[\leadsto \frac{0.5}{\frac{\color{blue}{1 \cdot y}}{{x}^{2}}} \]

      2. unpow2 49.6%

         \[\leadsto \frac{0.5}{\frac{1 \cdot y}{\color{blue}{x \cdot x}}} \]

      3. times-frac 51.3%

         \[\leadsto \frac{0.5}{\color{blue}{\frac{1}{x} \cdot \frac{y}{x}}} \]

   6. Applied egg-rr 51.3%

      \[\leadsto \frac{0.5}{\color{blue}{\frac{1}{x} \cdot \frac{y}{x}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 51.3%

         \[\leadsto \frac{0.5}{\color{blue}{\frac{1 \cdot \frac{y}{x}}{x}}} \]

      2. *-lft-identity 51.3%

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

   8. Simplified 51.3%

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

      1. associate-/l/ 49.6%

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

      2. unpow2 49.6%

         \[\leadsto \frac{0.5}{\frac{y}{\color{blue}{{x}^{2}}}} \]

      3. associate-/r/ 49.6%

         \[\leadsto \color{blue}{\frac{0.5}{y} \cdot {x}^{2}} \]

      4. metadata-eval 49.6%

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

      5. metadata-eval 49.6%

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

      6. metadata-eval 49.6%

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

      7. sqrt-pow2 49.4%

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

      8. associate-/r* 49.4%

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

      9. *-commutative 49.4%

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

      10. *-commutative 49.4%

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

      11. unpow2 49.4%

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

      12. associate-*l* 51.0%

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

      13. *-commutative 51.0%

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

      14. associate-/r* 51.0%

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

      15. sqrt-pow2 51.3%

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

      16. metadata-eval 51.3%

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

      17. metadata-eval 51.3%

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

      18. metadata-eval 51.3%

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

   10. Applied egg-rr 51.3%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+101} \lor \neg \left(y \leq 2150000000000\right):\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \]

Alternative 8: 52.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+101} \lor \neg \left(y \leq 1700000000000\right):\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.5}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= y -8.2e+101) (not (<= y 1700000000000.0)))
   (* 0.5 y)
   (/ (* x 0.5) (/ y x))))

C

x = abs(x);
double code(double x, double y, double z) {
	double tmp;
	if ((y <= -8.2e+101) || !(y <= 1700000000000.0)) {
		tmp = 0.5 * y;
	} else {
		tmp = (x * 0.5) / (y / x);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-8.2d+101)) .or. (.not. (y <= 1700000000000.0d0))) then
        tmp = 0.5d0 * y
    else
        tmp = (x * 0.5d0) / (y / x)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -8.2e+101) || !(y <= 1700000000000.0)) {
		tmp = 0.5 * y;
	} else {
		tmp = (x * 0.5) / (y / x);
	}
	return tmp;
}

Python

x = abs(x)
def code(x, y, z):
	tmp = 0
	if (y <= -8.2e+101) or not (y <= 1700000000000.0):
		tmp = 0.5 * y
	else:
		tmp = (x * 0.5) / (y / x)
	return tmp

Julia

x = abs(x)
function code(x, y, z)
	tmp = 0.0
	if ((y <= -8.2e+101) || !(y <= 1700000000000.0))
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(Float64(x * 0.5) / Float64(y / x));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -8.2e+101) || ~((y <= 1700000000000.0)))
		tmp = 0.5 * y;
	else
		tmp = (x * 0.5) / (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_, z_] := If[Or[LessEqual[y, -8.2e+101], N[Not[LessEqual[y, 1700000000000.0]], $MachinePrecision]], N[(0.5 * y), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.1999999999999999e101 or 1.7e12 < y

   1. Initial program 34.3%

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

   2. Taylor expanded in y around inf 70.9%

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

   if -8.1999999999999999e101 < y < 1.7e12

   1. Initial program 91.2%

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

   2. Taylor expanded in x around inf 49.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{y}} \]
   3. Step-by-step derivation

      1. associate-*r/ 49.7%

         \[\leadsto \color{blue}{\frac{0.5 \cdot {x}^{2}}{y}} \]

      2. associate-/l* 49.6%

         \[\leadsto \color{blue}{\frac{0.5}{\frac{y}{{x}^{2}}}} \]

   4. Simplified 49.6%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{y}{{x}^{2}}}} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 49.6%

         \[\leadsto \frac{0.5}{\frac{\color{blue}{1 \cdot y}}{{x}^{2}}} \]

      2. unpow2 49.6%

         \[\leadsto \frac{0.5}{\frac{1 \cdot y}{\color{blue}{x \cdot x}}} \]

      3. times-frac 51.3%

         \[\leadsto \frac{0.5}{\color{blue}{\frac{1}{x} \cdot \frac{y}{x}}} \]

   6. Applied egg-rr 51.3%

      \[\leadsto \frac{0.5}{\color{blue}{\frac{1}{x} \cdot \frac{y}{x}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 51.3%

         \[\leadsto \frac{0.5}{\color{blue}{\frac{1 \cdot \frac{y}{x}}{x}}} \]

      2. *-lft-identity 51.3%

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

   8. Simplified 51.3%

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

      1. associate-/l/ 49.6%

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

      2. unpow2 49.6%

         \[\leadsto \frac{0.5}{\frac{y}{\color{blue}{{x}^{2}}}} \]

      3. associate-/r/ 49.6%

         \[\leadsto \color{blue}{\frac{0.5}{y} \cdot {x}^{2}} \]

      4. metadata-eval 49.6%

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

      5. metadata-eval 49.6%

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

      6. metadata-eval 49.6%

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

      7. sqrt-pow2 49.4%

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

      8. associate-/r* 49.4%

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

      9. *-commutative 49.4%

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

      10. *-commutative 49.4%

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

      11. unpow2 49.4%

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

      12. associate-*l* 51.0%

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

      13. *-commutative 51.0%

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

      14. associate-/r* 51.0%

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

      15. sqrt-pow2 51.3%

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

      16. metadata-eval 51.3%

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

      17. metadata-eval 51.3%

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

      18. metadata-eval 51.3%

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

   10. Applied egg-rr 51.3%

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

       1. *-commutative 51.3%

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

       2. *-commutative 51.3%

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

       3. associate-*l/ 51.3%

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

       4. associate-*l/ 49.7%

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

       5. associate-*r/ 51.3%

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

       6. clear-num 51.3%

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

       7. un-div-inv 51.3%

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

   12. Applied egg-rr 51.3%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+101} \lor \neg \left(y \leq 1700000000000\right):\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.5}{\frac{y}{x}}\\ \end{array} \]

Alternative 9: 34.2% accurate, 5.0× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z) :precision binary64 (* 0.5 y))

C

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

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.5d0 * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_, z_] := N[(0.5 * y), $MachinePrecision]

Derivation

1. Initial program 72.1%

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

2. Taylor expanded in y around inf 33.1%

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

3. Final simplification 33.1%

   \[\leadsto 0.5 \cdot y \]

Developer target: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023310 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

  :herbie-target
  (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x)))

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