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

Percentage Accurate: 68.4% → 93.4%

Time: 11.2s

Alternatives: 10

Speedup: 0.5×

• 40.9% 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: 68.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 93.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_0 := \frac{{z\_m}^{2}}{y}\\ \mathbf{if}\;z\_m \cdot z\_m \leq 2 \cdot 10^{-30}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(-1, t\_0, \left(z\_m + y\right) \cdot \frac{y - z\_m}{y} + \left(t\_0 + \frac{{x}^{2}}{y}\right)\right)\\ \mathbf{elif}\;z\_m \cdot z\_m \leq 10^{+296}:\\ \;\;\;\;0.5 \cdot \left({z\_m}^{2} \cdot \left(\frac{y}{{z\_m}^{2}} + \mathsf{fma}\left(\frac{x}{y}, \frac{x}{{z\_m}^{2}}, \frac{-1}{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z\_m \cdot \frac{z\_m}{y}}{y} \cdot \left(y \cdot -0.5\right) + 0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (/ (pow z_m 2.0) y)))
   (if (<= (* z_m z_m) 2e-30)
     (*
      0.5
      (fma
       -1.0
       t_0
       (+ (* (+ z_m y) (/ (- y z_m) y)) (+ t_0 (/ (pow x 2.0) y)))))
     (if (<= (* z_m z_m) 1e+296)
       (*
        0.5
        (*
         (pow z_m 2.0)
         (+ (/ y (pow z_m 2.0)) (fma (/ x y) (/ x (pow z_m 2.0)) (/ -1.0 y)))))
       (+ (* (/ (* z_m (/ z_m y)) y) (* y -0.5)) (* 0.5 y))))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double t_0 = pow(z_m, 2.0) / y;
	double tmp;
	if ((z_m * z_m) <= 2e-30) {
		tmp = 0.5 * fma(-1.0, t_0, (((z_m + y) * ((y - z_m) / y)) + (t_0 + (pow(x, 2.0) / y))));
	} else if ((z_m * z_m) <= 1e+296) {
		tmp = 0.5 * (pow(z_m, 2.0) * ((y / pow(z_m, 2.0)) + fma((x / y), (x / pow(z_m, 2.0)), (-1.0 / y))));
	} else {
		tmp = (((z_m * (z_m / y)) / y) * (y * -0.5)) + (0.5 * y);
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m)
	t_0 = Float64((z_m ^ 2.0) / y)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 2e-30)
		tmp = Float64(0.5 * fma(-1.0, t_0, Float64(Float64(Float64(z_m + y) * Float64(Float64(y - z_m) / y)) + Float64(t_0 + Float64((x ^ 2.0) / y)))));
	elseif (Float64(z_m * z_m) <= 1e+296)
		tmp = Float64(0.5 * Float64((z_m ^ 2.0) * Float64(Float64(y / (z_m ^ 2.0)) + fma(Float64(x / y), Float64(x / (z_m ^ 2.0)), Float64(-1.0 / y)))));
	else
		tmp = Float64(Float64(Float64(Float64(z_m * Float64(z_m / y)) / y) * Float64(y * -0.5)) + Float64(0.5 * y));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[Power[z$95$m, 2.0], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 2e-30], N[(0.5 * N[(-1.0 * t$95$0 + N[(N[(N[(z$95$m + y), $MachinePrecision] * N[(N[(y - z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 + N[(N[Power[x, 2.0], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 1e+296], N[(0.5 * N[(N[Power[z$95$m, 2.0], $MachinePrecision] * N[(N[(y / N[Power[z$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(x / y), $MachinePrecision] * N[(x / N[Power[z$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(z$95$m * N[(z$95$m / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] * N[(y * -0.5), $MachinePrecision]), $MachinePrecision] + N[(0.5 * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 2e-30

   1. Initial program 80.5%

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

      1. remove-double-neg 80.5%

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

      2. distribute-lft-neg-out 80.5%

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

      3. distribute-frac-neg2 80.5%

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

      4. distribute-frac-neg 80.5%

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

      5. neg-mul-1 80.5%

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

      6. distribute-lft-neg-out 80.5%

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

      7. *-commutative 80.5%

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

      8. distribute-lft-neg-in 80.5%

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

      9. times-frac 80.5%

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

      10. metadata-eval 80.5%

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

      11. metadata-eval 80.5%

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

      12. associate--l+ 80.5%

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

      13. fma-define 80.5%

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

   3. Simplified 80.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. prod-diff 80.5%

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

      2. fma-neg 80.5%

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

      3. difference-of-squares 80.5%

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

      4. fma-define 80.5%

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

      5. pow2 80.5%

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

   6. Applied egg-rr 80.5%

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

   7. Taylor expanded in x around 0 80.5%

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

      1. fma-define 80.5%

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

      2. associate-/l* 96.9%

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

      3. +-commutative 96.9%

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

   9. Simplified 96.9%

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

   if 2e-30 < (*.f64 z z) < 9.99999999999999981e295

   1. Initial program 83.5%

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

      1. remove-double-neg 83.5%

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

      2. distribute-lft-neg-out 83.5%

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

      3. distribute-frac-neg2 83.5%

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

      4. distribute-frac-neg 83.5%

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

      5. neg-mul-1 83.5%

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

      6. distribute-lft-neg-out 83.5%

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

      7. *-commutative 83.5%

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

      8. distribute-lft-neg-in 83.5%

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

      9. times-frac 83.5%

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

      10. metadata-eval 83.5%

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

      11. metadata-eval 83.5%

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

      12. associate--l+ 83.5%

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

      13. fma-define 83.5%

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

   3. Simplified 83.5%

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

   5. Taylor expanded in z around inf 91.8%

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

      1. associate--l+ 91.8%

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

      2. unpow2 91.8%

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

      3. times-frac 98.1%

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

      4. fma-neg 98.1%

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

      5. distribute-neg-frac 98.1%

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

      6. metadata-eval 98.1%

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

   7. Simplified 98.1%

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

   if 9.99999999999999981e295 < (*.f64 z z)

   1. Initial program 48.1%

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

   3. Taylor expanded in y around inf 48.1%

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

      1. *-commutative 48.1%

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

   5. Simplified 48.1%

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

   6. Taylor expanded in x around 0 60.8%

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

      1. *-commutative 60.8%

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

   8. Simplified 60.8%

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

      1. +-commutative 60.8%

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

      2. distribute-rgt-in 60.8%

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

      3. associate-*l* 60.8%

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

      4. add-sqr-sqrt 60.8%

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

      5. pow2 60.8%

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

      6. sqrt-div 60.8%

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

      7. sqrt-pow1 81.4%

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

      8. metadata-eval 81.4%

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

      9. pow1 81.4%

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

      10. sqrt-pow1 93.6%

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

      11. metadata-eval 93.6%

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

      12. pow1 93.6%

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

   10. Applied egg-rr 93.6%

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

       1. unpow2 93.6%

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

       2. associate-*r/ 93.7%

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

   12. Applied egg-rr 93.7%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-30}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(-1, \frac{{z}^{2}}{y}, \left(z + y\right) \cdot \frac{y - z}{y} + \left(\frac{{z}^{2}}{y} + \frac{{x}^{2}}{y}\right)\right)\\ \mathbf{elif}\;z \cdot z \leq 10^{+296}:\\ \;\;\;\;0.5 \cdot \left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \mathsf{fma}\left(\frac{x}{y}, \frac{x}{{z}^{2}}, \frac{-1}{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \frac{z}{y}}{y} \cdot \left(y \cdot -0.5\right) + 0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{2 \cdot y}\\ \mathbf{if}\;t\_0 \leq 10^{+289}:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z\_m \cdot z\_m\right)}{y}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \left(x \cdot \frac{1}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z\_m \cdot \frac{z\_m}{y}}{y} \cdot \left(y \cdot -0.5\right) + 0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* 2.0 y))))
   (if (<= t_0 1e+289)
     (* 0.5 (/ (fma x x (- (* y y) (* z_m z_m))) y))
     (if (<= t_0 INFINITY)
       (* 0.5 (+ y (* x (* x (/ 1.0 y)))))
       (+ (* (/ (* z_m (/ z_m y)) y) (* y -0.5)) (* 0.5 y))))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (2.0 * y);
	double tmp;
	if (t_0 <= 1e+289) {
		tmp = 0.5 * (fma(x, x, ((y * y) - (z_m * z_m))) / y);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 0.5 * (y + (x * (x * (1.0 / y))));
	} else {
		tmp = (((z_m * (z_m / y)) / y) * (y * -0.5)) + (0.5 * y);
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(2.0 * y))
	tmp = 0.0
	if (t_0 <= 1e+289)
		tmp = Float64(0.5 * Float64(fma(x, x, Float64(Float64(y * y) - Float64(z_m * z_m))) / y));
	elseif (t_0 <= Inf)
		tmp = Float64(0.5 * Float64(y + Float64(x * Float64(x * Float64(1.0 / y)))));
	else
		tmp = Float64(Float64(Float64(Float64(z_m * Float64(z_m / y)) / y) * Float64(y * -0.5)) + Float64(0.5 * y));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(2.0 * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e+289], N[(0.5 * N[(N[(x * x + N[(N[(y * y), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(0.5 * N[(y + N[(x * N[(x * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(z$95$m * N[(z$95$m / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] * N[(y * -0.5), $MachinePrecision]), $MachinePrecision] + N[(0.5 * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 1.0000000000000001e289

   1. Initial program 87.6%

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

      1. remove-double-neg 87.6%

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

      2. distribute-lft-neg-out 87.6%

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

      3. distribute-frac-neg2 87.6%

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

      4. distribute-frac-neg 87.6%

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

      5. neg-mul-1 87.6%

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

      6. distribute-lft-neg-out 87.6%

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

      7. *-commutative 87.6%

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

      8. distribute-lft-neg-in 87.6%

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

      9. times-frac 87.6%

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

      10. metadata-eval 87.6%

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

      11. metadata-eval 87.6%

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

      12. associate--l+ 87.6%

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

      13. fma-define 87.6%

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

   3. Simplified 87.6%

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

   if 1.0000000000000001e289 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 71.9%

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

   3. Taylor expanded in z around 0 35.6%

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

      1. associate-*r/ 35.6%

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

      2. rem-square-sqrt 35.6%

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

      3. unpow2 35.6%

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

      4. unpow2 35.6%

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

      5. hypot-undefine 35.6%

         \[\leadsto \frac{0.5 \cdot \left(\color{blue}{\mathsf{hypot}\left(x, y\right)} \cdot \sqrt{{x}^{2} + {y}^{2}}\right)}{y} \]

      6. unpow2 35.6%

         \[\leadsto \frac{0.5 \cdot \left(\mathsf{hypot}\left(x, y\right) \cdot \sqrt{\color{blue}{x \cdot x} + {y}^{2}}\right)}{y} \]

      7. unpow2 35.6%

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

      8. hypot-undefine 35.6%

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

      9. unpow2 35.6%

         \[\leadsto \frac{0.5 \cdot \color{blue}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2}}}{y} \]

      10. associate-*l/ 35.6%

          \[\leadsto \color{blue}{\frac{0.5}{y} \cdot {\left(\mathsf{hypot}\left(x, y\right)\right)}^{2}} \]

      11. *-commutative 35.6%

          \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} \cdot \frac{0.5}{y}} \]

   5. Simplified 35.6%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(y, x\right)\right)}^{2} \cdot \frac{0.5}{y}} \]

   6. Taylor expanded in x around 0 56.8%

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

      1. distribute-lft-out 56.8%

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

   8. Simplified 56.8%

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

      1. div-inv 56.8%

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

      2. unpow2 56.8%

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

      3. associate-*l* 62.3%

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

   10. Applied egg-rr 62.3%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in y around inf 0.0%

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

      1. *-commutative 0.0%

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

   5. Simplified 0.0%

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

   6. Taylor expanded in x around 0 26.7%

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

      1. *-commutative 26.7%

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

   8. Simplified 26.7%

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

      1. +-commutative 26.7%

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

      2. distribute-rgt-in 26.7%

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

      3. associate-*l* 26.7%

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

      4. add-sqr-sqrt 26.7%

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

      5. pow2 26.7%

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

      6. sqrt-div 26.7%

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

      7. sqrt-pow1 61.1%

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

      8. metadata-eval 61.1%

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

      9. pow1 61.1%

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

      10. sqrt-pow1 86.7%

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

      11. metadata-eval 86.7%

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

      12. pow1 86.7%

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

   10. Applied egg-rr 86.7%

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

       1. unpow2 86.7%

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

       2. associate-*r/ 86.8%

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

   12. Applied egg-rr 86.8%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{2 \cdot y} \leq 10^{+289}:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{2 \cdot y} \leq \infty:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \left(x \cdot \frac{1}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \frac{z}{y}}{y} \cdot \left(y \cdot -0.5\right) + 0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{2 \cdot y}\\ \mathbf{if}\;t\_0 \leq 10^{+289}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \left(x \cdot \frac{1}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z\_m \cdot \frac{z\_m}{y}}{y} \cdot \left(y \cdot -0.5\right) + 0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* 2.0 y))))
   (if (<= t_0 1e+289)
     t_0
     (if (<= t_0 INFINITY)
       (* 0.5 (+ y (* x (* x (/ 1.0 y)))))
       (+ (* (/ (* z_m (/ z_m y)) y) (* y -0.5)) (* 0.5 y))))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (2.0 * y);
	double tmp;
	if (t_0 <= 1e+289) {
		tmp = t_0;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 0.5 * (y + (x * (x * (1.0 / y))));
	} else {
		tmp = (((z_m * (z_m / y)) / y) * (y * -0.5)) + (0.5 * y);
	}
	return tmp;
}

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	double t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (2.0 * y);
	double tmp;
	if (t_0 <= 1e+289) {
		tmp = t_0;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = 0.5 * (y + (x * (x * (1.0 / y))));
	} else {
		tmp = (((z_m * (z_m / y)) / y) * (y * -0.5)) + (0.5 * y);
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (2.0 * y)
	tmp = 0
	if t_0 <= 1e+289:
		tmp = t_0
	elif t_0 <= math.inf:
		tmp = 0.5 * (y + (x * (x * (1.0 / y))))
	else:
		tmp = (((z_m * (z_m / y)) / y) * (y * -0.5)) + (0.5 * y)
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(2.0 * y))
	tmp = 0.0
	if (t_0 <= 1e+289)
		tmp = t_0;
	elseif (t_0 <= Inf)
		tmp = Float64(0.5 * Float64(y + Float64(x * Float64(x * Float64(1.0 / y)))));
	else
		tmp = Float64(Float64(Float64(Float64(z_m * Float64(z_m / y)) / y) * Float64(y * -0.5)) + Float64(0.5 * y));
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m)
	t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (2.0 * y);
	tmp = 0.0;
	if (t_0 <= 1e+289)
		tmp = t_0;
	elseif (t_0 <= Inf)
		tmp = 0.5 * (y + (x * (x * (1.0 / y))));
	else
		tmp = (((z_m * (z_m / y)) / y) * (y * -0.5)) + (0.5 * y);
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(2.0 * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e+289], t$95$0, If[LessEqual[t$95$0, Infinity], N[(0.5 * N[(y + N[(x * N[(x * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(z$95$m * N[(z$95$m / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] * N[(y * -0.5), $MachinePrecision]), $MachinePrecision] + N[(0.5 * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 1.0000000000000001e289

   1. Initial program 87.6%

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

   if 1.0000000000000001e289 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 71.9%

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

   3. Taylor expanded in z around 0 35.6%

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

      1. associate-*r/ 35.6%

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

      2. rem-square-sqrt 35.6%

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

      3. unpow2 35.6%

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

      4. unpow2 35.6%

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

      5. hypot-undefine 35.6%

         \[\leadsto \frac{0.5 \cdot \left(\color{blue}{\mathsf{hypot}\left(x, y\right)} \cdot \sqrt{{x}^{2} + {y}^{2}}\right)}{y} \]

      6. unpow2 35.6%

         \[\leadsto \frac{0.5 \cdot \left(\mathsf{hypot}\left(x, y\right) \cdot \sqrt{\color{blue}{x \cdot x} + {y}^{2}}\right)}{y} \]

      7. unpow2 35.6%

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

      8. hypot-undefine 35.6%

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

      9. unpow2 35.6%

         \[\leadsto \frac{0.5 \cdot \color{blue}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2}}}{y} \]

      10. associate-*l/ 35.6%

          \[\leadsto \color{blue}{\frac{0.5}{y} \cdot {\left(\mathsf{hypot}\left(x, y\right)\right)}^{2}} \]

      11. *-commutative 35.6%

          \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} \cdot \frac{0.5}{y}} \]

   5. Simplified 35.6%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(y, x\right)\right)}^{2} \cdot \frac{0.5}{y}} \]

   6. Taylor expanded in x around 0 56.8%

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

      1. distribute-lft-out 56.8%

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

   8. Simplified 56.8%

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

      1. div-inv 56.8%

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

      2. unpow2 56.8%

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

      3. associate-*l* 62.3%

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

   10. Applied egg-rr 62.3%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in y around inf 0.0%

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

      1. *-commutative 0.0%

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

   5. Simplified 0.0%

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

   6. Taylor expanded in x around 0 26.7%

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

      1. *-commutative 26.7%

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

   8. Simplified 26.7%

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

      1. +-commutative 26.7%

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

      2. distribute-rgt-in 26.7%

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

      3. associate-*l* 26.7%

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

      4. add-sqr-sqrt 26.7%

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

      5. pow2 26.7%

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

      6. sqrt-div 26.7%

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

      7. sqrt-pow1 61.1%

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

      8. metadata-eval 61.1%

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

      9. pow1 61.1%

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

      10. sqrt-pow1 86.7%

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

      11. metadata-eval 86.7%

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

      12. pow1 86.7%

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

   10. Applied egg-rr 86.7%

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

       1. unpow2 86.7%

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

       2. associate-*r/ 86.8%

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

   12. Applied egg-rr 86.8%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{2 \cdot y} \leq 10^{+289}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{2 \cdot y}\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{2 \cdot y} \leq \infty:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \left(x \cdot \frac{1}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \frac{z}{y}}{y} \cdot \left(y \cdot -0.5\right) + 0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 5.1 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{\frac{x}{y}}{\frac{1}{x}}\right)\\ \mathbf{elif}\;z\_m \leq 4.7 \cdot 10^{+20} \lor \neg \left(z\_m \leq 10^{+36}\right):\\ \;\;\;\;\frac{z\_m \cdot \frac{z\_m}{y}}{y} \cdot \left(y \cdot -0.5\right) + 0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x}{\frac{y}{x}}\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= z_m 5.1e-20)
   (* 0.5 (+ y (/ (/ x y) (/ 1.0 x))))
   (if (or (<= z_m 4.7e+20) (not (<= z_m 1e+36)))
     (+ (* (/ (* z_m (/ z_m y)) y) (* y -0.5)) (* 0.5 y))
     (* 0.5 (+ y (/ x (/ y x)))))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double tmp;
	if (z_m <= 5.1e-20) {
		tmp = 0.5 * (y + ((x / y) / (1.0 / x)));
	} else if ((z_m <= 4.7e+20) || !(z_m <= 1e+36)) {
		tmp = (((z_m * (z_m / y)) / y) * (y * -0.5)) + (0.5 * y);
	} else {
		tmp = 0.5 * (y + (x / (y / x)));
	}
	return tmp;
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 5.1d-20) then
        tmp = 0.5d0 * (y + ((x / y) / (1.0d0 / x)))
    else if ((z_m <= 4.7d+20) .or. (.not. (z_m <= 1d+36))) then
        tmp = (((z_m * (z_m / y)) / y) * (y * (-0.5d0))) + (0.5d0 * y)
    else
        tmp = 0.5d0 * (y + (x / (y / x)))
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	double tmp;
	if (z_m <= 5.1e-20) {
		tmp = 0.5 * (y + ((x / y) / (1.0 / x)));
	} else if ((z_m <= 4.7e+20) || !(z_m <= 1e+36)) {
		tmp = (((z_m * (z_m / y)) / y) * (y * -0.5)) + (0.5 * y);
	} else {
		tmp = 0.5 * (y + (x / (y / x)));
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	tmp = 0
	if z_m <= 5.1e-20:
		tmp = 0.5 * (y + ((x / y) / (1.0 / x)))
	elif (z_m <= 4.7e+20) or not (z_m <= 1e+36):
		tmp = (((z_m * (z_m / y)) / y) * (y * -0.5)) + (0.5 * y)
	else:
		tmp = 0.5 * (y + (x / (y / x)))
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m)
	tmp = 0.0
	if (z_m <= 5.1e-20)
		tmp = Float64(0.5 * Float64(y + Float64(Float64(x / y) / Float64(1.0 / x))));
	elseif ((z_m <= 4.7e+20) || !(z_m <= 1e+36))
		tmp = Float64(Float64(Float64(Float64(z_m * Float64(z_m / y)) / y) * Float64(y * -0.5)) + Float64(0.5 * y));
	else
		tmp = Float64(0.5 * Float64(y + Float64(x / Float64(y / x))));
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m)
	tmp = 0.0;
	if (z_m <= 5.1e-20)
		tmp = 0.5 * (y + ((x / y) / (1.0 / x)));
	elseif ((z_m <= 4.7e+20) || ~((z_m <= 1e+36)))
		tmp = (((z_m * (z_m / y)) / y) * (y * -0.5)) + (0.5 * y);
	else
		tmp = 0.5 * (y + (x / (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := If[LessEqual[z$95$m, 5.1e-20], N[(0.5 * N[(y + N[(N[(x / y), $MachinePrecision] / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z$95$m, 4.7e+20], N[Not[LessEqual[z$95$m, 1e+36]], $MachinePrecision]], N[(N[(N[(N[(z$95$m * N[(z$95$m / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] * N[(y * -0.5), $MachinePrecision]), $MachinePrecision] + N[(0.5 * y), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y + N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 5.10000000000000019e-20

   1. Initial program 74.5%

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

   3. Taylor expanded in z around 0 55.5%

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

      1. associate-*r/ 55.5%

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

      2. rem-square-sqrt 55.5%

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

      3. unpow2 55.5%

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

      4. unpow2 55.5%

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

      5. hypot-undefine 55.5%

         \[\leadsto \frac{0.5 \cdot \left(\color{blue}{\mathsf{hypot}\left(x, y\right)} \cdot \sqrt{{x}^{2} + {y}^{2}}\right)}{y} \]

      6. unpow2 55.5%

         \[\leadsto \frac{0.5 \cdot \left(\mathsf{hypot}\left(x, y\right) \cdot \sqrt{\color{blue}{x \cdot x} + {y}^{2}}\right)}{y} \]

      7. unpow2 55.5%

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

      8. hypot-undefine 55.5%

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

      9. unpow2 55.5%

         \[\leadsto \frac{0.5 \cdot \color{blue}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2}}}{y} \]

      10. associate-*l/ 55.4%

          \[\leadsto \color{blue}{\frac{0.5}{y} \cdot {\left(\mathsf{hypot}\left(x, y\right)\right)}^{2}} \]

      11. *-commutative 55.4%

          \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} \cdot \frac{0.5}{y}} \]

   5. Simplified 55.4%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(y, x\right)\right)}^{2} \cdot \frac{0.5}{y}} \]

   6. Taylor expanded in x around 0 70.2%

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

      1. distribute-lft-out 70.2%

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

   8. Simplified 70.2%

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

      1. unpow2 70.2%

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

      2. *-un-lft-identity 70.2%

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

      3. times-frac 73.7%

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

   10. Applied egg-rr 73.7%

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

       1. clear-num 73.7%

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

       2. associate-*l/ 73.7%

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

       3. *-un-lft-identity 73.7%

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

   12. Applied egg-rr 73.7%

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

   if 5.10000000000000019e-20 < z < 4.7e20 or 1.00000000000000004e36 < z

   1. Initial program 68.1%

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

   3. Taylor expanded in y around inf 63.3%

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

      1. *-commutative 63.3%

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

   5. Simplified 63.3%

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

   6. Taylor expanded in x around 0 71.8%

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

      1. *-commutative 71.8%

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

   8. Simplified 71.8%

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

      1. +-commutative 71.8%

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

      2. distribute-rgt-in 71.8%

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

      3. associate-*l* 71.8%

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

      4. add-sqr-sqrt 71.7%

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

      5. pow2 71.7%

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

      6. sqrt-div 71.7%

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

      7. sqrt-pow1 80.7%

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

      8. metadata-eval 80.7%

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

      9. pow1 80.7%

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

      10. sqrt-pow1 85.3%

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

      11. metadata-eval 85.3%

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

      12. pow1 85.3%

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

   10. Applied egg-rr 85.3%

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

       1. unpow2 85.3%

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

       2. associate-*r/ 85.4%

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

   12. Applied egg-rr 85.4%

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

   if 4.7e20 < z < 1.00000000000000004e36

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0 99.5%

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

      1. associate-*r/ 99.5%

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

      2. rem-square-sqrt 99.5%

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

      3. unpow2 99.5%

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

      4. unpow2 99.5%

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

      5. hypot-undefine 99.5%

         \[\leadsto \frac{0.5 \cdot \left(\color{blue}{\mathsf{hypot}\left(x, y\right)} \cdot \sqrt{{x}^{2} + {y}^{2}}\right)}{y} \]

      6. unpow2 99.5%

         \[\leadsto \frac{0.5 \cdot \left(\mathsf{hypot}\left(x, y\right) \cdot \sqrt{\color{blue}{x \cdot x} + {y}^{2}}\right)}{y} \]

      7. unpow2 99.5%

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

      8. hypot-undefine 99.5%

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

      9. unpow2 99.5%

         \[\leadsto \frac{0.5 \cdot \color{blue}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2}}}{y} \]

      10. associate-*l/ 100.0%

          \[\leadsto \color{blue}{\frac{0.5}{y} \cdot {\left(\mathsf{hypot}\left(x, y\right)\right)}^{2}} \]

      11. *-commutative 100.0%

          \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} \cdot \frac{0.5}{y}} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(y, x\right)\right)}^{2} \cdot \frac{0.5}{y}} \]

   6. Taylor expanded in x around 0 99.5%

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

      1. distribute-lft-out 99.5%

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

   8. Simplified 99.5%

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

      1. unpow2 99.5%

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

      2. *-un-lft-identity 99.5%

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

      3. times-frac 99.5%

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

   10. Applied egg-rr 99.5%

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

       1. /-rgt-identity 99.5%

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

       2. clear-num 99.5%

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

       3. un-div-inv 100.0%

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

   12. Applied egg-rr 100.0%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.1 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{\frac{x}{y}}{\frac{1}{x}}\right)\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{+20} \lor \neg \left(z \leq 10^{+36}\right):\\ \;\;\;\;\frac{z \cdot \frac{z}{y}}{y} \cdot \left(y \cdot -0.5\right) + 0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x}{\frac{y}{x}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 5.2 \cdot 10^{-21}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{\frac{x}{y}}{\frac{1}{x}}\right)\\ \mathbf{elif}\;z\_m \leq 4.4 \cdot 10^{+19} \lor \neg \left(z\_m \leq 2.05 \cdot 10^{+36}\right):\\ \;\;\;\;0.5 \cdot y + \left(y \cdot -0.5\right) \cdot \frac{z\_m}{y \cdot \frac{y}{z\_m}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x}{\frac{y}{x}}\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= z_m 5.2e-21)
   (* 0.5 (+ y (/ (/ x y) (/ 1.0 x))))
   (if (or (<= z_m 4.4e+19) (not (<= z_m 2.05e+36)))
     (+ (* 0.5 y) (* (* y -0.5) (/ z_m (* y (/ y z_m)))))
     (* 0.5 (+ y (/ x (/ y x)))))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double tmp;
	if (z_m <= 5.2e-21) {
		tmp = 0.5 * (y + ((x / y) / (1.0 / x)));
	} else if ((z_m <= 4.4e+19) || !(z_m <= 2.05e+36)) {
		tmp = (0.5 * y) + ((y * -0.5) * (z_m / (y * (y / z_m))));
	} else {
		tmp = 0.5 * (y + (x / (y / x)));
	}
	return tmp;
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 5.2d-21) then
        tmp = 0.5d0 * (y + ((x / y) / (1.0d0 / x)))
    else if ((z_m <= 4.4d+19) .or. (.not. (z_m <= 2.05d+36))) then
        tmp = (0.5d0 * y) + ((y * (-0.5d0)) * (z_m / (y * (y / z_m))))
    else
        tmp = 0.5d0 * (y + (x / (y / x)))
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	double tmp;
	if (z_m <= 5.2e-21) {
		tmp = 0.5 * (y + ((x / y) / (1.0 / x)));
	} else if ((z_m <= 4.4e+19) || !(z_m <= 2.05e+36)) {
		tmp = (0.5 * y) + ((y * -0.5) * (z_m / (y * (y / z_m))));
	} else {
		tmp = 0.5 * (y + (x / (y / x)));
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	tmp = 0
	if z_m <= 5.2e-21:
		tmp = 0.5 * (y + ((x / y) / (1.0 / x)))
	elif (z_m <= 4.4e+19) or not (z_m <= 2.05e+36):
		tmp = (0.5 * y) + ((y * -0.5) * (z_m / (y * (y / z_m))))
	else:
		tmp = 0.5 * (y + (x / (y / x)))
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m)
	tmp = 0.0
	if (z_m <= 5.2e-21)
		tmp = Float64(0.5 * Float64(y + Float64(Float64(x / y) / Float64(1.0 / x))));
	elseif ((z_m <= 4.4e+19) || !(z_m <= 2.05e+36))
		tmp = Float64(Float64(0.5 * y) + Float64(Float64(y * -0.5) * Float64(z_m / Float64(y * Float64(y / z_m)))));
	else
		tmp = Float64(0.5 * Float64(y + Float64(x / Float64(y / x))));
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m)
	tmp = 0.0;
	if (z_m <= 5.2e-21)
		tmp = 0.5 * (y + ((x / y) / (1.0 / x)));
	elseif ((z_m <= 4.4e+19) || ~((z_m <= 2.05e+36)))
		tmp = (0.5 * y) + ((y * -0.5) * (z_m / (y * (y / z_m))));
	else
		tmp = 0.5 * (y + (x / (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := If[LessEqual[z$95$m, 5.2e-21], N[(0.5 * N[(y + N[(N[(x / y), $MachinePrecision] / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z$95$m, 4.4e+19], N[Not[LessEqual[z$95$m, 2.05e+36]], $MachinePrecision]], N[(N[(0.5 * y), $MachinePrecision] + N[(N[(y * -0.5), $MachinePrecision] * N[(z$95$m / N[(y * N[(y / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y + N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 5.20000000000000035e-21

   1. Initial program 74.5%

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

   3. Taylor expanded in z around 0 55.5%

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

      1. associate-*r/ 55.5%

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

      2. rem-square-sqrt 55.5%

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

      3. unpow2 55.5%

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

      4. unpow2 55.5%

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

      5. hypot-undefine 55.5%

         \[\leadsto \frac{0.5 \cdot \left(\color{blue}{\mathsf{hypot}\left(x, y\right)} \cdot \sqrt{{x}^{2} + {y}^{2}}\right)}{y} \]

      6. unpow2 55.5%

         \[\leadsto \frac{0.5 \cdot \left(\mathsf{hypot}\left(x, y\right) \cdot \sqrt{\color{blue}{x \cdot x} + {y}^{2}}\right)}{y} \]

      7. unpow2 55.5%

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

      8. hypot-undefine 55.5%

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

      9. unpow2 55.5%

         \[\leadsto \frac{0.5 \cdot \color{blue}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2}}}{y} \]

      10. associate-*l/ 55.4%

          \[\leadsto \color{blue}{\frac{0.5}{y} \cdot {\left(\mathsf{hypot}\left(x, y\right)\right)}^{2}} \]

      11. *-commutative 55.4%

          \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} \cdot \frac{0.5}{y}} \]

   5. Simplified 55.4%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(y, x\right)\right)}^{2} \cdot \frac{0.5}{y}} \]

   6. Taylor expanded in x around 0 70.2%

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

      1. distribute-lft-out 70.2%

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

   8. Simplified 70.2%

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

      1. unpow2 70.2%

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

      2. *-un-lft-identity 70.2%

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

      3. times-frac 73.7%

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

   10. Applied egg-rr 73.7%

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

       1. clear-num 73.7%

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

       2. associate-*l/ 73.7%

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

       3. *-un-lft-identity 73.7%

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

   12. Applied egg-rr 73.7%

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

   if 5.20000000000000035e-21 < z < 4.4e19 or 2.05000000000000006e36 < z

   1. Initial program 68.1%

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

   3. Taylor expanded in y around inf 63.3%

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

      1. *-commutative 63.3%

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

   5. Simplified 63.3%

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

   6. Taylor expanded in x around 0 71.8%

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

      1. *-commutative 71.8%

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

   8. Simplified 71.8%

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

      1. +-commutative 71.8%

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

      2. distribute-rgt-in 71.8%

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

      3. associate-*l* 71.8%

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

      4. add-sqr-sqrt 71.7%

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

      5. pow2 71.7%

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

      6. sqrt-div 71.7%

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

      7. sqrt-pow1 80.7%

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

      8. metadata-eval 80.7%

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

      9. pow1 80.7%

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

      10. sqrt-pow1 85.3%

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

      11. metadata-eval 85.3%

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

      12. pow1 85.3%

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

   10. Applied egg-rr 85.3%

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

       1. unpow2 85.3%

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

       2. clear-num 85.4%

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

       3. frac-times 85.4%

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

       4. *-un-lft-identity 85.4%

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

   12. Applied egg-rr 85.4%

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

   if 4.4e19 < z < 2.05000000000000006e36

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0 99.5%

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

      1. associate-*r/ 99.5%

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

      2. rem-square-sqrt 99.5%

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

      3. unpow2 99.5%

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

      4. unpow2 99.5%

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

      5. hypot-undefine 99.5%

         \[\leadsto \frac{0.5 \cdot \left(\color{blue}{\mathsf{hypot}\left(x, y\right)} \cdot \sqrt{{x}^{2} + {y}^{2}}\right)}{y} \]

      6. unpow2 99.5%

         \[\leadsto \frac{0.5 \cdot \left(\mathsf{hypot}\left(x, y\right) \cdot \sqrt{\color{blue}{x \cdot x} + {y}^{2}}\right)}{y} \]

      7. unpow2 99.5%

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

      8. hypot-undefine 99.5%

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

      9. unpow2 99.5%

         \[\leadsto \frac{0.5 \cdot \color{blue}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2}}}{y} \]

      10. associate-*l/ 100.0%

          \[\leadsto \color{blue}{\frac{0.5}{y} \cdot {\left(\mathsf{hypot}\left(x, y\right)\right)}^{2}} \]

      11. *-commutative 100.0%

          \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} \cdot \frac{0.5}{y}} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(y, x\right)\right)}^{2} \cdot \frac{0.5}{y}} \]

   6. Taylor expanded in x around 0 99.5%

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

      1. distribute-lft-out 99.5%

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

   8. Simplified 99.5%

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

      1. unpow2 99.5%

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

      2. *-un-lft-identity 99.5%

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

      3. times-frac 99.5%

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

   10. Applied egg-rr 99.5%

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

       1. /-rgt-identity 99.5%

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

       2. clear-num 99.5%

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

       3. un-div-inv 100.0%

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

   12. Applied egg-rr 100.0%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.2 \cdot 10^{-21}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{\frac{x}{y}}{\frac{1}{x}}\right)\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{+19} \lor \neg \left(z \leq 2.05 \cdot 10^{+36}\right):\\ \;\;\;\;0.5 \cdot y + \left(y \cdot -0.5\right) \cdot \frac{z}{y \cdot \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x}{\frac{y}{x}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 5.1 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{\frac{x}{y}}{\frac{1}{x}}\right)\\ \mathbf{elif}\;z\_m \leq 6.2 \cdot 10^{+20} \lor \neg \left(z\_m \leq 1.15 \cdot 10^{+36}\right):\\ \;\;\;\;y \cdot \left(0.5 + -0.5 \cdot \left(\frac{z\_m}{y} \cdot \frac{z\_m}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x}{\frac{y}{x}}\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= z_m 5.1e-20)
   (* 0.5 (+ y (/ (/ x y) (/ 1.0 x))))
   (if (or (<= z_m 6.2e+20) (not (<= z_m 1.15e+36)))
     (* y (+ 0.5 (* -0.5 (* (/ z_m y) (/ z_m y)))))
     (* 0.5 (+ y (/ x (/ y x)))))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double tmp;
	if (z_m <= 5.1e-20) {
		tmp = 0.5 * (y + ((x / y) / (1.0 / x)));
	} else if ((z_m <= 6.2e+20) || !(z_m <= 1.15e+36)) {
		tmp = y * (0.5 + (-0.5 * ((z_m / y) * (z_m / y))));
	} else {
		tmp = 0.5 * (y + (x / (y / x)));
	}
	return tmp;
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 5.1d-20) then
        tmp = 0.5d0 * (y + ((x / y) / (1.0d0 / x)))
    else if ((z_m <= 6.2d+20) .or. (.not. (z_m <= 1.15d+36))) then
        tmp = y * (0.5d0 + ((-0.5d0) * ((z_m / y) * (z_m / y))))
    else
        tmp = 0.5d0 * (y + (x / (y / x)))
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	double tmp;
	if (z_m <= 5.1e-20) {
		tmp = 0.5 * (y + ((x / y) / (1.0 / x)));
	} else if ((z_m <= 6.2e+20) || !(z_m <= 1.15e+36)) {
		tmp = y * (0.5 + (-0.5 * ((z_m / y) * (z_m / y))));
	} else {
		tmp = 0.5 * (y + (x / (y / x)));
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	tmp = 0
	if z_m <= 5.1e-20:
		tmp = 0.5 * (y + ((x / y) / (1.0 / x)))
	elif (z_m <= 6.2e+20) or not (z_m <= 1.15e+36):
		tmp = y * (0.5 + (-0.5 * ((z_m / y) * (z_m / y))))
	else:
		tmp = 0.5 * (y + (x / (y / x)))
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m)
	tmp = 0.0
	if (z_m <= 5.1e-20)
		tmp = Float64(0.5 * Float64(y + Float64(Float64(x / y) / Float64(1.0 / x))));
	elseif ((z_m <= 6.2e+20) || !(z_m <= 1.15e+36))
		tmp = Float64(y * Float64(0.5 + Float64(-0.5 * Float64(Float64(z_m / y) * Float64(z_m / y)))));
	else
		tmp = Float64(0.5 * Float64(y + Float64(x / Float64(y / x))));
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m)
	tmp = 0.0;
	if (z_m <= 5.1e-20)
		tmp = 0.5 * (y + ((x / y) / (1.0 / x)));
	elseif ((z_m <= 6.2e+20) || ~((z_m <= 1.15e+36)))
		tmp = y * (0.5 + (-0.5 * ((z_m / y) * (z_m / y))));
	else
		tmp = 0.5 * (y + (x / (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := If[LessEqual[z$95$m, 5.1e-20], N[(0.5 * N[(y + N[(N[(x / y), $MachinePrecision] / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z$95$m, 6.2e+20], N[Not[LessEqual[z$95$m, 1.15e+36]], $MachinePrecision]], N[(y * N[(0.5 + N[(-0.5 * N[(N[(z$95$m / y), $MachinePrecision] * N[(z$95$m / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y + N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 5.10000000000000019e-20

   1. Initial program 74.5%

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

   3. Taylor expanded in z around 0 55.5%

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

      1. associate-*r/ 55.5%

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

      2. rem-square-sqrt 55.5%

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

      3. unpow2 55.5%

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

      4. unpow2 55.5%

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

      5. hypot-undefine 55.5%

         \[\leadsto \frac{0.5 \cdot \left(\color{blue}{\mathsf{hypot}\left(x, y\right)} \cdot \sqrt{{x}^{2} + {y}^{2}}\right)}{y} \]

      6. unpow2 55.5%

         \[\leadsto \frac{0.5 \cdot \left(\mathsf{hypot}\left(x, y\right) \cdot \sqrt{\color{blue}{x \cdot x} + {y}^{2}}\right)}{y} \]

      7. unpow2 55.5%

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

      8. hypot-undefine 55.5%

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

      9. unpow2 55.5%

         \[\leadsto \frac{0.5 \cdot \color{blue}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2}}}{y} \]

      10. associate-*l/ 55.4%

          \[\leadsto \color{blue}{\frac{0.5}{y} \cdot {\left(\mathsf{hypot}\left(x, y\right)\right)}^{2}} \]

      11. *-commutative 55.4%

          \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} \cdot \frac{0.5}{y}} \]

   5. Simplified 55.4%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(y, x\right)\right)}^{2} \cdot \frac{0.5}{y}} \]

   6. Taylor expanded in x around 0 70.2%

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

      1. distribute-lft-out 70.2%

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

   8. Simplified 70.2%

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

      1. unpow2 70.2%

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

      2. *-un-lft-identity 70.2%

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

      3. times-frac 73.7%

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

   10. Applied egg-rr 73.7%

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

       1. clear-num 73.7%

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

       2. associate-*l/ 73.7%

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

       3. *-un-lft-identity 73.7%

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

   12. Applied egg-rr 73.7%

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

   if 5.10000000000000019e-20 < z < 6.2e20 or 1.14999999999999998e36 < z

   1. Initial program 68.1%

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

   3. Taylor expanded in y around inf 63.3%

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

      1. *-commutative 63.3%

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

   5. Simplified 63.3%

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

   6. Taylor expanded in x around 0 71.8%

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

      1. *-commutative 71.8%

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

   8. Simplified 71.8%

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

      1. unpow2 71.8%

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

      2. unpow2 71.8%

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

      3. times-frac 85.3%

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

   10. Applied egg-rr 85.3%

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

   if 6.2e20 < z < 1.14999999999999998e36

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0 99.5%

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

      1. associate-*r/ 99.5%

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

      2. rem-square-sqrt 99.5%

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

      3. unpow2 99.5%

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

      4. unpow2 99.5%

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

      5. hypot-undefine 99.5%

         \[\leadsto \frac{0.5 \cdot \left(\color{blue}{\mathsf{hypot}\left(x, y\right)} \cdot \sqrt{{x}^{2} + {y}^{2}}\right)}{y} \]

      6. unpow2 99.5%

         \[\leadsto \frac{0.5 \cdot \left(\mathsf{hypot}\left(x, y\right) \cdot \sqrt{\color{blue}{x \cdot x} + {y}^{2}}\right)}{y} \]

      7. unpow2 99.5%

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

      8. hypot-undefine 99.5%

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

      9. unpow2 99.5%

         \[\leadsto \frac{0.5 \cdot \color{blue}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2}}}{y} \]

      10. associate-*l/ 100.0%

          \[\leadsto \color{blue}{\frac{0.5}{y} \cdot {\left(\mathsf{hypot}\left(x, y\right)\right)}^{2}} \]

      11. *-commutative 100.0%

          \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} \cdot \frac{0.5}{y}} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(y, x\right)\right)}^{2} \cdot \frac{0.5}{y}} \]

   6. Taylor expanded in x around 0 99.5%

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

      1. distribute-lft-out 99.5%

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

   8. Simplified 99.5%

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

      1. unpow2 99.5%

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

      2. *-un-lft-identity 99.5%

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

      3. times-frac 99.5%

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

   10. Applied egg-rr 99.5%

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

       1. /-rgt-identity 99.5%

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

       2. clear-num 99.5%

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

       3. un-div-inv 100.0%

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

   12. Applied egg-rr 100.0%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.1 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{\frac{x}{y}}{\frac{1}{x}}\right)\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+20} \lor \neg \left(z \leq 1.15 \cdot 10^{+36}\right):\\ \;\;\;\;y \cdot \left(0.5 + -0.5 \cdot \left(\frac{z}{y} \cdot \frac{z}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x}{\frac{y}{x}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ 0.5 \cdot \left(y + \frac{\frac{x}{y}}{\frac{1}{x}}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m) :precision binary64 (* 0.5 (+ y (/ (/ x y) (/ 1.0 x)))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	return 0.5 * (y + ((x / y) / (1.0 / x)));
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    code = 0.5d0 * (y + ((x / y) / (1.0d0 / x)))
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	return 0.5 * (y + ((x / y) / (1.0 / x)));
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	return 0.5 * (y + ((x / y) / (1.0 / x)))

Julia

z_m = abs(z)
function code(x, y, z_m)
	return Float64(0.5 * Float64(y + Float64(Float64(x / y) / Float64(1.0 / x))))
end

MATLAB

z_m = abs(z);
function tmp = code(x, y, z_m)
	tmp = 0.5 * (y + ((x / y) / (1.0 / x)));
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := N[(0.5 * N[(y + N[(N[(x / y), $MachinePrecision] / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 z around 0 47.3%

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

   1. associate-*r/ 47.3%

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

   2. rem-square-sqrt 47.3%

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

   3. unpow2 47.3%

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

   4. unpow2 47.3%

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

   5. hypot-undefine 47.3%

      \[\leadsto \frac{0.5 \cdot \left(\color{blue}{\mathsf{hypot}\left(x, y\right)} \cdot \sqrt{{x}^{2} + {y}^{2}}\right)}{y} \]

   6. unpow2 47.3%

      \[\leadsto \frac{0.5 \cdot \left(\mathsf{hypot}\left(x, y\right) \cdot \sqrt{\color{blue}{x \cdot x} + {y}^{2}}\right)}{y} \]

   7. unpow2 47.3%

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

   8. hypot-undefine 47.3%

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

   9. unpow2 47.3%

      \[\leadsto \frac{0.5 \cdot \color{blue}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2}}}{y} \]

   10. associate-*l/ 47.2%

       \[\leadsto \color{blue}{\frac{0.5}{y} \cdot {\left(\mathsf{hypot}\left(x, y\right)\right)}^{2}} \]

   11. *-commutative 47.2%

       \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} \cdot \frac{0.5}{y}} \]

5. Simplified 47.2%

   \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(y, x\right)\right)}^{2} \cdot \frac{0.5}{y}} \]

6. Taylor expanded in x around 0 61.1%

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

   1. distribute-lft-out 61.1%

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

8. Simplified 61.1%

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

   1. unpow2 61.1%

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

   2. *-un-lft-identity 61.1%

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

   3. times-frac 64.4%

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

10. Applied egg-rr 64.4%

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

    1. clear-num 64.4%

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

    2. associate-*l/ 64.4%

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

    3. *-un-lft-identity 64.4%

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

12. Applied egg-rr 64.4%

    \[\leadsto 0.5 \cdot \left(y + \color{blue}{\frac{\frac{x}{y}}{\frac{1}{x}}}\right) \]
13. Add Preprocessing

Alternative 8: 67.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 z around 0 47.3%

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

   1. associate-*r/ 47.3%

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

   2. rem-square-sqrt 47.3%

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

   3. unpow2 47.3%

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

   4. unpow2 47.3%

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

   5. hypot-undefine 47.3%

      \[\leadsto \frac{0.5 \cdot \left(\color{blue}{\mathsf{hypot}\left(x, y\right)} \cdot \sqrt{{x}^{2} + {y}^{2}}\right)}{y} \]

   6. unpow2 47.3%

      \[\leadsto \frac{0.5 \cdot \left(\mathsf{hypot}\left(x, y\right) \cdot \sqrt{\color{blue}{x \cdot x} + {y}^{2}}\right)}{y} \]

   7. unpow2 47.3%

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

   8. hypot-undefine 47.3%

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

   9. unpow2 47.3%

      \[\leadsto \frac{0.5 \cdot \color{blue}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2}}}{y} \]

   10. associate-*l/ 47.2%

       \[\leadsto \color{blue}{\frac{0.5}{y} \cdot {\left(\mathsf{hypot}\left(x, y\right)\right)}^{2}} \]

   11. *-commutative 47.2%

       \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} \cdot \frac{0.5}{y}} \]

5. Simplified 47.2%

   \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(y, x\right)\right)}^{2} \cdot \frac{0.5}{y}} \]

6. Taylor expanded in x around 0 61.1%

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

   1. distribute-lft-out 61.1%

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

8. Simplified 61.1%

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

   1. unpow2 61.1%

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

   2. *-un-lft-identity 61.1%

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

   3. times-frac 64.4%

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

10. Applied egg-rr 64.4%

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

    1. /-rgt-identity 64.4%

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

    2. clear-num 64.4%

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

    3. un-div-inv 64.4%

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

12. Applied egg-rr 64.4%

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

Alternative 9: 35.0% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 34.0%

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

   1. *-commutative 34.0%

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

5. Simplified 34.0%

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

6. Final simplification 34.0%

   \[\leadsto 0.5 \cdot y \]
7. Add Preprocessing

Alternative 10: 1.8% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

   1. clear-num 73.2%

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

   2. inv-pow 73.2%

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

   3. associate-/l* 72.8%

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

   4. add-sqr-sqrt 72.8%

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

   5. pow2 72.8%

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

   6. hypot-define 72.8%

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

   7. pow2 72.8%

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

4. Applied egg-rr 72.8%

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

   1. unpow-1 72.8%

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

   2. associate-*r/ 73.2%

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

6. Simplified 73.2%

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

7. Taylor expanded in y around inf 33.9%

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

   1. add-sqr-sqrt 18.6%

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

   2. sqrt-unprod 13.1%

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

   3. associate-/r/ 13.1%

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

   4. metadata-eval 13.1%

      \[\leadsto \sqrt{\left(\color{blue}{0.5} \cdot y\right) \cdot \frac{1}{\frac{2}{y}}} \]

   5. associate-/r/ 13.1%

      \[\leadsto \sqrt{\left(0.5 \cdot y\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot y\right)}} \]

   6. metadata-eval 13.1%

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

   7. swap-sqr 13.1%

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

   8. metadata-eval 13.1%

      \[\leadsto \sqrt{\color{blue}{0.25} \cdot \left(y \cdot y\right)} \]

   9. metadata-eval 13.1%

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

   10. swap-sqr 13.1%

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

   11. sqrt-unprod 0.9%

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

   12. add-sqr-sqrt 1.9%

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

   13. *-commutative 1.9%

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

9. Applied egg-rr 1.9%

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

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 2024085 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

  :alt
  (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x)))

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