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

Percentage Accurate: 69.0% → 93.2%

Time: 12.1s

Alternatives: 9

Speedup: 1.1×

• 41.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.0% 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: 93.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+195}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \mathsf{fma}\left(x, \frac{x}{y}, y\right), -0.5 \cdot \frac{{z}^{2}}{y}\right)\\ \mathbf{elif}\;z \cdot z \leq 10^{+291}:\\ \;\;\;\;\frac{0.5}{y} \cdot \left({\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - {z}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+195)
   (fma 0.5 (fma x (/ x y) y) (* -0.5 (/ (pow z 2.0) y)))
   (if (<= (* z z) 1e+291)
     (* (/ 0.5 y) (- (pow (hypot x y) 2.0) (pow z 2.0)))
     (* 0.5 (- y (* z (/ z y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+195) {
		tmp = fma(0.5, fma(x, (x / y), y), (-0.5 * (pow(z, 2.0) / y)));
	} else if ((z * z) <= 1e+291) {
		tmp = (0.5 / y) * (pow(hypot(x, y), 2.0) - pow(z, 2.0));
	} else {
		tmp = 0.5 * (y - (z * (z / y)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+195)
		tmp = fma(0.5, fma(x, Float64(x / y), y), Float64(-0.5 * Float64((z ^ 2.0) / y)));
	elseif (Float64(z * z) <= 1e+291)
		tmp = Float64(Float64(0.5 / y) * Float64((hypot(x, y) ^ 2.0) - (z ^ 2.0)));
	else
		tmp = Float64(0.5 * Float64(y - Float64(z * Float64(z / y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+195], N[(0.5 * N[(x * N[(x / y), $MachinePrecision] + y), $MachinePrecision] + N[(-0.5 * N[(N[Power[z, 2.0], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 1e+291], N[(N[(0.5 / y), $MachinePrecision] * N[(N[Power[N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y - N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 1.99999999999999995e195

   1. Initial program 75.9%

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

      1. clear-num 75.7%

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

      2. inv-pow 75.7%

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

      3. *-commutative 75.7%

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

      4. *-un-lft-identity 75.7%

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

      5. times-frac 75.7%

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

      6. metadata-eval 75.7%

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

      7. add-sqr-sqrt 75.7%

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

      8. pow2 75.7%

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

      9. hypot-def 75.7%

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

      10. pow2 75.7%

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

   4. Applied egg-rr 75.7%

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

      1. unpow-1 75.7%

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

      2. associate-/r* 75.7%

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

      3. metadata-eval 75.7%

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

      4. associate-/l* 75.9%

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

      5. add-cube-cbrt 74.8%

         \[\leadsto \frac{0.5 \cdot \left({\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - {z}^{2}\right)}{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}} \]

      6. times-frac 74.8%

         \[\leadsto \color{blue}{\frac{0.5}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - {z}^{2}}{\sqrt[3]{y}}} \]

      7. pow2 74.8%

         \[\leadsto \frac{0.5}{\color{blue}{{\left(\sqrt[3]{y}\right)}^{2}}} \cdot \frac{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - {z}^{2}}{\sqrt[3]{y}} \]

   6. Applied egg-rr 74.8%

      \[\leadsto \color{blue}{\frac{0.5}{{\left(\sqrt[3]{y}\right)}^{2}} \cdot \frac{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - {z}^{2}}{\sqrt[3]{y}}} \]

   7. Taylor expanded in z around inf 89.3%

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

      1. +-commutative 89.3%

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

      2. fma-def 89.3%

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

      3. +-commutative 89.3%

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

      4. unpow2 89.3%

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

      5. associate-*l/ 96.5%

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

      6. *-commutative 96.5%

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

      7. fma-def 96.5%

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

   9. Simplified 96.5%

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

   if 1.99999999999999995e195 < (*.f64 z z) < 9.9999999999999996e290

   1. Initial program 95.2%

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

      1. div-sub 55.2%

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

      2. sub-neg 55.2%

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

      3. div-inv 55.2%

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

      4. add-sqr-sqrt 55.2%

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

      5. pow2 55.2%

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

      6. hypot-def 55.2%

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

      7. *-commutative 55.2%

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

      8. associate-/r* 55.2%

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

      9. metadata-eval 55.2%

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

      10. div-inv 55.2%

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

      11. pow2 55.2%

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

      12. *-commutative 55.2%

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

      13. associate-/r* 55.2%

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

      14. metadata-eval 55.2%

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

   4. Applied egg-rr 55.2%

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

      1. sub-neg 55.2%

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

      2. distribute-rgt-out-- 95.2%

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

   6. Simplified 95.2%

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

   if 9.9999999999999996e290 < (*.f64 z z)

   1. Initial program 54.5%

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

   3. Taylor expanded in x around 0 61.4%

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

      1. div-sub 61.4%

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

      2. unpow2 61.4%

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

      3. associate-/l* 66.8%

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

      4. *-inverses 66.8%

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

      5. /-rgt-identity 66.8%

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

   5. Simplified 66.8%

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

      1. unpow2 66.8%

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

      2. *-un-lft-identity 66.8%

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

      3. times-frac 93.2%

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

   7. Applied egg-rr 93.2%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+195}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \mathsf{fma}\left(x, \frac{x}{y}, y\right), -0.5 \cdot \frac{{z}^{2}}{y}\right)\\ \mathbf{elif}\;z \cdot z \leq 10^{+291}:\\ \;\;\;\;\frac{0.5}{y} \cdot \left({\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - {z}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+79}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y - z, z + y, x \cdot x\right)}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4e+79)
   (/ (fma (- y z) (+ z y) (* x x)) (* y 2.0))
   (* 0.5 (- y (* z (/ z y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4e+79) {
		tmp = fma((y - z), (z + y), (x * x)) / (y * 2.0);
	} else {
		tmp = 0.5 * (y - (z * (z / y)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4e+79)
		tmp = Float64(fma(Float64(y - z), Float64(z + y), Float64(x * x)) / Float64(y * 2.0));
	else
		tmp = Float64(0.5 * Float64(y - Float64(z * Float64(z / y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4e+79], N[(N[(N[(y - z), $MachinePrecision] * N[(z + y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y - N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.99999999999999987e79

   1. Initial program 79.0%

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

      1. associate--l+ 79.0%

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

      2. +-commutative 79.0%

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

      3. sqr-neg 79.0%

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

      4. difference-of-squares 79.2%

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

      5. fma-def 80.6%

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

      6. sub-neg 80.6%

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

      7. sub-neg 80.6%

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

      8. remove-double-neg 80.6%

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

   3. Simplified 80.6%

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

   if 3.99999999999999987e79 < y

   1. Initial program 40.4%

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

   3. Taylor expanded in x around 0 40.5%

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

      1. div-sub 40.5%

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

      2. unpow2 40.5%

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

      3. associate-/l* 71.0%

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

      4. *-inverses 71.0%

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

      5. /-rgt-identity 71.0%

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

   5. Simplified 71.0%

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

      1. unpow2 71.0%

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

      2. *-un-lft-identity 71.0%

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

      3. times-frac 89.1%

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

   7. Applied egg-rr 89.1%

      \[\leadsto 0.5 \cdot \left(y - \color{blue}{\frac{z}{1} \cdot \frac{z}{y}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+79}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y - z, z + y, x \cdot x\right)}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+79}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4e+79)
   (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))
   (* 0.5 (- y (* z (/ z y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4e+79) {
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	} else {
		tmp = 0.5 * (y - (z * (z / y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 4d+79) then
        tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0d0)
    else
        tmp = 0.5d0 * (y - (z * (z / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4e+79) {
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	} else {
		tmp = 0.5 * (y - (z * (z / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4e+79:
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0)
	else:
		tmp = 0.5 * (y - (z * (z / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4e+79)
		tmp = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0));
	else
		tmp = Float64(0.5 * Float64(y - Float64(z * Float64(z / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4e+79)
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	else
		tmp = 0.5 * (y - (z * (z / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4e+79], N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y - N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.99999999999999987e79

   1. Initial program 79.0%

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

   if 3.99999999999999987e79 < y

   1. Initial program 40.4%

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

   3. Taylor expanded in x around 0 40.5%

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

      1. div-sub 40.5%

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

      2. unpow2 40.5%

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

      3. associate-/l* 71.0%

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

      4. *-inverses 71.0%

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

      5. /-rgt-identity 71.0%

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

   5. Simplified 71.0%

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

      1. unpow2 71.0%

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

      2. *-un-lft-identity 71.0%

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

      3. times-frac 89.1%

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

   7. Applied egg-rr 89.1%

      \[\leadsto 0.5 \cdot \left(y - \color{blue}{\frac{z}{1} \cdot \frac{z}{y}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+79}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.7 \cdot 10^{+99}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z \cdot z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{2 \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 4.7e+99) (* 0.5 (- y (/ (* z z) y))) (/ x (* 2.0 (/ y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.7e+99) {
		tmp = 0.5 * (y - ((z * z) / y));
	} else {
		tmp = x / (2.0 * (y / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 4.7d+99) then
        tmp = 0.5d0 * (y - ((z * z) / y))
    else
        tmp = x / (2.0d0 * (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.7e+99) {
		tmp = 0.5 * (y - ((z * z) / y));
	} else {
		tmp = x / (2.0 * (y / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 4.7e+99:
		tmp = 0.5 * (y - ((z * z) / y))
	else:
		tmp = x / (2.0 * (y / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 4.7e+99)
		tmp = Float64(0.5 * Float64(y - Float64(Float64(z * z) / y)));
	else
		tmp = Float64(x / Float64(2.0 * Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 4.7e+99)
		tmp = 0.5 * (y - ((z * z) / y));
	else
		tmp = x / (2.0 * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 4.7e+99], N[(0.5 * N[(y - N[(N[(z * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.69999999999999982e99

   1. Initial program 73.5%

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

   3. Taylor expanded in x around 0 54.5%

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

      1. div-sub 54.6%

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

      2. unpow2 54.6%

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

      3. associate-/l* 70.0%

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

      4. *-inverses 70.0%

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

      5. /-rgt-identity 70.0%

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

   5. Simplified 70.0%

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

      1. unpow2 70.0%

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

   7. Applied egg-rr 70.0%

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

   if 4.69999999999999982e99 < x

   1. Initial program 67.2%

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

   3. Taylor expanded in x around inf 70.0%

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

      1. unpow2 70.0%

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

      2. times-frac 77.1%

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

   5. Applied egg-rr 77.1%

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

      1. clear-num 77.1%

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

      2. frac-times 77.1%

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

      3. *-un-lft-identity 77.1%

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

   7. Applied egg-rr 77.1%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.7 \cdot 10^{+99}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z \cdot z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{2 \cdot \frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{+107}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{2 \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 5.6e+107) (* 0.5 (- y (* z (/ z y)))) (/ x (* 2.0 (/ y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 5.6e+107) {
		tmp = 0.5 * (y - (z * (z / y)));
	} else {
		tmp = x / (2.0 * (y / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 5.6d+107) then
        tmp = 0.5d0 * (y - (z * (z / y)))
    else
        tmp = x / (2.0d0 * (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 5.6e+107) {
		tmp = 0.5 * (y - (z * (z / y)));
	} else {
		tmp = x / (2.0 * (y / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 5.6e+107:
		tmp = 0.5 * (y - (z * (z / y)))
	else:
		tmp = x / (2.0 * (y / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 5.6e+107)
		tmp = Float64(0.5 * Float64(y - Float64(z * Float64(z / y))));
	else
		tmp = Float64(x / Float64(2.0 * Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 5.6e+107)
		tmp = 0.5 * (y - (z * (z / y)));
	else
		tmp = x / (2.0 * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 5.6e+107], N[(0.5 * N[(y - N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.59999999999999969e107

   1. Initial program 73.2%

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

   3. Taylor expanded in x around 0 54.3%

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

      1. div-sub 54.3%

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

      2. unpow2 54.3%

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

      3. associate-/l* 69.7%

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

      4. *-inverses 69.7%

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

      5. /-rgt-identity 69.7%

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

   5. Simplified 69.7%

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

      1. unpow2 69.7%

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

      2. *-un-lft-identity 69.7%

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

      3. times-frac 76.2%

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

   7. Applied egg-rr 76.2%

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

   if 5.59999999999999969e107 < x

   1. Initial program 68.8%

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

   3. Taylor expanded in x around inf 71.8%

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

      1. unpow2 71.8%

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

      2. times-frac 79.1%

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

   5. Applied egg-rr 79.1%

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

      1. clear-num 79.2%

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

      2. frac-times 79.2%

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

      3. *-un-lft-identity 79.2%

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

   7. Applied egg-rr 79.2%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{+107}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{2 \cdot \frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 43.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.46:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.46) (* x (* x (/ 0.5 y))) (* 0.5 y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.46) {
		tmp = x * (x * (0.5 / y));
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 0.46d0) then
        tmp = x * (x * (0.5d0 / y))
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.46) {
		tmp = x * (x * (0.5 / y));
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 0.46:
		tmp = x * (x * (0.5 / y))
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 0.46)
		tmp = Float64(x * Float64(x * Float64(0.5 / y)));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 0.46)
		tmp = x * (x * (0.5 / y));
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 0.46], N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.46000000000000002

   1. Initial program 80.0%

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

   3. Taylor expanded in x around inf 35.1%

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

      1. div-inv 35.1%

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

      2. unpow2 35.1%

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

      3. *-commutative 35.1%

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

      4. associate-/r* 35.1%

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

      5. metadata-eval 35.1%

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

      6. associate-*l* 38.3%

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

   5. Applied egg-rr 38.3%

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

   if 0.46000000000000002 < y

   1. Initial program 45.9%

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

   3. Taylor expanded in y around inf 65.1%

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

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.46:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3000000000:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3000000000.0) (* (/ x y) (/ x 2.0)) (* 0.5 y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3000000000.0) {
		tmp = (x / y) * (x / 2.0);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 3000000000.0d0) then
        tmp = (x / y) * (x / 2.0d0)
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 3000000000.0) {
		tmp = (x / y) * (x / 2.0);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3000000000.0:
		tmp = (x / y) * (x / 2.0)
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3000000000.0)
		tmp = Float64(Float64(x / y) * Float64(x / 2.0));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3000000000.0)
		tmp = (x / y) * (x / 2.0);
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3000000000.0], N[(N[(x / y), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3e9

   1. Initial program 80.0%

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

   3. Taylor expanded in x around inf 35.1%

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

      1. unpow2 35.1%

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

      2. times-frac 38.3%

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

   5. Applied egg-rr 38.3%

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

   if 3e9 < y

   1. Initial program 45.9%

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

   3. Taylor expanded in y around inf 65.1%

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

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3000000000:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.034:\\ \;\;\;\;\frac{x}{2 \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.034) (/ x (* 2.0 (/ y x))) (* 0.5 y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.034) {
		tmp = x / (2.0 * (y / x));
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 0.034d0) then
        tmp = x / (2.0d0 * (y / x))
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.034) {
		tmp = x / (2.0 * (y / x));
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 0.034:
		tmp = x / (2.0 * (y / x))
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 0.034)
		tmp = Float64(x / Float64(2.0 * Float64(y / x)));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 0.034)
		tmp = x / (2.0 * (y / x));
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 0.034], N[(x / N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.034000000000000002

   1. Initial program 80.0%

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

   3. Taylor expanded in x around inf 35.1%

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

      1. unpow2 35.1%

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

      2. times-frac 38.3%

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

   5. Applied egg-rr 38.3%

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

      1. clear-num 38.3%

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

      2. frac-times 38.3%

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

      3. *-un-lft-identity 38.3%

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

   7. Applied egg-rr 38.3%

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

   if 0.034000000000000002 < y

   1. Initial program 45.9%

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

   3. Taylor expanded in y around inf 65.1%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.034:\\ \;\;\;\;\frac{x}{2 \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 34.0% accurate, 5.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (* 0.5 y))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(0.5 * y), $MachinePrecision]

Derivation

1. Initial program 72.5%

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

3. Taylor expanded in y around inf 34.2%

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

4. Final simplification 34.2%

   \[\leadsto 0.5 \cdot y \]
5. 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 2024036 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

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

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