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

Percentage Accurate: 68.9% → 93.3%

Time: 18.8s

Alternatives: 4

Speedup: 0.7×

• 41.6% 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: 68.9% 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: 93.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2} \leq -5 \cdot 10^{-98}:\\ \;\;\;\;\frac{1}{\frac{1}{z} \cdot \left(\frac{1}{z} \cdot \left(y\_m \cdot -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(x, \frac{x}{y\_m}, y\_m\right)\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0)) -5e-98)
    (/ 1.0 (* (/ 1.0 z) (* (/ 1.0 z) (* y_m -2.0))))
    (* 0.5 (fma x (/ x y_m) y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (((((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)) <= -5e-98) {
		tmp = 1.0 / ((1.0 / z) * ((1.0 / z) * (y_m * -2.0)));
	} else {
		tmp = 0.5 * fma(x, (x / y_m), y_m);
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0)) <= -5e-98)
		tmp = Float64(1.0 / Float64(Float64(1.0 / z) * Float64(Float64(1.0 / z) * Float64(y_m * -2.0))));
	else
		tmp = Float64(0.5 * fma(x, Float64(x / y_m), y_m));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], -5e-98], N[(1.0 / N[(N[(1.0 / z), $MachinePrecision] * N[(N[(1.0 / z), $MachinePrecision] * N[(y$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x * N[(x / y$95$m), $MachinePrecision] + y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -5.00000000000000018e-98

   1. Initial program 75.5%

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

   3. Taylor expanded in z around inf 27.7%

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

      1. associate-*r/ 27.7%

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

      2. metadata-eval 27.7%

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

      3. distribute-lft-neg-in 27.7%

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

      4. *-commutative 27.7%

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

      5. distribute-neg-frac 27.7%

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

      6. associate-*r/ 27.7%

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

      7. distribute-rgt-neg-in 27.7%

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

      8. distribute-neg-frac 27.7%

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

      9. metadata-eval 27.7%

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

   5. Simplified 27.7%

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

      1. associate-*r/ 27.7%

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

      2. clear-num 27.7%

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

   7. Applied egg-rr 27.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{{z}^{2} \cdot -0.5}}} \]
   8. Step-by-step derivation

      1. add-cube-cbrt 27.5%

         \[\leadsto \frac{1}{\color{blue}{\left(\sqrt[3]{\frac{y}{{z}^{2} \cdot -0.5}} \cdot \sqrt[3]{\frac{y}{{z}^{2} \cdot -0.5}}\right) \cdot \sqrt[3]{\frac{y}{{z}^{2} \cdot -0.5}}}} \]

      2. pow3 27.5%

         \[\leadsto \frac{1}{\color{blue}{{\left(\sqrt[3]{\frac{y}{{z}^{2} \cdot -0.5}}\right)}^{3}}} \]

      3. *-un-lft-identity 27.5%

         \[\leadsto \frac{1}{{\left(\sqrt[3]{\frac{\color{blue}{1 \cdot y}}{{z}^{2} \cdot -0.5}}\right)}^{3}} \]

      4. times-frac 27.6%

         \[\leadsto \frac{1}{{\left(\sqrt[3]{\color{blue}{\frac{1}{{z}^{2}} \cdot \frac{y}{-0.5}}}\right)}^{3}} \]

      5. pow-flip 27.6%

         \[\leadsto \frac{1}{{\left(\sqrt[3]{\color{blue}{{z}^{\left(-2\right)}} \cdot \frac{y}{-0.5}}\right)}^{3}} \]

      6. metadata-eval 27.6%

         \[\leadsto \frac{1}{{\left(\sqrt[3]{{z}^{\color{blue}{-2}} \cdot \frac{y}{-0.5}}\right)}^{3}} \]

   9. Applied egg-rr 27.6%

      \[\leadsto \frac{1}{\color{blue}{{\left(\sqrt[3]{{z}^{-2} \cdot \frac{y}{-0.5}}\right)}^{3}}} \]
   10. Step-by-step derivation

       1. rem-cube-cbrt 27.7%

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

       2. sqr-pow 27.7%

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

       3. associate-*l* 28.5%

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

       4. metadata-eval 28.5%

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

       5. unpow-1 28.5%

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

       6. metadata-eval 28.5%

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

       7. unpow-1 28.5%

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

       8. div-inv 28.5%

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

       9. metadata-eval 28.5%

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

   11. Applied egg-rr 28.5%

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

   if -5.00000000000000018e-98 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 59.2%

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

      1. remove-double-neg 59.2%

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

      2. distribute-lft-neg-out 59.2%

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

      3. distribute-frac-neg2 59.2%

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

      4. distribute-frac-neg 59.2%

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

      5. neg-mul-1 59.2%

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

      6. distribute-lft-neg-out 59.2%

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

      7. *-commutative 59.2%

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

      8. distribute-lft-neg-in 59.2%

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

      9. times-frac 59.2%

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

      10. metadata-eval 59.2%

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

      11. metadata-eval 59.2%

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

      12. associate--l+ 59.2%

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

      13. fma-define 64.0%

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

   3. Simplified 64.0%

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

   5. Taylor expanded in z around inf 49.7%

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

      1. associate--l+ 49.7%

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

      2. unpow2 49.7%

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

      3. times-frac 59.1%

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

      4. fma-neg 59.1%

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

      5. distribute-neg-frac 59.1%

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

      6. metadata-eval 59.1%

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

   7. Simplified 59.1%

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

      1. *-un-lft-identity 59.1%

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

      2. unpow2 59.1%

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

      3. times-frac 59.5%

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

   9. Applied egg-rr 59.5%

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

   10. Taylor expanded in z around 0 58.4%

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

       1. +-commutative 58.4%

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

       2. unpow2 58.4%

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

       3. associate-/l* 65.3%

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

       4. fma-define 65.3%

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

   12. Simplified 65.3%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -5 \cdot 10^{-98}:\\ \;\;\;\;\frac{1}{\frac{1}{z} \cdot \left(\frac{1}{z} \cdot \left(y \cdot -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(x, \frac{x}{y}, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 43.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 1.85 \cdot 10^{+49} \lor \neg \left(z \leq 1.5 \cdot 10^{+71}\right) \land z \leq 2.55 \cdot 10^{+124}:\\ \;\;\;\;y\_m \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{z} \cdot \left(\frac{1}{z} \cdot \left(y\_m \cdot -2\right)\right)}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (or (<= z 1.85e+49) (and (not (<= z 1.5e+71)) (<= z 2.55e+124)))
    (* y_m 0.5)
    (/ 1.0 (* (/ 1.0 z) (* (/ 1.0 z) (* y_m -2.0)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z <= 1.85e+49) || (!(z <= 1.5e+71) && (z <= 2.55e+124))) {
		tmp = y_m * 0.5;
	} else {
		tmp = 1.0 / ((1.0 / z) * ((1.0 / z) * (y_m * -2.0)));
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= 1.85d+49) .or. (.not. (z <= 1.5d+71)) .and. (z <= 2.55d+124)) then
        tmp = y_m * 0.5d0
    else
        tmp = 1.0d0 / ((1.0d0 / z) * ((1.0d0 / z) * (y_m * (-2.0d0))))
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z <= 1.85e+49) || (!(z <= 1.5e+71) && (z <= 2.55e+124))) {
		tmp = y_m * 0.5;
	} else {
		tmp = 1.0 / ((1.0 / z) * ((1.0 / z) * (y_m * -2.0)));
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if (z <= 1.85e+49) or (not (z <= 1.5e+71) and (z <= 2.55e+124)):
		tmp = y_m * 0.5
	else:
		tmp = 1.0 / ((1.0 / z) * ((1.0 / z) * (y_m * -2.0)))
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if ((z <= 1.85e+49) || (!(z <= 1.5e+71) && (z <= 2.55e+124)))
		tmp = Float64(y_m * 0.5);
	else
		tmp = Float64(1.0 / Float64(Float64(1.0 / z) * Float64(Float64(1.0 / z) * Float64(y_m * -2.0))));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if ((z <= 1.85e+49) || (~((z <= 1.5e+71)) && (z <= 2.55e+124)))
		tmp = y_m * 0.5;
	else
		tmp = 1.0 / ((1.0 / z) * ((1.0 / z) * (y_m * -2.0)));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[Or[LessEqual[z, 1.85e+49], And[N[Not[LessEqual[z, 1.5e+71]], $MachinePrecision], LessEqual[z, 2.55e+124]]], N[(y$95$m * 0.5), $MachinePrecision], N[(1.0 / N[(N[(1.0 / z), $MachinePrecision] * N[(N[(1.0 / z), $MachinePrecision] * N[(y$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.85000000000000009e49 or 1.50000000000000006e71 < z < 2.5499999999999999e124

   1. Initial program 67.1%

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

   3. Taylor expanded in y around inf 44.3%

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

      1. *-commutative 44.3%

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

   5. Simplified 44.3%

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

   if 1.85000000000000009e49 < z < 1.50000000000000006e71 or 2.5499999999999999e124 < z

   1. Initial program 62.4%

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

   3. Taylor expanded in z around inf 70.0%

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

      1. associate-*r/ 70.0%

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

      2. metadata-eval 70.0%

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

      3. distribute-lft-neg-in 70.0%

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

      4. *-commutative 70.0%

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

      5. distribute-neg-frac 70.0%

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

      6. associate-*r/ 70.0%

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

      7. distribute-rgt-neg-in 70.0%

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

      8. distribute-neg-frac 70.0%

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

      9. metadata-eval 70.0%

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

   5. Simplified 70.0%

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

      1. associate-*r/ 70.0%

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

      2. clear-num 70.0%

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

   7. Applied egg-rr 70.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{{z}^{2} \cdot -0.5}}} \]
   8. Step-by-step derivation

      1. add-cube-cbrt 69.9%

         \[\leadsto \frac{1}{\color{blue}{\left(\sqrt[3]{\frac{y}{{z}^{2} \cdot -0.5}} \cdot \sqrt[3]{\frac{y}{{z}^{2} \cdot -0.5}}\right) \cdot \sqrt[3]{\frac{y}{{z}^{2} \cdot -0.5}}}} \]

      2. pow3 69.9%

         \[\leadsto \frac{1}{\color{blue}{{\left(\sqrt[3]{\frac{y}{{z}^{2} \cdot -0.5}}\right)}^{3}}} \]

      3. *-un-lft-identity 69.9%

         \[\leadsto \frac{1}{{\left(\sqrt[3]{\frac{\color{blue}{1 \cdot y}}{{z}^{2} \cdot -0.5}}\right)}^{3}} \]

      4. times-frac 69.9%

         \[\leadsto \frac{1}{{\left(\sqrt[3]{\color{blue}{\frac{1}{{z}^{2}} \cdot \frac{y}{-0.5}}}\right)}^{3}} \]

      5. pow-flip 70.7%

         \[\leadsto \frac{1}{{\left(\sqrt[3]{\color{blue}{{z}^{\left(-2\right)}} \cdot \frac{y}{-0.5}}\right)}^{3}} \]

      6. metadata-eval 70.7%

         \[\leadsto \frac{1}{{\left(\sqrt[3]{{z}^{\color{blue}{-2}} \cdot \frac{y}{-0.5}}\right)}^{3}} \]

   9. Applied egg-rr 70.7%

      \[\leadsto \frac{1}{\color{blue}{{\left(\sqrt[3]{{z}^{-2} \cdot \frac{y}{-0.5}}\right)}^{3}}} \]
   10. Step-by-step derivation

       1. rem-cube-cbrt 70.8%

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

       2. sqr-pow 70.8%

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

       3. associate-*l* 75.5%

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

       4. metadata-eval 75.5%

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

       5. unpow-1 75.5%

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

       6. metadata-eval 75.5%

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

       7. unpow-1 75.5%

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

       8. div-inv 75.5%

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

       9. metadata-eval 75.5%

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

   11. Applied egg-rr 75.5%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.85 \cdot 10^{+49} \lor \neg \left(z \leq 1.5 \cdot 10^{+71}\right) \land z \leq 2.55 \cdot 10^{+124}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{z} \cdot \left(\frac{1}{z} \cdot \left(y \cdot -2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.05 \cdot 10^{+160}:\\ \;\;\;\;\frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 1.05e+160)
    (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))
    (* y_m 0.5))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.05e+160) {
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	} else {
		tmp = y_m * 0.5;
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 1.05d+160) then
        tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0d0)
    else
        tmp = y_m * 0.5d0
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.05e+160) {
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	} else {
		tmp = y_m * 0.5;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if y_m <= 1.05e+160:
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)
	else:
		tmp = y_m * 0.5
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 1.05e+160)
		tmp = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0));
	else
		tmp = Float64(y_m * 0.5);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (y_m <= 1.05e+160)
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	else
		tmp = y_m * 0.5;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 1.05e+160], N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], N[(y$95$m * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.04999999999999998e160

   1. Initial program 74.4%

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

   if 1.04999999999999998e160 < y

   1. Initial program 8.8%

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

   3. Taylor expanded in y around inf 85.1%

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

      1. *-commutative 85.1%

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

   5. Simplified 85.1%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{+160}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 34.2% accurate, 5.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(y\_m \cdot 0.5\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z) :precision binary64 (* y_s (* y_m 0.5)))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	return y_s * (y_m * 0.5);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = y_s * (y_m * 0.5d0)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	return y_s * (y_m * 0.5);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	return y_s * (y_m * 0.5)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	return Float64(y_s * Float64(y_m * 0.5))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp = code(y_s, x, y_m, z)
	tmp = y_s * (y_m * 0.5);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * N[(y$95$m * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.2%

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

3. Taylor expanded in y around inf 37.6%

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

   1. *-commutative 37.6%

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

5. Simplified 37.6%

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

6. Final simplification 37.6%

   \[\leadsto y \cdot 0.5 \]
7. 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 2024076 
(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)))