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

Percentage Accurate: 68.9% → 89.0%

Time: 9.1s

Alternatives: 7

Speedup: 0.9×

• 41.9% 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: 68.9% 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: 89.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.1 \cdot 10^{+145}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{{x\_m}^{2} - {z}^{2}}{y}\right)\\ \mathbf{elif}\;x\_m \leq 4.5 \cdot 10^{+171}:\\ \;\;\;\;y \cdot \left(0.5 + \mathsf{fma}\left(x\_m, \frac{x\_m}{2 \cdot {y}^{2}}, \frac{{z}^{2}}{{y}^{2} \cdot -2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x\_m + z\right) \cdot \left(x\_m - z\right)\right) \cdot \frac{0.5}{y}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z)
 :precision binary64
 (if (<= x_m 2.1e+145)
   (* 0.5 (+ y (/ (- (pow x_m 2.0) (pow z 2.0)) y)))
   (if (<= x_m 4.5e+171)
     (*
      y
      (+
       0.5
       (fma
        x_m
        (/ x_m (* 2.0 (pow y 2.0)))
        (/ (pow z 2.0) (* (pow y 2.0) -2.0)))))
     (* (* (+ x_m z) (- x_m z)) (/ 0.5 y)))))

C

x_m = fabs(x);
double code(double x_m, double y, double z) {
	double tmp;
	if (x_m <= 2.1e+145) {
		tmp = 0.5 * (y + ((pow(x_m, 2.0) - pow(z, 2.0)) / y));
	} else if (x_m <= 4.5e+171) {
		tmp = y * (0.5 + fma(x_m, (x_m / (2.0 * pow(y, 2.0))), (pow(z, 2.0) / (pow(y, 2.0) * -2.0))));
	} else {
		tmp = ((x_m + z) * (x_m - z)) * (0.5 / y);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y, z)
	tmp = 0.0
	if (x_m <= 2.1e+145)
		tmp = Float64(0.5 * Float64(y + Float64(Float64((x_m ^ 2.0) - (z ^ 2.0)) / y)));
	elseif (x_m <= 4.5e+171)
		tmp = Float64(y * Float64(0.5 + fma(x_m, Float64(x_m / Float64(2.0 * (y ^ 2.0))), Float64((z ^ 2.0) / Float64((y ^ 2.0) * -2.0)))));
	else
		tmp = Float64(Float64(Float64(x_m + z) * Float64(x_m - z)) * Float64(0.5 / y));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_] := If[LessEqual[x$95$m, 2.1e+145], N[(0.5 * N[(y + N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] - N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$95$m, 4.5e+171], N[(y * N[(0.5 + N[(x$95$m * N[(x$95$m / N[(2.0 * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[z, 2.0], $MachinePrecision] / N[(N[Power[y, 2.0], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m + z), $MachinePrecision] * N[(x$95$m - z), $MachinePrecision]), $MachinePrecision] * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.09999999999999989e145

   1. Initial program 72.5%

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

      1. remove-double-neg 72.5%

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

      2. distribute-lft-neg-out 72.5%

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

      3. distribute-frac-neg2 72.5%

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

      4. distribute-frac-neg 72.5%

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

      5. neg-mul-1 72.5%

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

      6. distribute-lft-neg-out 72.5%

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

      7. *-commutative 72.5%

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

      8. distribute-lft-neg-in 72.5%

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

      9. times-frac 72.5%

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

      10. metadata-eval 72.5%

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

      11. metadata-eval 72.5%

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

      12. associate--l+ 72.5%

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

      13. fma-define 73.5%

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

   3. Simplified 73.5%

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

   5. Taylor expanded in x around 0 81.8%

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

      1. associate--l+ 81.8%

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

      2. div-sub 85.4%

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

   7. Simplified 85.4%

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

   if 2.09999999999999989e145 < x < 4.49999999999999969e171

   1. Initial program 31.0%

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

      1. add-sqr-sqrt 31.0%

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

      2. pow2 31.0%

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

   4. Applied egg-rr 31.0%

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

      1. unpow2 31.0%

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

      2. add-sqr-sqrt 31.0%

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

      3. metadata-eval 31.0%

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

      4. div-inv 31.0%

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

      5. clear-num 31.0%

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

   6. Applied egg-rr 31.0%

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

      1. clear-num 31.0%

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

      2. div-inv 31.0%

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

      3. metadata-eval 31.0%

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

      4. add-sqr-sqrt 31.0%

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

      5. unpow2 31.0%

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

      6. pow-flip 31.0%

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

      7. metadata-eval 31.0%

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

   8. Applied egg-rr 31.0%

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

   9. Taylor expanded in y around inf 52.0%

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

       1. associate--l+ 52.0%

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

       2. unpow2 52.0%

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

       3. rem-square-sqrt 52.4%

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

       4. metadata-eval 52.4%

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

       5. unpow2 52.4%

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

       6. unpow2 52.4%

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

       7. rem-square-sqrt 52.4%

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

       8. associate-/l* 100.0%

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

       9. fmm-def 100.0%

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

       10. *-commutative 100.0%

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

   11. Simplified 100.0%

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

   if 4.49999999999999969e171 < x

   1. Initial program 73.7%

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

      1. remove-double-neg 73.7%

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

      2. distribute-lft-neg-out 73.7%

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

      3. distribute-frac-neg2 73.7%

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

      4. distribute-frac-neg 73.7%

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

      5. neg-mul-1 73.7%

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

      6. distribute-lft-neg-out 73.7%

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

      7. *-commutative 73.7%

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

      8. distribute-lft-neg-in 73.7%

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

      9. times-frac 73.7%

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

      10. metadata-eval 73.7%

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

      11. metadata-eval 73.7%

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

      12. associate--l+ 73.7%

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

      13. fma-define 77.1%

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

   3. Simplified 77.1%

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

   5. Taylor expanded in y around 0 73.7%

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

      1. associate-*r/ 73.7%

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

      2. *-commutative 73.7%

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

      3. associate-/l* 73.7%

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

   7. Simplified 73.7%

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

      1. pow2 73.7%

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

   9. Applied egg-rr 73.7%

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

       1. pow2 73.7%

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

       2. difference-of-squares 84.2%

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

   11. Applied egg-rr 84.2%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{+145}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{{x}^{2} - {z}^{2}}{y}\right)\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+171}:\\ \;\;\;\;y \cdot \left(0.5 + \mathsf{fma}\left(x, \frac{x}{2 \cdot {y}^{2}}, \frac{{z}^{2}}{{y}^{2} \cdot -2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + z\right) \cdot \left(x - z\right)\right) \cdot \frac{0.5}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 9 \cdot 10^{+157}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{{x\_m}^{2} - {z}^{2}}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x\_m \cdot \sqrt{\frac{0.5}{y}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z)
 :precision binary64
 (if (<= x_m 9e+157)
   (* 0.5 (+ y (/ (- (pow x_m 2.0) (pow z 2.0)) y)))
   (pow (* x_m (sqrt (/ 0.5 y))) 2.0)))

C

x_m = fabs(x);
double code(double x_m, double y, double z) {
	double tmp;
	if (x_m <= 9e+157) {
		tmp = 0.5 * (y + ((pow(x_m, 2.0) - pow(z, 2.0)) / y));
	} else {
		tmp = pow((x_m * sqrt((0.5 / y))), 2.0);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y, z)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 9d+157) then
        tmp = 0.5d0 * (y + (((x_m ** 2.0d0) - (z ** 2.0d0)) / y))
    else
        tmp = (x_m * sqrt((0.5d0 / y))) ** 2.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y, double z) {
	double tmp;
	if (x_m <= 9e+157) {
		tmp = 0.5 * (y + ((Math.pow(x_m, 2.0) - Math.pow(z, 2.0)) / y));
	} else {
		tmp = Math.pow((x_m * Math.sqrt((0.5 / y))), 2.0);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z):
	tmp = 0
	if x_m <= 9e+157:
		tmp = 0.5 * (y + ((math.pow(x_m, 2.0) - math.pow(z, 2.0)) / y))
	else:
		tmp = math.pow((x_m * math.sqrt((0.5 / y))), 2.0)
	return tmp

Julia

x_m = abs(x)
function code(x_m, y, z)
	tmp = 0.0
	if (x_m <= 9e+157)
		tmp = Float64(0.5 * Float64(y + Float64(Float64((x_m ^ 2.0) - (z ^ 2.0)) / y)));
	else
		tmp = Float64(x_m * sqrt(Float64(0.5 / y))) ^ 2.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y, z)
	tmp = 0.0;
	if (x_m <= 9e+157)
		tmp = 0.5 * (y + (((x_m ^ 2.0) - (z ^ 2.0)) / y));
	else
		tmp = (x_m * sqrt((0.5 / y))) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_] := If[LessEqual[x$95$m, 9e+157], N[(0.5 * N[(y + N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] - N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(x$95$m * N[Sqrt[N[(0.5 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.9999999999999997e157

   1. Initial program 72.1%

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

      1. remove-double-neg 72.1%

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

      2. distribute-lft-neg-out 72.1%

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

      3. distribute-frac-neg2 72.1%

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

      4. distribute-frac-neg 72.1%

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

      5. neg-mul-1 72.1%

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

      6. distribute-lft-neg-out 72.1%

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

      7. *-commutative 72.1%

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

      8. distribute-lft-neg-in 72.1%

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

      9. times-frac 72.1%

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

      10. metadata-eval 72.1%

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

      11. metadata-eval 72.1%

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

      12. associate--l+ 72.1%

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

      13. fma-define 73.0%

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

   3. Simplified 73.0%

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

   5. Taylor expanded in x around 0 81.6%

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

      1. associate--l+ 81.6%

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

      2. div-sub 85.2%

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

   7. Simplified 85.2%

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

   if 8.9999999999999997e157 < x

   1. Initial program 71.5%

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

      1. remove-double-neg 71.5%

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

      2. distribute-lft-neg-out 71.5%

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

      3. distribute-frac-neg2 71.5%

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

      4. distribute-frac-neg 71.5%

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

      5. neg-mul-1 71.5%

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

      6. distribute-lft-neg-out 71.5%

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

      7. *-commutative 71.5%

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

      8. distribute-lft-neg-in 71.5%

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

      9. times-frac 71.5%

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

      10. metadata-eval 71.5%

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

      11. metadata-eval 71.5%

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

      12. associate--l+ 71.5%

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

      13. fma-define 74.9%

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

   3. Simplified 74.9%

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

   5. Taylor expanded in x around inf 78.4%

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

      1. *-commutative 78.4%

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

      2. associate-*l/ 78.4%

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

      3. associate-*r/ 78.4%

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

   7. Simplified 78.4%

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

      1. add-sqr-sqrt 34.4%

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

      2. pow2 34.4%

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

      3. sqrt-prod 34.4%

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

      4. sqrt-pow1 43.4%

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

      5. metadata-eval 43.4%

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

      6. pow1 43.4%

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

   9. Applied egg-rr 43.4%

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

Alternative 3: 79.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(x\_m, x\_m, y \cdot y - z \cdot z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z)
 :precision binary64
 (if (<= y 1.35e+154)
   (* 0.5 (/ (fma x_m x_m (- (* y y) (* z z))) y))
   (* 0.5 y)))

C

x_m = fabs(x);
double code(double x_m, double y, double z) {
	double tmp;
	if (y <= 1.35e+154) {
		tmp = 0.5 * (fma(x_m, x_m, ((y * y) - (z * z))) / y);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y, z)
	tmp = 0.0
	if (y <= 1.35e+154)
		tmp = Float64(0.5 * Float64(fma(x_m, x_m, Float64(Float64(y * y) - Float64(z * z))) / y));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_] := If[LessEqual[y, 1.35e+154], N[(0.5 * N[(N[(x$95$m * x$95$m + N[(N[(y * y), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.35000000000000003e154

   1. Initial program 79.8%

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

      1. remove-double-neg 79.8%

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

      2. distribute-lft-neg-out 79.8%

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

      3. distribute-frac-neg2 79.8%

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

      4. distribute-frac-neg 79.8%

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

      5. neg-mul-1 79.8%

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

      6. distribute-lft-neg-out 79.8%

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

      7. *-commutative 79.8%

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

      8. distribute-lft-neg-in 79.8%

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

      9. times-frac 79.8%

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

      10. metadata-eval 79.8%

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

      11. metadata-eval 79.8%

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

      12. associate--l+ 79.8%

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

      13. fma-define 81.2%

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

   3. Simplified 81.2%

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

   if 1.35000000000000003e154 < y

   1. Initial program 8.2%

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

      1. remove-double-neg 8.2%

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

      2. distribute-lft-neg-out 8.2%

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

      3. distribute-frac-neg2 8.2%

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

      4. distribute-frac-neg 8.2%

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

      5. neg-mul-1 8.2%

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

      6. distribute-lft-neg-out 8.2%

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

      7. *-commutative 8.2%

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

      8. distribute-lft-neg-in 8.2%

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

      9. times-frac 8.2%

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

      10. metadata-eval 8.2%

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

      11. metadata-eval 8.2%

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

      12. associate--l+ 8.2%

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

      13. fma-define 8.2%

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

   3. Simplified 8.2%

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

   5. Taylor expanded in y around inf 69.0%

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

Alternative 4: 77.2% accurate, 0.7× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z)
 :precision binary64
 (if (<= y 1.35e+154)
   (/ (- (+ (* y y) (* x_m x_m)) (* z z)) (* y 2.0))
   (* 0.5 y)))

C

x_m = fabs(x);
double code(double x_m, double y, double z) {
	double tmp;
	if (y <= 1.35e+154) {
		tmp = (((y * y) + (x_m * x_m)) - (z * z)) / (y * 2.0);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y, z)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.35d+154) then
        tmp = (((y * y) + (x_m * x_m)) - (z * z)) / (y * 2.0d0)
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

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

Python

x_m = math.fabs(x)
def code(x_m, y, z):
	tmp = 0
	if y <= 1.35e+154:
		tmp = (((y * y) + (x_m * x_m)) - (z * z)) / (y * 2.0)
	else:
		tmp = 0.5 * y
	return tmp

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_] := If[LessEqual[y, 1.35e+154], N[(N[(N[(N[(y * y), $MachinePrecision] + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.35000000000000003e154

   1. Initial program 79.8%

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

   if 1.35000000000000003e154 < y

   1. Initial program 8.2%

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

      1. remove-double-neg 8.2%

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

      2. distribute-lft-neg-out 8.2%

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

      3. distribute-frac-neg2 8.2%

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

      4. distribute-frac-neg 8.2%

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

      5. neg-mul-1 8.2%

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

      6. distribute-lft-neg-out 8.2%

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

      7. *-commutative 8.2%

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

      8. distribute-lft-neg-in 8.2%

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

      9. times-frac 8.2%

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

      10. metadata-eval 8.2%

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

      11. metadata-eval 8.2%

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

      12. associate--l+ 8.2%

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

      13. fma-define 8.2%

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

   3. Simplified 8.2%

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

   5. Taylor expanded in y around inf 69.0%

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

4. Final simplification 78.7%

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

Alternative 5: 70.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{+94}:\\ \;\;\;\;\left(\left(x\_m + z\right) \cdot \left(x\_m - z\right)\right) \cdot \frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z)
 :precision binary64
 (if (<= y 1.1e+94) (* (* (+ x_m z) (- x_m z)) (/ 0.5 y)) (* 0.5 y)))

C

x_m = fabs(x);
double code(double x_m, double y, double z) {
	double tmp;
	if (y <= 1.1e+94) {
		tmp = ((x_m + z) * (x_m - z)) * (0.5 / y);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y, z)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.1d+94) then
        tmp = ((x_m + z) * (x_m - z)) * (0.5d0 / y)
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y, double z) {
	double tmp;
	if (y <= 1.1e+94) {
		tmp = ((x_m + z) * (x_m - z)) * (0.5 / y);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z):
	tmp = 0
	if y <= 1.1e+94:
		tmp = ((x_m + z) * (x_m - z)) * (0.5 / y)
	else:
		tmp = 0.5 * y
	return tmp

Julia

x_m = abs(x)
function code(x_m, y, z)
	tmp = 0.0
	if (y <= 1.1e+94)
		tmp = Float64(Float64(Float64(x_m + z) * Float64(x_m - z)) * Float64(0.5 / y));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y, z)
	tmp = 0.0;
	if (y <= 1.1e+94)
		tmp = ((x_m + z) * (x_m - z)) * (0.5 / y);
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_] := If[LessEqual[y, 1.1e+94], N[(N[(N[(x$95$m + z), $MachinePrecision] * N[(x$95$m - z), $MachinePrecision]), $MachinePrecision] * N[(0.5 / y), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.10000000000000006e94

   1. Initial program 79.7%

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

      1. remove-double-neg 79.7%

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

      2. distribute-lft-neg-out 79.7%

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

      3. distribute-frac-neg2 79.7%

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

      4. distribute-frac-neg 79.7%

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

      5. neg-mul-1 79.7%

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

      6. distribute-lft-neg-out 79.7%

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

      7. *-commutative 79.7%

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

      8. distribute-lft-neg-in 79.7%

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

      9. times-frac 79.7%

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

      10. metadata-eval 79.7%

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

      11. metadata-eval 79.7%

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

      12. associate--l+ 79.7%

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

      13. fma-define 81.1%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in y around 0 68.9%

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

      1. associate-*r/ 68.9%

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

      2. *-commutative 68.9%

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

      3. associate-/l* 68.8%

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

   7. Simplified 68.8%

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

      1. pow2 68.8%

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

   9. Applied egg-rr 68.8%

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

       1. pow2 68.8%

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

       2. difference-of-squares 74.0%

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

   11. Applied egg-rr 74.0%

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

   if 1.10000000000000006e94 < y

   1. Initial program 34.8%

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

      1. remove-double-neg 34.8%

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

      2. distribute-lft-neg-out 34.8%

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

      3. distribute-frac-neg2 34.8%

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

      4. distribute-frac-neg 34.8%

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

      5. neg-mul-1 34.8%

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

      6. distribute-lft-neg-out 34.8%

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

      7. *-commutative 34.8%

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

      8. distribute-lft-neg-in 34.8%

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

      9. times-frac 34.8%

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

      10. metadata-eval 34.8%

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

      11. metadata-eval 34.8%

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

      12. associate--l+ 34.8%

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

      13. fma-define 35.1%

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

   3. Simplified 35.1%

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

   5. Taylor expanded in y around inf 63.4%

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

Alternative 6: 41.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{+18}:\\ \;\;\;\;\frac{0.5}{y} \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z)
 :precision binary64
 (if (<= y 3.7e+18) (* (/ 0.5 y) (* x_m x_m)) (* 0.5 y)))

C

x_m = fabs(x);
double code(double x_m, double y, double z) {
	double tmp;
	if (y <= 3.7e+18) {
		tmp = (0.5 / y) * (x_m * x_m);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y, z)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 3.7d+18) then
        tmp = (0.5d0 / y) * (x_m * x_m)
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y, double z) {
	double tmp;
	if (y <= 3.7e+18) {
		tmp = (0.5 / y) * (x_m * x_m);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z):
	tmp = 0
	if y <= 3.7e+18:
		tmp = (0.5 / y) * (x_m * x_m)
	else:
		tmp = 0.5 * y
	return tmp

Julia

x_m = abs(x)
function code(x_m, y, z)
	tmp = 0.0
	if (y <= 3.7e+18)
		tmp = Float64(Float64(0.5 / y) * Float64(x_m * x_m));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y, z)
	tmp = 0.0;
	if (y <= 3.7e+18)
		tmp = (0.5 / y) * (x_m * x_m);
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_] := If[LessEqual[y, 3.7e+18], N[(N[(0.5 / y), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.7e18

   1. Initial program 79.6%

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

      1. remove-double-neg 79.6%

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

      2. distribute-lft-neg-out 79.6%

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

      3. distribute-frac-neg2 79.6%

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

      4. distribute-frac-neg 79.6%

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

      5. neg-mul-1 79.6%

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

      6. distribute-lft-neg-out 79.6%

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

      7. *-commutative 79.6%

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

      8. distribute-lft-neg-in 79.6%

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

      9. times-frac 79.6%

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

      10. metadata-eval 79.6%

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

      11. metadata-eval 79.6%

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

      12. associate--l+ 79.6%

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

      13. fma-define 80.6%

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

   3. Simplified 80.6%

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

   5. Taylor expanded in x around inf 39.6%

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

      1. *-commutative 39.6%

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

      2. associate-*l/ 39.6%

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

      3. associate-*r/ 39.5%

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

   7. Simplified 39.5%

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

      1. pow2 39.5%

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

   9. Applied egg-rr 39.5%

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

   if 3.7e18 < y

   1. Initial program 47.4%

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

      1. remove-double-neg 47.4%

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

      2. distribute-lft-neg-out 47.4%

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

      3. distribute-frac-neg2 47.4%

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

      4. distribute-frac-neg 47.4%

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

      5. neg-mul-1 47.4%

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

      6. distribute-lft-neg-out 47.4%

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

      7. *-commutative 47.4%

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

      8. distribute-lft-neg-in 47.4%

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

      9. times-frac 47.4%

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

      10. metadata-eval 47.4%

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

      11. metadata-eval 47.4%

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

      12. associate--l+ 47.4%

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

      13. fma-define 49.2%

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

   3. Simplified 49.2%

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

   5. Taylor expanded in y around inf 58.2%

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

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{+18}:\\ \;\;\;\;\frac{0.5}{y} \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 34.7% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.0%

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

   1. remove-double-neg 72.0%

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

   2. distribute-lft-neg-out 72.0%

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

   3. distribute-frac-neg2 72.0%

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

   4. distribute-frac-neg 72.0%

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

   5. neg-mul-1 72.0%

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

   6. distribute-lft-neg-out 72.0%

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

   7. *-commutative 72.0%

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

   8. distribute-lft-neg-in 72.0%

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

   9. times-frac 72.0%

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

   10. metadata-eval 72.0%

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

   11. metadata-eval 72.0%

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

   12. associate--l+ 72.0%

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

   13. fma-define 73.2%

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

3. Simplified 73.2%

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

5. Taylor expanded in y around inf 30.7%

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

Developer Target 1: 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 2024163 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* y 1/2) (* (* (/ 1/2 y) (+ z x)) (- z x))))

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