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

Percentage Accurate: 69.6% → 99.7%

Time: 11.3s

Alternatives: 5

Speedup: 1.1×

• 40.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.1× speedup

Math

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

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 (* 0.5 (+ y_m (* (/ (- x z) (sqrt y_m)) (/ (+ x z) (sqrt 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) {
	return y_s * (0.5 * (y_m + (((x - z) / sqrt(y_m)) * ((x + z) / sqrt(y_m)))));
}

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 * (0.5d0 * (y_m + (((x - z) / sqrt(y_m)) * ((x + z) / sqrt(y_m)))))
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 * (0.5 * (y_m + (((x - z) / Math.sqrt(y_m)) * ((x + z) / Math.sqrt(y_m)))));
}

Python

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

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	return Float64(y_s * Float64(0.5 * Float64(y_m + Float64(Float64(Float64(x - z) / sqrt(y_m)) * Float64(Float64(x + z) / sqrt(y_m))))))
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 * (0.5 * (y_m + (((x - z) / sqrt(y_m)) * ((x + z) / sqrt(y_m)))));
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[(0.5 * N[(y$95$m + N[(N[(N[(x - z), $MachinePrecision] / N[Sqrt[y$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(x + z), $MachinePrecision] / N[Sqrt[y$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.1%

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

   1. remove-double-neg 71.1%

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

   2. distribute-lft-neg-out 71.1%

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

   3. distribute-frac-neg2 71.1%

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

   4. distribute-frac-neg 71.1%

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

   5. neg-mul-1 71.1%

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

   6. distribute-lft-neg-out 71.1%

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

   7. *-commutative 71.1%

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

   8. distribute-lft-neg-in 71.1%

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

   9. times-frac 71.1%

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

   10. metadata-eval 71.1%

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

   11. metadata-eval 71.1%

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

   12. associate--l+ 71.1%

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

   13. fma-define 72.2%

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

3. Simplified 72.2%

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

5. Taylor expanded in x around 0 82.7%

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

   1. associate--l+ 82.7%

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

   2. div-sub 85.4%

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

7. Simplified 85.4%

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

   1. add-cbrt-cube 69.0%

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

   2. pow3 69.0%

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

9. Applied egg-rr 69.0%

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

    1. rem-cbrt-cube 85.4%

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

    2. unpow2 85.4%

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

    3. unpow2 85.4%

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

    4. difference-of-squares 89.4%

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

11. Applied egg-rr 89.4%

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

    1. *-commutative 89.4%

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

    2. add-sqr-sqrt 44.5%

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

    3. times-frac 48.7%

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

13. Applied egg-rr 48.7%

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

14. Final simplification 48.7%

    \[\leadsto 0.5 \cdot \left(y + \frac{x - z}{\sqrt{y}} \cdot \frac{x + z}{\sqrt{y}}\right) \]
15. Add Preprocessing

Alternative 2: 94.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2} \leq 0:\\ \;\;\;\;0.5 \cdot \left(y\_m - z \cdot \frac{z}{y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y\_m + x \cdot \frac{x}{y\_m}\right)\\ \end{array} \end{array} \]

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)) 0.0)
    (* 0.5 (- y_m (* z (/ z y_m))))
    (* 0.5 (+ y_m (* x (/ x 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)) <= 0.0) {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	} else {
		tmp = 0.5 * (y_m + (x * (x / y_m)));
	}
	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 (((((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0d0)) <= 0.0d0) then
        tmp = 0.5d0 * (y_m - (z * (z / y_m)))
    else
        tmp = 0.5d0 * (y_m + (x * (x / y_m)))
    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 (((((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)) <= 0.0) {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	} else {
		tmp = 0.5 * (y_m + (x * (x / y_m)));
	}
	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 ((((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)) <= 0.0:
		tmp = 0.5 * (y_m - (z * (z / y_m)))
	else:
		tmp = 0.5 * (y_m + (x * (x / 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)) <= 0.0)
		tmp = Float64(0.5 * Float64(y_m - Float64(z * Float64(z / y_m))));
	else
		tmp = Float64(0.5 * Float64(y_m + Float64(x * Float64(x / y_m))));
	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 (((((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)) <= 0.0)
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	else
		tmp = 0.5 * (y_m + (x * (x / y_m)));
	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[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], 0.0], N[(0.5 * N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y$95$m + N[(x * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $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))) < 0.0

   1. Initial program 74.0%

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

      1. remove-double-neg 74.0%

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

      2. distribute-lft-neg-out 74.0%

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

      3. distribute-frac-neg2 74.0%

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

      4. distribute-frac-neg 74.0%

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

      5. neg-mul-1 74.0%

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

      6. distribute-lft-neg-out 74.0%

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

      7. *-commutative 74.0%

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

      8. distribute-lft-neg-in 74.0%

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

      9. times-frac 74.0%

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

      10. metadata-eval 74.0%

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

      11. metadata-eval 74.0%

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

      12. associate--l+ 74.0%

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

      13. fma-define 74.0%

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

   3. Simplified 74.0%

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

   5. Taylor expanded in x around 0 85.3%

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

      1. associate--l+ 85.3%

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

      2. div-sub 90.3%

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

   7. Simplified 90.3%

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

      1. add-cbrt-cube 71.0%

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

      2. pow3 71.0%

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

   9. Applied egg-rr 71.0%

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

       1. rem-cbrt-cube 90.3%

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

       2. unpow2 90.3%

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

       3. unpow2 90.3%

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

       4. difference-of-squares 90.3%

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

   11. Applied egg-rr 90.3%

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

   12. Taylor expanded in x around 0 57.1%

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

       1. +-commutative 57.1%

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

       2. distribute-lft-in 54.6%

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

       3. associate-+l+ 54.8%

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

       4. *-commutative 54.8%

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

       5. associate-*l* 54.8%

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

       6. associate-*l/ 57.2%

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

       7. associate-*r/ 59.6%

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

       8. *-commutative 59.6%

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

       9. associate-*l* 59.6%

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

       10. *-commutative 59.6%

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

       11. associate-*r/ 61.2%

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

       12. associate-*l/ 58.8%

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

       13. *-commutative 58.8%

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

       14. mul-1-neg 58.8%

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

       15. distribute-frac-neg2 58.8%

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

       16. unpow2 58.8%

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

       17. associate-/l* 61.6%

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

       18. distribute-neg-frac2 61.6%

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

       19. mul-1-neg 61.6%

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

       20. distribute-lft-in 62.4%

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

       21. +-commutative 62.4%

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

   14. Simplified 66.6%

       \[\leadsto 0.5 \cdot \left(y + \color{blue}{z \cdot \left(\frac{z}{-y} + 0\right)}\right) \]

   if 0.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 68.4%

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

      1. remove-double-neg 68.4%

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

      2. distribute-lft-neg-out 68.4%

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

      3. distribute-frac-neg2 68.4%

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

      4. distribute-frac-neg 68.4%

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

      5. neg-mul-1 68.4%

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

      6. distribute-lft-neg-out 68.4%

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

      7. *-commutative 68.4%

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

      8. distribute-lft-neg-in 68.4%

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

      9. times-frac 68.4%

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

      10. metadata-eval 68.4%

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

      11. metadata-eval 68.4%

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

      12. associate--l+ 68.4%

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

      13. fma-define 70.6%

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

   3. Simplified 70.6%

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

   5. Taylor expanded in x around 0 80.3%

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

      1. associate--l+ 80.3%

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

      2. div-sub 81.1%

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

   7. Simplified 81.1%

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

      1. add-cbrt-cube 67.1%

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

      2. pow3 67.1%

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

   9. Applied egg-rr 67.1%

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

       1. rem-cbrt-cube 81.1%

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

       2. unpow2 81.1%

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

       3. unpow2 81.1%

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

       4. difference-of-squares 88.6%

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

   11. Applied egg-rr 88.6%

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

   12. Taylor expanded in z around 0 49.6%

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

       1. distribute-rgt-in 48.1%

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

       2. associate-+l+ 48.1%

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

       3. associate-*l* 48.1%

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

       4. *-commutative 48.1%

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

       5. associate-*r/ 48.9%

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

       6. associate-*l/ 52.6%

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

       7. associate-*l* 52.6%

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

       8. *-commutative 52.6%

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

       9. associate-*l/ 53.9%

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

       10. associate-*r/ 54.0%

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

       11. unpow2 54.0%

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

       12. associate-/l* 60.1%

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

       13. distribute-lft-in 61.6%

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

       14. +-commutative 61.6%

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

       15. +-commutative 61.6%

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

       16. distribute-lft-in 64.6%

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

       17. associate-+l+ 64.6%

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

       18. distribute-rgt1-in 64.6%

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

       19. metadata-eval 64.6%

           \[\leadsto 0.5 \cdot \left(y + x \cdot \left(\frac{x}{y} + \color{blue}{0} \cdot \frac{z}{y}\right)\right) \]

       20. mul0-lft 67.6%

           \[\leadsto 0.5 \cdot \left(y + x \cdot \left(\frac{x}{y} + \color{blue}{0}\right)\right) \]

   14. Simplified 67.6%

       \[\leadsto 0.5 \cdot \left(y + \color{blue}{x \cdot \left(\frac{x}{y} + 0\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq 0:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 2.05 \cdot 10^{+176}:\\ \;\;\;\;0.5 \cdot \left(y\_m + x \cdot \frac{x}{y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y\_m}\right)\\ \end{array} \end{array} \]

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 (<= z 2.05e+176)
    (* 0.5 (+ y_m (* x (/ x y_m))))
    (* z (* z (/ -0.5 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 (z <= 2.05e+176) {
		tmp = 0.5 * (y_m + (x * (x / y_m)));
	} else {
		tmp = z * (z * (-0.5 / y_m));
	}
	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 <= 2.05d+176) then
        tmp = 0.5d0 * (y_m + (x * (x / y_m)))
    else
        tmp = z * (z * ((-0.5d0) / y_m))
    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 <= 2.05e+176) {
		tmp = 0.5 * (y_m + (x * (x / y_m)));
	} else {
		tmp = z * (z * (-0.5 / y_m));
	}
	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 <= 2.05e+176:
		tmp = 0.5 * (y_m + (x * (x / y_m)))
	else:
		tmp = z * (z * (-0.5 / 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 (z <= 2.05e+176)
		tmp = Float64(0.5 * Float64(y_m + Float64(x * Float64(x / y_m))));
	else
		tmp = Float64(z * Float64(z * Float64(-0.5 / y_m)));
	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 <= 2.05e+176)
		tmp = 0.5 * (y_m + (x * (x / y_m)));
	else
		tmp = z * (z * (-0.5 / y_m));
	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[z, 2.05e+176], N[(0.5 * N[(y$95$m + N[(x * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(-0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.05e176

   1. Initial program 72.2%

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

      1. remove-double-neg 72.2%

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

      2. distribute-lft-neg-out 72.2%

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

      3. distribute-frac-neg2 72.2%

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

      4. distribute-frac-neg 72.2%

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

      5. neg-mul-1 72.2%

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

      6. distribute-lft-neg-out 72.2%

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

      7. *-commutative 72.2%

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

      8. distribute-lft-neg-in 72.2%

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

      9. times-frac 72.2%

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

      10. metadata-eval 72.2%

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

      11. metadata-eval 72.2%

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

      12. associate--l+ 72.2%

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

      13. fma-define 72.6%

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

   3. Simplified 72.6%

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

   5. Taylor expanded in x around 0 84.4%

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

      1. associate--l+ 84.4%

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

      2. div-sub 87.4%

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

   7. Simplified 87.4%

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

      1. add-cbrt-cube 69.4%

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

      2. pow3 69.4%

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

   9. Applied egg-rr 69.4%

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

       1. rem-cbrt-cube 87.4%

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

       2. unpow2 87.4%

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

       3. unpow2 87.4%

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

       4. difference-of-squares 90.5%

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

   11. Applied egg-rr 90.5%

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

   12. Taylor expanded in z around 0 55.6%

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

       1. distribute-rgt-in 54.8%

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

       2. associate-+l+ 54.8%

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

       3. associate-*l* 54.8%

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

       4. *-commutative 54.8%

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

       5. associate-*r/ 56.6%

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

       6. associate-*l/ 58.7%

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

       7. associate-*l* 58.7%

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

       8. *-commutative 58.7%

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

       9. associate-*l/ 60.7%

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

       10. associate-*r/ 60.8%

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

       11. unpow2 60.8%

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

       12. associate-/l* 67.4%

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

       13. distribute-lft-in 68.7%

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

       14. +-commutative 68.7%

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

       15. +-commutative 68.7%

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

       16. distribute-lft-in 71.3%

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

       17. associate-+l+ 71.2%

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

       18. distribute-rgt1-in 71.2%

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

       19. metadata-eval 71.2%

           \[\leadsto 0.5 \cdot \left(y + x \cdot \left(\frac{x}{y} + \color{blue}{0} \cdot \frac{z}{y}\right)\right) \]

       20. mul0-lft 73.8%

           \[\leadsto 0.5 \cdot \left(y + x \cdot \left(\frac{x}{y} + \color{blue}{0}\right)\right) \]

   14. Simplified 73.8%

       \[\leadsto 0.5 \cdot \left(y + \color{blue}{x \cdot \left(\frac{x}{y} + 0\right)}\right) \]

   if 2.05e176 < z

   1. Initial program 59.4%

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

   3. Taylor expanded in z around inf 73.7%

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

      1. associate-*r/ 73.7%

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

      2. metadata-eval 73.7%

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

      3. distribute-lft-neg-in 73.7%

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

      4. *-commutative 73.7%

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

      5. distribute-neg-frac 73.7%

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

      6. associate-*r/ 73.7%

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

      7. distribute-rgt-neg-in 73.7%

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

      8. distribute-neg-frac 73.7%

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

      9. metadata-eval 73.7%

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

   5. Simplified 73.7%

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

      1. add-cbrt-cube 73.7%

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

      2. pow3 73.7%

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

   7. Applied egg-rr 73.7%

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

      1. rem-cbrt-cube 73.7%

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

      2. *-commutative 73.7%

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

      3. unpow2 73.7%

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

      4. associate-*r* 86.3%

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

   9. Applied egg-rr 86.3%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.05 \cdot 10^{+176}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 3 \cdot 10^{+38}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\_m\\ \end{array} \end{array} \]

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 3e+38) (* z (* z (/ -0.5 y_m))) (* 0.5 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 (y_m <= 3e+38) {
		tmp = z * (z * (-0.5 / y_m));
	} else {
		tmp = 0.5 * y_m;
	}
	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 <= 3d+38) then
        tmp = z * (z * ((-0.5d0) / y_m))
    else
        tmp = 0.5d0 * y_m
    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 <= 3e+38) {
		tmp = z * (z * (-0.5 / y_m));
	} else {
		tmp = 0.5 * y_m;
	}
	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 <= 3e+38:
		tmp = z * (z * (-0.5 / y_m))
	else:
		tmp = 0.5 * 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 (y_m <= 3e+38)
		tmp = Float64(z * Float64(z * Float64(-0.5 / y_m)));
	else
		tmp = Float64(0.5 * y_m);
	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 <= 3e+38)
		tmp = z * (z * (-0.5 / y_m));
	else
		tmp = 0.5 * y_m;
	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, 3e+38], N[(z * N[(z * N[(-0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * y$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.0000000000000001e38

   1. Initial program 76.8%

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

   3. Taylor expanded in z around inf 35.5%

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

      1. associate-*r/ 35.5%

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

      2. metadata-eval 35.5%

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

      3. distribute-lft-neg-in 35.5%

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

      4. *-commutative 35.5%

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

      5. distribute-neg-frac 35.5%

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

      6. associate-*r/ 35.5%

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

      7. distribute-rgt-neg-in 35.5%

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

      8. distribute-neg-frac 35.5%

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

      9. metadata-eval 35.5%

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

   5. Simplified 35.5%

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

      1. add-cbrt-cube 28.2%

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

      2. pow3 28.2%

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

   7. Applied egg-rr 28.2%

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

      1. rem-cbrt-cube 35.5%

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

      2. *-commutative 35.5%

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

      3. unpow2 35.5%

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

      4. associate-*r* 37.7%

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

   9. Applied egg-rr 37.7%

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

   if 3.0000000000000001e38 < y

   1. Initial program 48.1%

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

   3. Taylor expanded in y around inf 68.6%

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

      1. *-commutative 68.6%

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

   5. Simplified 68.6%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{+38}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 34.2% accurate, 5.0× speedup

Math

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

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 (* 0.5 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) {
	return y_s * (0.5 * y_m);
}

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 * (0.5d0 * y_m)
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 * (0.5 * y_m);
}

Python

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

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	return Float64(y_s * Float64(0.5 * y_m))
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 * (0.5 * y_m);
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[(0.5 * y$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.1%

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

3. Taylor expanded in y around inf 36.9%

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

   1. *-commutative 36.9%

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

5. Simplified 36.9%

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

6. Final simplification 36.9%

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

Developer target: 99.8% 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 2024100 
(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)))