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

Percentage Accurate: 68.6% → 82.9%

Time: 14.5s

Alternatives: 8

Speedup: 1.1×

• 41.2% 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: 68.6% 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: 82.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \sqrt{y_m \cdot 2}\\ t_1 := \frac{-z}{t_0}\\ y_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+182}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{hypot}\left(x, y_m\right), \frac{\mathsf{hypot}\left(x, y_m\right)}{t_0}, z \cdot t_1\right) + \mathsf{fma}\left(t_1, z, z \cdot \frac{z}{t_0}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y_m - z \cdot \frac{z}{y_m}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (sqrt (* y_m 2.0))) (t_1 (/ (- z) t_0)))
   (*
    y_s
    (if (<= (* z z) 1e+182)
      (/
       (+
        (fma (hypot x y_m) (/ (hypot x y_m) t_0) (* z t_1))
        (fma t_1 z (* z (/ z t_0))))
       t_0)
      (* 0.5 (- y_m (* z (/ z 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 t_0 = sqrt((y_m * 2.0));
	double t_1 = -z / t_0;
	double tmp;
	if ((z * z) <= 1e+182) {
		tmp = (fma(hypot(x, y_m), (hypot(x, y_m) / t_0), (z * t_1)) + fma(t_1, z, (z * (z / t_0)))) / t_0;
	} else {
		tmp = 0.5 * (y_m - (z * (z / 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)
	t_0 = sqrt(Float64(y_m * 2.0))
	t_1 = Float64(Float64(-z) / t_0)
	tmp = 0.0
	if (Float64(z * z) <= 1e+182)
		tmp = Float64(Float64(fma(hypot(x, y_m), Float64(hypot(x, y_m) / t_0), Float64(z * t_1)) + fma(t_1, z, Float64(z * Float64(z / t_0)))) / t_0);
	else
		tmp = Float64(0.5 * Float64(y_m - Float64(z * Float64(z / y_m))));
	end
	return Float64(y_s * tmp)
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[Sqrt[N[(y$95$m * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[((-z) / t$95$0), $MachinePrecision]}, N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 1e+182], N[(N[(N[(N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision] * N[(N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision] / t$95$0), $MachinePrecision] + N[(z * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * z + N[(z * N[(z / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(0.5 * N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.0000000000000001e182

   1. Initial program 80.7%

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

      1. *-un-lft-identity 80.7%

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

      2. add-sqr-sqrt 48.2%

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

      3. times-frac 48.1%

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

      4. add-sqr-sqrt 48.1%

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

      5. pow2 48.1%

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

      6. hypot-def 48.1%

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

      7. pow2 48.1%

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

   4. Applied egg-rr 48.1%

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

      1. associate-*l/ 48.1%

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

      2. *-lft-identity 48.1%

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

   6. Simplified 48.1%

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

      1. div-sub 48.1%

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

      2. unpow2 48.1%

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

      3. *-un-lft-identity 48.1%

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

      4. times-frac 54.2%

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

      5. unpow2 54.2%

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

      6. *-un-lft-identity 54.2%

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

      7. times-frac 55.1%

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

      8. prod-diff 55.1%

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

   8. Applied egg-rr 55.1%

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

   if 1.0000000000000001e182 < (*.f64 z z)

   1. Initial program 52.4%

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

   3. Taylor expanded in x around 0 58.2%

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

      1. div-sub 58.2%

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

      2. sub-neg 58.2%

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

      3. unpow2 58.2%

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

      4. associate-/l* 67.1%

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

      5. *-inverses 67.1%

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

      6. /-rgt-identity 67.1%

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

      7. sub-neg 67.1%

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

   5. Simplified 67.1%

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

      1. unpow2 67.1%

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

      2. sqr-neg 67.1%

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

      3. div-inv 67.1%

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

      4. sqr-neg 67.1%

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

      5. associate-*l* 81.6%

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

      6. div-inv 81.6%

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

   7. Applied egg-rr 81.6%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+182}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{hypot}\left(x, y\right), \frac{\mathsf{hypot}\left(x, y\right)}{\sqrt{y \cdot 2}}, z \cdot \frac{-z}{\sqrt{y \cdot 2}}\right) + \mathsf{fma}\left(\frac{-z}{\sqrt{y \cdot 2}}, z, z \cdot \frac{z}{\sqrt{y \cdot 2}}\right)}{\sqrt{y \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 1.75 \cdot 10^{+162}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y_m - z, z + y_m, x \cdot x\right)}{y_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y_m - z \cdot \frac{z}{y_m}\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 1.75e+162)
    (/ (fma (- y_m z) (+ z y_m) (* x x)) (* y_m 2.0))
    (* 0.5 (- y_m (* z (/ z 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 <= 1.75e+162) {
		tmp = fma((y_m - z), (z + y_m), (x * x)) / (y_m * 2.0);
	} else {
		tmp = 0.5 * (y_m - (z * (z / 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 <= 1.75e+162)
		tmp = Float64(fma(Float64(y_m - z), Float64(z + y_m), Float64(x * x)) / Float64(y_m * 2.0));
	else
		tmp = Float64(0.5 * Float64(y_m - Float64(z * Float64(z / y_m))));
	end
	return Float64(y_s * tmp)
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 1.75e+162], N[(N[(N[(y$95$m - z), $MachinePrecision] * N[(z + y$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.75000000000000009e162

   1. Initial program 78.5%

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

      1. associate--l+ 78.5%

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

      2. +-commutative 78.5%

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

      3. sqr-neg 78.5%

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

      4. difference-of-squares 79.1%

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

      5. fma-def 81.3%

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

      6. sub-neg 81.3%

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

      7. sub-neg 81.3%

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

      8. remove-double-neg 81.3%

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

   3. Simplified 81.3%

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

   if 1.75000000000000009e162 < y

   1. Initial program 8.9%

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

   3. Taylor expanded in x around 0 8.9%

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

      1. div-sub 8.9%

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

      2. sub-neg 8.9%

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

      3. unpow2 8.9%

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

      4. associate-/l* 75.2%

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

      5. *-inverses 75.2%

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

      6. /-rgt-identity 75.2%

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

      7. sub-neg 75.2%

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

   5. Simplified 75.2%

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

      1. unpow2 75.2%

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

      2. sqr-neg 75.2%

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

      3. div-inv 75.2%

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

      4. sqr-neg 75.2%

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

      5. associate-*l* 89.8%

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

      6. div-inv 89.7%

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

   7. Applied egg-rr 89.7%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.75 \cdot 10^{+162}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y - z, z + y, x \cdot x\right)}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 1.2 \cdot 10^{+143}:\\ \;\;\;\;\frac{\left(x \cdot x + y_m \cdot y_m\right) - z \cdot z}{y_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y_m - z \cdot \frac{z}{y_m}\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 1.2e+143)
    (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))
    (* 0.5 (- y_m (* z (/ z 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 <= 1.2e+143) {
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	} else {
		tmp = 0.5 * (y_m - (z * (z / 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 <= 1.2d+143) then
        tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0d0)
    else
        tmp = 0.5d0 * (y_m - (z * (z / 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 <= 1.2e+143) {
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	} else {
		tmp = 0.5 * (y_m - (z * (z / 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 <= 1.2e+143:
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)
	else:
		tmp = 0.5 * (y_m - (z * (z / 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 <= 1.2e+143)
		tmp = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0));
	else
		tmp = Float64(0.5 * Float64(y_m - Float64(z * Float64(z / 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 <= 1.2e+143)
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	else
		tmp = 0.5 * (y_m - (z * (z / 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, 1.2e+143], N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.1999999999999999e143

   1. Initial program 79.5%

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

   if 1.1999999999999999e143 < y

   1. Initial program 18.7%

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

   3. Taylor expanded in x around 0 18.7%

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

      1. div-sub 18.7%

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

      2. sub-neg 18.7%

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

      3. unpow2 18.7%

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

      4. associate-/l* 72.5%

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

      5. *-inverses 72.5%

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

      6. /-rgt-identity 72.5%

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

      7. sub-neg 72.5%

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

   5. Simplified 72.5%

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

      1. unpow2 72.5%

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

      2. sqr-neg 72.5%

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

      3. div-inv 72.5%

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

      4. sqr-neg 72.5%

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

      5. associate-*l* 83.8%

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

      6. div-inv 83.8%

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

   7. Applied egg-rr 83.8%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{+143}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 9.2 \cdot 10^{+16}:\\ \;\;\;\;0.5 \cdot \left(y_m - z \cdot \frac{z}{y_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y_m} \cdot \frac{x}{2}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 9.2e+16) (* 0.5 (- y_m (* z (/ z y_m)))) (* (/ x y_m) (/ x 2.0)))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 9.2e+16) {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	} else {
		tmp = (x / y_m) * (x / 2.0);
	}
	return y_s * tmp;
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 9.2d+16) then
        tmp = 0.5d0 * (y_m - (z * (z / y_m)))
    else
        tmp = (x / y_m) * (x / 2.0d0)
    end if
    code = y_s * tmp
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 9.2e+16) {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	} else {
		tmp = (x / y_m) * (x / 2.0);
	}
	return y_s * tmp;
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if x <= 9.2e+16:
		tmp = 0.5 * (y_m - (z * (z / y_m)))
	else:
		tmp = (x / y_m) * (x / 2.0)
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 9.2e+16)
		tmp = Float64(0.5 * Float64(y_m - Float64(z * Float64(z / y_m))));
	else
		tmp = Float64(Float64(x / y_m) * Float64(x / 2.0));
	end
	return Float64(y_s * tmp)
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (x <= 9.2e+16)
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	else
		tmp = (x / y_m) * (x / 2.0);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 9.2e+16], N[(0.5 * N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y$95$m), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 9.2e16

   1. Initial program 73.3%

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

   3. Taylor expanded in x around 0 50.3%

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

      1. div-sub 50.3%

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

      2. sub-neg 50.3%

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

      3. unpow2 50.3%

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

      4. associate-/l* 64.2%

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

      5. *-inverses 64.2%

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

      6. /-rgt-identity 64.2%

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

      7. sub-neg 64.2%

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

   5. Simplified 64.2%

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

      1. unpow2 64.2%

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

      2. sqr-neg 64.2%

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

      3. div-inv 64.2%

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

      4. sqr-neg 64.2%

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

      5. associate-*l* 70.6%

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

      6. div-inv 70.7%

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

   7. Applied egg-rr 70.7%

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

   if 9.2e16 < x

   1. Initial program 62.9%

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

   3. Taylor expanded in x around inf 61.2%

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

      1. unpow2 61.2%

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

      2. times-frac 64.7%

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

   5. Applied egg-rr 64.7%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.2 \cdot 10^{+16}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 3.6 \cdot 10^{+16}:\\ \;\;\;\;0.5 \cdot \left(y_m - z \cdot \frac{z}{y_m}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{\frac{y_m}{x \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 3.6e+16)
    (* 0.5 (- y_m (* z (/ z y_m))))
    (* x (/ 1.0 (/ y_m (* x 0.5)))))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 3.6e+16) {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	} else {
		tmp = x * (1.0 / (y_m / (x * 0.5)));
	}
	return y_s * tmp;
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 3.6d+16) then
        tmp = 0.5d0 * (y_m - (z * (z / y_m)))
    else
        tmp = x * (1.0d0 / (y_m / (x * 0.5d0)))
    end if
    code = y_s * tmp
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 3.6e+16) {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	} else {
		tmp = x * (1.0 / (y_m / (x * 0.5)));
	}
	return y_s * tmp;
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if x <= 3.6e+16:
		tmp = 0.5 * (y_m - (z * (z / y_m)))
	else:
		tmp = x * (1.0 / (y_m / (x * 0.5)))
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 3.6e+16)
		tmp = Float64(0.5 * Float64(y_m - Float64(z * Float64(z / y_m))));
	else
		tmp = Float64(x * Float64(1.0 / Float64(y_m / Float64(x * 0.5))));
	end
	return Float64(y_s * tmp)
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (x <= 3.6e+16)
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	else
		tmp = x * (1.0 / (y_m / (x * 0.5)));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 3.6e+16], N[(0.5 * N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 / N[(y$95$m / N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 3.6e16

   1. Initial program 73.3%

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

   3. Taylor expanded in x around 0 50.3%

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

      1. div-sub 50.3%

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

      2. sub-neg 50.3%

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

      3. unpow2 50.3%

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

      4. associate-/l* 64.2%

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

      5. *-inverses 64.2%

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

      6. /-rgt-identity 64.2%

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

      7. sub-neg 64.2%

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

   5. Simplified 64.2%

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

      1. unpow2 64.2%

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

      2. sqr-neg 64.2%

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

      3. div-inv 64.2%

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

      4. sqr-neg 64.2%

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

      5. associate-*l* 70.6%

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

      6. div-inv 70.7%

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

   7. Applied egg-rr 70.7%

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

   if 3.6e16 < x

   1. Initial program 62.9%

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

   3. Taylor expanded in x around inf 61.2%

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

      1. div-inv 61.2%

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

      2. metadata-eval 61.2%

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

      3. div-inv 61.2%

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

      4. clear-num 61.2%

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

      5. unpow2 61.2%

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

      6. associate-*l* 64.6%

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

   5. Applied egg-rr 64.6%

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

      1. associate-*r/ 64.7%

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

      2. clear-num 64.6%

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{y}{x \cdot 0.5}}} \]

   7. Applied egg-rr 64.6%

      \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{y}{x \cdot 0.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.6 \cdot 10^{+16}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{\frac{y}{x \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 43.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2400000000000:\\ \;\;\;\;y_m \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y_m}\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= x 2400000000000.0) (* y_m 0.5) (* x (* x (/ 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 (x <= 2400000000000.0) {
		tmp = y_m * 0.5;
	} else {
		tmp = x * (x * (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 (x <= 2400000000000.0d0) then
        tmp = y_m * 0.5d0
    else
        tmp = x * (x * (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 (x <= 2400000000000.0) {
		tmp = y_m * 0.5;
	} else {
		tmp = x * (x * (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 x <= 2400000000000.0:
		tmp = y_m * 0.5
	else:
		tmp = x * (x * (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 (x <= 2400000000000.0)
		tmp = Float64(y_m * 0.5);
	else
		tmp = Float64(x * Float64(x * 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 (x <= 2400000000000.0)
		tmp = y_m * 0.5;
	else
		tmp = x * (x * (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[x, 2400000000000.0], N[(y$95$m * 0.5), $MachinePrecision], N[(x * N[(x * N[(0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.4e12

   1. Initial program 73.3%

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

   3. Taylor expanded in y around inf 40.2%

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

   if 2.4e12 < x

   1. Initial program 62.9%

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

   3. Taylor expanded in x around inf 61.2%

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

      1. div-inv 61.2%

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

      2. metadata-eval 61.2%

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

      3. div-inv 61.2%

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

      4. clear-num 61.2%

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

      5. unpow2 61.2%

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

      6. associate-*l* 64.6%

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

   5. Applied egg-rr 64.6%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2400000000000:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 92000000000:\\ \;\;\;\;y_m \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y_m} \cdot \frac{x}{2}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= x 92000000000.0) (* y_m 0.5) (* (/ x y_m) (/ x 2.0)))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 92000000000.0) {
		tmp = y_m * 0.5;
	} else {
		tmp = (x / y_m) * (x / 2.0);
	}
	return y_s * tmp;
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 92000000000.0d0) then
        tmp = y_m * 0.5d0
    else
        tmp = (x / y_m) * (x / 2.0d0)
    end if
    code = y_s * tmp
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 92000000000.0) {
		tmp = y_m * 0.5;
	} else {
		tmp = (x / y_m) * (x / 2.0);
	}
	return y_s * tmp;
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if x <= 92000000000.0:
		tmp = y_m * 0.5
	else:
		tmp = (x / y_m) * (x / 2.0)
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 92000000000.0)
		tmp = Float64(y_m * 0.5);
	else
		tmp = Float64(Float64(x / y_m) * Float64(x / 2.0));
	end
	return Float64(y_s * tmp)
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (x <= 92000000000.0)
		tmp = y_m * 0.5;
	else
		tmp = (x / y_m) * (x / 2.0);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 92000000000.0], N[(y$95$m * 0.5), $MachinePrecision], N[(N[(x / y$95$m), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 9.2e10

   1. Initial program 73.3%

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

   3. Taylor expanded in y around inf 40.2%

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

   if 9.2e10 < x

   1. Initial program 62.9%

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

   3. Taylor expanded in x around inf 61.2%

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

      1. unpow2 61.2%

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

      2. times-frac 64.7%

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

   5. Applied egg-rr 64.7%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 92000000000:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 34.9% accurate, 5.0× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z) :precision binary64 (* y_s (* y_m 0.5)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.2%

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

3. Taylor expanded in y around inf 35.1%

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

4. Final simplification 35.1%

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

Developer target: 99.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(y * 0.5), $MachinePrecision] - N[(N[(N[(0.5 / y), $MachinePrecision] * N[(z + x), $MachinePrecision]), $MachinePrecision] * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024021 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

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

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