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

Percentage Accurate: 69.0% → 90.5%

Time: 16.9s

Alternatives: 7

Speedup: 0.9×

• 41.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \sqrt{y_m \cdot 2}\\ y_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 7.5 \cdot 10^{+190}:\\ \;\;\;\;\frac{1}{t_0} \cdot \mathsf{fma}\left(\mathsf{hypot}\left(x, y_m\right), \frac{\mathsf{hypot}\left(x, y_m\right)}{t_0}, \frac{z}{t_0} \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y_m \cdot 0.5\\ \end{array} \end{array} \end{array} \]

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))))
   (*
    y_s
    (if (<= y_m 7.5e+190)
      (*
       (/ 1.0 t_0)
       (fma (hypot x y_m) (/ (hypot x y_m) t_0) (* (/ z t_0) (- z))))
      (* y_m 0.5)))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = sqrt((y_m * 2.0));
	double tmp;
	if (y_m <= 7.5e+190) {
		tmp = (1.0 / t_0) * fma(hypot(x, y_m), (hypot(x, y_m) / t_0), ((z / t_0) * -z));
	} else {
		tmp = y_m * 0.5;
	}
	return y_s * tmp;
}

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = sqrt(Float64(y_m * 2.0))
	tmp = 0.0
	if (y_m <= 7.5e+190)
		tmp = Float64(Float64(1.0 / t_0) * fma(hypot(x, y_m), Float64(hypot(x, y_m) / t_0), Float64(Float64(z / t_0) * Float64(-z))));
	else
		tmp = Float64(y_m * 0.5);
	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]}, N[(y$95$s * If[LessEqual[y$95$m, 7.5e+190], N[(N[(1.0 / t$95$0), $MachinePrecision] * 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[(N[(z / t$95$0), $MachinePrecision] * (-z)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * 0.5), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.4999999999999994e190

   1. Initial program 74.6%

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

      1. *-un-lft-identity 74.6%

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

      2. add-sqr-sqrt 37.8%

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

      3. times-frac 37.8%

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

      4. add-sqr-sqrt 37.8%

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

      5. pow2 37.8%

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

      6. hypot-def 37.8%

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

      7. pow2 37.8%

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

   3. Applied egg-rr 37.8%

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

      1. div-sub 36.5%

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

      2. unpow2 36.5%

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

      3. *-un-lft-identity 36.5%

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

      4. times-frac 40.9%

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

      5. unpow2 40.9%

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

      6. *-un-lft-identity 40.9%

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

      7. times-frac 42.4%

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

      8. prod-diff 35.7%

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

   5. Applied egg-rr 35.7%

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

   6. Taylor expanded in z around 0 36.1%

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

      1. distribute-lft1-in 36.1%

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

      2. metadata-eval 36.1%

         \[\leadsto \frac{1}{\sqrt{y \cdot 2}} \cdot \left(\mathsf{fma}\left(\frac{\mathsf{hypot}\left(x, y\right)}{1}, \frac{\mathsf{hypot}\left(x, y\right)}{\sqrt{y \cdot 2}}, -\frac{z}{\sqrt{y \cdot 2}} \cdot \frac{z}{1}\right) + {z}^{2} \cdot \left(\color{blue}{0} \cdot \left(\sqrt{\frac{1}{y}} \cdot \frac{1}{\sqrt{2}}\right)\right)\right) \]

      3. mul0-lft 36.1%

         \[\leadsto \frac{1}{\sqrt{y \cdot 2}} \cdot \left(\mathsf{fma}\left(\frac{\mathsf{hypot}\left(x, y\right)}{1}, \frac{\mathsf{hypot}\left(x, y\right)}{\sqrt{y \cdot 2}}, -\frac{z}{\sqrt{y \cdot 2}} \cdot \frac{z}{1}\right) + {z}^{2} \cdot \color{blue}{0}\right) \]

   8. Simplified 36.1%

      \[\leadsto \frac{1}{\sqrt{y \cdot 2}} \cdot \left(\mathsf{fma}\left(\frac{\mathsf{hypot}\left(x, y\right)}{1}, \frac{\mathsf{hypot}\left(x, y\right)}{\sqrt{y \cdot 2}}, -\frac{z}{\sqrt{y \cdot 2}} \cdot \frac{z}{1}\right) + \color{blue}{{z}^{2} \cdot 0}\right) \]

   9. Taylor expanded in z around 0 44.1%

      \[\leadsto \frac{1}{\sqrt{y \cdot 2}} \cdot \left(\mathsf{fma}\left(\frac{\mathsf{hypot}\left(x, y\right)}{1}, \frac{\mathsf{hypot}\left(x, y\right)}{\sqrt{y \cdot 2}}, -\frac{z}{\sqrt{y \cdot 2}} \cdot \frac{z}{1}\right) + \color{blue}{0}\right) \]

   if 7.4999999999999994e190 < y

   1. Initial program 4.2%

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

   2. Taylor expanded in y around inf 81.8%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{+190}:\\ \;\;\;\;\frac{1}{\sqrt{y \cdot 2}} \cdot \mathsf{fma}\left(\mathsf{hypot}\left(x, y\right), \frac{\mathsf{hypot}\left(x, y\right)}{\sqrt{y \cdot 2}}, \frac{z}{\sqrt{y \cdot 2}} \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 2: 88.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 2.8 \cdot 10^{+149}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y_m - z, y_m + z, x \cdot x\right)}{y_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;y_m \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 2.8e+149)
    (/ (fma (- y_m z) (+ y_m z) (* x x)) (* y_m 2.0))
    (* y_m 0.5))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 2.8e+149) {
		tmp = fma((y_m - z), (y_m + z), (x * x)) / (y_m * 2.0);
	} else {
		tmp = y_m * 0.5;
	}
	return y_s * tmp;
}

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 2.8e+149)
		tmp = Float64(fma(Float64(y_m - z), Float64(y_m + z), Float64(x * x)) / Float64(y_m * 2.0));
	else
		tmp = Float64(y_m * 0.5);
	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, 2.8e+149], N[(N[(N[(y$95$m - z), $MachinePrecision] * N[(y$95$m + z), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], N[(y$95$m * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.7999999999999999e149

   1. Initial program 77.8%

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

      1. associate--l+ 77.8%

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

      2. +-commutative 77.8%

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

      3. sqr-neg 77.8%

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

      4. difference-of-squares 79.3%

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

      5. fma-def 82.4%

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

      6. sub-neg 82.4%

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

      7. sub-neg 82.4%

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

      8. remove-double-neg 82.4%

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

   3. Simplified 82.4%

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

   if 2.7999999999999999e149 < y

   1. Initial program 10.9%

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

   2. Taylor expanded in y around inf 78.0%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{+149}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y - z, y + z, x \cdot x\right)}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 3: 54.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\left(y_m - z\right) \cdot \left(y_m + z\right)}{y_m \cdot 2}\\ y_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 5.7 \cdot 10^{+31}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+75}:\\ \;\;\;\;\frac{x \cdot \frac{x}{2}}{y_m}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+107}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{2} \cdot \frac{x}{y_m}\\ \end{array} \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (/ (* (- y_m z) (+ y_m z)) (* y_m 2.0))))
   (*
    y_s
    (if (<= x 5.7e+31)
      t_0
      (if (<= x 5e+75)
        (/ (* x (/ x 2.0)) y_m)
        (if (<= x 1.25e+107) t_0 (* (/ x 2.0) (/ 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 t_0 = ((y_m - z) * (y_m + z)) / (y_m * 2.0);
	double tmp;
	if (x <= 5.7e+31) {
		tmp = t_0;
	} else if (x <= 5e+75) {
		tmp = (x * (x / 2.0)) / y_m;
	} else if (x <= 1.25e+107) {
		tmp = t_0;
	} else {
		tmp = (x / 2.0) * (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) :: t_0
    real(8) :: tmp
    t_0 = ((y_m - z) * (y_m + z)) / (y_m * 2.0d0)
    if (x <= 5.7d+31) then
        tmp = t_0
    else if (x <= 5d+75) then
        tmp = (x * (x / 2.0d0)) / y_m
    else if (x <= 1.25d+107) then
        tmp = t_0
    else
        tmp = (x / 2.0d0) * (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 t_0 = ((y_m - z) * (y_m + z)) / (y_m * 2.0);
	double tmp;
	if (x <= 5.7e+31) {
		tmp = t_0;
	} else if (x <= 5e+75) {
		tmp = (x * (x / 2.0)) / y_m;
	} else if (x <= 1.25e+107) {
		tmp = t_0;
	} else {
		tmp = (x / 2.0) * (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):
	t_0 = ((y_m - z) * (y_m + z)) / (y_m * 2.0)
	tmp = 0
	if x <= 5.7e+31:
		tmp = t_0
	elif x <= 5e+75:
		tmp = (x * (x / 2.0)) / y_m
	elif x <= 1.25e+107:
		tmp = t_0
	else:
		tmp = (x / 2.0) * (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)
	t_0 = Float64(Float64(Float64(y_m - z) * Float64(y_m + z)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (x <= 5.7e+31)
		tmp = t_0;
	elseif (x <= 5e+75)
		tmp = Float64(Float64(x * Float64(x / 2.0)) / y_m);
	elseif (x <= 1.25e+107)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / 2.0) * 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)
	t_0 = ((y_m - z) * (y_m + z)) / (y_m * 2.0);
	tmp = 0.0;
	if (x <= 5.7e+31)
		tmp = t_0;
	elseif (x <= 5e+75)
		tmp = (x * (x / 2.0)) / y_m;
	elseif (x <= 1.25e+107)
		tmp = t_0;
	else
		tmp = (x / 2.0) * (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_] := Block[{t$95$0 = N[(N[(N[(y$95$m - z), $MachinePrecision] * N[(y$95$m + z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[x, 5.7e+31], t$95$0, If[LessEqual[x, 5e+75], N[(N[(x * N[(x / 2.0), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], If[LessEqual[x, 1.25e+107], t$95$0, N[(N[(x / 2.0), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if x < 5.7e31 or 5.0000000000000002e75 < x < 1.25e107

   1. Initial program 73.3%

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

      1. associate--l+ 73.3%

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

      2. +-commutative 73.3%

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

      3. sqr-neg 73.3%

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

      4. difference-of-squares 76.0%

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

      5. fma-def 77.0%

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

      6. sub-neg 77.0%

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

      7. sub-neg 77.0%

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

      8. remove-double-neg 77.0%

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

   3. Simplified 77.0%

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

   4. Taylor expanded in x around 0 55.0%

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

   if 5.7e31 < x < 5.0000000000000002e75

   1. Initial program 76.4%

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

   2. Taylor expanded in x around inf 63.5%

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

      1. unpow2 63.5%

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

      2. times-frac 63.3%

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

   4. Applied egg-rr 63.3%

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

      1. *-commutative 63.3%

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

      2. associate-*r/ 63.5%

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

   6. Simplified 63.5%

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

   if 1.25e107 < x

   1. Initial program 54.2%

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

   2. Taylor expanded in x around inf 64.3%

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

      1. unpow2 64.3%

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

      2. times-frac 73.1%

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

   4. Applied egg-rr 73.1%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.7 \cdot 10^{+31}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot \left(y + z\right)}{y \cdot 2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+75}:\\ \;\;\;\;\frac{x \cdot \frac{x}{2}}{y}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+107}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot \left(y + z\right)}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{2} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 4: 85.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 2.8 \cdot 10^{+149}:\\ \;\;\;\;\frac{\left(x \cdot x + y_m \cdot y_m\right) - z \cdot z}{y_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;y_m \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 2.8e+149)
    (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))
    (* y_m 0.5))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 2.8e+149) {
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	} else {
		tmp = y_m * 0.5;
	}
	return y_s * tmp;
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 2.8d+149) then
        tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0d0)
    else
        tmp = y_m * 0.5d0
    end if
    code = y_s * tmp
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 2.8e+149) {
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	} else {
		tmp = y_m * 0.5;
	}
	return y_s * tmp;
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if y_m <= 2.8e+149:
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)
	else:
		tmp = y_m * 0.5
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 2.8e+149)
		tmp = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0));
	else
		tmp = Float64(y_m * 0.5);
	end
	return Float64(y_s * tmp)
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (y_m <= 2.8e+149)
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	else
		tmp = y_m * 0.5;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 2.8e+149], N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], N[(y$95$m * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.7999999999999999e149

   1. Initial program 77.8%

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

   if 2.7999999999999999e149 < y

   1. Initial program 10.9%

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

   2. Taylor expanded in y around inf 78.0%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{+149}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 5: 54.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 3.45 \cdot 10^{+50}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y_m}\right)\\ \mathbf{else}:\\ \;\;\;\;y_m \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= y_m 3.45e+50) (* x (* x (/ 0.5 y_m))) (* y_m 0.5))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 3.45e+50) {
		tmp = x * (x * (0.5 / y_m));
	} else {
		tmp = y_m * 0.5;
	}
	return y_s * tmp;
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 3.45d+50) then
        tmp = x * (x * (0.5d0 / y_m))
    else
        tmp = y_m * 0.5d0
    end if
    code = y_s * tmp
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 3.45e+50) {
		tmp = x * (x * (0.5 / y_m));
	} else {
		tmp = y_m * 0.5;
	}
	return y_s * tmp;
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if y_m <= 3.45e+50:
		tmp = x * (x * (0.5 / y_m))
	else:
		tmp = y_m * 0.5
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 3.45e+50)
		tmp = Float64(x * Float64(x * Float64(0.5 / y_m)));
	else
		tmp = Float64(y_m * 0.5);
	end
	return Float64(y_s * tmp)
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (y_m <= 3.45e+50)
		tmp = x * (x * (0.5 / y_m));
	else
		tmp = y_m * 0.5;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 3.45e+50], N[(x * N[(x * N[(0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.45000000000000016e50

   1. Initial program 78.8%

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

   2. Taylor expanded in x around inf 38.9%

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

      1. div-inv 38.9%

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

      2. unpow2 38.9%

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

      3. *-commutative 38.9%

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

      4. associate-/r* 38.9%

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

      5. metadata-eval 38.9%

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

      6. associate-*l* 40.7%

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

   4. Applied egg-rr 40.7%

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

   if 3.45000000000000016e50 < y

   1. Initial program 38.1%

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

   2. Taylor expanded in y around inf 62.9%

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

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.45 \cdot 10^{+50}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 6: 54.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 3.8 \cdot 10^{+50}:\\ \;\;\;\;\frac{x}{2 \cdot \frac{y_m}{x}}\\ \mathbf{else}:\\ \;\;\;\;y_m \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= y_m 3.8e+50) (/ x (* 2.0 (/ y_m x))) (* y_m 0.5))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 3.8e+50) {
		tmp = x / (2.0 * (y_m / x));
	} else {
		tmp = y_m * 0.5;
	}
	return y_s * tmp;
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 3.8d+50) then
        tmp = x / (2.0d0 * (y_m / x))
    else
        tmp = y_m * 0.5d0
    end if
    code = y_s * tmp
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 3.8e+50) {
		tmp = x / (2.0 * (y_m / x));
	} else {
		tmp = y_m * 0.5;
	}
	return y_s * tmp;
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if y_m <= 3.8e+50:
		tmp = x / (2.0 * (y_m / x))
	else:
		tmp = y_m * 0.5
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 3.8e+50)
		tmp = Float64(x / Float64(2.0 * Float64(y_m / x)));
	else
		tmp = Float64(y_m * 0.5);
	end
	return Float64(y_s * tmp)
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (y_m <= 3.8e+50)
		tmp = x / (2.0 * (y_m / x));
	else
		tmp = y_m * 0.5;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 3.8e+50], N[(x / N[(2.0 * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.79999999999999987e50

   1. Initial program 78.8%

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

   2. Taylor expanded in x around inf 38.9%

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

      1. unpow2 38.9%

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

      2. times-frac 40.7%

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

   4. Applied egg-rr 40.7%

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

      1. clear-num 40.6%

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

      2. frac-times 40.7%

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

      3. *-un-lft-identity 40.7%

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

   6. Applied egg-rr 40.7%

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

   if 3.79999999999999987e50 < y

   1. Initial program 38.1%

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

   2. Taylor expanded in y around inf 62.9%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{+50}:\\ \;\;\;\;\frac{x}{2 \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 7: 34.2% accurate, 5.0× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 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 70.2%

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

2. Taylor expanded in y around inf 31.9%

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

3. Final simplification 31.9%

   \[\leadsto y \cdot 0.5 \]

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 2023322 
(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)))