Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F

Average Error: 11.3 → 4.7

Time: 5.2s

Precision: binary64

Math

\[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]

↓

\[\begin{array}{l} \mathbf{if}\;y \le 4.1136023726528613 \cdot 10^{-4} \lor \neg \left(y \le 2.8316162764436968 \cdot 10^{120}\right):\\ \;\;\;\;\frac{\left({\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x} \cdot {\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}\right) \cdot {\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\frac{x}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}{\sqrt[3]{x + y}}\right)}^{x}}{x}\\ \end{array}\]

C

double code(double x, double y) {
	return ((double) (((double) exp(((double) (x * ((double) log(((double) (x / ((double) (x + y)))))))))) / x));
}

↓

double code(double x, double y) {
	double VAR;
	if (((y <= 0.00041136023726528613) || !(y <= 2.831616276443697e+120))) {
		VAR = ((double) (((double) (((double) (((double) pow(((double) (((double) cbrt(x)) / ((double) cbrt(((double) (x + y)))))), x)) * ((double) pow(((double) (((double) cbrt(x)) / ((double) cbrt(((double) (x + y)))))), x)))) * ((double) pow(((double) (((double) cbrt(x)) / ((double) cbrt(((double) (x + y)))))), x)))) / x));
	} else {
		VAR = ((double) (((double) pow(((double) (((double) (x / ((double) (((double) cbrt(((double) (x + y)))) * ((double) cbrt(((double) (x + y)))))))) / ((double) cbrt(((double) (x + y)))))), x)) / x));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Target

Original	11.3
Target	7.9
Herbie	4.7

\[\begin{array}{l} \mathbf{if}\;y \lt -3.73118442066479561 \cdot 10^{94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y \lt 2.81795924272828789 \cdot 10^{37}:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y \lt 2.347387415166998 \cdot 10^{178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if y < 4.1136023726528613e-4 or 2.8316162764436968e120 < y

   1. Initial program 8.3

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]

   2. Simplified 8.3

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt 28.5

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

   5. Applied add-cube-cbrt 8.3

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

   6. Applied times-frac 8.3

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

   7. Applied unpow-prod-down 3.7

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right)}^{x} \cdot {\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}}{x}\]
   8. Using strategy rm

   9. Applied times-frac 3.7

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

   10. Applied unpow-prod-down 2.6

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

   if 4.1136023726528613e-4 < y < 2.8316162764436968e120

   1. Initial program 36.3

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]

   2. Simplified 36.2

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt 23.0

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

   5. Applied associate-/r* 23.1

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

4. Final simplification 4.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 4.1136023726528613 \cdot 10^{-4} \lor \neg \left(y \le 2.8316162764436968 \cdot 10^{120}\right):\\ \;\;\;\;\frac{\left({\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x} \cdot {\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}\right) \cdot {\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\frac{x}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}{\sqrt[3]{x + y}}\right)}^{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020161 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F"
  :precision binary64

  :herbie-target
  (if (< y -3.7311844206647956e+94) (/ (exp (/ -1.0 y)) x) (if (< y 2.817959242728288e+37) (/ (pow (/ x (+ y x)) x) x) (if (< y 2.347387415166998e+178) (log (exp (/ (pow (/ x (+ y x)) x) x))) (/ (exp (/ -1.0 y)) x))))

  (/ (exp (* x (log (/ x (+ x y))))) x))