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

Average Error: 11.3 → 5.1

Time: 5.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq 285.81500652502143 \lor \neg \left(y \leq 1.6006945835005084 \cdot 10^{+89}\right):\\ \;\;\;\;\frac{e^{x \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}\right) \cdot 2\right)} \cdot {\left(\frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}\right)}^{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \end{array}\]

C

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

↓

double code(double x, double y) {
	double VAR;
	if (((y <= 285.81500652502143) || !(y <= 1.6006945835005084e+89))) {
		VAR = (((double) (((double) exp(((double) (x * ((double) (((double) log((((double) cbrt(x)) / ((double) cbrt(((double) (y + x))))))) * 2.0)))))) * ((double) pow((((double) cbrt(x)) / ((double) cbrt(((double) (y + x))))), x)))) / x);
	} else {
		VAR = ((double) log(((double) exp((((double) pow((x / ((double) (y + x))), x)) / x)))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Target

Original	11.3
Target	8.4
Herbie	5.1

\[\begin{array}{l} \mathbf{if}\;y < -3.7311844206647956 \cdot 10^{+94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y < 2.817959242728288 \cdot 10^{+37}:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y < 2.347387415166998 \cdot 10^{+178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if y < 285.81500652502143 or 1.6006945835005084e89 < y

   1. Initial program 9.2

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

   2. Simplified 9.2

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

   4. Applied add-cube-cbrt 28.1

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

   5. Applied add-cube-cbrt 9.2

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

   6. Applied times-frac 9.2

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

   7. Applied unpow-prod-down 4.5

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

   8. Simplified 4.5

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

   10. Applied add-exp-log 45.1

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

   11. Applied add-exp-log 33.6

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

   12. Applied div-exp 33.6

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

   13. Applied add-exp-log 45.1

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

   14. Applied add-exp-log 33.6

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

   15. Applied div-exp 33.6

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

   16. Applied prod-exp 33.6

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

   17. Applied pow-exp 33.2

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

   18. Simplified 3.5

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

   if 285.81500652502143 < y < 1.6006945835005084e89

   1. Initial program 36.2

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

   2. Simplified 36.2

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

   4. Applied add-log-exp 24.0

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

4. Final simplification 5.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 285.81500652502143 \lor \neg \left(y \leq 1.6006945835005084 \cdot 10^{+89}\right):\\ \;\;\;\;\frac{e^{x \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}\right) \cdot 2\right)} \cdot {\left(\frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}\right)}^{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(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))