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

Average Error: 11.1 → 0.4

Time: 7.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -7.0366427384808783 \cdot 10^{48} \lor \neg \left(x \le 9.41291569922092328 \cdot 10^{-13}\right):\\ \;\;\;\;\frac{e^{-1 \cdot y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(0 + 2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{\frac{x}{\frac{{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{1}}}\\ \end{array}\]

C

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

↓

double code(double x, double y) {
	double temp;
	if (((x <= -7.036642738480878e+48) || !(x <= 9.412915699220923e-13))) {
		temp = (exp((-1.0 * y)) / x);
	} else {
		temp = (exp((x * (0.0 + (2.0 * log((cbrt(x) / cbrt((x + y)))))))) / (x / (pow((cbrt(x) / cbrt((x + y))), x) / 1.0)));
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Target

Original	11.1
Target	8.0
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if x < -7.036642738480878e+48 or 9.412915699220923e-13 < x

   1. Initial program 11.4

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

   2. Taylor expanded around inf 0.5

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

   if -7.036642738480878e+48 < x < 9.412915699220923e-13

   1. Initial program 10.9

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 13.6

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

   4. Applied add-cube-cbrt 10.9

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

   5. Applied times-frac 10.9

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

   6. Applied log-prod 2.4

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

   7. Applied distribute-lft-in 2.4

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

   8. Applied exp-sum 2.4

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

   9. Applied associate-/l* 2.4

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

   10. Simplified 2.4

       \[\leadsto \frac{e^{x \cdot \log \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right)}}{\color{blue}{\frac{x}{\frac{{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{1}}}}\]
   11. Using strategy rm

   12. Applied *-un-lft-identity 2.4

       \[\leadsto \frac{e^{x \cdot \log \color{blue}{\left(1 \cdot \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right)}}}{\frac{x}{\frac{{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{1}}}\]

   13. Applied log-prod 2.4

       \[\leadsto \frac{e^{x \cdot \color{blue}{\left(\log 1 + \log \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right)\right)}}}{\frac{x}{\frac{{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{1}}}\]

   14. Simplified 2.4

       \[\leadsto \frac{e^{x \cdot \left(\color{blue}{0} + \log \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right)\right)}}{\frac{x}{\frac{{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{1}}}\]

   15. Simplified 0.2

       \[\leadsto \frac{e^{x \cdot \left(0 + \color{blue}{2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}\right)}}{\frac{x}{\frac{{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{1}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7.0366427384808783 \cdot 10^{48} \lor \neg \left(x \le 9.41291569922092328 \cdot 10^{-13}\right):\\ \;\;\;\;\frac{e^{-1 \cdot y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(0 + 2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{\frac{x}{\frac{{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{1}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(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 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 y)) x))))

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