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

Average Error: 6.2 → 0.7

Time: 8.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -9.2285838323510326 \cdot 10^{106}:\\ \;\;\;\;x + \frac{e^{-1 \cdot z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) - \log \left(\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}\right)\right) \cdot y}}{\frac{y}{\frac{{\left(\sqrt{\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}}\right)}^{y} \cdot {\left(\sqrt{\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}}\right)}^{y}}{1}}}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z) {
	double temp;
	if ((y <= -9.228583832351033e+106)) {
		temp = (x + (exp((-1.0 * z)) / y));
	} else {
		temp = (x + (exp(((log((cbrt(y) * cbrt(y))) - log((cbrt((z + y)) * cbrt((z + y))))) * y)) / (y / ((pow(sqrt((cbrt(y) / cbrt((z + y)))), y) * pow(sqrt((cbrt(y) / cbrt((z + y)))), y)) / 1.0))));
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.2
Target	1.1
Herbie	0.7

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.1154157598 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if y < -9.228583832351033e+106

   1. Initial program 2.4

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

   2. Taylor expanded around inf 0.0

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

   if -9.228583832351033e+106 < y

   1. Initial program 7.0

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

   3. Applied add-cube-cbrt 16.8

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

   4. Applied add-cube-cbrt 7.0

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

   5. Applied times-frac 7.0

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

   6. Applied log-prod 2.0

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

   7. Applied distribute-rgt-in 2.0

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

   8. Applied exp-sum 2.0

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

   9. Applied associate-/l* 2.0

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

   10. Simplified 2.0

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

   12. Applied add-exp-log 15.0

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

   13. Applied add-exp-log 2.0

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

   14. Applied div-exp 2.0

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

   15. Applied rem-log-exp 0.8

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

   17. Applied add-sqr-sqrt 0.8

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

   18. Applied unpow-prod-down 0.8

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -9.2285838323510326 \cdot 10^{106}:\\ \;\;\;\;x + \frac{e^{-1 \cdot z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) - \log \left(\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}\right)\right) \cdot y}}{\frac{y}{\frac{{\left(\sqrt{\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}}\right)}^{y} \cdot {\left(\sqrt{\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}}\right)}^{y}}{1}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, G"
  :precision binary64

  :herbie-target
  (if (< (/ y (+ z y)) 7.1154157597908e-315) (+ x (/ (exp (/ -1 z)) y)) (+ x (/ (exp (log (pow (/ y (+ y z)) y))) y)))

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