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

Average Error: 6.1 → 0.9

Time: 19.0s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r227244 = x;
        double r227245 = y;
        double r227246 = z;
        double r227247 = r227246 + r227245;
        double r227248 = r227245 / r227247;
        double r227249 = log(r227248);
        double r227250 = r227245 * r227249;
        double r227251 = exp(r227250);
        double r227252 = r227251 / r227245;
        double r227253 = r227244 + r227252;
        return r227253;
}

↓

double f(double x, double y, double z) {
        double r227254 = x;
        double r227255 = 2.0;
        double r227256 = y;
        double r227257 = cbrt(r227256);
        double r227258 = z;
        double r227259 = r227258 + r227256;
        double r227260 = cbrt(r227259);
        double r227261 = r227257 / r227260;
        double r227262 = log(r227261);
        double r227263 = r227255 * r227262;
        double r227264 = r227263 * r227256;
        double r227265 = exp(r227264);
        double r227266 = pow(r227261, r227256);
        double r227267 = r227265 * r227266;
        double r227268 = r227267 / r227256;
        double r227269 = r227254 + r227268;
        return r227269;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.1
Target	0.9
Herbie	0.9

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.115415759790762719541517221498726780517 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

1. Initial program 6.1

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

2. Simplified 6.1

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

4. Applied add-cube-cbrt 19.4

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

5. Applied add-cube-cbrt 6.1

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

6. Applied times-frac 6.1

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

7. Applied unpow-prod-down 2.0

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

9. Applied add-exp-log 39.6

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

10. Applied add-exp-log 39.6

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

11. Applied prod-exp 39.7

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

12. Applied add-exp-log 39.6

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

13. Applied add-exp-log 35.8

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

14. Applied prod-exp 31.6

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

15. Applied div-exp 31.6

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

16. Applied pow-exp 31.1

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

17. Simplified 0.9

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

18. Final simplification 0.9

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

Reproduce

herbie shell --seed 2019208 +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)) 7.1154157598e-315) (+ x (/ (exp (/ -1 z)) y)) (+ x (/ (exp (log (pow (/ y (+ y z)) y))) y)))

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