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

Average Error: 6.0 → 0.6

Time: 20.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -5.528608899369224207630348298103310615572 \cdot 10^{60}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{y \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right)} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}{y}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r389066 = x;
        double r389067 = y;
        double r389068 = z;
        double r389069 = r389068 + r389067;
        double r389070 = r389067 / r389069;
        double r389071 = log(r389070);
        double r389072 = r389067 * r389071;
        double r389073 = exp(r389072);
        double r389074 = r389073 / r389067;
        double r389075 = r389066 + r389074;
        return r389075;
}

↓

double f(double x, double y, double z) {
        double r389076 = y;
        double r389077 = -5.528608899369224e+60;
        bool r389078 = r389076 <= r389077;
        double r389079 = x;
        double r389080 = z;
        double r389081 = -r389080;
        double r389082 = exp(r389081);
        double r389083 = r389082 / r389076;
        double r389084 = r389079 + r389083;
        double r389085 = 2.0;
        double r389086 = cbrt(r389076);
        double r389087 = r389080 + r389076;
        double r389088 = cbrt(r389087);
        double r389089 = r389086 / r389088;
        double r389090 = log(r389089);
        double r389091 = r389085 * r389090;
        double r389092 = r389076 * r389091;
        double r389093 = exp(r389092);
        double r389094 = pow(r389089, r389076);
        double r389095 = r389093 * r389094;
        double r389096 = r389095 / r389076;
        double r389097 = r389079 + r389096;
        double r389098 = r389078 ? r389084 : r389097;
        return r389098;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.0
Target	1.0
Herbie	0.6

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

Derivation

1. Split input into 2 regimes

2. if y < -5.528608899369224e+60

   1. Initial program 2.4

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

   2. Simplified 2.4

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

   3. Taylor expanded around inf 0.1

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

   if -5.528608899369224e+60 < y

   1. Initial program 6.9

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

   2. Simplified 6.9

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

   4. Applied add-cube-cbrt 15.0

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

   5. Applied add-cube-cbrt 6.9

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

   6. Applied times-frac 6.9

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

   7. Applied unpow-prod-down 1.9

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

   9. Applied add-exp-log 33.0

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

   10. Applied add-exp-log 33.0

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

   11. Applied prod-exp 33.1

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

   12. Applied add-exp-log 33.0

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

   13. Applied add-exp-log 27.9

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

   14. Applied prod-exp 22.8

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

   15. Applied div-exp 22.8

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

   16. Applied pow-exp 22.2

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

   17. Simplified 0.7

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -5.528608899369224207630348298103310615572 \cdot 10^{60}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{y \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right)} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(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)))