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

Average Error: 6.3 → 2.0

Time: 23.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le 2798330066458407314915328 \lor \neg \left(z \le 4.76689686693070055018888846744602636955 \cdot 10^{118}\right):\\ \;\;\;\;x + \frac{\left({\left(\left|\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right|\right)}^{y} \cdot {\left(\left|\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right|\right)}^{y}\right) \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}}\right)}^{y} \cdot e^{\frac{-1}{3} \cdot z}}{y}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r317420 = x;
        double r317421 = y;
        double r317422 = z;
        double r317423 = r317422 + r317421;
        double r317424 = r317421 / r317423;
        double r317425 = log(r317424);
        double r317426 = r317421 * r317425;
        double r317427 = exp(r317426);
        double r317428 = r317427 / r317421;
        double r317429 = r317420 + r317428;
        return r317429;
}

↓

double f(double x, double y, double z) {
        double r317430 = z;
        double r317431 = 2.7983300664584073e+24;
        bool r317432 = r317430 <= r317431;
        double r317433 = 4.7668968669307006e+118;
        bool r317434 = r317430 <= r317433;
        double r317435 = !r317434;
        bool r317436 = r317432 || r317435;
        double r317437 = x;
        double r317438 = y;
        double r317439 = cbrt(r317438);
        double r317440 = r317430 + r317438;
        double r317441 = cbrt(r317440);
        double r317442 = r317439 / r317441;
        double r317443 = fabs(r317442);
        double r317444 = pow(r317443, r317438);
        double r317445 = r317444 * r317444;
        double r317446 = pow(r317442, r317438);
        double r317447 = r317445 * r317446;
        double r317448 = r317447 / r317438;
        double r317449 = r317437 + r317448;
        double r317450 = r317439 * r317439;
        double r317451 = r317441 * r317441;
        double r317452 = r317450 / r317451;
        double r317453 = pow(r317452, r317438);
        double r317454 = -0.3333333333333333;
        double r317455 = r317454 * r317430;
        double r317456 = exp(r317455);
        double r317457 = r317453 * r317456;
        double r317458 = r317457 / r317438;
        double r317459 = r317437 + r317458;
        double r317460 = r317436 ? r317449 : r317459;
        return r317460;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.3
Target	1.1
Herbie	2.0

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

Derivation

1. Split input into 2 regimes

2. if z < 2.7983300664584073e+24 or 4.7668968669307006e+118 < z

   1. Initial program 5.9

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

   2. Simplified 5.9

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

   4. Applied add-cube-cbrt 19.3

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

   5. Applied add-cube-cbrt 5.9

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

   6. Applied times-frac 5.9

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

   7. Applied unpow-prod-down 1.8

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

   9. Applied add-sqr-sqrt 1.8

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

   10. Applied unpow-prod-down 1.8

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

   11. Simplified 1.8

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

   12. Simplified 0.7

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

   if 2.7983300664584073e+24 < z < 4.7668968669307006e+118

   1. Initial program 10.9

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

   2. Simplified 10.9

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

   4. Applied add-cube-cbrt 17.9

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

   5. Applied add-cube-cbrt 10.9

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

   6. Applied times-frac 10.9

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

   7. Applied unpow-prod-down 6.3

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

   8. Taylor expanded around inf 18.4

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

   9. Simplified 18.4

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

4. Final simplification 2.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le 2798330066458407314915328 \lor \neg \left(z \le 4.76689686693070055018888846744602636955 \cdot 10^{118}\right):\\ \;\;\;\;x + \frac{\left({\left(\left|\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right|\right)}^{y} \cdot {\left(\left|\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right|\right)}^{y}\right) \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}}\right)}^{y} \cdot e^{\frac{-1}{3} \cdot z}}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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)))