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

Average Error: 11.3 → 0.2

Time: 7.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -2.77596381646906558 \cdot 10^{45} \lor \neg \left(x \le 1.48784866468363561 \cdot 10^{-4}\right):\\ \;\;\;\;\frac{e^{-1 \cdot y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(2 \cdot \left(2 \cdot \log \left(\sqrt[3]{\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}}\right) + \log \left(\sqrt[3]{\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}}\right)\right)\right) \cdot x} \cdot {\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{x}\\ \end{array}\]

C

double f(double x, double y) {
        double r464398 = x;
        double r464399 = y;
        double r464400 = r464398 + r464399;
        double r464401 = r464398 / r464400;
        double r464402 = log(r464401);
        double r464403 = r464398 * r464402;
        double r464404 = exp(r464403);
        double r464405 = r464404 / r464398;
        return r464405;
}

↓

double f(double x, double y) {
        double r464406 = x;
        double r464407 = -2.7759638164690656e+45;
        bool r464408 = r464406 <= r464407;
        double r464409 = 0.00014878486646836356;
        bool r464410 = r464406 <= r464409;
        double r464411 = !r464410;
        bool r464412 = r464408 || r464411;
        double r464413 = -1.0;
        double r464414 = y;
        double r464415 = r464413 * r464414;
        double r464416 = exp(r464415);
        double r464417 = r464416 / r464406;
        double r464418 = 2.0;
        double r464419 = cbrt(r464406);
        double r464420 = r464406 + r464414;
        double r464421 = cbrt(r464420);
        double r464422 = r464419 / r464421;
        double r464423 = cbrt(r464422);
        double r464424 = log(r464423);
        double r464425 = r464418 * r464424;
        double r464426 = r464425 + r464424;
        double r464427 = r464418 * r464426;
        double r464428 = r464427 * r464406;
        double r464429 = exp(r464428);
        double r464430 = pow(r464422, r464406);
        double r464431 = r464429 * r464430;
        double r464432 = r464431 / r464406;
        double r464433 = r464412 ? r464417 : r464432;
        return r464433;
}

Error

Bits error versus x

Bits error versus y

Target

Original	11.3
Target	8.0
Herbie	0.2

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

Derivation

1. Split input into 2 regimes

2. if x < -2.7759638164690656e+45 or 0.00014878486646836356 < x

   1. Initial program 11.8

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

   2. Simplified 11.8

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

   3. Taylor expanded around inf 0.2

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

   4. Simplified 0.2

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

   if -2.7759638164690656e+45 < x < 0.00014878486646836356

   1. Initial program 10.8

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

   2. Simplified 10.8

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

   4. Applied add-cube-cbrt 13.1

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

   5. Applied add-cube-cbrt 10.8

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

   6. Applied times-frac 10.8

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

   7. Applied unpow-prod-down 2.1

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

   9. Applied add-exp-log 35.0

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

   10. Applied add-exp-log 35.0

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

   11. Applied prod-exp 35.0

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

   12. Applied add-exp-log 35.0

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

   13. Applied add-exp-log 35.0

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

   14. Applied prod-exp 35.0

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

   15. Applied div-exp 35.0

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

   16. Applied pow-exp 34.1

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

   17. Simplified 0.2

       \[\leadsto \frac{e^{\color{blue}{\left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) \cdot x}} \cdot {\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{x}\]
   18. Using strategy rm

   19. Applied add-cube-cbrt 0.2

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

   20. Applied log-prod 0.2

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

   21. Simplified 0.2

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.77596381646906558 \cdot 10^{45} \lor \neg \left(x \le 1.48784866468363561 \cdot 10^{-4}\right):\\ \;\;\;\;\frac{e^{-1 \cdot y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(2 \cdot \left(2 \cdot \log \left(\sqrt[3]{\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}}\right) + \log \left(\sqrt[3]{\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}}\right)\right)\right) \cdot x} \cdot {\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(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))