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

Average Error: 11.1 → 0.1

Time: 17.4s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -8.99548382710316602 \cdot 10^{46} \lor \neg \left(x \le 3.95485830747834965\right):\\ \;\;\;\;e^{-y} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) \cdot 2\right)} \cdot {\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{x}\\ \end{array}\]

C

double f(double x, double y) {
        double r461678 = x;
        double r461679 = y;
        double r461680 = r461678 + r461679;
        double r461681 = r461678 / r461680;
        double r461682 = log(r461681);
        double r461683 = r461678 * r461682;
        double r461684 = exp(r461683);
        double r461685 = r461684 / r461678;
        return r461685;
}

↓

double f(double x, double y) {
        double r461686 = x;
        double r461687 = -8.995483827103166e+46;
        bool r461688 = r461686 <= r461687;
        double r461689 = 3.9548583074783497;
        bool r461690 = r461686 <= r461689;
        double r461691 = !r461690;
        bool r461692 = r461688 || r461691;
        double r461693 = y;
        double r461694 = -r461693;
        double r461695 = exp(r461694);
        double r461696 = 1.0;
        double r461697 = r461696 / r461686;
        double r461698 = r461695 * r461697;
        double r461699 = cbrt(r461686);
        double r461700 = r461686 + r461693;
        double r461701 = cbrt(r461700);
        double r461702 = r461699 / r461701;
        double r461703 = log(r461702);
        double r461704 = 2.0;
        double r461705 = r461703 * r461704;
        double r461706 = r461686 * r461705;
        double r461707 = exp(r461706);
        double r461708 = pow(r461702, r461686);
        double r461709 = r461707 * r461708;
        double r461710 = r461709 / r461686;
        double r461711 = r461692 ? r461698 : r461710;
        return r461711;
}

Error

Bits error versus x

Bits error versus y

Target

Original	11.1
Target	8.0
Herbie	0.1

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

Derivation

1. Split input into 2 regimes

2. if x < -8.995483827103166e+46 or 3.9548583074783497 < x

   1. Initial program 11.1

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

   2. Simplified 11.1

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

   3. Taylor expanded around inf 0.1

      \[\leadsto \frac{\color{blue}{e^{-y}}}{x}\]
   4. Using strategy rm

   5. Applied div-inv 0.1

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

   if -8.995483827103166e+46 < x < 3.9548583074783497

   1. Initial program 11.1

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

   2. Simplified 11.1

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

   4. Applied add-cube-cbrt 13.4

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

   5. Applied add-cube-cbrt 11.1

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

   6. Applied times-frac 11.1

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

   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 34.8

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

   10. Applied add-exp-log 34.8

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

   11. Applied prod-exp 34.8

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

   12. Applied add-exp-log 34.8

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

   13. Applied add-exp-log 34.8

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

   14. Applied prod-exp 34.8

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

   15. Applied div-exp 34.8

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

   16. Applied pow-exp 33.7

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

   17. Simplified 0.2

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8.99548382710316602 \cdot 10^{46} \lor \neg \left(x \le 3.95485830747834965\right):\\ \;\;\;\;e^{-y} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) \cdot 2\right)} \cdot {\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 +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))