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

Average Error: 10.9 → 0.0

Time: 22.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -72249487565711714357348322366193664:\\ \;\;\;\;\frac{1}{x} \cdot e^{-y}\\ \mathbf{elif}\;x \le 95.6145486156965489499270915985107421875:\\ \;\;\;\;e^{x \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}\right) + \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}\right)\right)\right)} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot e^{-y}\\ \end{array}\]

C

double f(double x, double y) {
        double r14578637 = x;
        double r14578638 = y;
        double r14578639 = r14578637 + r14578638;
        double r14578640 = r14578637 / r14578639;
        double r14578641 = log(r14578640);
        double r14578642 = r14578637 * r14578641;
        double r14578643 = exp(r14578642);
        double r14578644 = r14578643 / r14578637;
        return r14578644;
}

↓

double f(double x, double y) {
        double r14578645 = x;
        double r14578646 = -7.224948756571171e+34;
        bool r14578647 = r14578645 <= r14578646;
        double r14578648 = 1.0;
        double r14578649 = r14578648 / r14578645;
        double r14578650 = y;
        double r14578651 = -r14578650;
        double r14578652 = exp(r14578651);
        double r14578653 = r14578649 * r14578652;
        double r14578654 = 95.61454861569655;
        bool r14578655 = r14578645 <= r14578654;
        double r14578656 = cbrt(r14578645);
        double r14578657 = r14578650 + r14578645;
        double r14578658 = cbrt(r14578657);
        double r14578659 = r14578656 / r14578658;
        double r14578660 = log(r14578659);
        double r14578661 = r14578660 + r14578660;
        double r14578662 = r14578660 + r14578661;
        double r14578663 = r14578645 * r14578662;
        double r14578664 = exp(r14578663);
        double r14578665 = r14578664 * r14578649;
        double r14578666 = r14578655 ? r14578665 : r14578653;
        double r14578667 = r14578647 ? r14578653 : r14578666;
        return r14578667;
}

Error

Bits error versus x

Bits error versus y

Target

Original	10.9
Target	7.6
Herbie	0.0

\[\begin{array}{l} \mathbf{if}\;y \lt -3.73118442066479561492798134439269393419 \cdot 10^{94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y \lt 28179592427282878868860376020282245120:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y \lt 2.347387415166997963747840232163110922613 \cdot 10^{178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if x < -7.224948756571171e+34 or 95.61454861569655 < x

   1. Initial program 11.2

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

   2. Taylor expanded around inf 0.0

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

   3. Simplified 0.0

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

   5. Applied div-inv 0.0

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

   if -7.224948756571171e+34 < x < 95.61454861569655

   1. Initial program 10.5

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]
   2. Using strategy rm

   3. Applied div-inv 10.5

      \[\leadsto \color{blue}{e^{x \cdot \log \left(\frac{x}{x + y}\right)} \cdot \frac{1}{x}}\]
   4. Using strategy rm

   5. Applied add-cube-cbrt 12.1

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

   6. Applied add-cube-cbrt 10.5

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

   7. Applied times-frac 10.5

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

   8. Applied log-prod 1.8

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

   9. Simplified 0.1

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -72249487565711714357348322366193664:\\ \;\;\;\;\frac{1}{x} \cdot e^{-y}\\ \mathbf{elif}\;x \le 95.6145486156965489499270915985107421875:\\ \;\;\;\;e^{x \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}\right) + \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}\right)\right)\right)} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot e^{-y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F"

  :herbie-target
  (if (< y -3.7311844206647956e+94) (/ (exp (/ -1.0 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 (/ -1.0 y)) x))))

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