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

Average Error: 11.0 → 4.1

Time: 8.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le 0.80535578456303214:\\ \;\;\;\;\frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}{x}\\ \mathbf{elif}\;y \le 9.9908906795322668 \cdot 10^{98}:\\ \;\;\;\;\frac{e^{x \cdot \log \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{x + y}{\sqrt[3]{x}}}\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}} \cdot \sqrt[3]{\sqrt[3]{x + y}}}\right)}}{x}\\ \end{array}\]

C

double f(double x, double y) {
        double r342704 = x;
        double r342705 = y;
        double r342706 = r342704 + r342705;
        double r342707 = r342704 / r342706;
        double r342708 = log(r342707);
        double r342709 = r342704 * r342708;
        double r342710 = exp(r342709);
        double r342711 = r342710 / r342704;
        return r342711;
}

↓

double f(double x, double y) {
        double r342712 = y;
        double r342713 = 0.8053557845630321;
        bool r342714 = r342712 <= r342713;
        double r342715 = x;
        double r342716 = 2.0;
        double r342717 = cbrt(r342715);
        double r342718 = r342715 + r342712;
        double r342719 = cbrt(r342718);
        double r342720 = r342717 / r342719;
        double r342721 = log(r342720);
        double r342722 = r342716 * r342721;
        double r342723 = r342715 * r342722;
        double r342724 = r342715 * r342721;
        double r342725 = r342723 + r342724;
        double r342726 = exp(r342725);
        double r342727 = r342726 / r342715;
        double r342728 = 9.990890679532267e+98;
        bool r342729 = r342712 <= r342728;
        double r342730 = r342717 * r342717;
        double r342731 = r342718 / r342717;
        double r342732 = r342730 / r342731;
        double r342733 = log(r342732);
        double r342734 = r342715 * r342733;
        double r342735 = exp(r342734);
        double r342736 = r342735 / r342715;
        double r342737 = r342719 * r342719;
        double r342738 = cbrt(r342737);
        double r342739 = cbrt(r342719);
        double r342740 = r342738 * r342739;
        double r342741 = r342717 / r342740;
        double r342742 = log(r342741);
        double r342743 = r342715 * r342742;
        double r342744 = r342723 + r342743;
        double r342745 = exp(r342744);
        double r342746 = r342745 / r342715;
        double r342747 = r342729 ? r342736 : r342746;
        double r342748 = r342714 ? r342727 : r342747;
        return r342748;
}

Error

Bits error versus x

Bits error versus y

Target

Original	11.0
Target	7.7
Herbie	4.1

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

Derivation

1. Split input into 3 regimes

2. if y < 0.8053557845630321

   1. Initial program 4.4

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

   3. Applied add-cube-cbrt 27.9

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

   4. Applied add-cube-cbrt 4.4

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

   5. Applied times-frac 4.4

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

   6. Applied log-prod 1.6

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

   7. Applied distribute-lft-in 1.6

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

   8. Simplified 0.7

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

   if 0.8053557845630321 < y < 9.990890679532267e+98

   1. Initial program 35.7

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

   3. Applied add-cube-cbrt 21.4

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

   4. Applied associate-/l* 21.3

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

   if 9.990890679532267e+98 < y

   1. Initial program 29.7

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

   3. Applied add-cube-cbrt 23.9

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

   4. Applied add-cube-cbrt 29.8

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

   5. Applied times-frac 29.8

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

   6. Applied log-prod 18.5

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

   7. Applied distribute-lft-in 18.5

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

   8. Simplified 14.5

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

   10. Applied add-cube-cbrt 12.6

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

   11. Applied cbrt-prod 11.1

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

4. Final simplification 4.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 0.80535578456303214:\\ \;\;\;\;\frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}{x}\\ \mathbf{elif}\;y \le 9.9908906795322668 \cdot 10^{98}:\\ \;\;\;\;\frac{e^{x \cdot \log \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{x + y}{\sqrt[3]{x}}}\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}} \cdot \sqrt[3]{\sqrt[3]{x + y}}}\right)}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 +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))