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

Average Error: 11.4 → 1.4

Time: 5.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -3.04718936956828664 \cdot 10^{55}:\\ \;\;\;\;\frac{1}{x \cdot e^{y}}\\ \mathbf{elif}\;x \le 8.4817809066997952 \cdot 10^{-11}:\\ \;\;\;\;\frac{e^{x \cdot \log \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right) + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-1 \cdot y}}{x}\\ \end{array}\]

C

double f(double x, double y) {
        double r352850 = x;
        double r352851 = y;
        double r352852 = r352850 + r352851;
        double r352853 = r352850 / r352852;
        double r352854 = log(r352853);
        double r352855 = r352850 * r352854;
        double r352856 = exp(r352855);
        double r352857 = r352856 / r352850;
        return r352857;
}

↓

double f(double x, double y) {
        double r352858 = x;
        double r352859 = -3.0471893695682866e+55;
        bool r352860 = r352858 <= r352859;
        double r352861 = 1.0;
        double r352862 = y;
        double r352863 = exp(r352862);
        double r352864 = r352858 * r352863;
        double r352865 = r352861 / r352864;
        double r352866 = 8.481780906699795e-11;
        bool r352867 = r352858 <= r352866;
        double r352868 = cbrt(r352858);
        double r352869 = r352868 * r352868;
        double r352870 = r352858 + r352862;
        double r352871 = cbrt(r352870);
        double r352872 = r352871 * r352871;
        double r352873 = r352869 / r352872;
        double r352874 = log(r352873);
        double r352875 = r352858 * r352874;
        double r352876 = r352868 / r352871;
        double r352877 = log(r352876);
        double r352878 = r352858 * r352877;
        double r352879 = r352875 + r352878;
        double r352880 = exp(r352879);
        double r352881 = r352880 / r352858;
        double r352882 = -1.0;
        double r352883 = r352882 * r352862;
        double r352884 = exp(r352883);
        double r352885 = r352884 / r352858;
        double r352886 = r352867 ? r352881 : r352885;
        double r352887 = r352860 ? r352865 : r352886;
        return r352887;
}

Error

Bits error versus x

Bits error versus y

Target

Original	11.4
Target	7.7
Herbie	1.4

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

Derivation

1. Split input into 3 regimes

2. if x < -3.0471893695682866e+55

   1. Initial program 14.4

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

   2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot y}}{x}}\]
   3. Using strategy rm

   4. Applied clear-num 0.0

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

   5. Simplified 0.0

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

   if -3.0471893695682866e+55 < x < 8.481780906699795e-11

   1. Initial program 10.9

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

   3. Applied add-cube-cbrt 14.1

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

   4. Applied add-cube-cbrt 10.9

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

   5. Applied times-frac 10.9

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

   6. Applied log-prod 2.3

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

   7. Applied distribute-lft-in 2.3

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

   if 8.481780906699795e-11 < x

   1. Initial program 10.0

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

   2. Taylor expanded around inf 0.8

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot y}}{x}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.04718936956828664 \cdot 10^{55}:\\ \;\;\;\;\frac{1}{x \cdot e^{y}}\\ \mathbf{elif}\;x \le 8.4817809066997952 \cdot 10^{-11}:\\ \;\;\;\;\frac{e^{x \cdot \log \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right) + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-1 \cdot y}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 +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))