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

Average Error: 11.2 → 0.1

Time: 13.3s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -3.98796217281238501 \cdot 10^{42}:\\ \;\;\;\;\frac{1}{e^{y} \cdot x}\\ \mathbf{elif}\;x \le 6065.9896509303089:\\ \;\;\;\;\left({\left(\left|\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right|\right)}^{x} \cdot {\left(\left|\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right|\right)}^{x}\right) \cdot \frac{{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \end{array}\]

C

double f(double x, double y) {
        double r365782 = x;
        double r365783 = y;
        double r365784 = r365782 + r365783;
        double r365785 = r365782 / r365784;
        double r365786 = log(r365785);
        double r365787 = r365782 * r365786;
        double r365788 = exp(r365787);
        double r365789 = r365788 / r365782;
        return r365789;
}

↓

double f(double x, double y) {
        double r365790 = x;
        double r365791 = -3.987962172812385e+42;
        bool r365792 = r365790 <= r365791;
        double r365793 = 1.0;
        double r365794 = y;
        double r365795 = exp(r365794);
        double r365796 = r365795 * r365790;
        double r365797 = r365793 / r365796;
        double r365798 = 6065.989650930309;
        bool r365799 = r365790 <= r365798;
        double r365800 = cbrt(r365790);
        double r365801 = r365790 + r365794;
        double r365802 = cbrt(r365801);
        double r365803 = r365800 / r365802;
        double r365804 = fabs(r365803);
        double r365805 = pow(r365804, r365790);
        double r365806 = r365805 * r365805;
        double r365807 = pow(r365803, r365790);
        double r365808 = r365807 / r365790;
        double r365809 = r365806 * r365808;
        double r365810 = -r365794;
        double r365811 = exp(r365810);
        double r365812 = r365811 / r365790;
        double r365813 = r365799 ? r365809 : r365812;
        double r365814 = r365792 ? r365797 : r365813;
        return r365814;
}

Error

Bits error versus x

Bits error versus y

Target

Original	11.2
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 3 regimes

2. if x < -3.987962172812385e+42

   1. Initial program 14.1

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

   2. Simplified 14.1

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

   3. Taylor expanded around inf 0.0

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

   5. Applied neg-sub0 0.0

      \[\leadsto \frac{e^{\color{blue}{0 - y}}}{x}\]

   6. Applied exp-diff 0.0

      \[\leadsto \frac{\color{blue}{\frac{e^{0}}{e^{y}}}}{x}\]

   7. Applied associate-/l/ 0.1

      \[\leadsto \color{blue}{\frac{e^{0}}{x \cdot e^{y}}}\]

   8. Simplified 0.1

      \[\leadsto \frac{e^{0}}{\color{blue}{e^{y} \cdot x}}\]

   if -3.987962172812385e+42 < x < 6065.989650930309

   1. Initial program 10.6

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

   2. Simplified 10.6

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

   4. Applied *-un-lft-identity 10.6

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

   5. Applied add-cube-cbrt 12.9

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

   6. Applied add-cube-cbrt 10.6

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

   7. Applied times-frac 10.6

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

   8. Applied unpow-prod-down 2.5

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

   9. Applied times-frac 2.5

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

   10. Simplified 2.5

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

   12. Applied add-sqr-sqrt 2.5

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

   13. Applied unpow-prod-down 2.5

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

   14. Simplified 2.5

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

   15. Simplified 0.1

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

   if 6065.989650930309 < x

   1. Initial program 10.3

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

   2. Simplified 10.3

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

   3. Taylor expanded around inf 0.0

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.98796217281238501 \cdot 10^{42}:\\ \;\;\;\;\frac{1}{e^{y} \cdot x}\\ \mathbf{elif}\;x \le 6065.9896509303089:\\ \;\;\;\;\left({\left(\left|\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right|\right)}^{x} \cdot {\left(\left|\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right|\right)}^{x}\right) \cdot \frac{{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +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))