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

Average Error: 11.4 → 0.7

Time: 7.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -2.930752839587006 \cdot 10^{86} \lor \neg \left(x \le 2.56432330242902842 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{e^{-1 \cdot y}}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) \cdot x} \cdot \frac{{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{x}\\ \end{array}\]

C

double f(double x, double y) {
        double r389609 = x;
        double r389610 = y;
        double r389611 = r389609 + r389610;
        double r389612 = r389609 / r389611;
        double r389613 = log(r389612);
        double r389614 = r389609 * r389613;
        double r389615 = exp(r389614);
        double r389616 = r389615 / r389609;
        return r389616;
}

↓

double f(double x, double y) {
        double r389617 = x;
        double r389618 = -2.930752839587006e+86;
        bool r389619 = r389617 <= r389618;
        double r389620 = 2.5643233024290284e-23;
        bool r389621 = r389617 <= r389620;
        double r389622 = !r389621;
        bool r389623 = r389619 || r389622;
        double r389624 = -1.0;
        double r389625 = y;
        double r389626 = r389624 * r389625;
        double r389627 = exp(r389626);
        double r389628 = r389627 / r389617;
        double r389629 = 2.0;
        double r389630 = cbrt(r389617);
        double r389631 = r389617 + r389625;
        double r389632 = cbrt(r389631);
        double r389633 = r389630 / r389632;
        double r389634 = log(r389633);
        double r389635 = r389629 * r389634;
        double r389636 = r389635 * r389617;
        double r389637 = exp(r389636);
        double r389638 = pow(r389633, r389617);
        double r389639 = r389638 / r389617;
        double r389640 = r389637 * r389639;
        double r389641 = r389623 ? r389628 : r389640;
        return r389641;
}

Error

Bits error versus x

Bits error versus y

Target

Original	11.4
Target	8.0
Herbie	0.7

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

Derivation

1. Split input into 2 regimes

2. if x < -2.930752839587006e+86 or 2.5643233024290284e-23 < x

   1. Initial program 11.4

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

   2. Simplified 11.4

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

   3. Taylor expanded around inf 0.8

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

   4. Simplified 0.8

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

   if -2.930752839587006e+86 < x < 2.5643233024290284e-23

   1. Initial program 11.4

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

   2. Simplified 11.4

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

   4. Applied *-un-lft-identity 11.4

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

   5. Applied add-cube-cbrt 16.2

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

   6. Applied add-cube-cbrt 11.4

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

   7. Applied times-frac 11.4

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

   8. Applied unpow-prod-down 2.8

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

   9. Applied times-frac 2.8

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

   10. Simplified 2.8

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

   12. Applied add-exp-log 38.4

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

   13. Applied add-exp-log 38.4

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

   14. Applied prod-exp 38.4

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

   15. Applied add-exp-log 38.4

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

   16. Applied add-exp-log 38.4

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

   17. Applied prod-exp 38.4

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

   18. Applied div-exp 38.4

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

   19. Applied pow-exp 37.3

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

   20. Simplified 0.5

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.930752839587006 \cdot 10^{86} \lor \neg \left(x \le 2.56432330242902842 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{e^{-1 \cdot y}}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) \cdot x} \cdot \frac{{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 +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))