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

Average Error: 11.2 → 0.1

Time: 20.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -13.81174464235228427355650637764483690262 \lor \neg \left(x \le 0.1204014281115151213663594376157561782748\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right)}^{x} \cdot \frac{{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}\right)}^{x} \cdot {\left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{x + y}}}\right)}^{x}}{x}\\ \end{array}\]

C

double f(double x, double y) {
        double r246993 = x;
        double r246994 = y;
        double r246995 = r246993 + r246994;
        double r246996 = r246993 / r246995;
        double r246997 = log(r246996);
        double r246998 = r246993 * r246997;
        double r246999 = exp(r246998);
        double r247000 = r246999 / r246993;
        return r247000;
}

↓

double f(double x, double y) {
        double r247001 = x;
        double r247002 = -13.811744642352284;
        bool r247003 = r247001 <= r247002;
        double r247004 = 0.12040142811151512;
        bool r247005 = r247001 <= r247004;
        double r247006 = !r247005;
        bool r247007 = r247003 || r247006;
        double r247008 = y;
        double r247009 = -r247008;
        double r247010 = exp(r247009);
        double r247011 = r247010 / r247001;
        double r247012 = 1.0;
        double r247013 = r247001 + r247008;
        double r247014 = cbrt(r247013);
        double r247015 = r247014 * r247014;
        double r247016 = r247012 / r247015;
        double r247017 = pow(r247016, r247001);
        double r247018 = cbrt(r247001);
        double r247019 = r247018 * r247018;
        double r247020 = cbrt(r247015);
        double r247021 = r247019 / r247020;
        double r247022 = pow(r247021, r247001);
        double r247023 = cbrt(r247014);
        double r247024 = r247018 / r247023;
        double r247025 = pow(r247024, r247001);
        double r247026 = r247022 * r247025;
        double r247027 = r247026 / r247001;
        double r247028 = r247017 * r247027;
        double r247029 = r247007 ? r247011 : r247028;
        return r247029;
}

Error

Bits error versus x

Bits error versus y

Target

Original	11.2
Target	8.2
Herbie	0.1

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

Derivation

1. Split input into 2 regimes

2. if x < -13.811744642352284 or 0.12040142811151512 < x

   1. Initial program 10.9

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

   2. Simplified 10.9

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

   3. Taylor expanded around inf 0.1

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

   if -13.811744642352284 < x < 0.12040142811151512

   1. Initial program 11.6

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

   2. Simplified 11.6

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

   4. Applied *-un-lft-identity 11.6

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

   5. Applied add-cube-cbrt 11.6

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

   6. Applied *-un-lft-identity 11.6

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

   7. Applied times-frac 11.6

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

   8. Applied unpow-prod-down 3.3

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

   9. Applied times-frac 3.3

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

   10. Simplified 3.3

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

   12. Applied add-cube-cbrt 3.3

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

   13. Applied cbrt-prod 3.3

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

   14. Applied add-cube-cbrt 3.3

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

   15. Applied times-frac 3.3

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

   16. Applied unpow-prod-down 0.0

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -13.81174464235228427355650637764483690262 \lor \neg \left(x \le 0.1204014281115151213663594376157561782748\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right)}^{x} \cdot \frac{{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}\right)}^{x} \cdot {\left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{x + y}}}\right)}^{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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))