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

Average Error: 11.2 → 0.1

Time: 21.0s

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 r269565 = x;
        double r269566 = y;
        double r269567 = r269565 + r269566;
        double r269568 = r269565 / r269567;
        double r269569 = log(r269568);
        double r269570 = r269565 * r269569;
        double r269571 = exp(r269570);
        double r269572 = r269571 / r269565;
        return r269572;
}

↓

double f(double x, double y) {
        double r269573 = x;
        double r269574 = -13.811744642352284;
        bool r269575 = r269573 <= r269574;
        double r269576 = 0.12040142811151512;
        bool r269577 = r269573 <= r269576;
        double r269578 = !r269577;
        bool r269579 = r269575 || r269578;
        double r269580 = y;
        double r269581 = -r269580;
        double r269582 = exp(r269581);
        double r269583 = r269582 / r269573;
        double r269584 = 1.0;
        double r269585 = r269573 + r269580;
        double r269586 = cbrt(r269585);
        double r269587 = r269586 * r269586;
        double r269588 = r269584 / r269587;
        double r269589 = pow(r269588, r269573);
        double r269590 = cbrt(r269573);
        double r269591 = r269590 * r269590;
        double r269592 = cbrt(r269587);
        double r269593 = r269591 / r269592;
        double r269594 = pow(r269593, r269573);
        double r269595 = cbrt(r269586);
        double r269596 = r269590 / r269595;
        double r269597 = pow(r269596, r269573);
        double r269598 = r269594 * r269597;
        double r269599 = r269598 / r269573;
        double r269600 = r269589 * r269599;
        double r269601 = r269579 ? r269583 : r269600;
        return r269601;
}

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))