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

Average Error: 11.3 → 5.6

Time: 17.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le 2.081714116143433 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;y \le 2.5159502131974188 \cdot 10^{+194}:\\ \;\;\;\;\frac{e^{\left(\log \left(\frac{\sqrt[3]{\frac{x}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}}{\sqrt[3]{\sqrt[3]{x + y}}}\right) + \log \left(\frac{\left(\sqrt[3]{x} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}\right) \cdot \sqrt[3]{\frac{x}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}}{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}\right)\right) \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\log \left(\frac{\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}}{\sqrt[3]{\sqrt{x + y}}}\right) + \log \left(\frac{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + y}}}{\sqrt[3]{\sqrt{x + y}}}\right)\right)}}{x}\\ \end{array}\]

C

double f(double x, double y) {
        double r15399441 = x;
        double r15399442 = y;
        double r15399443 = r15399441 + r15399442;
        double r15399444 = r15399441 / r15399443;
        double r15399445 = log(r15399444);
        double r15399446 = r15399441 * r15399445;
        double r15399447 = exp(r15399446);
        double r15399448 = r15399447 / r15399441;
        return r15399448;
}

↓

double f(double x, double y) {
        double r15399449 = y;
        double r15399450 = 2.081714116143433e-21;
        bool r15399451 = r15399449 <= r15399450;
        double r15399452 = 1.0;
        double r15399453 = x;
        double r15399454 = r15399452 / r15399453;
        double r15399455 = 2.5159502131974188e+194;
        bool r15399456 = r15399449 <= r15399455;
        double r15399457 = r15399453 + r15399449;
        double r15399458 = cbrt(r15399457);
        double r15399459 = r15399458 * r15399458;
        double r15399460 = r15399453 / r15399459;
        double r15399461 = cbrt(r15399460);
        double r15399462 = cbrt(r15399458);
        double r15399463 = r15399461 / r15399462;
        double r15399464 = log(r15399463);
        double r15399465 = cbrt(r15399453);
        double r15399466 = r15399452 / r15399459;
        double r15399467 = cbrt(r15399466);
        double r15399468 = r15399465 * r15399467;
        double r15399469 = r15399468 * r15399461;
        double r15399470 = cbrt(r15399459);
        double r15399471 = r15399469 / r15399470;
        double r15399472 = log(r15399471);
        double r15399473 = r15399464 + r15399472;
        double r15399474 = r15399473 * r15399453;
        double r15399475 = exp(r15399474);
        double r15399476 = r15399475 / r15399453;
        double r15399477 = r15399465 / r15399458;
        double r15399478 = sqrt(r15399457);
        double r15399479 = cbrt(r15399478);
        double r15399480 = r15399477 / r15399479;
        double r15399481 = log(r15399480);
        double r15399482 = r15399465 * r15399465;
        double r15399483 = r15399482 / r15399458;
        double r15399484 = r15399483 / r15399479;
        double r15399485 = log(r15399484);
        double r15399486 = r15399481 + r15399485;
        double r15399487 = r15399453 * r15399486;
        double r15399488 = exp(r15399487);
        double r15399489 = r15399488 / r15399453;
        double r15399490 = r15399456 ? r15399476 : r15399489;
        double r15399491 = r15399451 ? r15399454 : r15399490;
        return r15399491;
}

Error

Bits error versus x

Bits error versus y

Target

Original	11.3
Target	8.5
Herbie	5.6

\[\begin{array}{l} \mathbf{if}\;y \lt -3.7311844206647956 \cdot 10^{+94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y \lt 2.817959242728288 \cdot 10^{+37}:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y \lt 2.347387415166998 \cdot 10^{+178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if y < 2.081714116143433e-21

   1. Initial program 4.2

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

   2. Taylor expanded around inf 0.8

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

   if 2.081714116143433e-21 < y < 2.5159502131974188e+194

   1. Initial program 32.4

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

   3. Applied add-cube-cbrt 23.8

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

   4. Applied associate-/r* 23.8

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

   6. Applied add-cube-cbrt 24.1

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

   7. Applied cbrt-prod 24.0

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

   8. Applied add-cube-cbrt 23.8

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

   9. Applied times-frac 23.7

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

   10. Applied log-prod 21.3

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

   12. Applied div-inv 21.4

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

   13. Applied cbrt-prod 21.8

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

   if 2.5159502131974188e+194 < y

   1. Initial program 27.5

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

   3. Applied add-cube-cbrt 24.5

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

   4. Applied associate-/r* 24.6

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

   6. Applied add-sqr-sqrt 28.0

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

   7. Applied cbrt-prod 27.7

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

   8. Applied add-cube-cbrt 28.4

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

   9. Applied times-frac 28.4

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

   10. Applied times-frac 28.3

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

   11. Applied log-prod 11.9

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

4. Final simplification 5.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 2.081714116143433 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;y \le 2.5159502131974188 \cdot 10^{+194}:\\ \;\;\;\;\frac{e^{\left(\log \left(\frac{\sqrt[3]{\frac{x}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}}{\sqrt[3]{\sqrt[3]{x + y}}}\right) + \log \left(\frac{\left(\sqrt[3]{x} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}\right) \cdot \sqrt[3]{\frac{x}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}}{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}\right)\right) \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\log \left(\frac{\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}}{\sqrt[3]{\sqrt{x + y}}}\right) + \log \left(\frac{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + y}}}{\sqrt[3]{\sqrt{x + y}}}\right)\right)}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F"

  :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))