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

Average Error: 10.9 → 0.0

Time: 20.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -72249487565711714357348322366193664:\\ \;\;\;\;\frac{1}{x} \cdot e^{-y}\\ \mathbf{elif}\;x \le 95.6145486156965489499270915985107421875:\\ \;\;\;\;e^{x \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}\right) + \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}\right)\right)\right)} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot e^{-y}\\ \end{array}\]

C

double f(double x, double y) {
        double r22398368 = x;
        double r22398369 = y;
        double r22398370 = r22398368 + r22398369;
        double r22398371 = r22398368 / r22398370;
        double r22398372 = log(r22398371);
        double r22398373 = r22398368 * r22398372;
        double r22398374 = exp(r22398373);
        double r22398375 = r22398374 / r22398368;
        return r22398375;
}

↓

double f(double x, double y) {
        double r22398376 = x;
        double r22398377 = -7.224948756571171e+34;
        bool r22398378 = r22398376 <= r22398377;
        double r22398379 = 1.0;
        double r22398380 = r22398379 / r22398376;
        double r22398381 = y;
        double r22398382 = -r22398381;
        double r22398383 = exp(r22398382);
        double r22398384 = r22398380 * r22398383;
        double r22398385 = 95.61454861569655;
        bool r22398386 = r22398376 <= r22398385;
        double r22398387 = cbrt(r22398376);
        double r22398388 = r22398381 + r22398376;
        double r22398389 = cbrt(r22398388);
        double r22398390 = r22398387 / r22398389;
        double r22398391 = log(r22398390);
        double r22398392 = r22398391 + r22398391;
        double r22398393 = r22398391 + r22398392;
        double r22398394 = r22398376 * r22398393;
        double r22398395 = exp(r22398394);
        double r22398396 = r22398395 * r22398380;
        double r22398397 = r22398386 ? r22398396 : r22398384;
        double r22398398 = r22398378 ? r22398384 : r22398397;
        return r22398398;
}

Error

Bits error versus x

Bits error versus y

Target

Original	10.9
Target	7.6
Herbie	0.0

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

Derivation

1. Split input into 2 regimes

2. if x < -7.224948756571171e+34 or 95.61454861569655 < x

   1. Initial program 11.2

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

   2. Taylor expanded around inf 0.0

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

   3. Simplified 0.0

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

   5. Applied div-inv 0.0

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

   if -7.224948756571171e+34 < x < 95.61454861569655

   1. Initial program 10.5

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

   3. Applied div-inv 10.5

      \[\leadsto \color{blue}{e^{x \cdot \log \left(\frac{x}{x + y}\right)} \cdot \frac{1}{x}}\]
   4. Using strategy rm

   5. Applied add-cube-cbrt 12.1

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

   6. Applied add-cube-cbrt 10.5

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

   7. Applied times-frac 10.5

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

   8. Applied log-prod 1.8

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

   9. Simplified 0.1

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -72249487565711714357348322366193664:\\ \;\;\;\;\frac{1}{x} \cdot e^{-y}\\ \mathbf{elif}\;x \le 95.6145486156965489499270915985107421875:\\ \;\;\;\;e^{x \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}\right) + \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}\right)\right)\right)} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot e^{-y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F"

  :herbie-target
  (if (< y -3.7311844206647956e+94) (/ (exp (/ -1.0 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 (/ -1.0 y)) x))))

  (/ (exp (* x (log (/ x (+ x y))))) x))