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

Average Error: 11.2 → 0.1

Time: 11.2s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -3.98796217281238501 \cdot 10^{42}:\\ \;\;\;\;\frac{1}{e^{y} \cdot x}\\ \mathbf{elif}\;x \le 6065.9896509303089:\\ \;\;\;\;\left({\left(\left|\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right|\right)}^{x} \cdot {\left(\left|\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right|\right)}^{x}\right) \cdot \frac{{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \end{array}\]

C

double f(double x, double y) {
        double r381282 = x;
        double r381283 = y;
        double r381284 = r381282 + r381283;
        double r381285 = r381282 / r381284;
        double r381286 = log(r381285);
        double r381287 = r381282 * r381286;
        double r381288 = exp(r381287);
        double r381289 = r381288 / r381282;
        return r381289;
}

↓

double f(double x, double y) {
        double r381290 = x;
        double r381291 = -3.987962172812385e+42;
        bool r381292 = r381290 <= r381291;
        double r381293 = 1.0;
        double r381294 = y;
        double r381295 = exp(r381294);
        double r381296 = r381295 * r381290;
        double r381297 = r381293 / r381296;
        double r381298 = 6065.989650930309;
        bool r381299 = r381290 <= r381298;
        double r381300 = cbrt(r381290);
        double r381301 = r381290 + r381294;
        double r381302 = cbrt(r381301);
        double r381303 = r381300 / r381302;
        double r381304 = fabs(r381303);
        double r381305 = pow(r381304, r381290);
        double r381306 = r381305 * r381305;
        double r381307 = pow(r381303, r381290);
        double r381308 = r381307 / r381290;
        double r381309 = r381306 * r381308;
        double r381310 = -r381294;
        double r381311 = exp(r381310);
        double r381312 = r381311 / r381290;
        double r381313 = r381299 ? r381309 : r381312;
        double r381314 = r381292 ? r381297 : r381313;
        return r381314;
}

Error

Bits error versus x

Bits error versus y

Target

Original	11.2
Target	8.0
Herbie	0.1

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

Derivation

1. Split input into 3 regimes

2. if x < -3.987962172812385e+42

   1. Initial program 14.1

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

   2. Simplified 14.1

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

   3. Taylor expanded around inf 0.0

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

   5. Applied clear-num 0.1

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

   6. Simplified 0.1

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

   if -3.987962172812385e+42 < x < 6065.989650930309

   1. Initial program 10.6

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

   2. Simplified 10.6

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

   4. Applied *-un-lft-identity 10.6

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

   5. Applied add-cube-cbrt 12.9

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

   6. Applied add-cube-cbrt 10.6

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

   7. Applied times-frac 10.6

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

   8. Applied unpow-prod-down 2.5

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

   9. Applied times-frac 2.5

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

   10. Simplified 2.5

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

   12. Applied add-sqr-sqrt 2.5

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

   13. Applied unpow-prod-down 2.5

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

   14. Simplified 2.5

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

   15. Simplified 0.1

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

   if 6065.989650930309 < x

   1. Initial program 10.3

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

   2. Simplified 10.3

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

   3. Taylor expanded around inf 0.0

      \[\leadsto \frac{\color{blue}{e^{-y}}}{x}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.98796217281238501 \cdot 10^{42}:\\ \;\;\;\;\frac{1}{e^{y} \cdot x}\\ \mathbf{elif}\;x \le 6065.9896509303089:\\ \;\;\;\;\left({\left(\left|\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right|\right)}^{x} \cdot {\left(\left|\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right|\right)}^{x}\right) \cdot \frac{{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +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))