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

Average Error: 10.9 → 5.2

Time: 13.9s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le 51088118733.370209:\\ \;\;\;\;\frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) \cdot x}}{x}\\ \mathbf{elif}\;y \le 5.09320930132939825 \cdot 10^{119}:\\ \;\;\;\;\frac{e^{x \cdot \log \left(\frac{x}{x \cdot x - y \cdot y} \cdot \left(x - y\right)\right)}}{x}\\ \mathbf{elif}\;y \le 1.17402130880148184 \cdot 10^{142}:\\ \;\;\;\;\frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{x}\right)\right)}{\sqrt[3]{x + y}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{x}\right)\right)}{\sqrt[3]{x + y}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) \cdot x}}{x}\\ \end{array}\]

C

double f(double x, double y) {
        double r377274 = x;
        double r377275 = y;
        double r377276 = r377274 + r377275;
        double r377277 = r377274 / r377276;
        double r377278 = log(r377277);
        double r377279 = r377274 * r377278;
        double r377280 = exp(r377279);
        double r377281 = r377280 / r377274;
        return r377281;
}

↓

double f(double x, double y) {
        double r377282 = y;
        double r377283 = 51088118733.37021;
        bool r377284 = r377282 <= r377283;
        double r377285 = x;
        double r377286 = 2.0;
        double r377287 = cbrt(r377285);
        double r377288 = r377285 + r377282;
        double r377289 = cbrt(r377288);
        double r377290 = r377287 / r377289;
        double r377291 = log(r377290);
        double r377292 = r377286 * r377291;
        double r377293 = r377285 * r377292;
        double r377294 = r377291 * r377285;
        double r377295 = r377293 + r377294;
        double r377296 = exp(r377295);
        double r377297 = r377296 / r377285;
        double r377298 = 5.093209301329398e+119;
        bool r377299 = r377282 <= r377298;
        double r377300 = r377285 * r377285;
        double r377301 = r377282 * r377282;
        double r377302 = r377300 - r377301;
        double r377303 = r377285 / r377302;
        double r377304 = r377285 - r377282;
        double r377305 = r377303 * r377304;
        double r377306 = log(r377305);
        double r377307 = r377285 * r377306;
        double r377308 = exp(r377307);
        double r377309 = r377308 / r377285;
        double r377310 = 1.1740213088014818e+142;
        bool r377311 = r377282 <= r377310;
        double r377312 = expm1(r377287);
        double r377313 = log1p(r377312);
        double r377314 = r377313 / r377289;
        double r377315 = log(r377314);
        double r377316 = r377286 * r377315;
        double r377317 = r377285 * r377316;
        double r377318 = r377317 + r377294;
        double r377319 = exp(r377318);
        double r377320 = r377319 / r377285;
        double r377321 = log1p(r377287);
        double r377322 = expm1(r377321);
        double r377323 = r377322 / r377289;
        double r377324 = log(r377323);
        double r377325 = r377286 * r377324;
        double r377326 = r377285 * r377325;
        double r377327 = r377326 + r377294;
        double r377328 = exp(r377327);
        double r377329 = r377328 / r377285;
        double r377330 = r377311 ? r377320 : r377329;
        double r377331 = r377299 ? r377309 : r377330;
        double r377332 = r377284 ? r377297 : r377331;
        return r377332;
}

Error

Bits error versus x

Bits error versus y

Target

Original	10.9
Target	8.2
Herbie	5.2

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

Derivation

1. Split input into 4 regimes

2. if y < 51088118733.37021

   1. Initial program 4.7

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

   3. Applied add-cube-cbrt 28.8

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

   4. Applied add-cube-cbrt 4.7

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

   5. Applied times-frac 4.7

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

   6. Applied log-prod 2.0

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

   7. Applied distribute-lft-in 2.0

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

   8. Simplified 1.4

      \[\leadsto \frac{e^{\color{blue}{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)} + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}{x}\]

   9. Simplified 1.4

      \[\leadsto \frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + \color{blue}{\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) \cdot x}}}{x}\]

   if 51088118733.37021 < y < 5.093209301329398e+119

   1. Initial program 34.9

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

   3. Applied flip-+ 24.2

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

   4. Applied associate-/r/ 27.6

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

   if 5.093209301329398e+119 < y < 1.1740213088014818e+142

   1. Initial program 33.7

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

   3. Applied add-cube-cbrt 24.3

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

   4. Applied add-cube-cbrt 33.7

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

   5. Applied times-frac 33.7

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

   6. Applied log-prod 28.3

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

   7. Applied distribute-lft-in 28.3

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

   8. Simplified 28.3

      \[\leadsto \frac{e^{\color{blue}{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)} + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}{x}\]

   9. Simplified 28.3

      \[\leadsto \frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + \color{blue}{\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) \cdot x}}}{x}\]
   10. Using strategy rm

   11. Applied log1p-expm1-u 25.0

       \[\leadsto \frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{x}\right)\right)}}{\sqrt[3]{x + y}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) \cdot x}}{x}\]

   if 1.1740213088014818e+142 < y

   1. Initial program 28.6

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

   3. Applied add-cube-cbrt 25.2

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

   4. Applied add-cube-cbrt 28.9

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

   5. Applied times-frac 28.9

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

   6. Applied log-prod 15.4

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

   7. Applied distribute-lft-in 15.4

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

   8. Simplified 10.0

      \[\leadsto \frac{e^{\color{blue}{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)} + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}{x}\]

   9. Simplified 10.0

      \[\leadsto \frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + \color{blue}{\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) \cdot x}}}{x}\]
   10. Using strategy rm

   11. Applied expm1-log1p-u 9.4

       \[\leadsto \frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{x}\right)\right)}}{\sqrt[3]{x + y}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) \cdot x}}{x}\]
3. Recombined 4 regimes into one program.

4. Final simplification 5.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 51088118733.370209:\\ \;\;\;\;\frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) \cdot x}}{x}\\ \mathbf{elif}\;y \le 5.09320930132939825 \cdot 10^{119}:\\ \;\;\;\;\frac{e^{x \cdot \log \left(\frac{x}{x \cdot x - y \cdot y} \cdot \left(x - y\right)\right)}}{x}\\ \mathbf{elif}\;y \le 1.17402130880148184 \cdot 10^{142}:\\ \;\;\;\;\frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{x}\right)\right)}{\sqrt[3]{x + y}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{x}\right)\right)}{\sqrt[3]{x + y}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) \cdot x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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))