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

Average Error: 11.3 → 0.9

Time: 14.0s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -7.15064441356617753 \lor \neg \left(x \le 6.441277885744034\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) \cdot 2\right) \cdot x}}{\sqrt[3]{x} \cdot \left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right)} \cdot \frac{{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{\sqrt[3]{x}}\\ \end{array}\]

C

double f(double x, double y) {
        double r541200 = x;
        double r541201 = y;
        double r541202 = r541200 + r541201;
        double r541203 = r541200 / r541202;
        double r541204 = log(r541203);
        double r541205 = r541200 * r541204;
        double r541206 = exp(r541205);
        double r541207 = r541206 / r541200;
        return r541207;
}

↓

double f(double x, double y) {
        double r541208 = x;
        double r541209 = -7.1506444135661775;
        bool r541210 = r541208 <= r541209;
        double r541211 = 6.441277885744034;
        bool r541212 = r541208 <= r541211;
        double r541213 = !r541212;
        bool r541214 = r541210 || r541213;
        double r541215 = y;
        double r541216 = -r541215;
        double r541217 = exp(r541216);
        double r541218 = r541217 / r541208;
        double r541219 = cbrt(r541208);
        double r541220 = r541208 + r541215;
        double r541221 = cbrt(r541220);
        double r541222 = r541219 / r541221;
        double r541223 = log(r541222);
        double r541224 = 2.0;
        double r541225 = r541223 * r541224;
        double r541226 = r541225 * r541208;
        double r541227 = exp(r541226);
        double r541228 = cbrt(r541219);
        double r541229 = r541228 * r541228;
        double r541230 = r541229 * r541228;
        double r541231 = r541219 * r541230;
        double r541232 = r541227 / r541231;
        double r541233 = pow(r541222, r541208);
        double r541234 = r541233 / r541219;
        double r541235 = r541232 * r541234;
        double r541236 = r541214 ? r541218 : r541235;
        return r541236;
}

Error

Bits error versus x

Bits error versus y

Target

Original	11.3
Target	8.2
Herbie	0.9

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

Derivation

1. Split input into 2 regimes

2. if x < -7.1506444135661775 or 6.441277885744034 < 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 0.0

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

   if -7.1506444135661775 < x < 6.441277885744034

   1. Initial program 11.7

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

   2. Simplified 11.7

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

   4. Applied add-cube-cbrt 12.8

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

   5. Applied add-cube-cbrt 12.8

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

   6. Applied add-cube-cbrt 12.8

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

   7. Applied times-frac 12.8

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

   8. Applied unpow-prod-down 4.0

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

   9. Applied times-frac 4.0

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

   11. Applied add-exp-log 33.8

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

   12. Applied add-exp-log 33.8

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

   13. Applied prod-exp 33.8

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

   14. Applied add-exp-log 33.8

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

   15. Applied add-exp-log 33.8

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

   16. Applied prod-exp 33.8

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

   17. Applied div-exp 33.8

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

   18. Applied pow-exp 32.4

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

   19. Simplified 1.5

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

   21. Applied add-cube-cbrt 1.9

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7.15064441356617753 \lor \neg \left(x \le 6.441277885744034\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) \cdot 2\right) \cdot x}}{\sqrt[3]{x} \cdot \left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right)} \cdot \frac{{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{\sqrt[3]{x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +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))