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

Average Error: 11.2 → 0.1

Time: 6.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -9.6059130117611643 \cdot 10^{60} \lor \neg \left(x \le 18.070714455582099\right):\\ \;\;\;\;\frac{e^{-1 \cdot y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left({\left(\left|\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right|\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(\left|\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right|\right)}^{\left(\frac{x}{2}\right)}\right) \cdot {\left(\left|\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right|\right)}^{\left(2 \cdot \frac{x}{2}\right)}\right) \cdot {\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{x}\\ \end{array}\]

C

double f(double x, double y) {
        double r447846 = x;
        double r447847 = y;
        double r447848 = r447846 + r447847;
        double r447849 = r447846 / r447848;
        double r447850 = log(r447849);
        double r447851 = r447846 * r447850;
        double r447852 = exp(r447851);
        double r447853 = r447852 / r447846;
        return r447853;
}

↓

double f(double x, double y) {
        double r447854 = x;
        double r447855 = -9.605913011761164e+60;
        bool r447856 = r447854 <= r447855;
        double r447857 = 18.0707144555821;
        bool r447858 = r447854 <= r447857;
        double r447859 = !r447858;
        bool r447860 = r447856 || r447859;
        double r447861 = -1.0;
        double r447862 = y;
        double r447863 = r447861 * r447862;
        double r447864 = exp(r447863);
        double r447865 = r447864 / r447854;
        double r447866 = cbrt(r447854);
        double r447867 = r447854 + r447862;
        double r447868 = cbrt(r447867);
        double r447869 = r447866 / r447868;
        double r447870 = fabs(r447869);
        double r447871 = 2.0;
        double r447872 = r447854 / r447871;
        double r447873 = pow(r447870, r447872);
        double r447874 = r447873 * r447873;
        double r447875 = r447871 * r447872;
        double r447876 = pow(r447870, r447875);
        double r447877 = r447874 * r447876;
        double r447878 = pow(r447869, r447854);
        double r447879 = r447877 * r447878;
        double r447880 = r447879 / r447854;
        double r447881 = r447860 ? r447865 : r447880;
        return r447881;
}

Error

Bits error versus x

Bits error versus y

Target

Original	11.2
Target	7.6
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 2 regimes

2. if x < -9.605913011761164e+60 or 18.0707144555821 < x

   1. Initial program 12.3

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

   2. Simplified 12.3

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

   3. Taylor expanded around inf 0.0

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

   4. Simplified 0.0

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

   if -9.605913011761164e+60 < x < 18.0707144555821

   1. Initial program 10.1

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

   2. Simplified 10.1

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

   4. Applied add-cube-cbrt 13.1

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

   5. Applied add-cube-cbrt 10.1

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

   6. Applied times-frac 10.1

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

   7. Applied unpow-prod-down 2.3

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

   9. Applied add-sqr-sqrt 2.3

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

   10. Applied unpow-prod-down 2.4

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

   11. Simplified 2.4

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

   12. Simplified 0.2

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

   14. Applied sqr-pow 0.2

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

   15. Simplified 0.2

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

   16. Simplified 0.2

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9.6059130117611643 \cdot 10^{60} \lor \neg \left(x \le 18.070714455582099\right):\\ \;\;\;\;\frac{e^{-1 \cdot y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left({\left(\left|\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right|\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(\left|\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right|\right)}^{\left(\frac{x}{2}\right)}\right) \cdot {\left(\left|\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right|\right)}^{\left(2 \cdot \frac{x}{2}\right)}\right) \cdot {\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 
(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))