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

Average Error: 11.5 → 5.6

Time: 39.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le 2.081714116143433 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;y \le 2.5159502131974188 \cdot 10^{+194}:\\ \;\;\;\;\frac{e^{\left(\log \left(\frac{\sqrt[3]{\frac{x}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}}{\sqrt[3]{\sqrt[3]{x + y}}}\right) + \log \left(\frac{\left(\sqrt[3]{x} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}\right) \cdot \sqrt[3]{\frac{x}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}}{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}\right)\right) \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\log \left(\frac{\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}}{\sqrt[3]{\sqrt{x + y}}}\right) + \log \left(\frac{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + y}}}{\sqrt[3]{\sqrt{x + y}}}\right)\right)}}{x}\\ \end{array}\]

C

double f(double x, double y) {
        double r19574066 = x;
        double r19574067 = y;
        double r19574068 = r19574066 + r19574067;
        double r19574069 = r19574066 / r19574068;
        double r19574070 = log(r19574069);
        double r19574071 = r19574066 * r19574070;
        double r19574072 = exp(r19574071);
        double r19574073 = r19574072 / r19574066;
        return r19574073;
}

↓

double f(double x, double y) {
        double r19574074 = y;
        double r19574075 = 2.081714116143433e-21;
        bool r19574076 = r19574074 <= r19574075;
        double r19574077 = 1.0;
        double r19574078 = x;
        double r19574079 = r19574077 / r19574078;
        double r19574080 = 2.5159502131974188e+194;
        bool r19574081 = r19574074 <= r19574080;
        double r19574082 = r19574078 + r19574074;
        double r19574083 = cbrt(r19574082);
        double r19574084 = r19574083 * r19574083;
        double r19574085 = r19574078 / r19574084;
        double r19574086 = cbrt(r19574085);
        double r19574087 = cbrt(r19574083);
        double r19574088 = r19574086 / r19574087;
        double r19574089 = log(r19574088);
        double r19574090 = cbrt(r19574078);
        double r19574091 = r19574077 / r19574084;
        double r19574092 = cbrt(r19574091);
        double r19574093 = r19574090 * r19574092;
        double r19574094 = r19574093 * r19574086;
        double r19574095 = cbrt(r19574084);
        double r19574096 = r19574094 / r19574095;
        double r19574097 = log(r19574096);
        double r19574098 = r19574089 + r19574097;
        double r19574099 = r19574098 * r19574078;
        double r19574100 = exp(r19574099);
        double r19574101 = r19574100 / r19574078;
        double r19574102 = r19574090 / r19574083;
        double r19574103 = sqrt(r19574082);
        double r19574104 = cbrt(r19574103);
        double r19574105 = r19574102 / r19574104;
        double r19574106 = log(r19574105);
        double r19574107 = r19574090 * r19574090;
        double r19574108 = r19574107 / r19574083;
        double r19574109 = r19574108 / r19574104;
        double r19574110 = log(r19574109);
        double r19574111 = r19574106 + r19574110;
        double r19574112 = r19574078 * r19574111;
        double r19574113 = exp(r19574112);
        double r19574114 = r19574113 / r19574078;
        double r19574115 = r19574081 ? r19574101 : r19574114;
        double r19574116 = r19574076 ? r19574079 : r19574115;
        return r19574116;
}

Error

Bits error versus x

Bits error versus y

Target

Original	11.5
Target	8.6
Herbie	5.6

\[\begin{array}{l} \mathbf{if}\;y \lt -3.7311844206647956 \cdot 10^{+94}:\\ \;\;\;\;\frac{e^{\frac{-1.0}{y}}}{x}\\ \mathbf{elif}\;y \lt 2.817959242728288 \cdot 10^{+37}:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y \lt 2.347387415166998 \cdot 10^{+178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1.0}{y}}}{x}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if y < 2.081714116143433e-21

   1. Initial program 4.5

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

   2. Taylor expanded around inf 0.8

      \[\leadsto \frac{e^{\color{blue}{0}}}{x}\]

   if 2.081714116143433e-21 < y < 2.5159502131974188e+194

   1. Initial program 32.4

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

   3. Applied add-cube-cbrt 24.0

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

   4. Applied associate-/r* 24.1

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

   6. Applied add-cube-cbrt 24.3

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

   7. Applied cbrt-prod 24.2

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

   8. Applied add-cube-cbrt 24.0

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

   9. Applied times-frac 23.9

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

   10. Applied log-prod 21.5

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

   12. Applied div-inv 21.6

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

   13. Applied cbrt-prod 22.0

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

   if 2.5159502131974188e+194 < y

   1. Initial program 27.5

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

   3. Applied add-cube-cbrt 24.5

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

   4. Applied associate-/r* 24.6

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

   6. Applied add-sqr-sqrt 28.1

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

   7. Applied cbrt-prod 27.8

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

   8. Applied add-cube-cbrt 28.5

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

   9. Applied times-frac 28.5

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

   10. Applied times-frac 28.4

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

   11. Applied log-prod 12.0

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

4. Final simplification 5.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 2.081714116143433 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;y \le 2.5159502131974188 \cdot 10^{+194}:\\ \;\;\;\;\frac{e^{\left(\log \left(\frac{\sqrt[3]{\frac{x}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}}{\sqrt[3]{\sqrt[3]{x + y}}}\right) + \log \left(\frac{\left(\sqrt[3]{x} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}\right) \cdot \sqrt[3]{\frac{x}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}}{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}\right)\right) \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\log \left(\frac{\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}}{\sqrt[3]{\sqrt{x + y}}}\right) + \log \left(\frac{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + y}}}{\sqrt[3]{\sqrt{x + y}}}\right)\right)}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(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))