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

Average Error: 11.4 → 0.3

Time: 4.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.00731808408035590930024172351409106340725 \lor \neg \left(x \le 7.905402982476203987971530295908451080322\right):\\ \;\;\;\;\frac{e^{-1 \cdot y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{1}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right)}^{x} \cdot \left({\left(\frac{1}{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}\right)}^{x} \cdot {\left(\frac{x}{\sqrt[3]{\sqrt[3]{x + y}}}\right)}^{x}\right)}{x}\\ \end{array}\]

C

double f(double x, double y) {
        double r336919 = x;
        double r336920 = y;
        double r336921 = r336919 + r336920;
        double r336922 = r336919 / r336921;
        double r336923 = log(r336922);
        double r336924 = r336919 * r336923;
        double r336925 = exp(r336924);
        double r336926 = r336925 / r336919;
        return r336926;
}

↓

double f(double x, double y) {
        double r336927 = x;
        double r336928 = -0.007318084080355909;
        bool r336929 = r336927 <= r336928;
        double r336930 = 7.905402982476204;
        bool r336931 = r336927 <= r336930;
        double r336932 = !r336931;
        bool r336933 = r336929 || r336932;
        double r336934 = -1.0;
        double r336935 = y;
        double r336936 = r336934 * r336935;
        double r336937 = exp(r336936);
        double r336938 = r336937 / r336927;
        double r336939 = 1.0;
        double r336940 = r336927 + r336935;
        double r336941 = cbrt(r336940);
        double r336942 = r336941 * r336941;
        double r336943 = r336939 / r336942;
        double r336944 = pow(r336943, r336927);
        double r336945 = cbrt(r336942);
        double r336946 = r336939 / r336945;
        double r336947 = pow(r336946, r336927);
        double r336948 = cbrt(r336941);
        double r336949 = r336927 / r336948;
        double r336950 = pow(r336949, r336927);
        double r336951 = r336947 * r336950;
        double r336952 = r336944 * r336951;
        double r336953 = r336952 / r336927;
        double r336954 = r336933 ? r336938 : r336953;
        return r336954;
}

Error

Bits error versus x

Bits error versus y

Target

Original	11.4
Target	8.0
Herbie	0.3

\[\begin{array}{l} \mathbf{if}\;y \lt -3.73118442066479561492798134439269393419 \cdot 10^{94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y \lt 28179592427282878868860376020282245120:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y \lt 2.347387415166997963747840232163110922613 \cdot 10^{178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if x < -0.007318084080355909 or 7.905402982476204 < x

   1. Initial program 10.8

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

   2. Simplified 10.8

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

   3. Taylor expanded around inf 0.2

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

   4. Simplified 0.2

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

   if -0.007318084080355909 < x < 7.905402982476204

   1. Initial program 12.0

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

   2. Simplified 12.0

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

   4. Applied add-cube-cbrt 12.0

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

   5. Applied *-un-lft-identity 12.0

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

   6. Applied times-frac 12.0

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

   7. Applied unpow-prod-down 3.0

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

   9. Applied add-cube-cbrt 3.0

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

   10. Applied cbrt-prod 3.0

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

   11. Applied *-un-lft-identity 3.0

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

   12. Applied times-frac 3.0

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

   13. Applied unpow-prod-down 0.4

       \[\leadsto \frac{{\left(\frac{1}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right)}^{x} \cdot \color{blue}{\left({\left(\frac{1}{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}\right)}^{x} \cdot {\left(\frac{x}{\sqrt[3]{\sqrt[3]{x + y}}}\right)}^{x}\right)}}{x}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.00731808408035590930024172351409106340725 \lor \neg \left(x \le 7.905402982476203987971530295908451080322\right):\\ \;\;\;\;\frac{e^{-1 \cdot y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{1}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right)}^{x} \cdot \left({\left(\frac{1}{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}\right)}^{x} \cdot {\left(\frac{x}{\sqrt[3]{\sqrt[3]{x + y}}}\right)}^{x}\right)}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 +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))