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

Average Error: 10.7 → 0.1

Time: 19.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.82478823432214161812668457202583340491 \cdot 10^{60} \lor \neg \left(x \le 3.919013061791306373748966507264412939548\right):\\ \;\;\;\;\frac{1}{x} \cdot e^{-y}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}\right)}^{x} \cdot {\left(\frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}\right)}^{x}\right) \cdot \frac{{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}\right)}^{x}}{x}\\ \end{array}\]

C

double f(double x, double y) {
        double r329943 = x;
        double r329944 = y;
        double r329945 = r329943 + r329944;
        double r329946 = r329943 / r329945;
        double r329947 = log(r329946);
        double r329948 = r329943 * r329947;
        double r329949 = exp(r329948);
        double r329950 = r329949 / r329943;
        return r329950;
}

↓

double f(double x, double y) {
        double r329951 = x;
        double r329952 = -1.8247882343221416e+60;
        bool r329953 = r329951 <= r329952;
        double r329954 = 3.9190130617913064;
        bool r329955 = r329951 <= r329954;
        double r329956 = !r329955;
        bool r329957 = r329953 || r329956;
        double r329958 = 1.0;
        double r329959 = r329958 / r329951;
        double r329960 = y;
        double r329961 = -r329960;
        double r329962 = exp(r329961);
        double r329963 = r329959 * r329962;
        double r329964 = cbrt(r329951);
        double r329965 = r329960 + r329951;
        double r329966 = cbrt(r329965);
        double r329967 = r329964 / r329966;
        double r329968 = pow(r329967, r329951);
        double r329969 = r329968 * r329968;
        double r329970 = r329968 / r329951;
        double r329971 = r329969 * r329970;
        double r329972 = r329957 ? r329963 : r329971;
        return r329972;
}

Error

Bits error versus x

Bits error versus y

Target

Original	10.7
Target	8.0
Herbie	0.1

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

Derivation

1. Split input into 2 regimes

2. if x < -1.8247882343221416e+60 or 3.9190130617913064 < x

   1. Initial program 11.7

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

   2. Simplified 11.7

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

   3. Taylor expanded around inf 0.0

      \[\leadsto \frac{\color{blue}{e^{-y}}}{x}\]
   4. Using strategy rm

   5. Applied div-inv 0.0

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

   if -1.8247882343221416e+60 < x < 3.9190130617913064

   1. Initial program 9.8

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

   2. Simplified 9.8

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

   4. Applied *-un-lft-identity 9.8

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

   5. Applied add-cube-cbrt 12.6

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

   6. Applied add-cube-cbrt 9.8

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

   7. Applied times-frac 9.8

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

   8. Applied unpow-prod-down 2.0

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

   9. Applied times-frac 2.0

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

   10. Simplified 2.0

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

   12. Applied unpow-prod-down 0.2

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.82478823432214161812668457202583340491 \cdot 10^{60} \lor \neg \left(x \le 3.919013061791306373748966507264412939548\right):\\ \;\;\;\;\frac{1}{x} \cdot e^{-y}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}\right)}^{x} \cdot {\left(\frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}\right)}^{x}\right) \cdot \frac{{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}\right)}^{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(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))