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

Average Error: 6.0 → 1.5

Time: 5.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le 1609.50674842192324831557925790548324585 \lor \neg \left(z \le 3.956447898588514434000900140654028393961 \cdot 10^{51}\right):\\ \;\;\;\;x + \frac{e^{y \cdot \left(1 \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right) + \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-1 \cdot z}}{y}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r376924 = x;
        double r376925 = y;
        double r376926 = z;
        double r376927 = r376926 + r376925;
        double r376928 = r376925 / r376927;
        double r376929 = log(r376928);
        double r376930 = r376925 * r376929;
        double r376931 = exp(r376930);
        double r376932 = r376931 / r376925;
        double r376933 = r376924 + r376932;
        return r376933;
}

↓

double f(double x, double y, double z) {
        double r376934 = z;
        double r376935 = 1609.5067484219232;
        bool r376936 = r376934 <= r376935;
        double r376937 = 3.9564478985885144e+51;
        bool r376938 = r376934 <= r376937;
        double r376939 = !r376938;
        bool r376940 = r376936 || r376939;
        double r376941 = x;
        double r376942 = y;
        double r376943 = 1.0;
        double r376944 = 2.0;
        double r376945 = cbrt(r376942);
        double r376946 = r376934 + r376942;
        double r376947 = cbrt(r376946);
        double r376948 = r376945 / r376947;
        double r376949 = log(r376948);
        double r376950 = r376944 * r376949;
        double r376951 = r376943 * r376950;
        double r376952 = r376951 + r376949;
        double r376953 = r376942 * r376952;
        double r376954 = exp(r376953);
        double r376955 = r376954 / r376942;
        double r376956 = r376941 + r376955;
        double r376957 = -1.0;
        double r376958 = r376957 * r376934;
        double r376959 = exp(r376958);
        double r376960 = r376959 / r376942;
        double r376961 = r376941 + r376960;
        double r376962 = r376940 ? r376956 : r376961;
        return r376962;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.0
Target	1.1
Herbie	1.5

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.115415759790762719541517221498726780517 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < 1609.5067484219232 or 3.9564478985885144e+51 < z

   1. Initial program 5.8

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

   3. Applied add-cube-cbrt 19.5

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

   4. Applied add-cube-cbrt 5.8

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

   5. Applied times-frac 5.8

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

   6. Applied log-prod 1.8

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

   8. Applied pow1 1.8

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

   9. Applied log-pow 1.8

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

   10. Simplified 0.8

       \[\leadsto x + \frac{e^{y \cdot \left(1 \cdot \color{blue}{\left(2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right)} + \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right)}}{y}\]

   if 1609.5067484219232 < z < 3.9564478985885144e+51

   1. Initial program 9.2

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

   2. Taylor expanded around inf 15.9

      \[\leadsto \color{blue}{x + \frac{e^{-1 \cdot z}}{y}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le 1609.50674842192324831557925790548324585 \lor \neg \left(z \le 3.956447898588514434000900140654028393961 \cdot 10^{51}\right):\\ \;\;\;\;x + \frac{e^{y \cdot \left(1 \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right) + \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-1 \cdot z}}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, G"
  :precision binary64

  :herbie-target
  (if (< (/ y (+ z y)) 7.1154157597908e-315) (+ x (/ (exp (/ -1 z)) y)) (+ x (/ (exp (log (pow (/ y (+ y z)) y))) y)))

  (+ x (/ (exp (* y (log (/ y (+ z y))))) y)))