Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2

Average Error: 2.0 → 1.4

Time: 17.7s

Precision: 64

Math

\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]

↓

\[{\left(\frac{1}{{a}^{1}}\right)}^{1} \cdot \frac{x}{e^{\log \left(\frac{1}{z}\right) \cdot y + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)} \cdot y}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r95968 = x;
        double r95969 = y;
        double r95970 = z;
        double r95971 = log(r95970);
        double r95972 = r95969 * r95971;
        double r95973 = t;
        double r95974 = 1.0;
        double r95975 = r95973 - r95974;
        double r95976 = a;
        double r95977 = log(r95976);
        double r95978 = r95975 * r95977;
        double r95979 = r95972 + r95978;
        double r95980 = b;
        double r95981 = r95979 - r95980;
        double r95982 = exp(r95981);
        double r95983 = r95968 * r95982;
        double r95984 = r95983 / r95969;
        return r95984;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r95985 = 1.0;
        double r95986 = a;
        double r95987 = 1.0;
        double r95988 = pow(r95986, r95987);
        double r95989 = r95985 / r95988;
        double r95990 = pow(r95989, r95987);
        double r95991 = x;
        double r95992 = z;
        double r95993 = r95985 / r95992;
        double r95994 = log(r95993);
        double r95995 = y;
        double r95996 = r95994 * r95995;
        double r95997 = r95985 / r95986;
        double r95998 = log(r95997);
        double r95999 = t;
        double r96000 = r95998 * r95999;
        double r96001 = b;
        double r96002 = r96000 + r96001;
        double r96003 = r95996 + r96002;
        double r96004 = exp(r96003);
        double r96005 = r96004 * r95995;
        double r96006 = r95991 / r96005;
        double r96007 = r95990 * r96006;
        return r96007;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Derivation

1. Initial program 2.0

   \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]

2. Taylor expanded around inf 2.0

   \[\leadsto \frac{x \cdot \color{blue}{e^{1 \cdot \log \left(\frac{1}{a}\right) - \left(y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)\right)}}}{y}\]

3. Simplified 1.3

   \[\leadsto \frac{x \cdot \color{blue}{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}{y}\]
4. Using strategy rm

5. Applied add-cube-cbrt 1.3

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

6. Applied sqr-pow 1.3

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

7. Applied times-frac 1.3

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

8. Applied associate-*r* 1.3

   \[\leadsto \frac{\color{blue}{\left(x \cdot \frac{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt[3]{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}} \cdot \sqrt[3]{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}\right) \cdot \frac{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt[3]{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}}{y}\]
9. Using strategy rm

10. Applied associate-/l* 0.9

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

11. Taylor expanded around inf 1.4

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

12. Final simplification 1.4

    \[\leadsto {\left(\frac{1}{{a}^{1}}\right)}^{1} \cdot \frac{x}{e^{\log \left(\frac{1}{z}\right) \cdot y + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)} \cdot y}\]

Reproduce

herbie shell --seed 2020024 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2"
  :precision binary64
  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1) (log a))) b))) y))