Average Error: 0.1 → 0.1
Time: 12.8s
Precision: 64
\[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]
↓
\[\left(\left(\left(\left(\mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{y}\right), x \cdot \log \left({\left(\frac{1}{y}\right)}^{\frac{-1}{3}}\right)\right) + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]
\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i
↓
\left(\left(\left(\left(\mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{y}\right), x \cdot \log \left({\left(\frac{1}{y}\right)}^{\frac{-1}{3}}\right)\right) + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r61883 = x;
        double r61884 = y;
        double r61885 = log(r61884);
        double r61886 = r61883 * r61885;
        double r61887 = z;
        double r61888 = r61886 + r61887;
        double r61889 = t;
        double r61890 = r61888 + r61889;
        double r61891 = a;
        double r61892 = r61890 + r61891;
        double r61893 = b;
        double r61894 = 0.5;
        double r61895 = r61893 - r61894;
        double r61896 = c;
        double r61897 = log(r61896);
        double r61898 = r61895 * r61897;
        double r61899 = r61892 + r61898;
        double r61900 = i;
        double r61901 = r61884 * r61900;
        double r61902 = r61899 + r61901;
        return r61902;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r61903 = x;
        double r61904 = 2.0;
        double r61905 = y;
        double r61906 = cbrt(r61905);
        double r61907 = log(r61906);
        double r61908 = r61904 * r61907;
        double r61909 = 1.0;
        double r61910 = r61909 / r61905;
        double r61911 = -0.3333333333333333;
        double r61912 = pow(r61910, r61911);
        double r61913 = log(r61912);
        double r61914 = r61903 * r61913;
        double r61915 = fma(r61903, r61908, r61914);
        double r61916 = z;
        double r61917 = r61915 + r61916;
        double r61918 = t;
        double r61919 = r61917 + r61918;
        double r61920 = a;
        double r61921 = r61919 + r61920;
        double r61922 = b;
        double r61923 = 0.5;
        double r61924 = r61922 - r61923;
        double r61925 = c;
        double r61926 = log(r61925);
        double r61927 = r61924 * r61926;
        double r61928 = r61921 + r61927;
        double r61929 = i;
        double r61930 = r61905 * r61929;
        double r61931 = r61928 + r61930;
        return r61931;
}

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

Bits error versus c

Bits error versus i

Derivation

  1. Initial program 0.1

    \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]
  2. Using strategy rm

  3. Applied add-cube-cbrt0.1

    \[\leadsto \left(\left(\left(\left(x \cdot \log \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)} + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]

  4. Applied log-prod0.1

    \[\leadsto \left(\left(\left(\left(x \cdot \color{blue}{\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right)} + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]

  5. Applied distribute-lft-in0.1

    \[\leadsto \left(\left(\left(\left(\color{blue}{\left(x \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + x \cdot \log \left(\sqrt[3]{y}\right)\right)} + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]

  6. Simplified0.1

    \[\leadsto \left(\left(\left(\left(\left(\color{blue}{x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right)} + x \cdot \log \left(\sqrt[3]{y}\right)\right) + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]

  7. Taylor expanded around inf 0.1

    \[\leadsto \left(\left(\left(\left(\left(x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + x \cdot \log \color{blue}{\left({\left(\frac{1}{y}\right)}^{\frac{-1}{3}}\right)}\right) + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]
  8. Using strategy rm

  9. Applied fma-def0.1

    \[\leadsto \left(\left(\left(\left(\color{blue}{\mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{y}\right), x \cdot \log \left({\left(\frac{1}{y}\right)}^{\frac{-1}{3}}\right)\right)} + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]

  10. Final simplification0.1

    \[\leadsto \left(\left(\left(\left(\mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{y}\right), x \cdot \log \left({\left(\frac{1}{y}\right)}^{\frac{-1}{3}}\right)\right) + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x y z t a b c i)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, B"
  :precision binary64
  (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i)))