Average Error: 29.2 → 29.2
Time: 9.0s
Precision: 64
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
↓
\[\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.764470499998\right), y, 230661.510616000014\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}
↓
\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.764470499998\right), y, 230661.510616000014\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r81880 = x;
        double r81881 = y;
        double r81882 = r81880 * r81881;
        double r81883 = z;
        double r81884 = r81882 + r81883;
        double r81885 = r81884 * r81881;
        double r81886 = 27464.7644705;
        double r81887 = r81885 + r81886;
        double r81888 = r81887 * r81881;
        double r81889 = 230661.510616;
        double r81890 = r81888 + r81889;
        double r81891 = r81890 * r81881;
        double r81892 = t;
        double r81893 = r81891 + r81892;
        double r81894 = a;
        double r81895 = r81881 + r81894;
        double r81896 = r81895 * r81881;
        double r81897 = b;
        double r81898 = r81896 + r81897;
        double r81899 = r81898 * r81881;
        double r81900 = c;
        double r81901 = r81899 + r81900;
        double r81902 = r81901 * r81881;
        double r81903 = i;
        double r81904 = r81902 + r81903;
        double r81905 = r81893 / r81904;
        return r81905;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r81906 = x;
        double r81907 = y;
        double r81908 = z;
        double r81909 = fma(r81906, r81907, r81908);
        double r81910 = 27464.7644705;
        double r81911 = fma(r81909, r81907, r81910);
        double r81912 = 230661.510616;
        double r81913 = fma(r81911, r81907, r81912);
        double r81914 = r81913 * r81907;
        double r81915 = t;
        double r81916 = r81914 + r81915;
        double r81917 = a;
        double r81918 = r81907 + r81917;
        double r81919 = r81918 * r81907;
        double r81920 = b;
        double r81921 = r81919 + r81920;
        double r81922 = r81921 * r81907;
        double r81923 = c;
        double r81924 = r81922 + r81923;
        double r81925 = r81924 * r81907;
        double r81926 = i;
        double r81927 = r81925 + r81926;
        double r81928 = r81916 / r81927;
        return r81928;
}

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 29.2

    \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
  2. Using strategy rm

  3. Applied *-un-lft-identity29.2

    \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot \color{blue}{\left(1 \cdot y\right)} + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]

  4. Applied associate-*r*29.2

    \[\leadsto \frac{\color{blue}{\left(\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot 1\right) \cdot y} + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]

  5. Simplified29.2

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.764470499998\right), y, 230661.510616000014\right)} \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]

  6. Final simplification29.2

    \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.764470499998\right), y, 230661.510616000014\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]

Reproduce

herbie shell --seed 2020033 +o rules:numerics
(FPCore (x y z t a b c i)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2"
  :precision binary64
  (/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))