Average Error: 29.0 → 29.1
Time: 8.3s
Precision: 64
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
↓
\[\left(\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right) \cdot 1}\]
\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}
↓
\left(\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right) \cdot 1}
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r58822 = x;
        double r58823 = y;
        double r58824 = r58822 * r58823;
        double r58825 = z;
        double r58826 = r58824 + r58825;
        double r58827 = r58826 * r58823;
        double r58828 = 27464.7644705;
        double r58829 = r58827 + r58828;
        double r58830 = r58829 * r58823;
        double r58831 = 230661.510616;
        double r58832 = r58830 + r58831;
        double r58833 = r58832 * r58823;
        double r58834 = t;
        double r58835 = r58833 + r58834;
        double r58836 = a;
        double r58837 = r58823 + r58836;
        double r58838 = r58837 * r58823;
        double r58839 = b;
        double r58840 = r58838 + r58839;
        double r58841 = r58840 * r58823;
        double r58842 = c;
        double r58843 = r58841 + r58842;
        double r58844 = r58843 * r58823;
        double r58845 = i;
        double r58846 = r58844 + r58845;
        double r58847 = r58835 / r58846;
        return r58847;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r58848 = x;
        double r58849 = y;
        double r58850 = r58848 * r58849;
        double r58851 = z;
        double r58852 = r58850 + r58851;
        double r58853 = r58852 * r58849;
        double r58854 = 27464.7644705;
        double r58855 = r58853 + r58854;
        double r58856 = r58855 * r58849;
        double r58857 = 230661.510616;
        double r58858 = r58856 + r58857;
        double r58859 = r58858 * r58849;
        double r58860 = t;
        double r58861 = r58859 + r58860;
        double r58862 = 1.0;
        double r58863 = a;
        double r58864 = r58849 + r58863;
        double r58865 = b;
        double r58866 = fma(r58864, r58849, r58865);
        double r58867 = c;
        double r58868 = fma(r58866, r58849, r58867);
        double r58869 = i;
        double r58870 = fma(r58868, r58849, r58869);
        double r58871 = r58870 * r58862;
        double r58872 = r58862 / r58871;
        double r58873 = r58861 * r58872;
        return r58873;
}

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 29.0

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

  3. Applied div-inv29.1

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

  4. Simplified29.1

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

  5. Final simplification29.1

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

Reproduce

herbie shell --seed 2020100 +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)))