Average Error: 29.3 → 29.3
Time: 19.4s
Precision: 64
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
↓
\[\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.764470499998\right), y, 230661.510616000014\right), y, t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right)}\]
\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}
↓
\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.764470499998\right), y, 230661.510616000014\right), 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)}
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r84856 = x;
        double r84857 = y;
        double r84858 = r84856 * r84857;
        double r84859 = z;
        double r84860 = r84858 + r84859;
        double r84861 = r84860 * r84857;
        double r84862 = 27464.7644705;
        double r84863 = r84861 + r84862;
        double r84864 = r84863 * r84857;
        double r84865 = 230661.510616;
        double r84866 = r84864 + r84865;
        double r84867 = r84866 * r84857;
        double r84868 = t;
        double r84869 = r84867 + r84868;
        double r84870 = a;
        double r84871 = r84857 + r84870;
        double r84872 = r84871 * r84857;
        double r84873 = b;
        double r84874 = r84872 + r84873;
        double r84875 = r84874 * r84857;
        double r84876 = c;
        double r84877 = r84875 + r84876;
        double r84878 = r84877 * r84857;
        double r84879 = i;
        double r84880 = r84878 + r84879;
        double r84881 = r84869 / r84880;
        return r84881;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r84882 = x;
        double r84883 = y;
        double r84884 = z;
        double r84885 = fma(r84882, r84883, r84884);
        double r84886 = 27464.7644705;
        double r84887 = fma(r84885, r84883, r84886);
        double r84888 = 230661.510616;
        double r84889 = fma(r84887, r84883, r84888);
        double r84890 = t;
        double r84891 = fma(r84889, r84883, r84890);
        double r84892 = 1.0;
        double r84893 = a;
        double r84894 = r84883 + r84893;
        double r84895 = b;
        double r84896 = fma(r84894, r84883, r84895);
        double r84897 = c;
        double r84898 = fma(r84896, r84883, r84897);
        double r84899 = i;
        double r84900 = fma(r84898, r84883, r84899);
        double r84901 = r84892 / r84900;
        double r84902 = r84891 * r84901;
        return r84902;
}

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 29.3

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

  2. Simplified29.3

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

  4. Applied div-inv29.3

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

  5. Final simplification29.3

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

Reproduce

herbie shell --seed 2020045 +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)))