Average Error: 28.7 → 29.0
Time: 8.8s
Precision: 64
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
↓
\[\frac{1}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644704999984242022037506103515625\right), y, 230661.5106160000141244381666183471679688\right), y, t\right)}}\]
\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}
↓
\frac{1}{\frac{\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, 27464.7644704999984242022037506103515625\right), y, 230661.5106160000141244381666183471679688\right), y, t\right)}}
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r44906 = x;
        double r44907 = y;
        double r44908 = r44906 * r44907;
        double r44909 = z;
        double r44910 = r44908 + r44909;
        double r44911 = r44910 * r44907;
        double r44912 = 27464.7644705;
        double r44913 = r44911 + r44912;
        double r44914 = r44913 * r44907;
        double r44915 = 230661.510616;
        double r44916 = r44914 + r44915;
        double r44917 = r44916 * r44907;
        double r44918 = t;
        double r44919 = r44917 + r44918;
        double r44920 = a;
        double r44921 = r44907 + r44920;
        double r44922 = r44921 * r44907;
        double r44923 = b;
        double r44924 = r44922 + r44923;
        double r44925 = r44924 * r44907;
        double r44926 = c;
        double r44927 = r44925 + r44926;
        double r44928 = r44927 * r44907;
        double r44929 = i;
        double r44930 = r44928 + r44929;
        double r44931 = r44919 / r44930;
        return r44931;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r44932 = 1.0;
        double r44933 = y;
        double r44934 = a;
        double r44935 = r44933 + r44934;
        double r44936 = r44935 * r44933;
        double r44937 = b;
        double r44938 = r44936 + r44937;
        double r44939 = r44938 * r44933;
        double r44940 = c;
        double r44941 = r44939 + r44940;
        double r44942 = r44941 * r44933;
        double r44943 = i;
        double r44944 = r44942 + r44943;
        double r44945 = x;
        double r44946 = z;
        double r44947 = fma(r44945, r44933, r44946);
        double r44948 = 27464.7644705;
        double r44949 = fma(r44947, r44933, r44948);
        double r44950 = 230661.510616;
        double r44951 = fma(r44949, r44933, r44950);
        double r44952 = t;
        double r44953 = fma(r44951, r44933, r44952);
        double r44954 = r44944 / r44953;
        double r44955 = r44932 / r44954;
        return r44955;
}

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 28.7

    \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
  2. Using strategy rm

  3. Applied *-un-lft-identity28.7

    \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\color{blue}{1 \cdot \left(\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i\right)}}\]

  4. Applied associate-/r*28.7

    \[\leadsto \color{blue}{\frac{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}{1}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}}\]

  5. Simplified28.7

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644704999984242022037506103515625\right), y, 230661.5106160000141244381666183471679688\right), y, t\right)}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
  6. Using strategy rm

  7. Applied clear-num29.0

    \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644704999984242022037506103515625\right), y, 230661.5106160000141244381666183471679688\right), y, t\right)}}}\]

  8. Final simplification29.0

    \[\leadsto \frac{1}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644704999984242022037506103515625\right), y, 230661.5106160000141244381666183471679688\right), y, t\right)}}\]

Reproduce

herbie shell --seed 2019353 +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)))