Average Error: 28.7 → 29.0
Time: 8.5s
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 r91912 = x;
        double r91913 = y;
        double r91914 = r91912 * r91913;
        double r91915 = z;
        double r91916 = r91914 + r91915;
        double r91917 = r91916 * r91913;
        double r91918 = 27464.7644705;
        double r91919 = r91917 + r91918;
        double r91920 = r91919 * r91913;
        double r91921 = 230661.510616;
        double r91922 = r91920 + r91921;
        double r91923 = r91922 * r91913;
        double r91924 = t;
        double r91925 = r91923 + r91924;
        double r91926 = a;
        double r91927 = r91913 + r91926;
        double r91928 = r91927 * r91913;
        double r91929 = b;
        double r91930 = r91928 + r91929;
        double r91931 = r91930 * r91913;
        double r91932 = c;
        double r91933 = r91931 + r91932;
        double r91934 = r91933 * r91913;
        double r91935 = i;
        double r91936 = r91934 + r91935;
        double r91937 = r91925 / r91936;
        return r91937;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r91938 = 1.0;
        double r91939 = y;
        double r91940 = a;
        double r91941 = r91939 + r91940;
        double r91942 = r91941 * r91939;
        double r91943 = b;
        double r91944 = r91942 + r91943;
        double r91945 = r91944 * r91939;
        double r91946 = c;
        double r91947 = r91945 + r91946;
        double r91948 = r91947 * r91939;
        double r91949 = i;
        double r91950 = r91948 + r91949;
        double r91951 = x;
        double r91952 = z;
        double r91953 = fma(r91951, r91939, r91952);
        double r91954 = 27464.7644705;
        double r91955 = fma(r91953, r91939, r91954);
        double r91956 = 230661.510616;
        double r91957 = fma(r91955, r91939, r91956);
        double r91958 = t;
        double r91959 = fma(r91957, r91939, r91958);
        double r91960 = r91950 / r91959;
        double r91961 = r91938 / r91960;
        return r91961;
}

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)))