Average Error: 29.1 → 28.3
Time: 9.7s
Precision: 64
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \le 3.62450759398618954484228470238219178564 \cdot 10^{294}:\\ \;\;\;\;\left(\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right) \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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}
↓
\begin{array}{l}
\mathbf{if}\;\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} \le 3.62450759398618954484228470238219178564 \cdot 10^{294}:\\
\;\;\;\;\left(\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\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}\\

\mathbf{else}:\\
\;\;\;\;0\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r86909 = x;
        double r86910 = y;
        double r86911 = r86909 * r86910;
        double r86912 = z;
        double r86913 = r86911 + r86912;
        double r86914 = r86913 * r86910;
        double r86915 = 27464.7644705;
        double r86916 = r86914 + r86915;
        double r86917 = r86916 * r86910;
        double r86918 = 230661.510616;
        double r86919 = r86917 + r86918;
        double r86920 = r86919 * r86910;
        double r86921 = t;
        double r86922 = r86920 + r86921;
        double r86923 = a;
        double r86924 = r86910 + r86923;
        double r86925 = r86924 * r86910;
        double r86926 = b;
        double r86927 = r86925 + r86926;
        double r86928 = r86927 * r86910;
        double r86929 = c;
        double r86930 = r86928 + r86929;
        double r86931 = r86930 * r86910;
        double r86932 = i;
        double r86933 = r86931 + r86932;
        double r86934 = r86922 / r86933;
        return r86934;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r86935 = x;
        double r86936 = y;
        double r86937 = r86935 * r86936;
        double r86938 = z;
        double r86939 = r86937 + r86938;
        double r86940 = r86939 * r86936;
        double r86941 = 27464.7644705;
        double r86942 = r86940 + r86941;
        double r86943 = r86942 * r86936;
        double r86944 = 230661.510616;
        double r86945 = r86943 + r86944;
        double r86946 = r86945 * r86936;
        double r86947 = t;
        double r86948 = r86946 + r86947;
        double r86949 = a;
        double r86950 = r86936 + r86949;
        double r86951 = r86950 * r86936;
        double r86952 = b;
        double r86953 = r86951 + r86952;
        double r86954 = r86953 * r86936;
        double r86955 = c;
        double r86956 = r86954 + r86955;
        double r86957 = r86956 * r86936;
        double r86958 = i;
        double r86959 = r86957 + r86958;
        double r86960 = r86948 / r86959;
        double r86961 = 3.6245075939861895e+294;
        bool r86962 = r86960 <= r86961;
        double r86963 = 1.0;
        double r86964 = fma(r86950, r86936, r86952);
        double r86965 = fma(r86964, r86936, r86955);
        double r86966 = fma(r86965, r86936, r86958);
        double r86967 = r86966 * r86963;
        double r86968 = r86963 / r86967;
        double r86969 = r86948 * r86968;
        double r86970 = 0.0;
        double r86971 = r86962 ? r86969 : r86970;
        return r86971;
}

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. Split input into 2 regimes

  2. if (/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)) < 3.6245075939861895e+294

    1. Initial program 5.1

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

    3. Applied div-inv5.2

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

    4. Simplified5.2

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

    if 3.6245075939861895e+294 < (/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i))

    1. Initial program 63.8

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

    2. Taylor expanded around 0 61.7

      \[\leadsto \color{blue}{0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification28.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \le 3.62450759398618954484228470238219178564 \cdot 10^{294}:\\ \;\;\;\;\left(\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right) \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 +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)))