Average Error: 29.0 → 28.1
Time: 9.6s
Precision: 64
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \le 5.142889239312206 \cdot 10^{306}:\\ \;\;\;\;\left(\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right) \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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}
↓
\begin{array}{l}
\mathbf{if}\;\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} \le 5.142889239312206 \cdot 10^{306}:\\
\;\;\;\;\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}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r85030 = x;
        double r85031 = y;
        double r85032 = r85030 * r85031;
        double r85033 = z;
        double r85034 = r85032 + r85033;
        double r85035 = r85034 * r85031;
        double r85036 = 27464.7644705;
        double r85037 = r85035 + r85036;
        double r85038 = r85037 * r85031;
        double r85039 = 230661.510616;
        double r85040 = r85038 + r85039;
        double r85041 = r85040 * r85031;
        double r85042 = t;
        double r85043 = r85041 + r85042;
        double r85044 = a;
        double r85045 = r85031 + r85044;
        double r85046 = r85045 * r85031;
        double r85047 = b;
        double r85048 = r85046 + r85047;
        double r85049 = r85048 * r85031;
        double r85050 = c;
        double r85051 = r85049 + r85050;
        double r85052 = r85051 * r85031;
        double r85053 = i;
        double r85054 = r85052 + r85053;
        double r85055 = r85043 / r85054;
        return r85055;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r85056 = x;
        double r85057 = y;
        double r85058 = r85056 * r85057;
        double r85059 = z;
        double r85060 = r85058 + r85059;
        double r85061 = r85060 * r85057;
        double r85062 = 27464.7644705;
        double r85063 = r85061 + r85062;
        double r85064 = r85063 * r85057;
        double r85065 = 230661.510616;
        double r85066 = r85064 + r85065;
        double r85067 = r85066 * r85057;
        double r85068 = t;
        double r85069 = r85067 + r85068;
        double r85070 = a;
        double r85071 = r85057 + r85070;
        double r85072 = r85071 * r85057;
        double r85073 = b;
        double r85074 = r85072 + r85073;
        double r85075 = r85074 * r85057;
        double r85076 = c;
        double r85077 = r85075 + r85076;
        double r85078 = r85077 * r85057;
        double r85079 = i;
        double r85080 = r85078 + r85079;
        double r85081 = r85069 / r85080;
        double r85082 = 5.142889239312206e+306;
        bool r85083 = r85081 <= r85082;
        double r85084 = 1.0;
        double r85085 = fma(r85071, r85057, r85073);
        double r85086 = fma(r85085, r85057, r85076);
        double r85087 = fma(r85086, r85057, r85079);
        double r85088 = r85087 * r85084;
        double r85089 = r85084 / r85088;
        double r85090 = r85069 * r85089;
        double r85091 = 0.0;
        double r85092 = r85083 ? r85090 : r85091;
        return r85092;
}

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)) < 5.142889239312206e+306

    1. Initial program 5.2

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

    3. Applied div-inv5.4

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

    4. Simplified5.3

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

    if 5.142889239312206e+306 < (/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i))

    1. Initial program 64.0

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

    2. Taylor expanded around 0 61.8

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

  4. Final simplification28.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \le 5.142889239312206 \cdot 10^{306}:\\ \;\;\;\;\left(\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right) \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 +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)))