Average Error: 28.6 → 28.6
Time: 29.0s
Precision: 64
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
↓
\[\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(y, \mathsf{fma}\left(y + a, y, b\right), c\right), y, i\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644704999984242022037506103515625\right), 230661.5106160000141244381666183471679688\right), y, t\right)\]
\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}{\mathsf{fma}\left(\mathsf{fma}\left(y, \mathsf{fma}\left(y + a, y, b\right), c\right), y, i\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644704999984242022037506103515625\right), 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 r68782 = x;
        double r68783 = y;
        double r68784 = r68782 * r68783;
        double r68785 = z;
        double r68786 = r68784 + r68785;
        double r68787 = r68786 * r68783;
        double r68788 = 27464.7644705;
        double r68789 = r68787 + r68788;
        double r68790 = r68789 * r68783;
        double r68791 = 230661.510616;
        double r68792 = r68790 + r68791;
        double r68793 = r68792 * r68783;
        double r68794 = t;
        double r68795 = r68793 + r68794;
        double r68796 = a;
        double r68797 = r68783 + r68796;
        double r68798 = r68797 * r68783;
        double r68799 = b;
        double r68800 = r68798 + r68799;
        double r68801 = r68800 * r68783;
        double r68802 = c;
        double r68803 = r68801 + r68802;
        double r68804 = r68803 * r68783;
        double r68805 = i;
        double r68806 = r68804 + r68805;
        double r68807 = r68795 / r68806;
        return r68807;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r68808 = 1.0;
        double r68809 = y;
        double r68810 = a;
        double r68811 = r68809 + r68810;
        double r68812 = b;
        double r68813 = fma(r68811, r68809, r68812);
        double r68814 = c;
        double r68815 = fma(r68809, r68813, r68814);
        double r68816 = i;
        double r68817 = fma(r68815, r68809, r68816);
        double r68818 = r68808 / r68817;
        double r68819 = x;
        double r68820 = z;
        double r68821 = fma(r68809, r68819, r68820);
        double r68822 = 27464.7644705;
        double r68823 = fma(r68809, r68821, r68822);
        double r68824 = 230661.510616;
        double r68825 = fma(r68809, r68823, r68824);
        double r68826 = t;
        double r68827 = fma(r68825, r68809, r68826);
        double r68828 = r68818 * r68827;
        return r68828;
}

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 28.6

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

  2. Simplified28.6

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

  4. Applied div-inv28.6

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

  5. Simplified28.6

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

  6. Final simplification28.6

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

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (x y z t a b c i)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2"
  (/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))