Average Error: 63.0 → 0
Time: 17.7s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\mathsf{fma}\left(\frac{1}{n}, \frac{1}{2} - \frac{\frac{1}{6}}{n}, \log n\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\mathsf{fma}\left(\frac{1}{n}, \frac{1}{2} - \frac{\frac{1}{6}}{n}, \log n\right)
double f(double n) {
        double r2041974 = n;
        double r2041975 = 1.0;
        double r2041976 = r2041974 + r2041975;
        double r2041977 = log(r2041976);
        double r2041978 = r2041976 * r2041977;
        double r2041979 = log(r2041974);
        double r2041980 = r2041974 * r2041979;
        double r2041981 = r2041978 - r2041980;
        double r2041982 = r2041981 - r2041975;
        return r2041982;
}
↓
double f(double n) {
        double r2041983 = 1.0;
        double r2041984 = n;
        double r2041985 = r2041983 / r2041984;
        double r2041986 = 0.5;
        double r2041987 = 0.16666666666666666;
        double r2041988 = r2041987 / r2041984;
        double r2041989 = r2041986 - r2041988;
        double r2041990 = log(r2041984);
        double r2041991 = fma(r2041985, r2041989, r2041990);
        return r2041991;
}

Error

Bits error versus n

Target

Original63.0
Target0.0
Herbie0
\[\log \left(n + 1\right) - \left(\frac{1}{2 \cdot n} - \left(\frac{1}{3 \cdot \left(n \cdot n\right)} - \frac{4}{{n}^{3}}\right)\right)\]

Derivation

  1. Initial program 63.0

    \[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]

  2. Simplified61.8

    \[\leadsto \color{blue}{\mathsf{fma}\left(n, \mathsf{log1p}\left(n\right), \mathsf{log1p}\left(n\right) - \mathsf{fma}\left(n, \log n, 1\right)\right)}\]

  3. Taylor expanded around inf 0.0

    \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{1}{n} - \left(\frac{1}{6} \cdot \frac{1}{{n}^{2}} + \log \left(\frac{1}{n}\right)\right)}\]

  4. Simplified0

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{n}, \frac{1}{2} - \frac{\frac{1}{6}}{n}, \log n\right)}\]

  5. Final simplification0

    \[\leadsto \mathsf{fma}\left(\frac{1}{n}, \frac{1}{2} - \frac{\frac{1}{6}}{n}, \log n\right)\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (n)
  :name "logs (example 3.8)"
  :pre (> n 6.8e+15)

  :herbie-target
  (- (log (+ n 1)) (- (/ 1 (* 2 n)) (- (/ 1 (* 3 (* n n))) (/ 4 (pow n 3)))))

  (- (- (* (+ n 1) (log (+ n 1))) (* n (log n))) 1))