Average Error: 63.0 → 0
Time: 11.2s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\frac{\frac{1}{2}}{n} - \mathsf{fma}\left(\left(\log n\right), -1, \left(\frac{\frac{1}{6}}{n \cdot n}\right)\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\frac{\frac{1}{2}}{n} - \mathsf{fma}\left(\left(\log n\right), -1, \left(\frac{\frac{1}{6}}{n \cdot n}\right)\right)
double f(double n) {
        double r1365856 = n;
        double r1365857 = 1.0;
        double r1365858 = r1365856 + r1365857;
        double r1365859 = log(r1365858);
        double r1365860 = r1365858 * r1365859;
        double r1365861 = log(r1365856);
        double r1365862 = r1365856 * r1365861;
        double r1365863 = r1365860 - r1365862;
        double r1365864 = r1365863 - r1365857;
        return r1365864;
}
↓
double f(double n) {
        double r1365865 = 0.5;
        double r1365866 = n;
        double r1365867 = r1365865 / r1365866;
        double r1365868 = log(r1365866);
        double r1365869 = -1.0;
        double r1365870 = 0.16666666666666666;
        double r1365871 = r1365866 * r1365866;
        double r1365872 = r1365870 / r1365871;
        double r1365873 = fma(r1365868, r1365869, r1365872);
        double r1365874 = r1365867 - r1365873;
        return r1365874;
}

Error

Bits error versus n

Target

Original63.0
Target0
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. Simplified44.2

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

  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

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

  5. Final simplification0

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

Reproduce

herbie shell --seed 2019128 +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))