Average Error: 63.0 → 0
Time: 11.9s
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 r1215721 = n;
        double r1215722 = 1.0;
        double r1215723 = r1215721 + r1215722;
        double r1215724 = log(r1215723);
        double r1215725 = r1215723 * r1215724;
        double r1215726 = log(r1215721);
        double r1215727 = r1215721 * r1215726;
        double r1215728 = r1215725 - r1215727;
        double r1215729 = r1215728 - r1215722;
        return r1215729;
}
↓
double f(double n) {
        double r1215730 = 1.0;
        double r1215731 = n;
        double r1215732 = r1215730 / r1215731;
        double r1215733 = 0.5;
        double r1215734 = 0.16666666666666666;
        double r1215735 = r1215734 / r1215731;
        double r1215736 = r1215733 - r1215735;
        double r1215737 = log(r1215731);
        double r1215738 = fma(r1215732, r1215736, r1215737);
        return r1215738;
}

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. Simplified61.9

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

  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 2019152 +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))