Average Error: 63.0 → 0
Time: 16.3s
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 r1166574 = n;
        double r1166575 = 1.0;
        double r1166576 = r1166574 + r1166575;
        double r1166577 = log(r1166576);
        double r1166578 = r1166576 * r1166577;
        double r1166579 = log(r1166574);
        double r1166580 = r1166574 * r1166579;
        double r1166581 = r1166578 - r1166580;
        double r1166582 = r1166581 - r1166575;
        return r1166582;
}
↓
double f(double n) {
        double r1166583 = 1.0;
        double r1166584 = n;
        double r1166585 = r1166583 / r1166584;
        double r1166586 = 0.5;
        double r1166587 = 0.16666666666666666;
        double r1166588 = r1166587 / r1166584;
        double r1166589 = r1166586 - r1166588;
        double r1166590 = log(r1166584);
        double r1166591 = fma(r1166585, r1166589, r1166590);
        return r1166591;
}

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 2019163 +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))