Average Error: 63.0 → 0
Time: 2.8s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\mathsf{fma}\left(\frac{1}{n}, 0.5 - \frac{0.1666666666666666851703837437526090070605}{n}, \log n \cdot 1\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\mathsf{fma}\left(\frac{1}{n}, 0.5 - \frac{0.1666666666666666851703837437526090070605}{n}, \log n \cdot 1\right)
double f(double n) {
        double r80612 = n;
        double r80613 = 1.0;
        double r80614 = r80612 + r80613;
        double r80615 = log(r80614);
        double r80616 = r80614 * r80615;
        double r80617 = log(r80612);
        double r80618 = r80612 * r80617;
        double r80619 = r80616 - r80618;
        double r80620 = r80619 - r80613;
        return r80620;
}
↓
double f(double n) {
        double r80621 = 1.0;
        double r80622 = n;
        double r80623 = r80621 / r80622;
        double r80624 = 0.5;
        double r80625 = 0.16666666666666669;
        double r80626 = r80625 / r80622;
        double r80627 = r80624 - r80626;
        double r80628 = log(r80622);
        double r80629 = 1.0;
        double r80630 = r80628 * r80629;
        double r80631 = fma(r80623, r80627, r80630);
        return r80631;
}

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. Simplified61.9

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

  3. Taylor expanded around inf 0.0

    \[\leadsto \color{blue}{0.5 \cdot \frac{1}{n} - \left(1 \cdot \log \left(\frac{1}{n}\right) + 0.1666666666666666851703837437526090070605 \cdot \frac{1}{{n}^{2}}\right)}\]

  4. Simplified0

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{n}, 0.5 - \frac{0.1666666666666666851703837437526090070605}{n}, \log n \cdot 1\right)}\]

  5. Final simplification0

    \[\leadsto \mathsf{fma}\left(\frac{1}{n}, 0.5 - \frac{0.1666666666666666851703837437526090070605}{n}, \log n \cdot 1\right)\]

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (n)
  :name "logs (example 3.8)"
  :precision binary64
  :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))