Average Error: 63.0 → 0
Time: 11.6s
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 r1365505 = n;
        double r1365506 = 1.0;
        double r1365507 = r1365505 + r1365506;
        double r1365508 = log(r1365507);
        double r1365509 = r1365507 * r1365508;
        double r1365510 = log(r1365505);
        double r1365511 = r1365505 * r1365510;
        double r1365512 = r1365509 - r1365511;
        double r1365513 = r1365512 - r1365506;
        return r1365513;
}
↓
double f(double n) {
        double r1365514 = 1.0;
        double r1365515 = n;
        double r1365516 = r1365514 / r1365515;
        double r1365517 = 0.5;
        double r1365518 = 0.16666666666666666;
        double r1365519 = r1365518 / r1365515;
        double r1365520 = r1365517 - r1365519;
        double r1365521 = log(r1365515);
        double r1365522 = fma(r1365516, r1365520, r1365521);
        return r1365522;
}

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 2019151 +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))