Average Error: 63.0 → 0
Time: 14.9s
Precision: 64
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\frac{\frac{1}{2}}{n} - \left(\frac{\frac{1}{6}}{n \cdot n} - \log n\right)\]
double f(double n) {
        double r3988251 = n;
        double r3988252 = 1.0;
        double r3988253 = r3988251 + r3988252;
        double r3988254 = log(r3988253);
        double r3988255 = r3988253 * r3988254;
        double r3988256 = log(r3988251);
        double r3988257 = r3988251 * r3988256;
        double r3988258 = r3988255 - r3988257;
        double r3988259 = r3988258 - r3988252;
        return r3988259;
}
↓
double f(double n) {
        double r3988260 = 0.5;
        double r3988261 = n;
        double r3988262 = r3988260 / r3988261;
        double r3988263 = 0.16666666666666666;
        double r3988264 = r3988261 * r3988261;
        double r3988265 = r3988263 / r3988264;
        double r3988266 = log(r3988261);
        double r3988267 = r3988265 - r3988266;
        double r3988268 = r3988262 - r3988267;
        return r3988268;
}


\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\frac{\frac{1}{2}}{n} - \left(\frac{\frac{1}{6}}{n \cdot n} - \log n\right)

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. Simplified62.0

    \[\leadsto \color{blue}{(n \cdot \left(\log_* (1 + n)\right) + \left(\log_* (1 + n)\right))_* - (n \cdot \left(\log n\right) + 1)_*}\]

  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

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

  5. Final simplification0

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

Reproduce

herbie shell --seed 2019101 +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))