Average Error: 63.0 → 0
Time: 21.5s
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 r3430441 = n;
        double r3430442 = 1.0;
        double r3430443 = r3430441 + r3430442;
        double r3430444 = log(r3430443);
        double r3430445 = r3430443 * r3430444;
        double r3430446 = log(r3430441);
        double r3430447 = r3430441 * r3430446;
        double r3430448 = r3430445 - r3430447;
        double r3430449 = r3430448 - r3430442;
        return r3430449;
}
↓
double f(double n) {
        double r3430450 = 0.5;
        double r3430451 = n;
        double r3430452 = r3430450 / r3430451;
        double r3430453 = 0.16666666666666666;
        double r3430454 = r3430451 * r3430451;
        double r3430455 = r3430453 / r3430454;
        double r3430456 = log(r3430451);
        double r3430457 = r3430455 - r3430456;
        double r3430458 = r3430452 - r3430457;
        return r3430458;
}


\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. Taylor expanded around -inf 62.0

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

  3. Simplified0.0

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

  4. Taylor expanded around -inf 62.0

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

  5. Simplified0

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

  6. Final simplification0

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

Reproduce

herbie shell --seed 2019102 
(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))