Average Error: 63.0 → 0.0
Time: 18.4s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(\frac{\frac{-1}{6}}{n \cdot n} + \left(\log n + \frac{\frac{1}{2}}{n}\right)\right) + 1\right) - 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\left(\frac{\frac{-1}{6}}{n \cdot n} + \left(\log n + \frac{\frac{1}{2}}{n}\right)\right) + 1\right) - 1
double f(double n) {
        double r2241909 = n;
        double r2241910 = 1.0;
        double r2241911 = r2241909 + r2241910;
        double r2241912 = log(r2241911);
        double r2241913 = r2241911 * r2241912;
        double r2241914 = log(r2241909);
        double r2241915 = r2241909 * r2241914;
        double r2241916 = r2241913 - r2241915;
        double r2241917 = r2241916 - r2241910;
        return r2241917;
}
↓
double f(double n) {
        double r2241918 = -0.16666666666666666;
        double r2241919 = n;
        double r2241920 = r2241919 * r2241919;
        double r2241921 = r2241918 / r2241920;
        double r2241922 = log(r2241919);
        double r2241923 = 0.5;
        double r2241924 = r2241923 / r2241919;
        double r2241925 = r2241922 + r2241924;
        double r2241926 = r2241921 + r2241925;
        double r2241927 = 1.0;
        double r2241928 = r2241926 + r2241927;
        double r2241929 = r2241928 - r2241927;
        return r2241929;
}

Error

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original63.0
Target0
Herbie0.0
\[\log \left(n + 1\right) - \left(\frac{1}{2 \cdot n} - \left(\frac{1}{3 \cdot \left(n \cdot n\right)} - \frac{4}{{n}^{3}}\right)\right)\]

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 0.0

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019142 
(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))