Average Error: 63.0 → 0.0
Time: 17.8s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(\frac{\frac{1}{2}}{n} + 1\right) + \left(\log n + \frac{\frac{-1}{6}}{n \cdot n}\right)\right) - 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\left(\frac{\frac{1}{2}}{n} + 1\right) + \left(\log n + \frac{\frac{-1}{6}}{n \cdot n}\right)\right) - 1
double f(double n) {
        double r2437928 = n;
        double r2437929 = 1.0;
        double r2437930 = r2437928 + r2437929;
        double r2437931 = log(r2437930);
        double r2437932 = r2437930 * r2437931;
        double r2437933 = log(r2437928);
        double r2437934 = r2437928 * r2437933;
        double r2437935 = r2437932 - r2437934;
        double r2437936 = r2437935 - r2437929;
        return r2437936;
}
↓
double f(double n) {
        double r2437937 = 0.5;
        double r2437938 = n;
        double r2437939 = r2437937 / r2437938;
        double r2437940 = 1.0;
        double r2437941 = r2437939 + r2437940;
        double r2437942 = log(r2437938);
        double r2437943 = -0.16666666666666666;
        double r2437944 = r2437938 * r2437938;
        double r2437945 = r2437943 / r2437944;
        double r2437946 = r2437942 + r2437945;
        double r2437947 = r2437941 + r2437946;
        double r2437948 = r2437947 - r2437940;
        return r2437948;
}

Error

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original63.0
Target0.0
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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019137 
(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))