Average Error: 63.0 → 0.0
Time: 15.6s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(\frac{\frac{-1}{6}}{n \cdot n} + \left(\frac{\frac{1}{2}}{n} + 1\right)\right) + \log n\right) - 1\]
\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(\frac{\frac{1}{2}}{n} + 1\right)\right) + \log n\right) - 1
double f(double n) {
        double r3419930 = n;
        double r3419931 = 1.0;
        double r3419932 = r3419930 + r3419931;
        double r3419933 = log(r3419932);
        double r3419934 = r3419932 * r3419933;
        double r3419935 = log(r3419930);
        double r3419936 = r3419930 * r3419935;
        double r3419937 = r3419934 - r3419936;
        double r3419938 = r3419937 - r3419931;
        return r3419938;
}
↓
double f(double n) {
        double r3419939 = -0.16666666666666666;
        double r3419940 = n;
        double r3419941 = r3419940 * r3419940;
        double r3419942 = r3419939 / r3419941;
        double r3419943 = 0.5;
        double r3419944 = r3419943 / r3419940;
        double r3419945 = 1.0;
        double r3419946 = r3419944 + r3419945;
        double r3419947 = r3419942 + r3419946;
        double r3419948 = log(r3419940);
        double r3419949 = r3419947 + r3419948;
        double r3419950 = r3419949 - r3419945;
        return r3419950;
}

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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019163 
(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))