Average Error: 63.0 → 0.0
Time: 15.6s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(\left(\frac{\frac{1}{2}}{n} + \log n\right) + 1\right) + \frac{\frac{-1}{6}}{n \cdot n}\right) - 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\left(\left(\frac{\frac{1}{2}}{n} + \log n\right) + 1\right) + \frac{\frac{-1}{6}}{n \cdot n}\right) - 1
double f(double n) {
        double r1896875 = n;
        double r1896876 = 1.0;
        double r1896877 = r1896875 + r1896876;
        double r1896878 = log(r1896877);
        double r1896879 = r1896877 * r1896878;
        double r1896880 = log(r1896875);
        double r1896881 = r1896875 * r1896880;
        double r1896882 = r1896879 - r1896881;
        double r1896883 = r1896882 - r1896876;
        return r1896883;
}
↓
double f(double n) {
        double r1896884 = 0.5;
        double r1896885 = n;
        double r1896886 = r1896884 / r1896885;
        double r1896887 = log(r1896885);
        double r1896888 = r1896886 + r1896887;
        double r1896889 = 1.0;
        double r1896890 = r1896888 + r1896889;
        double r1896891 = -0.16666666666666666;
        double r1896892 = r1896885 * r1896885;
        double r1896893 = r1896891 / r1896892;
        double r1896894 = r1896890 + r1896893;
        double r1896895 = r1896894 - r1896889;
        return r1896895;
}

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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019138 
(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))