Average Error: 63.0 → 0.0
Time: 10.3s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(\frac{\frac{1}{2}}{n} + 1\right) - \left(\frac{\frac{1}{6}}{n \cdot n} - \log n\right)\right) - 1\]
\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(\frac{\frac{1}{6}}{n \cdot n} - \log n\right)\right) - 1
double f(double n) {
        double r1440875 = n;
        double r1440876 = 1.0;
        double r1440877 = r1440875 + r1440876;
        double r1440878 = log(r1440877);
        double r1440879 = r1440877 * r1440878;
        double r1440880 = log(r1440875);
        double r1440881 = r1440875 * r1440880;
        double r1440882 = r1440879 - r1440881;
        double r1440883 = r1440882 - r1440876;
        return r1440883;
}
↓
double f(double n) {
        double r1440884 = 0.5;
        double r1440885 = n;
        double r1440886 = r1440884 / r1440885;
        double r1440887 = 1.0;
        double r1440888 = r1440886 + r1440887;
        double r1440889 = 0.16666666666666666;
        double r1440890 = r1440885 * r1440885;
        double r1440891 = r1440889 / r1440890;
        double r1440892 = log(r1440885);
        double r1440893 = r1440891 - r1440892;
        double r1440894 = r1440888 - r1440893;
        double r1440895 = r1440894 - r1440887;
        return r1440895;
}

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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019156 
(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))