Average Error: 63.0 → 0.0
Time: 21.2s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\frac{\frac{1}{2}}{n} + \left(1 - \left(\frac{\frac{1}{6}}{n \cdot n} - \log n\right)\right)\right) - 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\frac{\frac{1}{2}}{n} + \left(1 - \left(\frac{\frac{1}{6}}{n \cdot n} - \log n\right)\right)\right) - 1
double f(double n) {
        double r2704840 = n;
        double r2704841 = 1.0;
        double r2704842 = r2704840 + r2704841;
        double r2704843 = log(r2704842);
        double r2704844 = r2704842 * r2704843;
        double r2704845 = log(r2704840);
        double r2704846 = r2704840 * r2704845;
        double r2704847 = r2704844 - r2704846;
        double r2704848 = r2704847 - r2704841;
        return r2704848;
}
↓
double f(double n) {
        double r2704849 = 0.5;
        double r2704850 = n;
        double r2704851 = r2704849 / r2704850;
        double r2704852 = 1.0;
        double r2704853 = 0.16666666666666666;
        double r2704854 = r2704850 * r2704850;
        double r2704855 = r2704853 / r2704854;
        double r2704856 = log(r2704850);
        double r2704857 = r2704855 - r2704856;
        double r2704858 = r2704852 - r2704857;
        double r2704859 = r2704851 + r2704858;
        double r2704860 = r2704859 - r2704852;
        return r2704860;
}

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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019134 
(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))