Average Error: 63.0 → 0
Time: 28.5s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(\frac{0.5}{n} + 1\right) - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) + \left(\log n \cdot 1 - 1\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\left(\frac{0.5}{n} + 1\right) - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) + \left(\log n \cdot 1 - 1\right)
double f(double n) {
        double r51884 = n;
        double r51885 = 1.0;
        double r51886 = r51884 + r51885;
        double r51887 = log(r51886);
        double r51888 = r51886 * r51887;
        double r51889 = log(r51884);
        double r51890 = r51884 * r51889;
        double r51891 = r51888 - r51890;
        double r51892 = r51891 - r51885;
        return r51892;
}
↓
double f(double n) {
        double r51893 = 0.5;
        double r51894 = n;
        double r51895 = r51893 / r51894;
        double r51896 = 1.0;
        double r51897 = r51895 + r51896;
        double r51898 = 0.16666666666666669;
        double r51899 = r51894 * r51894;
        double r51900 = r51898 / r51899;
        double r51901 = r51897 - r51900;
        double r51902 = log(r51894);
        double r51903 = r51902 * r51896;
        double r51904 = r51903 - r51896;
        double r51905 = r51901 + r51904;
        return r51905;
}

Error

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original63.0
Target0
Herbie0
\[\log \left(n + 1\right) - \left(\frac{1}{2 \cdot n} - \left(\frac{1}{3 \cdot \left(n \cdot n\right)} - \frac{4}{{n}^{3}}\right)\right)\]

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

    \[\leadsto \color{blue}{\left(\left(0.5 \cdot \frac{1}{n} + 1\right) - \left(1 \cdot \log \left(\frac{1}{n}\right) + 0.1666666666666666851703837437526090070605 \cdot \frac{1}{{n}^{2}}\right)\right)} - 1\]

  3. Simplified0.0

    \[\leadsto \color{blue}{\left(\left(\left(\frac{0.5}{n} + 1\right) - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) + \log n \cdot 1\right)} - 1\]
  4. Using strategy rm

  5. Applied associate--l+0

    \[\leadsto \color{blue}{\left(\left(\frac{0.5}{n} + 1\right) - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) + \left(\log n \cdot 1 - 1\right)}\]

  6. Final simplification0

    \[\leadsto \left(\left(\frac{0.5}{n} + 1\right) - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) + \left(\log n \cdot 1 - 1\right)\]

Reproduce

herbie shell --seed 2019323 
(FPCore (n)
  :name "logs (example 3.8)"
  :precision binary64
  :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))