Average Error: 63.0 → 0.0
Time: 16.9s
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 r4014095 = n;
        double r4014096 = 1.0;
        double r4014097 = r4014095 + r4014096;
        double r4014098 = log(r4014097);
        double r4014099 = r4014097 * r4014098;
        double r4014100 = log(r4014095);
        double r4014101 = r4014095 * r4014100;
        double r4014102 = r4014099 - r4014101;
        double r4014103 = r4014102 - r4014096;
        return r4014103;
}
↓
double f(double n) {
        double r4014104 = 0.5;
        double r4014105 = n;
        double r4014106 = r4014104 / r4014105;
        double r4014107 = 1.0;
        double r4014108 = r4014106 + r4014107;
        double r4014109 = 0.16666666666666666;
        double r4014110 = r4014105 * r4014105;
        double r4014111 = r4014109 / r4014110;
        double r4014112 = log(r4014105);
        double r4014113 = r4014111 - r4014112;
        double r4014114 = r4014108 - r4014113;
        double r4014115 = r4014114 - r4014107;
        return r4014115;
}

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

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

  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 2019158 
(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))