Average Error: 63.0 → 0.0
Time: 17.5s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(\frac{\frac{1}{2}}{n} + 1\right) + \left(\log n + \frac{\frac{-1}{6}}{n \cdot n}\right)\right) - 1\]
\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(\log n + \frac{\frac{-1}{6}}{n \cdot n}\right)\right) - 1
double f(double n) {
        double r2034528 = n;
        double r2034529 = 1.0;
        double r2034530 = r2034528 + r2034529;
        double r2034531 = log(r2034530);
        double r2034532 = r2034530 * r2034531;
        double r2034533 = log(r2034528);
        double r2034534 = r2034528 * r2034533;
        double r2034535 = r2034532 - r2034534;
        double r2034536 = r2034535 - r2034529;
        return r2034536;
}
↓
double f(double n) {
        double r2034537 = 0.5;
        double r2034538 = n;
        double r2034539 = r2034537 / r2034538;
        double r2034540 = 1.0;
        double r2034541 = r2034539 + r2034540;
        double r2034542 = log(r2034538);
        double r2034543 = -0.16666666666666666;
        double r2034544 = r2034538 * r2034538;
        double r2034545 = r2034543 / r2034544;
        double r2034546 = r2034542 + r2034545;
        double r2034547 = r2034541 + r2034546;
        double r2034548 = r2034547 - r2034540;
        return r2034548;
}

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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019135 
(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))