Average Error: 63.0 → 0.0
Time: 15.5s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(\left(\frac{\frac{1}{2}}{n} + \log n\right) + 1\right) + \frac{\frac{-1}{6}}{n \cdot n}\right) - 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\left(\left(\frac{\frac{1}{2}}{n} + \log n\right) + 1\right) + \frac{\frac{-1}{6}}{n \cdot n}\right) - 1
double f(double n) {
        double r2028506 = n;
        double r2028507 = 1.0;
        double r2028508 = r2028506 + r2028507;
        double r2028509 = log(r2028508);
        double r2028510 = r2028508 * r2028509;
        double r2028511 = log(r2028506);
        double r2028512 = r2028506 * r2028511;
        double r2028513 = r2028510 - r2028512;
        double r2028514 = r2028513 - r2028507;
        return r2028514;
}
↓
double f(double n) {
        double r2028515 = 0.5;
        double r2028516 = n;
        double r2028517 = r2028515 / r2028516;
        double r2028518 = log(r2028516);
        double r2028519 = r2028517 + r2028518;
        double r2028520 = 1.0;
        double r2028521 = r2028519 + r2028520;
        double r2028522 = -0.16666666666666666;
        double r2028523 = r2028516 * r2028516;
        double r2028524 = r2028522 / r2028523;
        double r2028525 = r2028521 + r2028524;
        double r2028526 = r2028525 - r2028520;
        return r2028526;
}

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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019139 
(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))