Average Error: 63.0 → 0
Time: 15.6s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\frac{\frac{1}{2}}{n} - \left(\frac{\frac{1}{6}}{n \cdot n} - \log n\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\frac{\frac{1}{2}}{n} - \left(\frac{\frac{1}{6}}{n \cdot n} - \log n\right)
double f(double n) {
        double r5637628 = n;
        double r5637629 = 1.0;
        double r5637630 = r5637628 + r5637629;
        double r5637631 = log(r5637630);
        double r5637632 = r5637630 * r5637631;
        double r5637633 = log(r5637628);
        double r5637634 = r5637628 * r5637633;
        double r5637635 = r5637632 - r5637634;
        double r5637636 = r5637635 - r5637629;
        return r5637636;
}
↓
double f(double n) {
        double r5637637 = 0.5;
        double r5637638 = n;
        double r5637639 = r5637637 / r5637638;
        double r5637640 = 0.16666666666666666;
        double r5637641 = r5637638 * r5637638;
        double r5637642 = r5637640 / r5637641;
        double r5637643 = log(r5637638);
        double r5637644 = r5637642 - r5637643;
        double r5637645 = r5637639 - r5637644;
        return r5637645;
}

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. Simplified62.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(n, \left(\mathsf{log1p}\left(n\right)\right), \left(\mathsf{log1p}\left(n\right)\right)\right) - \mathsf{fma}\left(n, \left(\log n\right), 1\right)}\]

  3. Taylor expanded around inf 0.0

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

  4. Simplified0

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

  5. Final simplification0

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

Reproduce

herbie shell --seed 2019120 +o rules:numerics
(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))