Average Error: 63.0 → 0
Time: 14.5s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\frac{\frac{1}{2}}{n} - \left(\frac{\frac{1}{6}}{n \cdot n} - \log n\right)\]
\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 r4969791 = n;
        double r4969792 = 1.0;
        double r4969793 = r4969791 + r4969792;
        double r4969794 = log(r4969793);
        double r4969795 = r4969793 * r4969794;
        double r4969796 = log(r4969791);
        double r4969797 = r4969791 * r4969796;
        double r4969798 = r4969795 - r4969797;
        double r4969799 = r4969798 - r4969792;
        return r4969799;
}
↓
double f(double n) {
        double r4969800 = 0.5;
        double r4969801 = n;
        double r4969802 = r4969800 / r4969801;
        double r4969803 = 0.16666666666666666;
        double r4969804 = r4969801 * r4969801;
        double r4969805 = r4969803 / r4969804;
        double r4969806 = log(r4969801);
        double r4969807 = r4969805 - r4969806;
        double r4969808 = r4969802 - r4969807;
        return r4969808;
}

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. Simplified44.2

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

  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 2019124 +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))