Average Error: 63.0 → 0
Time: 13.0s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\log n + \frac{\frac{1}{2}}{n}\right) - \frac{\frac{\frac{1}{6}}{n}}{n}\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\log n + \frac{\frac{1}{2}}{n}\right) - \frac{\frac{\frac{1}{6}}{n}}{n}
double f(double n) {
        double r2562766 = n;
        double r2562767 = 1.0;
        double r2562768 = r2562766 + r2562767;
        double r2562769 = log(r2562768);
        double r2562770 = r2562768 * r2562769;
        double r2562771 = log(r2562766);
        double r2562772 = r2562766 * r2562771;
        double r2562773 = r2562770 - r2562772;
        double r2562774 = r2562773 - r2562767;
        return r2562774;
}
↓
double f(double n) {
        double r2562775 = n;
        double r2562776 = log(r2562775);
        double r2562777 = 0.5;
        double r2562778 = r2562777 / r2562775;
        double r2562779 = r2562776 + r2562778;
        double r2562780 = 0.16666666666666666;
        double r2562781 = r2562780 / r2562775;
        double r2562782 = r2562781 / r2562775;
        double r2562783 = r2562779 - r2562782;
        return r2562783;
}

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

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

  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

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

  5. Final simplification0

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

Reproduce

herbie shell --seed 2019134 +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))