Average Error: 63.0 → 0.0
Time: 18.4s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(1 + \frac{\frac{-1}{6}}{n \cdot n}\right) + \left(\frac{\frac{1}{2}}{n} + \log n\right)\right) - 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\left(1 + \frac{\frac{-1}{6}}{n \cdot n}\right) + \left(\frac{\frac{1}{2}}{n} + \log n\right)\right) - 1
double f(double n) {
        double r1882714 = n;
        double r1882715 = 1.0;
        double r1882716 = r1882714 + r1882715;
        double r1882717 = log(r1882716);
        double r1882718 = r1882716 * r1882717;
        double r1882719 = log(r1882714);
        double r1882720 = r1882714 * r1882719;
        double r1882721 = r1882718 - r1882720;
        double r1882722 = r1882721 - r1882715;
        return r1882722;
}
↓
double f(double n) {
        double r1882723 = 1.0;
        double r1882724 = -0.16666666666666666;
        double r1882725 = n;
        double r1882726 = r1882725 * r1882725;
        double r1882727 = r1882724 / r1882726;
        double r1882728 = r1882723 + r1882727;
        double r1882729 = 0.5;
        double r1882730 = r1882729 / r1882725;
        double r1882731 = log(r1882725);
        double r1882732 = r1882730 + r1882731;
        double r1882733 = r1882728 + r1882732;
        double r1882734 = r1882733 - r1882723;
        return r1882734;
}

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. Simplified44.2

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

  3. Taylor expanded around inf 0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019129 +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))