Average Error: 63.0 → 0
Time: 9.8s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\log n + \left(\frac{\frac{1}{2}}{n} - \frac{\frac{1}{6}}{n \cdot n}\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\log n + \left(\frac{\frac{1}{2}}{n} - \frac{\frac{1}{6}}{n \cdot n}\right)
double f(double n) {
        double r1130558 = n;
        double r1130559 = 1.0;
        double r1130560 = r1130558 + r1130559;
        double r1130561 = log(r1130560);
        double r1130562 = r1130560 * r1130561;
        double r1130563 = log(r1130558);
        double r1130564 = r1130558 * r1130563;
        double r1130565 = r1130562 - r1130564;
        double r1130566 = r1130565 - r1130559;
        return r1130566;
}
↓
double f(double n) {
        double r1130567 = n;
        double r1130568 = log(r1130567);
        double r1130569 = 0.5;
        double r1130570 = r1130569 / r1130567;
        double r1130571 = 0.16666666666666666;
        double r1130572 = r1130567 * r1130567;
        double r1130573 = r1130571 / r1130572;
        double r1130574 = r1130570 - r1130573;
        double r1130575 = r1130568 + r1130574;
        return r1130575;
}

Error

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original63.0
Target0.0
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. Simplified61.9

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

  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

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

  5. Final simplification0

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

Reproduce

herbie shell --seed 2019156 +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))