Average Error: 63.0 → 0
Time: 11.1s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\log n - \frac{\frac{1}{6}}{n \cdot n}\right) + \frac{\frac{1}{2}}{n}\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\log n - \frac{\frac{1}{6}}{n \cdot n}\right) + \frac{\frac{1}{2}}{n}
double f(double n) {
        double r1200657 = n;
        double r1200658 = 1.0;
        double r1200659 = r1200657 + r1200658;
        double r1200660 = log(r1200659);
        double r1200661 = r1200659 * r1200660;
        double r1200662 = log(r1200657);
        double r1200663 = r1200657 * r1200662;
        double r1200664 = r1200661 - r1200663;
        double r1200665 = r1200664 - r1200658;
        return r1200665;
}
↓
double f(double n) {
        double r1200666 = n;
        double r1200667 = log(r1200666);
        double r1200668 = 0.16666666666666666;
        double r1200669 = r1200666 * r1200666;
        double r1200670 = r1200668 / r1200669;
        double r1200671 = r1200667 - r1200670;
        double r1200672 = 0.5;
        double r1200673 = r1200672 / r1200666;
        double r1200674 = r1200671 + r1200673;
        return r1200674;
}

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. 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

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

  5. Final simplification0

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

Reproduce

herbie shell --seed 2019153 +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))