Average Error: 63.0 → 0
Time: 20.4s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\log n + \frac{\frac{-1}{6}}{n \cdot n}\right) + \frac{\frac{1}{2}}{n}\]
\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 r2017366 = n;
        double r2017367 = 1.0;
        double r2017368 = r2017366 + r2017367;
        double r2017369 = log(r2017368);
        double r2017370 = r2017368 * r2017369;
        double r2017371 = log(r2017366);
        double r2017372 = r2017366 * r2017371;
        double r2017373 = r2017370 - r2017372;
        double r2017374 = r2017373 - r2017367;
        return r2017374;
}
↓
double f(double n) {
        double r2017375 = n;
        double r2017376 = log(r2017375);
        double r2017377 = -0.16666666666666666;
        double r2017378 = r2017375 * r2017375;
        double r2017379 = r2017377 / r2017378;
        double r2017380 = r2017376 + r2017379;
        double r2017381 = 0.5;
        double r2017382 = r2017381 / r2017375;
        double r2017383 = r2017380 + r2017382;
        return r2017383;
}

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

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

  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 2019146 +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))