Average Error: 63.0 → 0
Time: 13.4s
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 r1513528 = n;
        double r1513529 = 1.0;
        double r1513530 = r1513528 + r1513529;
        double r1513531 = log(r1513530);
        double r1513532 = r1513530 * r1513531;
        double r1513533 = log(r1513528);
        double r1513534 = r1513528 * r1513533;
        double r1513535 = r1513532 - r1513534;
        double r1513536 = r1513535 - r1513529;
        return r1513536;
}
↓
double f(double n) {
        double r1513537 = n;
        double r1513538 = log(r1513537);
        double r1513539 = -0.16666666666666666;
        double r1513540 = r1513537 * r1513537;
        double r1513541 = r1513539 / r1513540;
        double r1513542 = r1513538 + r1513541;
        double r1513543 = 0.5;
        double r1513544 = r1513543 / r1513537;
        double r1513545 = r1513542 + r1513544;
        return r1513545;
}

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

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

  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

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

  5. Final simplification0

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

Reproduce

herbie shell --seed 2019133 +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))