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 r1237223 = n;
        double r1237224 = 1.0;
        double r1237225 = r1237223 + r1237224;
        double r1237226 = log(r1237225);
        double r1237227 = r1237225 * r1237226;
        double r1237228 = log(r1237223);
        double r1237229 = r1237223 * r1237228;
        double r1237230 = r1237227 - r1237229;
        double r1237231 = r1237230 - r1237224;
        return r1237231;
}
↓
double f(double n) {
        double r1237232 = n;
        double r1237233 = log(r1237232);
        double r1237234 = 0.16666666666666666;
        double r1237235 = r1237232 * r1237232;
        double r1237236 = r1237234 / r1237235;
        double r1237237 = r1237233 - r1237236;
        double r1237238 = 0.5;
        double r1237239 = r1237238 / r1237232;
        double r1237240 = r1237237 + r1237239;
        return r1237240;
}

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))