Average Error: 63.0 → 0.0
Time: 14.7s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\frac{\frac{1}{2}}{n} - \left(\frac{\frac{1}{6}}{n \cdot n} - \log n\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\frac{\frac{1}{2}}{n} - \left(\frac{\frac{1}{6}}{n \cdot n} - \log n\right)
double f(double n) {
        double r13911253 = n;
        double r13911254 = 1.0;
        double r13911255 = r13911253 + r13911254;
        double r13911256 = log(r13911255);
        double r13911257 = r13911255 * r13911256;
        double r13911258 = log(r13911253);
        double r13911259 = r13911253 * r13911258;
        double r13911260 = r13911257 - r13911259;
        double r13911261 = r13911260 - r13911254;
        return r13911261;
}
↓
double f(double n) {
        double r13911262 = 0.5;
        double r13911263 = n;
        double r13911264 = r13911262 / r13911263;
        double r13911265 = 0.16666666666666666;
        double r13911266 = r13911263 * r13911263;
        double r13911267 = r13911265 / r13911266;
        double r13911268 = log(r13911263);
        double r13911269 = r13911267 - r13911268;
        double r13911270 = r13911264 - r13911269;
        return r13911270;
}

Error

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original63.0
Target0
Herbie0.0
\[\log \left(n + 1\right) - \left(\frac{1}{2 \cdot n} - \left(\frac{1}{3 \cdot \left(n \cdot n\right)} - \frac{4}{{n}^{3}}\right)\right)\]

Derivation

  1. Initial program 63.0

    \[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]

  2. Simplified62.0

    \[\leadsto \color{blue}{(n \cdot \left(\log_* (1 + n)\right) + \left(\log_* (1 + n)\right))_* - (n \cdot \left(\log n\right) + 1)_*}\]

  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. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019112 +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))