Average Error: 63.0 → 0.0
Time: 17.6s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(1 - \left(\frac{\frac{\frac{1}{6}}{n}}{n} - \log n\right)\right) + \frac{\frac{1}{2}}{n}\right) - 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\left(1 - \left(\frac{\frac{\frac{1}{6}}{n}}{n} - \log n\right)\right) + \frac{\frac{1}{2}}{n}\right) - 1
double f(double n) {
        double r2439242 = n;
        double r2439243 = 1.0;
        double r2439244 = r2439242 + r2439243;
        double r2439245 = log(r2439244);
        double r2439246 = r2439244 * r2439245;
        double r2439247 = log(r2439242);
        double r2439248 = r2439242 * r2439247;
        double r2439249 = r2439246 - r2439248;
        double r2439250 = r2439249 - r2439243;
        return r2439250;
}
↓
double f(double n) {
        double r2439251 = 1.0;
        double r2439252 = 0.16666666666666666;
        double r2439253 = n;
        double r2439254 = r2439252 / r2439253;
        double r2439255 = r2439254 / r2439253;
        double r2439256 = log(r2439253);
        double r2439257 = r2439255 - r2439256;
        double r2439258 = r2439251 - r2439257;
        double r2439259 = 0.5;
        double r2439260 = r2439259 / r2439253;
        double r2439261 = r2439258 + r2439260;
        double r2439262 = r2439261 - r2439251;
        return r2439262;
}

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. Taylor expanded around inf 0.0

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019133 
(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))