Average Error: 63.0 → 0.0
Time: 13.4s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(\left(1 + \frac{0.5}{n}\right) - \frac{0.16666666666666669}{n \cdot n}\right) + \log n \cdot 1\right) - 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\left(\left(1 + \frac{0.5}{n}\right) - \frac{0.16666666666666669}{n \cdot n}\right) + \log n \cdot 1\right) - 1
double f(double n) {
        double r57292 = n;
        double r57293 = 1.0;
        double r57294 = r57292 + r57293;
        double r57295 = log(r57294);
        double r57296 = r57294 * r57295;
        double r57297 = log(r57292);
        double r57298 = r57292 * r57297;
        double r57299 = r57296 - r57298;
        double r57300 = r57299 - r57293;
        return r57300;
}
↓
double f(double n) {
        double r57301 = 1.0;
        double r57302 = 0.5;
        double r57303 = n;
        double r57304 = r57302 / r57303;
        double r57305 = r57301 + r57304;
        double r57306 = 0.16666666666666669;
        double r57307 = r57303 * r57303;
        double r57308 = r57306 / r57307;
        double r57309 = r57305 - r57308;
        double r57310 = log(r57303);
        double r57311 = r57310 * r57301;
        double r57312 = r57309 + r57311;
        double r57313 = r57312 - r57301;
        return r57313;
}

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

    \[\leadsto \color{blue}{\left(\left(0.5 \cdot \frac{1}{n} + 1\right) - \left(1 \cdot \log \left(\frac{1}{n}\right) + 0.16666666666666669 \cdot \frac{1}{{n}^{2}}\right)\right)} - 1\]

  3. Simplified0.0

    \[\leadsto \color{blue}{\left(\left(\left(1 + \frac{0.5}{n}\right) - \frac{0.16666666666666669}{n \cdot n}\right) + \log n \cdot 1\right)} - 1\]

  4. Final simplification0.0

    \[\leadsto \left(\left(\left(1 + \frac{0.5}{n}\right) - \frac{0.16666666666666669}{n \cdot n}\right) + \log n \cdot 1\right) - 1\]

Reproduce

herbie shell --seed 2020047 
(FPCore (n)
  :name "logs (example 3.8)"
  :precision binary64
  :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))