Average Error: 63.0 → 0.0
Time: 17.7s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(1 - \left(\frac{\frac{1}{6}}{n \cdot n} - \log n\right)\right) + \frac{\frac{1}{2}}{n}\right) - 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\left(1 - \left(\frac{\frac{1}{6}}{n \cdot n} - \log n\right)\right) + \frac{\frac{1}{2}}{n}\right) - 1
double f(double n) {
        double r2301425 = n;
        double r2301426 = 1.0;
        double r2301427 = r2301425 + r2301426;
        double r2301428 = log(r2301427);
        double r2301429 = r2301427 * r2301428;
        double r2301430 = log(r2301425);
        double r2301431 = r2301425 * r2301430;
        double r2301432 = r2301429 - r2301431;
        double r2301433 = r2301432 - r2301426;
        return r2301433;
}
↓
double f(double n) {
        double r2301434 = 1.0;
        double r2301435 = 0.16666666666666666;
        double r2301436 = n;
        double r2301437 = r2301436 * r2301436;
        double r2301438 = r2301435 / r2301437;
        double r2301439 = log(r2301436);
        double r2301440 = r2301438 - r2301439;
        double r2301441 = r2301434 - r2301440;
        double r2301442 = 0.5;
        double r2301443 = r2301442 / r2301436;
        double r2301444 = r2301441 + r2301443;
        double r2301445 = r2301444 - r2301434;
        return r2301445;
}

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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019144 
(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))