Average Error: 63.0 → 0.0
Time: 3.4s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(1 - \left(1 \cdot \log \left(\frac{1}{n}\right) + 0.16666666666666669 \cdot \frac{1}{{n}^{2}}\right)\right) + \frac{0.5}{n}\right) - 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\left(1 - \left(1 \cdot \log \left(\frac{1}{n}\right) + 0.16666666666666669 \cdot \frac{1}{{n}^{2}}\right)\right) + \frac{0.5}{n}\right) - 1
double f(double n) {
        double r50579 = n;
        double r50580 = 1.0;
        double r50581 = r50579 + r50580;
        double r50582 = log(r50581);
        double r50583 = r50581 * r50582;
        double r50584 = log(r50579);
        double r50585 = r50579 * r50584;
        double r50586 = r50583 - r50585;
        double r50587 = r50586 - r50580;
        return r50587;
}
↓
double f(double n) {
        double r50588 = 1.0;
        double r50589 = 1.0;
        double r50590 = n;
        double r50591 = r50589 / r50590;
        double r50592 = log(r50591);
        double r50593 = r50588 * r50592;
        double r50594 = 0.16666666666666669;
        double r50595 = 2.0;
        double r50596 = pow(r50590, r50595);
        double r50597 = r50589 / r50596;
        double r50598 = r50594 * r50597;
        double r50599 = r50593 + r50598;
        double r50600 = r50588 - r50599;
        double r50601 = 0.5;
        double r50602 = r50601 / r50590;
        double r50603 = r50600 + r50602;
        double r50604 = r50603 - r50588;
        return r50604;
}

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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2020056 
(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))