Average Error: 63.0 → 0.0
Time: 13.2s
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 r1850459 = n;
        double r1850460 = 1.0;
        double r1850461 = r1850459 + r1850460;
        double r1850462 = log(r1850461);
        double r1850463 = r1850461 * r1850462;
        double r1850464 = log(r1850459);
        double r1850465 = r1850459 * r1850464;
        double r1850466 = r1850463 - r1850465;
        double r1850467 = r1850466 - r1850460;
        return r1850467;
}
↓
double f(double n) {
        double r1850468 = 1.0;
        double r1850469 = -0.16666666666666666;
        double r1850470 = n;
        double r1850471 = r1850470 * r1850470;
        double r1850472 = r1850469 / r1850471;
        double r1850473 = log(r1850470);
        double r1850474 = r1850472 + r1850473;
        double r1850475 = r1850468 + r1850474;
        double r1850476 = 0.5;
        double r1850477 = r1850476 / r1850470;
        double r1850478 = r1850475 + r1850477;
        double r1850479 = r1850478 - r1850468;
        return r1850479;
}

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 62.0

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

  3. Simplified0.0

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

  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 2019119 
(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))