\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]

↓

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

double f(double n) {
        double r5057598 = n;
        double r5057599 = 1.0;
        double r5057600 = r5057598 + r5057599;
        double r5057601 = log(r5057600);
        double r5057602 = r5057600 * r5057601;
        double r5057603 = log(r5057598);
        double r5057604 = r5057598 * r5057603;
        double r5057605 = r5057602 - r5057604;
        double r5057606 = r5057605 - r5057599;
        return r5057606;
}

↓

double f(double n) {
        double r5057607 = 1.0;
        double r5057608 = -0.5;
        double r5057609 = n;
        double r5057610 = r5057608 / r5057609;
        double r5057611 = log(r5057609);
        double r5057612 = r5057610 - r5057611;
        double r5057613 = 0.16666666666666666;
        double r5057614 = r5057609 * r5057609;
        double r5057615 = r5057613 / r5057614;
        double r5057616 = r5057612 + r5057615;
        double r5057617 = r5057607 - r5057616;
        double r5057618 = r5057617 - r5057607;
        return r5057618;
}

\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1

↓

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

Original	63.0
Target	0
Herbie	0.0

Derivation

Initial program 63.0
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
Taylor expanded around inf 0.0
\[\leadsto \color{blue}{\left(\left(1 + \frac{1}{2} \cdot \frac{1}{n}\right) - \left(\frac{1}{6} \cdot \frac{1}{{n}^{2}} + \log \left(\frac{1}{n}\right)\right)\right)} - 1\]
Simplified0.0
\[\leadsto \color{blue}{\left(1 - \left(\frac{\frac{1}{6}}{n \cdot n} + \left(\frac{\frac{-1}{2}}{n} - \log n\right)\right)\right)} - 1\]
Final simplification0.0
\[\leadsto \left(1 - \left(\left(\frac{\frac{-1}{2}}{n} - \log n\right) + \frac{\frac{1}{6}}{n \cdot n}\right)\right) - 1\]

Reproduce

herbie shell --seed 2019101 
(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))

Error

Target

Derivation

Reproduce