Average Error: 63.0 → 0.0
Time: 13.8s
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 r5343739 = n;
        double r5343740 = 1.0;
        double r5343741 = r5343739 + r5343740;
        double r5343742 = log(r5343741);
        double r5343743 = r5343741 * r5343742;
        double r5343744 = log(r5343739);
        double r5343745 = r5343739 * r5343744;
        double r5343746 = r5343743 - r5343745;
        double r5343747 = r5343746 - r5343740;
        return r5343747;
}
↓
double f(double n) {
        double r5343748 = 1.0;
        double r5343749 = -0.16666666666666666;
        double r5343750 = n;
        double r5343751 = r5343750 * r5343750;
        double r5343752 = r5343749 / r5343751;
        double r5343753 = log(r5343750);
        double r5343754 = r5343752 + r5343753;
        double r5343755 = r5343748 + r5343754;
        double r5343756 = 0.5;
        double r5343757 = r5343756 / r5343750;
        double r5343758 = r5343755 + r5343757;
        double r5343759 = r5343758 - r5343748;
        return r5343759;
}

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 2019107 
(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))