Average Error: 63.0 → 0
Time: 18.4s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\log n + \frac{\frac{-1}{6}}{n \cdot n}\right) + \frac{\frac{1}{2}}{n}\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\log n + \frac{\frac{-1}{6}}{n \cdot n}\right) + \frac{\frac{1}{2}}{n}
double f(double n) {
        double r2658692 = n;
        double r2658693 = 1.0;
        double r2658694 = r2658692 + r2658693;
        double r2658695 = log(r2658694);
        double r2658696 = r2658694 * r2658695;
        double r2658697 = log(r2658692);
        double r2658698 = r2658692 * r2658697;
        double r2658699 = r2658696 - r2658698;
        double r2658700 = r2658699 - r2658693;
        return r2658700;
}
↓
double f(double n) {
        double r2658701 = n;
        double r2658702 = log(r2658701);
        double r2658703 = -0.16666666666666666;
        double r2658704 = r2658701 * r2658701;
        double r2658705 = r2658703 / r2658704;
        double r2658706 = r2658702 + r2658705;
        double r2658707 = 0.5;
        double r2658708 = r2658707 / r2658701;
        double r2658709 = r2658706 + r2658708;
        return r2658709;
}

Error

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original63.0
Target0
Herbie0
\[\log \left(n + 1\right) - \left(\frac{1}{2 \cdot n} - \left(\frac{1}{3 \cdot \left(n \cdot n\right)} - \frac{4}{{n}^{3}}\right)\right)\]

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

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

  4. Taylor expanded around 0 0

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

  5. Simplified0

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

  6. Final simplification0

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

Reproduce

herbie shell --seed 2019141 
(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))