Average Error: 63.0 → 0.0
Time: 12.9s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(\left(1 + \frac{0.5}{n}\right) - \frac{0.16666666666666669}{n \cdot n}\right) + \log n \cdot 1\right) - 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\left(\left(1 + \frac{0.5}{n}\right) - \frac{0.16666666666666669}{n \cdot n}\right) + \log n \cdot 1\right) - 1
double f(double n) {
        double r58746 = n;
        double r58747 = 1.0;
        double r58748 = r58746 + r58747;
        double r58749 = log(r58748);
        double r58750 = r58748 * r58749;
        double r58751 = log(r58746);
        double r58752 = r58746 * r58751;
        double r58753 = r58750 - r58752;
        double r58754 = r58753 - r58747;
        return r58754;
}
↓
double f(double n) {
        double r58755 = 1.0;
        double r58756 = 0.5;
        double r58757 = n;
        double r58758 = r58756 / r58757;
        double r58759 = r58755 + r58758;
        double r58760 = 0.16666666666666669;
        double r58761 = r58757 * r58757;
        double r58762 = r58760 / r58761;
        double r58763 = r58759 - r58762;
        double r58764 = log(r58757);
        double r58765 = r58764 * r58755;
        double r58766 = r58763 + r58765;
        double r58767 = r58766 - r58755;
        return r58767;
}

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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2020042 
(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))