Average Error: 63.0 → 0.0
Time: 13.3s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(1 - \left(\left(\frac{\frac{-1}{2}}{n} - \log n\right) + \frac{\frac{1}{6}}{n \cdot n}\right)\right) - 1\]
\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 r5343796 = n;
        double r5343797 = 1.0;
        double r5343798 = r5343796 + r5343797;
        double r5343799 = log(r5343798);
        double r5343800 = r5343798 * r5343799;
        double r5343801 = log(r5343796);
        double r5343802 = r5343796 * r5343801;
        double r5343803 = r5343800 - r5343802;
        double r5343804 = r5343803 - r5343797;
        return r5343804;
}
↓
double f(double n) {
        double r5343805 = 1.0;
        double r5343806 = -0.5;
        double r5343807 = n;
        double r5343808 = r5343806 / r5343807;
        double r5343809 = log(r5343807);
        double r5343810 = r5343808 - r5343809;
        double r5343811 = 0.16666666666666666;
        double r5343812 = r5343807 * r5343807;
        double r5343813 = r5343811 / r5343812;
        double r5343814 = r5343810 + r5343813;
        double r5343815 = r5343805 - r5343814;
        double r5343816 = r5343815 - r5343805;
        return r5343816;
}

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

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

  3. Simplified0.0

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

  4. 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 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))