Average Error: 63.0 → 0.0
Time: 18.1s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(\frac{\frac{-1}{6}}{n \cdot n} + \left(\log n + \frac{\frac{1}{2}}{n}\right)\right) + 1\right) - 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\left(\frac{\frac{-1}{6}}{n \cdot n} + \left(\log n + \frac{\frac{1}{2}}{n}\right)\right) + 1\right) - 1
double f(double n) {
        double r2046156 = n;
        double r2046157 = 1.0;
        double r2046158 = r2046156 + r2046157;
        double r2046159 = log(r2046158);
        double r2046160 = r2046158 * r2046159;
        double r2046161 = log(r2046156);
        double r2046162 = r2046156 * r2046161;
        double r2046163 = r2046160 - r2046162;
        double r2046164 = r2046163 - r2046157;
        return r2046164;
}
↓
double f(double n) {
        double r2046165 = -0.16666666666666666;
        double r2046166 = n;
        double r2046167 = r2046166 * r2046166;
        double r2046168 = r2046165 / r2046167;
        double r2046169 = log(r2046166);
        double r2046170 = 0.5;
        double r2046171 = r2046170 / r2046166;
        double r2046172 = r2046169 + r2046171;
        double r2046173 = r2046168 + r2046172;
        double r2046174 = 1.0;
        double r2046175 = r2046173 + r2046174;
        double r2046176 = r2046175 - r2046174;
        return r2046176;
}

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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019146 
(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))