Average Error: 63.0 → 0.0
Time: 17.7s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(\left(1 + \frac{\frac{-1}{6}}{n \cdot n}\right) + \frac{\frac{1}{2}}{n}\right) + \log n\right) - 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\left(\left(1 + \frac{\frac{-1}{6}}{n \cdot n}\right) + \frac{\frac{1}{2}}{n}\right) + \log n\right) - 1
double f(double n) {
        double r2623160 = n;
        double r2623161 = 1.0;
        double r2623162 = r2623160 + r2623161;
        double r2623163 = log(r2623162);
        double r2623164 = r2623162 * r2623163;
        double r2623165 = log(r2623160);
        double r2623166 = r2623160 * r2623165;
        double r2623167 = r2623164 - r2623166;
        double r2623168 = r2623167 - r2623161;
        return r2623168;
}
↓
double f(double n) {
        double r2623169 = 1.0;
        double r2623170 = -0.16666666666666666;
        double r2623171 = n;
        double r2623172 = r2623171 * r2623171;
        double r2623173 = r2623170 / r2623172;
        double r2623174 = r2623169 + r2623173;
        double r2623175 = 0.5;
        double r2623176 = r2623175 / r2623171;
        double r2623177 = r2623174 + r2623176;
        double r2623178 = log(r2623171);
        double r2623179 = r2623177 + r2623178;
        double r2623180 = r2623179 - r2623169;
        return r2623180;
}

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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019163 
(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))