Average Error: 63.0 → 0
Time: 39.2s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\frac{\frac{1}{2}}{n} - \left(\frac{\frac{1}{6}}{n \cdot n} - \log n\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\frac{\frac{1}{2}}{n} - \left(\frac{\frac{1}{6}}{n \cdot n} - \log n\right)
double f(double n) {
        double r10355130 = n;
        double r10355131 = 1.0;
        double r10355132 = r10355130 + r10355131;
        double r10355133 = log(r10355132);
        double r10355134 = r10355132 * r10355133;
        double r10355135 = log(r10355130);
        double r10355136 = r10355130 * r10355135;
        double r10355137 = r10355134 - r10355136;
        double r10355138 = r10355137 - r10355131;
        return r10355138;
}
↓
double f(double n) {
        double r10355139 = 0.5;
        double r10355140 = n;
        double r10355141 = r10355139 / r10355140;
        double r10355142 = 0.16666666666666666;
        double r10355143 = r10355140 * r10355140;
        double r10355144 = r10355142 / r10355143;
        double r10355145 = log(r10355140);
        double r10355146 = r10355144 - r10355145;
        double r10355147 = r10355141 - r10355146;
        return r10355147;
}

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

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

  4. Taylor expanded around -inf 62.0

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

  5. Simplified0

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

  6. Final simplification0

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

Reproduce

herbie shell --seed 2019128 
(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))