Average Error: 63.0 → 0
Time: 19.8s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\log n + \frac{\frac{-1}{6}}{n \cdot n}\right) + \frac{\frac{1}{2}}{n}\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\log n + \frac{\frac{-1}{6}}{n \cdot n}\right) + \frac{\frac{1}{2}}{n}
double f(double n) {
        double r2434040 = n;
        double r2434041 = 1.0;
        double r2434042 = r2434040 + r2434041;
        double r2434043 = log(r2434042);
        double r2434044 = r2434042 * r2434043;
        double r2434045 = log(r2434040);
        double r2434046 = r2434040 * r2434045;
        double r2434047 = r2434044 - r2434046;
        double r2434048 = r2434047 - r2434041;
        return r2434048;
}
↓
double f(double n) {
        double r2434049 = n;
        double r2434050 = log(r2434049);
        double r2434051 = -0.16666666666666666;
        double r2434052 = r2434049 * r2434049;
        double r2434053 = r2434051 / r2434052;
        double r2434054 = r2434050 + r2434053;
        double r2434055 = 0.5;
        double r2434056 = r2434055 / r2434049;
        double r2434057 = r2434054 + r2434056;
        return r2434057;
}

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

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

  4. Taylor expanded around 0 0

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

  5. Simplified0

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

  6. Final simplification0

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

Reproduce

herbie shell --seed 2019132 
(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))