Average Error: 63.0 → 0.0
Time: 11.0s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(\left(\frac{0.5}{n} + 1\right) + 1 \cdot \log n\right) - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) - 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\left(\left(\frac{0.5}{n} + 1\right) + 1 \cdot \log n\right) - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) - 1
double f(double n) {
        double r27051 = n;
        double r27052 = 1.0;
        double r27053 = r27051 + r27052;
        double r27054 = log(r27053);
        double r27055 = r27053 * r27054;
        double r27056 = log(r27051);
        double r27057 = r27051 * r27056;
        double r27058 = r27055 - r27057;
        double r27059 = r27058 - r27052;
        return r27059;
}
↓
double f(double n) {
        double r27060 = 0.5;
        double r27061 = n;
        double r27062 = r27060 / r27061;
        double r27063 = 1.0;
        double r27064 = r27062 + r27063;
        double r27065 = log(r27061);
        double r27066 = r27063 * r27065;
        double r27067 = r27064 + r27066;
        double r27068 = 0.16666666666666669;
        double r27069 = r27061 * r27061;
        double r27070 = r27068 / r27069;
        double r27071 = r27067 - r27070;
        double r27072 = r27071 - r27063;
        return r27072;
}

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

    \[\leadsto \color{blue}{\left(\left(0.5 \cdot \frac{1}{n} + 1\right) - \left(1 \cdot \log \left(\frac{1}{n}\right) + 0.1666666666666666851703837437526090070605 \cdot \frac{1}{{n}^{2}}\right)\right)} - 1\]

  3. Simplified0.0

    \[\leadsto \color{blue}{\left(\left(\left(\frac{0.5}{n} + 1\right) + 1 \cdot \log n\right) - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right)} - 1\]

  4. Final simplification0.0

    \[\leadsto \left(\left(\left(\frac{0.5}{n} + 1\right) + 1 \cdot \log n\right) - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) - 1\]

Reproduce

herbie shell --seed 2019305 
(FPCore (n)
  :name "logs (example 3.8)"
  :precision binary64
  :pre (> n 6.8e15)

  :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))