Average Error: 63.0 → 0
Time: 15.6s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\frac{0.5}{n} - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) + \log n \cdot 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\frac{0.5}{n} - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) + \log n \cdot 1
double f(double n) {
        double r2936027 = n;
        double r2936028 = 1.0;
        double r2936029 = r2936027 + r2936028;
        double r2936030 = log(r2936029);
        double r2936031 = r2936029 * r2936030;
        double r2936032 = log(r2936027);
        double r2936033 = r2936027 * r2936032;
        double r2936034 = r2936031 - r2936033;
        double r2936035 = r2936034 - r2936028;
        return r2936035;
}
↓
double f(double n) {
        double r2936036 = 0.5;
        double r2936037 = n;
        double r2936038 = r2936036 / r2936037;
        double r2936039 = 0.16666666666666669;
        double r2936040 = r2936037 * r2936037;
        double r2936041 = r2936039 / r2936040;
        double r2936042 = r2936038 - r2936041;
        double r2936043 = log(r2936037);
        double r2936044 = 1.0;
        double r2936045 = r2936043 * r2936044;
        double r2936046 = r2936042 + r2936045;
        return r2936046;
}

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. Simplified62.0

    \[\leadsto \color{blue}{\log \left(1 + n\right) \cdot \left(1 + n\right) - \mathsf{fma}\left(n, \log n, 1\right)}\]

  3. Taylor expanded around inf 0.0

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

  4. Simplified0

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

  5. Final simplification0

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

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (n)
  :name "logs (example 3.8)"
  :pre (> n 6.8e+15)

  :herbie-target
  (- (log (+ n 1.0)) (- (/ 1.0 (* 2.0 n)) (- (/ 1.0 (* 3.0 (* n n))) (/ 4.0 (pow n 3.0)))))

  (- (- (* (+ n 1.0) (log (+ n 1.0))) (* n (log n))) 1.0))