Average Error: 63.0 → 0.0
Time: 13.7s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(1 - \left(\left(\frac{\frac{-1}{2}}{n} - \log n\right) + \frac{\frac{1}{6}}{n \cdot n}\right)\right) - 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(1 - \left(\left(\frac{\frac{-1}{2}}{n} - \log n\right) + \frac{\frac{1}{6}}{n \cdot n}\right)\right) - 1
double f(double n) {
        double r2798045 = n;
        double r2798046 = 1.0;
        double r2798047 = r2798045 + r2798046;
        double r2798048 = log(r2798047);
        double r2798049 = r2798047 * r2798048;
        double r2798050 = log(r2798045);
        double r2798051 = r2798045 * r2798050;
        double r2798052 = r2798049 - r2798051;
        double r2798053 = r2798052 - r2798046;
        return r2798053;
}
↓
double f(double n) {
        double r2798054 = 1.0;
        double r2798055 = -0.5;
        double r2798056 = n;
        double r2798057 = r2798055 / r2798056;
        double r2798058 = log(r2798056);
        double r2798059 = r2798057 - r2798058;
        double r2798060 = 0.16666666666666666;
        double r2798061 = r2798056 * r2798056;
        double r2798062 = r2798060 / r2798061;
        double r2798063 = r2798059 + r2798062;
        double r2798064 = r2798054 - r2798063;
        double r2798065 = r2798064 - r2798054;
        return r2798065;
}

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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019121 
(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))