Average Error: 63.0 → 0
Time: 39.7s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1.0\right) \cdot \log \left(n + 1.0\right) - n \cdot \log n\right) - 1.0\]
↓
\[\left(\log n \cdot 1.0 - \frac{0.16666666666666669}{n \cdot n}\right) + \frac{0.5}{n}\]
\left(\left(n + 1.0\right) \cdot \log \left(n + 1.0\right) - n \cdot \log n\right) - 1.0
↓
\left(\log n \cdot 1.0 - \frac{0.16666666666666669}{n \cdot n}\right) + \frac{0.5}{n}
double f(double n) {
        double r2471897 = n;
        double r2471898 = 1.0;
        double r2471899 = r2471897 + r2471898;
        double r2471900 = log(r2471899);
        double r2471901 = r2471899 * r2471900;
        double r2471902 = log(r2471897);
        double r2471903 = r2471897 * r2471902;
        double r2471904 = r2471901 - r2471903;
        double r2471905 = r2471904 - r2471898;
        return r2471905;
}
↓
double f(double n) {
        double r2471906 = n;
        double r2471907 = log(r2471906);
        double r2471908 = 1.0;
        double r2471909 = r2471907 * r2471908;
        double r2471910 = 0.16666666666666669;
        double r2471911 = r2471906 * r2471906;
        double r2471912 = r2471910 / r2471911;
        double r2471913 = r2471909 - r2471912;
        double r2471914 = 0.5;
        double r2471915 = r2471914 / r2471906;
        double r2471916 = r2471913 + r2471915;
        return r2471916;
}

Error

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original63.0
Target0
Herbie0
\[\log \left(n + 1.0\right) - \left(\frac{1.0}{2.0 \cdot n} - \left(\frac{1.0}{3.0 \cdot \left(n \cdot n\right)} - \frac{4.0}{{n}^{3.0}}\right)\right)\]

Derivation

  1. Initial program 63.0

    \[\left(\left(n + 1.0\right) \cdot \log \left(n + 1.0\right) - n \cdot \log n\right) - 1.0\]

  2. Taylor expanded around inf 0.0

    \[\leadsto \color{blue}{\left(\left(1.0 + 0.5 \cdot \frac{1}{n}\right) - \left(0.16666666666666669 \cdot \frac{1}{{n}^{2}} + 1.0 \cdot \log \left(\frac{1}{n}\right)\right)\right)} - 1.0\]

  3. Simplified0.0

    \[\leadsto \color{blue}{\left(\left(\left(1.0 + \frac{0.5}{n}\right) - \frac{0.16666666666666669}{n \cdot n}\right) + 1.0 \cdot \log n\right)} - 1.0\]

  4. Taylor expanded around inf 0.0

    \[\leadsto \color{blue}{0.5 \cdot \frac{1}{n} - \left(0.16666666666666669 \cdot \frac{1}{{n}^{2}} + 1.0 \cdot \log \left(\frac{1}{n}\right)\right)}\]

  5. Simplified0

    \[\leadsto \color{blue}{\frac{0.5}{n} + \left(\log n \cdot 1.0 - \frac{0.16666666666666669}{n \cdot n}\right)}\]

  6. Final simplification0

    \[\leadsto \left(\log n \cdot 1.0 - \frac{0.16666666666666669}{n \cdot n}\right) + \frac{0.5}{n}\]

Reproduce

herbie shell --seed 2019165 
(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))