Average Error: 63.0 → 0
Time: 22.1s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\log n + \left(\frac{\frac{-1}{6}}{n \cdot n} + \frac{\frac{1}{2}}{n}\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\log n + \left(\frac{\frac{-1}{6}}{n \cdot n} + \frac{\frac{1}{2}}{n}\right)
double f(double n) {
        double r1583995 = n;
        double r1583996 = 1.0;
        double r1583997 = r1583995 + r1583996;
        double r1583998 = log(r1583997);
        double r1583999 = r1583997 * r1583998;
        double r1584000 = log(r1583995);
        double r1584001 = r1583995 * r1584000;
        double r1584002 = r1583999 - r1584001;
        double r1584003 = r1584002 - r1583996;
        return r1584003;
}
↓
double f(double n) {
        double r1584004 = n;
        double r1584005 = log(r1584004);
        double r1584006 = -0.16666666666666666;
        double r1584007 = r1584004 * r1584004;
        double r1584008 = r1584006 / r1584007;
        double r1584009 = 0.5;
        double r1584010 = r1584009 / r1584004;
        double r1584011 = r1584008 + r1584010;
        double r1584012 = r1584005 + r1584011;
        return r1584012;
}

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

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

  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

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

  6. Final simplification0

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

Reproduce

herbie shell --seed 2019168 
(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))