Average Error: 63.0 → 0.0
Time: 16.5s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(\frac{\frac{1}{2}}{n} + 1\right) + \left(\log n + \frac{\frac{-1}{6}}{n \cdot n}\right)\right) - 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\left(\frac{\frac{1}{2}}{n} + 1\right) + \left(\log n + \frac{\frac{-1}{6}}{n \cdot n}\right)\right) - 1
double f(double n) {
        double r1358121 = n;
        double r1358122 = 1.0;
        double r1358123 = r1358121 + r1358122;
        double r1358124 = log(r1358123);
        double r1358125 = r1358123 * r1358124;
        double r1358126 = log(r1358121);
        double r1358127 = r1358121 * r1358126;
        double r1358128 = r1358125 - r1358127;
        double r1358129 = r1358128 - r1358122;
        return r1358129;
}
↓
double f(double n) {
        double r1358130 = 0.5;
        double r1358131 = n;
        double r1358132 = r1358130 / r1358131;
        double r1358133 = 1.0;
        double r1358134 = r1358132 + r1358133;
        double r1358135 = log(r1358131);
        double r1358136 = -0.16666666666666666;
        double r1358137 = r1358131 * r1358131;
        double r1358138 = r1358136 / r1358137;
        double r1358139 = r1358135 + r1358138;
        double r1358140 = r1358134 + r1358139;
        double r1358141 = r1358140 - r1358133;
        return r1358141;
}

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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019130 
(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))