Average Error: 63.0 → 0.0
Time: 17.1s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(\frac{\frac{1}{2}}{n} + 1\right) + \left(\log n + \frac{\frac{-1}{6}}{n \cdot n}\right)\right) - 1\]
\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 r989247 = n;
        double r989248 = 1.0;
        double r989249 = r989247 + r989248;
        double r989250 = log(r989249);
        double r989251 = r989249 * r989250;
        double r989252 = log(r989247);
        double r989253 = r989247 * r989252;
        double r989254 = r989251 - r989253;
        double r989255 = r989254 - r989248;
        return r989255;
}
↓
double f(double n) {
        double r989256 = 0.5;
        double r989257 = n;
        double r989258 = r989256 / r989257;
        double r989259 = 1.0;
        double r989260 = r989258 + r989259;
        double r989261 = log(r989257);
        double r989262 = -0.16666666666666666;
        double r989263 = r989257 * r989257;
        double r989264 = r989262 / r989263;
        double r989265 = r989261 + r989264;
        double r989266 = r989260 + r989265;
        double r989267 = r989266 - r989259;
        return r989267;
}

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 2019129 
(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))