Average Error: 63.0 → 0.0
Time: 23.0s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\frac{\frac{1}{2}}{n} - \left(\frac{\frac{1}{6}}{n \cdot n} - \log n\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\frac{\frac{1}{2}}{n} - \left(\frac{\frac{1}{6}}{n \cdot n} - \log n\right)
double f(double n) {
        double r3873392 = n;
        double r3873393 = 1.0;
        double r3873394 = r3873392 + r3873393;
        double r3873395 = log(r3873394);
        double r3873396 = r3873394 * r3873395;
        double r3873397 = log(r3873392);
        double r3873398 = r3873392 * r3873397;
        double r3873399 = r3873396 - r3873398;
        double r3873400 = r3873399 - r3873393;
        return r3873400;
}
↓
double f(double n) {
        double r3873401 = 0.5;
        double r3873402 = n;
        double r3873403 = r3873401 / r3873402;
        double r3873404 = 0.16666666666666666;
        double r3873405 = r3873402 * r3873402;
        double r3873406 = r3873404 / r3873405;
        double r3873407 = log(r3873402);
        double r3873408 = r3873406 - r3873407;
        double r3873409 = r3873403 - r3873408;
        return r3873409;
}

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. Taylor expanded around -inf 62.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019112 
(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))