Average Error: 63.0 → 0.0
Time: 22.9s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(\log n \cdot 1 + \left(\frac{0.5}{n} + 1\right)\right) - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) - 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\left(\log n \cdot 1 + \left(\frac{0.5}{n} + 1\right)\right) - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) - 1
double f(double n) {
        double r37400 = n;
        double r37401 = 1.0;
        double r37402 = r37400 + r37401;
        double r37403 = log(r37402);
        double r37404 = r37402 * r37403;
        double r37405 = log(r37400);
        double r37406 = r37400 * r37405;
        double r37407 = r37404 - r37406;
        double r37408 = r37407 - r37401;
        return r37408;
}
↓
double f(double n) {
        double r37409 = n;
        double r37410 = log(r37409);
        double r37411 = 1.0;
        double r37412 = r37410 * r37411;
        double r37413 = 0.5;
        double r37414 = r37413 / r37409;
        double r37415 = r37414 + r37411;
        double r37416 = r37412 + r37415;
        double r37417 = 0.16666666666666669;
        double r37418 = r37409 * r37409;
        double r37419 = r37417 / r37418;
        double r37420 = r37416 - r37419;
        double r37421 = r37420 - r37411;
        return r37421;
}

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

    \[\leadsto \color{blue}{\left(\left(0.5 \cdot \frac{1}{n} + 1\right) - \left(1 \cdot \log \left(\frac{1}{n}\right) + 0.1666666666666666851703837437526090070605 \cdot \frac{1}{{n}^{2}}\right)\right)} - 1\]

  3. Simplified0.0

    \[\leadsto \color{blue}{\left(\left(\log n \cdot 1 + \left(\frac{0.5}{n} + 1\right)\right) - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right)} - 1\]

  4. Final simplification0.0

    \[\leadsto \left(\left(\log n \cdot 1 + \left(\frac{0.5}{n} + 1\right)\right) - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) - 1\]

Reproduce

herbie shell --seed 2019306 
(FPCore (n)
  :name "logs (example 3.8)"
  :precision binary64
  :pre (> n 6.8e15)

  :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))