Average Error: 63.0 → 0.0
Time: 11.8s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(\left(\frac{0.5}{n} + 1\right) + 1 \cdot \log n\right) - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) - 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\left(\left(\frac{0.5}{n} + 1\right) + 1 \cdot \log n\right) - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) - 1
double f(double n) {
        double r82495 = n;
        double r82496 = 1.0;
        double r82497 = r82495 + r82496;
        double r82498 = log(r82497);
        double r82499 = r82497 * r82498;
        double r82500 = log(r82495);
        double r82501 = r82495 * r82500;
        double r82502 = r82499 - r82501;
        double r82503 = r82502 - r82496;
        return r82503;
}
↓
double f(double n) {
        double r82504 = 0.5;
        double r82505 = n;
        double r82506 = r82504 / r82505;
        double r82507 = 1.0;
        double r82508 = r82506 + r82507;
        double r82509 = log(r82505);
        double r82510 = r82507 * r82509;
        double r82511 = r82508 + r82510;
        double r82512 = 0.16666666666666669;
        double r82513 = r82505 * r82505;
        double r82514 = r82512 / r82513;
        double r82515 = r82511 - r82514;
        double r82516 = r82515 - r82507;
        return r82516;
}

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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019212 
(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))