Average Error: 63.0 → 0.0
Time: 20.6s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(\left(\frac{0.5}{n} + 1\right) - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) + \log n \cdot 1\right) - 1\]
\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) - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) + \log n \cdot 1\right) - 1
double f(double n) {
        double r41455 = n;
        double r41456 = 1.0;
        double r41457 = r41455 + r41456;
        double r41458 = log(r41457);
        double r41459 = r41457 * r41458;
        double r41460 = log(r41455);
        double r41461 = r41455 * r41460;
        double r41462 = r41459 - r41461;
        double r41463 = r41462 - r41456;
        return r41463;
}
↓
double f(double n) {
        double r41464 = 0.5;
        double r41465 = n;
        double r41466 = r41464 / r41465;
        double r41467 = 1.0;
        double r41468 = r41466 + r41467;
        double r41469 = 0.16666666666666669;
        double r41470 = r41465 * r41465;
        double r41471 = r41469 / r41470;
        double r41472 = r41468 - r41471;
        double r41473 = log(r41465);
        double r41474 = r41473 * r41467;
        double r41475 = r41472 + r41474;
        double r41476 = r41475 - r41467;
        return r41476;
}

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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019325 
(FPCore (n)
  :name "logs (example 3.8)"
  :precision binary64
  :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))