Average Error: 63.0 → 0
Time: 6.2s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\log n \cdot 1 + \frac{0.5}{n}\right) - \frac{0.1666666666666666851703837437526090070605}{{n}^{2}}\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\log n \cdot 1 + \frac{0.5}{n}\right) - \frac{0.1666666666666666851703837437526090070605}{{n}^{2}}
double f(double n) {
        double r71415 = n;
        double r71416 = 1.0;
        double r71417 = r71415 + r71416;
        double r71418 = log(r71417);
        double r71419 = r71417 * r71418;
        double r71420 = log(r71415);
        double r71421 = r71415 * r71420;
        double r71422 = r71419 - r71421;
        double r71423 = r71422 - r71416;
        return r71423;
}
↓
double f(double n) {
        double r71424 = n;
        double r71425 = log(r71424);
        double r71426 = 1.0;
        double r71427 = r71425 * r71426;
        double r71428 = 0.5;
        double r71429 = r71428 / r71424;
        double r71430 = r71427 + r71429;
        double r71431 = 0.16666666666666669;
        double r71432 = 2.0;
        double r71433 = pow(r71424, r71432);
        double r71434 = r71431 / r71433;
        double r71435 = r71430 - r71434;
        return r71435;
}

Error

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original63.0
Target0
Herbie0
\[\log \left(n + 1\right) - \left(\frac{1}{2 \cdot n} - \left(\frac{1}{3 \cdot \left(n \cdot n\right)} - \frac{4}{{n}^{3}}\right)\right)\]

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

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

  4. Final simplification0

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

Reproduce

herbie shell --seed 2019308 
(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))