Average Error: 63.0 → 0.0
Time: 3.4s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(1 - \left(1 \cdot \log \left(\frac{1}{n}\right) + 0.16666666666666669 \cdot \frac{1}{{n}^{2}}\right)\right) + \frac{0.5}{n}\right) - 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\left(1 - \left(1 \cdot \log \left(\frac{1}{n}\right) + 0.16666666666666669 \cdot \frac{1}{{n}^{2}}\right)\right) + \frac{0.5}{n}\right) - 1
double f(double n) {
        double r74545 = n;
        double r74546 = 1.0;
        double r74547 = r74545 + r74546;
        double r74548 = log(r74547);
        double r74549 = r74547 * r74548;
        double r74550 = log(r74545);
        double r74551 = r74545 * r74550;
        double r74552 = r74549 - r74551;
        double r74553 = r74552 - r74546;
        return r74553;
}
↓
double f(double n) {
        double r74554 = 1.0;
        double r74555 = 1.0;
        double r74556 = n;
        double r74557 = r74555 / r74556;
        double r74558 = log(r74557);
        double r74559 = r74554 * r74558;
        double r74560 = 0.16666666666666669;
        double r74561 = 2.0;
        double r74562 = pow(r74556, r74561);
        double r74563 = r74555 / r74562;
        double r74564 = r74560 * r74563;
        double r74565 = r74559 + r74564;
        double r74566 = r74554 - r74565;
        double r74567 = 0.5;
        double r74568 = r74567 / r74556;
        double r74569 = r74566 + r74568;
        double r74570 = r74569 - r74554;
        return r74570;
}

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

    \[\leadsto \color{blue}{\left(\left(0.5 \cdot \frac{1}{n} + 1\right) - \left(1 \cdot \log \left(\frac{1}{n}\right) + 0.16666666666666669 \cdot \frac{1}{{n}^{2}}\right)\right)} - 1\]

  3. Simplified0.0

    \[\leadsto \color{blue}{\left(\left(1 - \left(1 \cdot \log \left(\frac{1}{n}\right) + 0.16666666666666669 \cdot \frac{1}{{n}^{2}}\right)\right) + \frac{0.5}{n}\right)} - 1\]

  4. Final simplification0.0

    \[\leadsto \left(\left(1 - \left(1 \cdot \log \left(\frac{1}{n}\right) + 0.16666666666666669 \cdot \frac{1}{{n}^{2}}\right)\right) + \frac{0.5}{n}\right) - 1\]

Reproduce

herbie shell --seed 2020018 
(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))