Average Error: 63.0 → 0.0
Time: 11.1s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(1 - \left(\frac{0.16666666666666669}{n \cdot n} - \log n \cdot 1\right)\right) + \frac{0.5}{n}\right) - 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\left(1 - \left(\frac{0.16666666666666669}{n \cdot n} - \log n \cdot 1\right)\right) + \frac{0.5}{n}\right) - 1
double f(double n) {
        double r66642 = n;
        double r66643 = 1.0;
        double r66644 = r66642 + r66643;
        double r66645 = log(r66644);
        double r66646 = r66644 * r66645;
        double r66647 = log(r66642);
        double r66648 = r66642 * r66647;
        double r66649 = r66646 - r66648;
        double r66650 = r66649 - r66643;
        return r66650;
}
↓
double f(double n) {
        double r66651 = 1.0;
        double r66652 = 0.16666666666666669;
        double r66653 = n;
        double r66654 = r66653 * r66653;
        double r66655 = r66652 / r66654;
        double r66656 = log(r66653);
        double r66657 = r66656 * r66651;
        double r66658 = r66655 - r66657;
        double r66659 = r66651 - r66658;
        double r66660 = 0.5;
        double r66661 = r66660 / r66653;
        double r66662 = r66659 + r66661;
        double r66663 = r66662 - r66651;
        return r66663;
}

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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2020047 
(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))