Average Error: 63.0 → 0.0
Time: 11.4s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(\left(\frac{0.5}{n} + 1\right) + \log n \cdot 1\right) - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) - 1\]
\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) + \log n \cdot 1\right) - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) - 1
double f(double n) {
        double r64585 = n;
        double r64586 = 1.0;
        double r64587 = r64585 + r64586;
        double r64588 = log(r64587);
        double r64589 = r64587 * r64588;
        double r64590 = log(r64585);
        double r64591 = r64585 * r64590;
        double r64592 = r64589 - r64591;
        double r64593 = r64592 - r64586;
        return r64593;
}
↓
double f(double n) {
        double r64594 = 0.5;
        double r64595 = n;
        double r64596 = r64594 / r64595;
        double r64597 = 1.0;
        double r64598 = r64596 + r64597;
        double r64599 = log(r64595);
        double r64600 = r64599 * r64597;
        double r64601 = r64598 + r64600;
        double r64602 = 0.16666666666666669;
        double r64603 = r64595 * r64595;
        double r64604 = r64602 / r64603;
        double r64605 = r64601 - r64604;
        double r64606 = r64605 - r64597;
        return r64606;
}

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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019209 
(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))