Average Error: 63.0 → 0.0
Time: 12.2s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(1 \cdot \log n + \left(\left(1 + \frac{0.5}{n}\right) - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right)\right) - 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(1 \cdot \log n + \left(\left(1 + \frac{0.5}{n}\right) - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right)\right) - 1
double f(double n) {
        double r2281719 = n;
        double r2281720 = 1.0;
        double r2281721 = r2281719 + r2281720;
        double r2281722 = log(r2281721);
        double r2281723 = r2281721 * r2281722;
        double r2281724 = log(r2281719);
        double r2281725 = r2281719 * r2281724;
        double r2281726 = r2281723 - r2281725;
        double r2281727 = r2281726 - r2281720;
        return r2281727;
}
↓
double f(double n) {
        double r2281728 = 1.0;
        double r2281729 = n;
        double r2281730 = log(r2281729);
        double r2281731 = r2281728 * r2281730;
        double r2281732 = 0.5;
        double r2281733 = r2281732 / r2281729;
        double r2281734 = r2281728 + r2281733;
        double r2281735 = 0.16666666666666669;
        double r2281736 = r2281729 * r2281729;
        double r2281737 = r2281735 / r2281736;
        double r2281738 = r2281734 - r2281737;
        double r2281739 = r2281731 + r2281738;
        double r2281740 = r2281739 - r2281728;
        return r2281740;
}

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

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019174 
(FPCore (n)
  :name "logs (example 3.8)"
  :pre (> n 6.8e+15)

  :herbie-target
  (- (log (+ n 1.0)) (- (/ 1.0 (* 2.0 n)) (- (/ 1.0 (* 3.0 (* n n))) (/ 4.0 (pow n 3.0)))))

  (- (- (* (+ n 1.0) (log (+ n 1.0))) (* n (log n))) 1.0))