Average Error: 63.0 → 0.0
Time: 12.4s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(1 \cdot \log n + \left(\left(1 - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) + \frac{0.5}{n}\right)\right) - 1\]
\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{0.1666666666666666851703837437526090070605}{n \cdot n}\right) + \frac{0.5}{n}\right)\right) - 1
double f(double n) {
        double r4761920 = n;
        double r4761921 = 1.0;
        double r4761922 = r4761920 + r4761921;
        double r4761923 = log(r4761922);
        double r4761924 = r4761922 * r4761923;
        double r4761925 = log(r4761920);
        double r4761926 = r4761920 * r4761925;
        double r4761927 = r4761924 - r4761926;
        double r4761928 = r4761927 - r4761921;
        return r4761928;
}
↓
double f(double n) {
        double r4761929 = 1.0;
        double r4761930 = n;
        double r4761931 = log(r4761930);
        double r4761932 = r4761929 * r4761931;
        double r4761933 = 0.16666666666666669;
        double r4761934 = r4761930 * r4761930;
        double r4761935 = r4761933 / r4761934;
        double r4761936 = r4761929 - r4761935;
        double r4761937 = 0.5;
        double r4761938 = r4761937 / r4761930;
        double r4761939 = r4761936 + r4761938;
        double r4761940 = r4761932 + r4761939;
        double r4761941 = r4761940 - r4761929;
        return r4761941;
}

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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019172 
(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))