Average Error: 63.0 → 0
Time: 7.3s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\log n \cdot 1 + \frac{\frac{1}{2}}{n}\right) - \frac{\frac{3002399751580331}{18014398509481984}}{{n}^{2}}\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\log n \cdot 1 + \frac{\frac{1}{2}}{n}\right) - \frac{\frac{3002399751580331}{18014398509481984}}{{n}^{2}}
double f(double n) {
        double r48923 = n;
        double r48924 = 1.0;
        double r48925 = r48923 + r48924;
        double r48926 = log(r48925);
        double r48927 = r48925 * r48926;
        double r48928 = log(r48923);
        double r48929 = r48923 * r48928;
        double r48930 = r48927 - r48929;
        double r48931 = r48930 - r48924;
        return r48931;
}
↓
double f(double n) {
        double r48932 = n;
        double r48933 = log(r48932);
        double r48934 = 1.0;
        double r48935 = r48933 * r48934;
        double r48936 = 2.0;
        double r48937 = r48934 / r48936;
        double r48938 = r48937 / r48932;
        double r48939 = r48935 + r48938;
        double r48940 = 3002399751580331.0;
        double r48941 = 18014398509481984.0;
        double r48942 = r48940 / r48941;
        double r48943 = 2.0;
        double r48944 = pow(r48932, r48943);
        double r48945 = r48942 / r48944;
        double r48946 = r48939 - r48945;
        return r48946;
}

Error

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original63.0
Target0
Herbie0
\[\log \left(n + 1\right) - \left(\frac{1}{2 \cdot n} - \left(\frac{1}{3 \cdot \left(n \cdot n\right)} - \frac{4}{{n}^{3}}\right)\right)\]

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

    \[\leadsto \color{blue}{\left(1 + \left(\left(\frac{\frac{1}{2}}{n} + \frac{-\frac{3002399751580331}{18014398509481984}}{{n}^{2}}\right) + \log n \cdot 1\right)\right)} - 1\]

  4. Final simplification0

    \[\leadsto \left(\log n \cdot 1 + \frac{\frac{1}{2}}{n}\right) - \frac{\frac{3002399751580331}{18014398509481984}}{{n}^{2}}\]

Reproduce

herbie shell --seed 197574269 
(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))