Average Error: 63.0 → 0.0
Time: 4.9s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[1 - \left(\left(1 + \left(\frac{0.16666666666666669}{{n}^{2}} + 1 \cdot \log \left(\frac{1}{n}\right)\right)\right) - \frac{0.5}{n}\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
1 - \left(\left(1 + \left(\frac{0.16666666666666669}{{n}^{2}} + 1 \cdot \log \left(\frac{1}{n}\right)\right)\right) - \frac{0.5}{n}\right)
double f(double n) {
        double r61964 = n;
        double r61965 = 1.0;
        double r61966 = r61964 + r61965;
        double r61967 = log(r61966);
        double r61968 = r61966 * r61967;
        double r61969 = log(r61964);
        double r61970 = r61964 * r61969;
        double r61971 = r61968 - r61970;
        double r61972 = r61971 - r61965;
        return r61972;
}
↓
double f(double n) {
        double r61973 = 1.0;
        double r61974 = 0.16666666666666669;
        double r61975 = n;
        double r61976 = 2.0;
        double r61977 = pow(r61975, r61976);
        double r61978 = r61974 / r61977;
        double r61979 = 1.0;
        double r61980 = r61979 / r61975;
        double r61981 = log(r61980);
        double r61982 = r61973 * r61981;
        double r61983 = r61978 + r61982;
        double r61984 = r61973 + r61983;
        double r61985 = 0.5;
        double r61986 = r61985 / r61975;
        double r61987 = r61984 - r61986;
        double r61988 = r61973 - r61987;
        return r61988;
}

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

    \[\leadsto \color{blue}{\left(\left(1 - \left(1 \cdot \log \left(\frac{1}{n}\right) + 0.16666666666666669 \cdot \frac{1}{{n}^{2}}\right)\right) + \frac{0.5}{n}\right)} - 1\]
  4. Using strategy rm

  5. Applied associate-+l-0.0

    \[\leadsto \color{blue}{\left(1 - \left(\left(1 \cdot \log \left(\frac{1}{n}\right) + 0.16666666666666669 \cdot \frac{1}{{n}^{2}}\right) - \frac{0.5}{n}\right)\right)} - 1\]

  6. Applied associate--l-0.0

    \[\leadsto \color{blue}{1 - \left(\left(\left(1 \cdot \log \left(\frac{1}{n}\right) + 0.16666666666666669 \cdot \frac{1}{{n}^{2}}\right) - \frac{0.5}{n}\right) + 1\right)}\]

  7. Simplified0.0

    \[\leadsto 1 - \color{blue}{\left(\left(1 + \left(\frac{0.16666666666666669}{{n}^{2}} + 1 \cdot \log \left(\frac{1}{n}\right)\right)\right) - \frac{0.5}{n}\right)}\]

  8. Final simplification0.0

    \[\leadsto 1 - \left(\left(1 + \left(\frac{0.16666666666666669}{{n}^{2}} + 1 \cdot \log \left(\frac{1}{n}\right)\right)\right) - \frac{0.5}{n}\right)\]

Reproduce

herbie shell --seed 2020060 
(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))