Average Error: 63.0 → 0.0
Time: 16.2s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(1 + \left(\log n \cdot 1 - \frac{0.16666666666666669}{n \cdot n}\right)\right) + \frac{0.5}{n}\right) - 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\left(1 + \left(\log n \cdot 1 - \frac{0.16666666666666669}{n \cdot n}\right)\right) + \frac{0.5}{n}\right) - 1
double f(double n) {
        double r51959 = n;
        double r51960 = 1.0;
        double r51961 = r51959 + r51960;
        double r51962 = log(r51961);
        double r51963 = r51961 * r51962;
        double r51964 = log(r51959);
        double r51965 = r51959 * r51964;
        double r51966 = r51963 - r51965;
        double r51967 = r51966 - r51960;
        return r51967;
}
↓
double f(double n) {
        double r51968 = 1.0;
        double r51969 = n;
        double r51970 = log(r51969);
        double r51971 = r51970 * r51968;
        double r51972 = 0.16666666666666669;
        double r51973 = r51969 * r51969;
        double r51974 = r51972 / r51973;
        double r51975 = r51971 - r51974;
        double r51976 = r51968 + r51975;
        double r51977 = 0.5;
        double r51978 = r51977 / r51969;
        double r51979 = r51976 + r51978;
        double r51980 = r51979 - r51968;
        return r51980;
}

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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2020043 
(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))