Average Error: 63.0 → 0.0
Time: 16.4s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(\left(\frac{0.5}{n} + 1\right) + 1 \cdot \log n\right) - \frac{0.16666666666666669}{n \cdot n}\right) - 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\left(\left(\frac{0.5}{n} + 1\right) + 1 \cdot \log n\right) - \frac{0.16666666666666669}{n \cdot n}\right) - 1
double f(double n) {
        double r28958 = n;
        double r28959 = 1.0;
        double r28960 = r28958 + r28959;
        double r28961 = log(r28960);
        double r28962 = r28960 * r28961;
        double r28963 = log(r28958);
        double r28964 = r28958 * r28963;
        double r28965 = r28962 - r28964;
        double r28966 = r28965 - r28959;
        return r28966;
}
↓
double f(double n) {
        double r28967 = 0.5;
        double r28968 = n;
        double r28969 = r28967 / r28968;
        double r28970 = 1.0;
        double r28971 = r28969 + r28970;
        double r28972 = log(r28968);
        double r28973 = r28970 * r28972;
        double r28974 = r28971 + r28973;
        double r28975 = 0.16666666666666669;
        double r28976 = r28968 * r28968;
        double r28977 = r28975 / r28976;
        double r28978 = r28974 - r28977;
        double r28979 = r28978 - r28970;
        return r28979;
}

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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019199 
(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))