Average Error: 63.0 → 0.0
Time: 11.8s
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 r46809 = n;
        double r46810 = 1.0;
        double r46811 = r46809 + r46810;
        double r46812 = log(r46811);
        double r46813 = r46811 * r46812;
        double r46814 = log(r46809);
        double r46815 = r46809 * r46814;
        double r46816 = r46813 - r46815;
        double r46817 = r46816 - r46810;
        return r46817;
}
↓
double f(double n) {
        double r46818 = 0.5;
        double r46819 = n;
        double r46820 = r46818 / r46819;
        double r46821 = 1.0;
        double r46822 = r46820 + r46821;
        double r46823 = log(r46819);
        double r46824 = r46821 * r46823;
        double r46825 = r46822 + r46824;
        double r46826 = 0.16666666666666669;
        double r46827 = r46819 * r46819;
        double r46828 = r46826 / r46827;
        double r46829 = r46825 - r46828;
        double r46830 = r46829 - r46821;
        return r46830;
}

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 2019198 
(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))