Average Error: 63.0 → 0
Time: 16.5s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\frac{0.5}{n} + \left(1 \cdot \log n - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\frac{0.5}{n} + \left(1 \cdot \log n - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right)
double f(double n) {
        double r51776 = n;
        double r51777 = 1.0;
        double r51778 = r51776 + r51777;
        double r51779 = log(r51778);
        double r51780 = r51778 * r51779;
        double r51781 = log(r51776);
        double r51782 = r51776 * r51781;
        double r51783 = r51780 - r51782;
        double r51784 = r51783 - r51777;
        return r51784;
}
↓
double f(double n) {
        double r51785 = 0.5;
        double r51786 = n;
        double r51787 = r51785 / r51786;
        double r51788 = 1.0;
        double r51789 = log(r51786);
        double r51790 = r51788 * r51789;
        double r51791 = 0.16666666666666669;
        double r51792 = r51786 * r51786;
        double r51793 = r51791 / r51792;
        double r51794 = r51790 - r51793;
        double r51795 = r51787 + r51794;
        return r51795;
}

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

    \[\leadsto \color{blue}{\left(\left(\left(\frac{0.5}{n} + 1\right) + \log n \cdot 1\right) - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right)} - 1\]

  4. Taylor expanded around 0 0

    \[\leadsto \color{blue}{\left(0.5 \cdot \frac{1}{n} + 1 \cdot \log n\right) - 0.1666666666666666851703837437526090070605 \cdot \frac{1}{{n}^{2}}}\]

  5. Simplified0

    \[\leadsto \color{blue}{\frac{0.5}{n} + \left(1 \cdot \log n - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right)}\]

  6. Final simplification0

    \[\leadsto \frac{0.5}{n} + \left(1 \cdot \log n - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right)\]

Reproduce

herbie shell --seed 2019208 
(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))