Average Error: 63.0 → 0
Time: 4.8s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[1 \cdot \log n + \left(0.5 \cdot \frac{1}{n} - \frac{0.1666666666666666851703837437526090070605}{{n}^{2}}\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
1 \cdot \log n + \left(0.5 \cdot \frac{1}{n} - \frac{0.1666666666666666851703837437526090070605}{{n}^{2}}\right)
double f(double n) {
        double r172733 = n;
        double r172734 = 1.0;
        double r172735 = r172733 + r172734;
        double r172736 = log(r172735);
        double r172737 = r172735 * r172736;
        double r172738 = log(r172733);
        double r172739 = r172733 * r172738;
        double r172740 = r172737 - r172739;
        double r172741 = r172740 - r172734;
        return r172741;
}
↓
double f(double n) {
        double r172742 = 1.0;
        double r172743 = n;
        double r172744 = log(r172743);
        double r172745 = r172742 * r172744;
        double r172746 = 0.5;
        double r172747 = 1.0;
        double r172748 = r172747 / r172743;
        double r172749 = r172746 * r172748;
        double r172750 = 0.16666666666666669;
        double r172751 = 2.0;
        double r172752 = pow(r172743, r172751);
        double r172753 = r172750 / r172752;
        double r172754 = r172749 - r172753;
        double r172755 = r172745 + r172754;
        return r172755;
}

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

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

  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

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

  6. Final simplification0

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

Reproduce

herbie shell --seed 2019356 
(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))