Average Error: 63.0 → 0
Time: 15.0s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(1 \cdot \log n + \frac{0.5}{n}\right) - \frac{0.1666666666666666851703837437526090070605}{{n}^{2}}\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(1 \cdot \log n + \frac{0.5}{n}\right) - \frac{0.1666666666666666851703837437526090070605}{{n}^{2}}
double f(double n) {
        double r32738 = n;
        double r32739 = 1.0;
        double r32740 = r32738 + r32739;
        double r32741 = log(r32740);
        double r32742 = r32740 * r32741;
        double r32743 = log(r32738);
        double r32744 = r32738 * r32743;
        double r32745 = r32742 - r32744;
        double r32746 = r32745 - r32739;
        return r32746;
}
↓
double f(double n) {
        double r32747 = 1.0;
        double r32748 = n;
        double r32749 = log(r32748);
        double r32750 = r32747 * r32749;
        double r32751 = 0.5;
        double r32752 = r32751 / r32748;
        double r32753 = r32750 + r32752;
        double r32754 = 0.16666666666666669;
        double r32755 = 2.0;
        double r32756 = pow(r32748, r32755);
        double r32757 = r32754 / r32756;
        double r32758 = r32753 - r32757;
        return r32758;
}

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. Using strategy rm

  5. Applied associate-+l-0.0

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

  6. Applied associate--l-0.0

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

  7. Simplified0.0

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

  8. Final simplification0

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

Reproduce

herbie shell --seed 2019297 
(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))