Average Error: 63.0 → 0
Time: 25.1s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\log n + \frac{\frac{-1}{6}}{n \cdot n}\right) + \frac{\frac{1}{2}}{n}\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\log n + \frac{\frac{-1}{6}}{n \cdot n}\right) + \frac{\frac{1}{2}}{n}
double f(double n) {
        double r2083648 = n;
        double r2083649 = 1.0;
        double r2083650 = r2083648 + r2083649;
        double r2083651 = log(r2083650);
        double r2083652 = r2083650 * r2083651;
        double r2083653 = log(r2083648);
        double r2083654 = r2083648 * r2083653;
        double r2083655 = r2083652 - r2083654;
        double r2083656 = r2083655 - r2083649;
        return r2083656;
}
↓
double f(double n) {
        double r2083657 = n;
        double r2083658 = log(r2083657);
        double r2083659 = -0.16666666666666666;
        double r2083660 = r2083657 * r2083657;
        double r2083661 = r2083659 / r2083660;
        double r2083662 = r2083658 + r2083661;
        double r2083663 = 0.5;
        double r2083664 = r2083663 / r2083657;
        double r2083665 = r2083662 + r2083664;
        return r2083665;
}

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

    \[\leadsto \color{blue}{\left(\left(1 + \frac{1}{2} \cdot \frac{1}{n}\right) - \left(\frac{1}{6} \cdot \frac{1}{{n}^{2}} + \log \left(\frac{1}{n}\right)\right)\right)} - 1\]

  3. Simplified0.0

    \[\leadsto \color{blue}{\left(\frac{\frac{1}{2}}{n} + \left(1 - \left(\frac{\frac{1}{6}}{n \cdot n} - \log n\right)\right)\right)} - 1\]

  4. Taylor expanded around inf 0.0

    \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{1}{n} - \left(\frac{1}{6} \cdot \frac{1}{{n}^{2}} + \log \left(\frac{1}{n}\right)\right)}\]

  5. Simplified0

    \[\leadsto \color{blue}{\frac{\frac{1}{2}}{n} + \left(\log n + \frac{\frac{-1}{6}}{n \cdot n}\right)}\]

  6. Final simplification0

    \[\leadsto \left(\log n + \frac{\frac{-1}{6}}{n \cdot n}\right) + \frac{\frac{1}{2}}{n}\]

Reproduce

herbie shell --seed 2019149 
(FPCore (n)
  :name "logs (example 3.8)"
  :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))