Average Error: 63.0 → 0.0
Time: 20.1s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[-1 + \left(\left(\log n - \frac{\frac{1}{6}}{n \cdot n}\right) + \left(\frac{\frac{1}{2}}{n} + 1\right)\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
-1 + \left(\left(\log n - \frac{\frac{1}{6}}{n \cdot n}\right) + \left(\frac{\frac{1}{2}}{n} + 1\right)\right)
double f(double n) {
        double r1842554 = n;
        double r1842555 = 1.0;
        double r1842556 = r1842554 + r1842555;
        double r1842557 = log(r1842556);
        double r1842558 = r1842556 * r1842557;
        double r1842559 = log(r1842554);
        double r1842560 = r1842554 * r1842559;
        double r1842561 = r1842558 - r1842560;
        double r1842562 = r1842561 - r1842555;
        return r1842562;
}
↓
double f(double n) {
        double r1842563 = -1.0;
        double r1842564 = n;
        double r1842565 = log(r1842564);
        double r1842566 = 0.16666666666666666;
        double r1842567 = r1842564 * r1842564;
        double r1842568 = r1842566 / r1842567;
        double r1842569 = r1842565 - r1842568;
        double r1842570 = 0.5;
        double r1842571 = r1842570 / r1842564;
        double r1842572 = 1.0;
        double r1842573 = r1842571 + r1842572;
        double r1842574 = r1842569 + r1842573;
        double r1842575 = r1842563 + r1842574;
        return r1842575;
}

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. Simplified44.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(n, \mathsf{log1p}\left(n\right) - \log n, \mathsf{log1p}\left(n\right)\right) + -1}\]

  3. Taylor expanded around inf 0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(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))