Average Error: 63.0 → 0.0
Time: 3.9s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(1 - \left(1 \cdot \left(-1 \cdot \log n\right) + 0.16666666666666669 \cdot \frac{1}{{n}^{2}}\right)\right) + \frac{0.5}{n}\right) - 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\left(1 - \left(1 \cdot \left(-1 \cdot \log n\right) + 0.16666666666666669 \cdot \frac{1}{{n}^{2}}\right)\right) + \frac{0.5}{n}\right) - 1
double f(double n) {
        double r72437 = n;
        double r72438 = 1.0;
        double r72439 = r72437 + r72438;
        double r72440 = log(r72439);
        double r72441 = r72439 * r72440;
        double r72442 = log(r72437);
        double r72443 = r72437 * r72442;
        double r72444 = r72441 - r72443;
        double r72445 = r72444 - r72438;
        return r72445;
}
↓
double f(double n) {
        double r72446 = 1.0;
        double r72447 = -1.0;
        double r72448 = n;
        double r72449 = log(r72448);
        double r72450 = r72447 * r72449;
        double r72451 = r72446 * r72450;
        double r72452 = 0.16666666666666669;
        double r72453 = 1.0;
        double r72454 = 2.0;
        double r72455 = pow(r72448, r72454);
        double r72456 = r72453 / r72455;
        double r72457 = r72452 * r72456;
        double r72458 = r72451 + r72457;
        double r72459 = r72446 - r72458;
        double r72460 = 0.5;
        double r72461 = r72460 / r72448;
        double r72462 = r72459 + r72461;
        double r72463 = r72462 - r72446;
        return r72463;
}

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. Taylor expanded around inf 0.0

    \[\leadsto \color{blue}{\left(\left(0.5 \cdot \frac{1}{n} + 1\right) - \left(1 \cdot \log \left(\frac{1}{n}\right) + 0.16666666666666669 \cdot \frac{1}{{n}^{2}}\right)\right)} - 1\]

  3. Simplified0.0

    \[\leadsto \color{blue}{\left(\left(1 - \left(1 \cdot \log \left(\frac{1}{n}\right) + 0.16666666666666669 \cdot \frac{1}{{n}^{2}}\right)\right) + \frac{0.5}{n}\right)} - 1\]
  4. Using strategy rm

  5. Applied inv-pow0.0

    \[\leadsto \left(\left(1 - \left(1 \cdot \log \color{blue}{\left({n}^{-1}\right)} + 0.16666666666666669 \cdot \frac{1}{{n}^{2}}\right)\right) + \frac{0.5}{n}\right) - 1\]

  6. Applied log-pow0.0

    \[\leadsto \left(\left(1 - \left(1 \cdot \color{blue}{\left(-1 \cdot \log n\right)} + 0.16666666666666669 \cdot \frac{1}{{n}^{2}}\right)\right) + \frac{0.5}{n}\right) - 1\]

  7. Final simplification0.0

    \[\leadsto \left(\left(1 - \left(1 \cdot \left(-1 \cdot \log n\right) + 0.16666666666666669 \cdot \frac{1}{{n}^{2}}\right)\right) + \frac{0.5}{n}\right) - 1\]

Reproduce

herbie shell --seed 2020021 
(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))