Average Error: 63.0 → 0
Time: 15.2s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\frac{\frac{1}{2}}{n} - \left(\frac{\frac{1}{6}}{n \cdot n} - \log n\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\frac{\frac{1}{2}}{n} - \left(\frac{\frac{1}{6}}{n \cdot n} - \log n\right)
double f(double n) {
        double r5321438 = n;
        double r5321439 = 1.0;
        double r5321440 = r5321438 + r5321439;
        double r5321441 = log(r5321440);
        double r5321442 = r5321440 * r5321441;
        double r5321443 = log(r5321438);
        double r5321444 = r5321438 * r5321443;
        double r5321445 = r5321442 - r5321444;
        double r5321446 = r5321445 - r5321439;
        return r5321446;
}
↓
double f(double n) {
        double r5321447 = 0.5;
        double r5321448 = n;
        double r5321449 = r5321447 / r5321448;
        double r5321450 = 0.16666666666666666;
        double r5321451 = r5321448 * r5321448;
        double r5321452 = r5321450 / r5321451;
        double r5321453 = log(r5321448);
        double r5321454 = r5321452 - r5321453;
        double r5321455 = r5321449 - r5321454;
        return r5321455;
}

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. Simplified62.0

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

  3. Taylor expanded around inf 0.0

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

  4. Simplified0

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

  5. Final simplification0

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

Reproduce

herbie shell --seed 2019121 +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))