Average Error: 63.0 → 0
Time: 2.8s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\mathsf{fma}\left(\frac{1}{n}, 0.5 - \frac{0.16666666666666669}{n}, \log n \cdot 1\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\mathsf{fma}\left(\frac{1}{n}, 0.5 - \frac{0.16666666666666669}{n}, \log n \cdot 1\right)
double f(double n) {
        double r105858 = n;
        double r105859 = 1.0;
        double r105860 = r105858 + r105859;
        double r105861 = log(r105860);
        double r105862 = r105860 * r105861;
        double r105863 = log(r105858);
        double r105864 = r105858 * r105863;
        double r105865 = r105862 - r105864;
        double r105866 = r105865 - r105859;
        return r105866;
}
↓
double f(double n) {
        double r105867 = 1.0;
        double r105868 = n;
        double r105869 = r105867 / r105868;
        double r105870 = 0.5;
        double r105871 = 0.16666666666666669;
        double r105872 = r105871 / r105868;
        double r105873 = r105870 - r105872;
        double r105874 = log(r105868);
        double r105875 = 1.0;
        double r105876 = r105874 * r105875;
        double r105877 = fma(r105869, r105873, r105876);
        return r105877;
}

Error

Bits error versus n

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. Simplified61.9

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

  3. Taylor expanded around inf 0.0

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

  4. Simplified0

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{n}, 0.5 - \frac{0.16666666666666669}{n}, \log n \cdot 1\right)}\]

  5. Final simplification0

    \[\leadsto \mathsf{fma}\left(\frac{1}{n}, 0.5 - \frac{0.16666666666666669}{n}, \log n \cdot 1\right)\]

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(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))