Average Error: 63.0 → 0
Time: 17.7s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\frac{0.5}{n} - \mathsf{fma}\left(1, -\log n, \frac{0.16666666666666669}{n \cdot n}\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\frac{0.5}{n} - \mathsf{fma}\left(1, -\log n, \frac{0.16666666666666669}{n \cdot n}\right)
double f(double n) {
        double r82855 = n;
        double r82856 = 1.0;
        double r82857 = r82855 + r82856;
        double r82858 = log(r82857);
        double r82859 = r82857 * r82858;
        double r82860 = log(r82855);
        double r82861 = r82855 * r82860;
        double r82862 = r82859 - r82861;
        double r82863 = r82862 - r82856;
        return r82863;
}
↓
double f(double n) {
        double r82864 = 0.5;
        double r82865 = n;
        double r82866 = r82864 / r82865;
        double r82867 = 1.0;
        double r82868 = log(r82865);
        double r82869 = -r82868;
        double r82870 = 0.16666666666666669;
        double r82871 = r82865 * r82865;
        double r82872 = r82870 / r82871;
        double r82873 = fma(r82867, r82869, r82872);
        double r82874 = r82866 - r82873;
        return r82874;
}

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. Simplified62.0

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

  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

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

  5. Final simplification0

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

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (n)
  :name "logs (example 3.8)"
  :pre (> n 6.8e+15)

  :herbie-target
  (- (log (+ n 1.0)) (- (/ 1.0 (* 2.0 n)) (- (/ 1.0 (* 3.0 (* n n))) (/ 4.0 (pow n 3.0)))))

  (- (- (* (+ n 1.0) (log (+ n 1.0))) (* n (log n))) 1.0))