Average Error: 63.0 → 0
Time: 21.5s
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 r129327 = n;
        double r129328 = 1.0;
        double r129329 = r129327 + r129328;
        double r129330 = log(r129329);
        double r129331 = r129329 * r129330;
        double r129332 = log(r129327);
        double r129333 = r129327 * r129332;
        double r129334 = r129331 - r129333;
        double r129335 = r129334 - r129328;
        return r129335;
}
↓
double f(double n) {
        double r129336 = 0.5;
        double r129337 = n;
        double r129338 = r129336 / r129337;
        double r129339 = 1.0;
        double r129340 = log(r129337);
        double r129341 = -r129340;
        double r129342 = 0.16666666666666669;
        double r129343 = r129337 * r129337;
        double r129344 = r129342 / r129343;
        double r129345 = fma(r129339, r129341, r129344);
        double r129346 = r129338 - r129345;
        return r129346;
}

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 2020046 +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))