Average Error: 63.0 → 0.0
Time: 3.1s
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 r69455 = n;
        double r69456 = 1.0;
        double r69457 = r69455 + r69456;
        double r69458 = log(r69457);
        double r69459 = r69457 * r69458;
        double r69460 = log(r69455);
        double r69461 = r69455 * r69460;
        double r69462 = r69459 - r69461;
        double r69463 = r69462 - r69456;
        return r69463;
}
↓
double f(double n) {
        double r69464 = 1.0;
        double r69465 = n;
        double r69466 = r69464 / r69465;
        double r69467 = 0.5;
        double r69468 = 0.16666666666666669;
        double r69469 = r69468 / r69465;
        double r69470 = r69467 - r69469;
        double r69471 = log(r69465);
        double r69472 = 1.0;
        double r69473 = r69471 * r69472;
        double r69474 = fma(r69466, r69470, r69473);
        return r69474;
}

Error

Bits error versus n

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. 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. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020049 +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))