Average Error: 63.0 → 0
Time: 2.8s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\mathsf{fma}\left(\frac{1}{n}, 0.5 - \frac{0.1666666666666666851703837437526090070605}{n}, \log n \cdot 1\right)\]
\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{0.1666666666666666851703837437526090070605}{n}, \log n \cdot 1\right)
double f(double n) {
        double r72327 = n;
        double r72328 = 1.0;
        double r72329 = r72327 + r72328;
        double r72330 = log(r72329);
        double r72331 = r72329 * r72330;
        double r72332 = log(r72327);
        double r72333 = r72327 * r72332;
        double r72334 = r72331 - r72333;
        double r72335 = r72334 - r72328;
        return r72335;
}
↓
double f(double n) {
        double r72336 = 1.0;
        double r72337 = n;
        double r72338 = r72336 / r72337;
        double r72339 = 0.5;
        double r72340 = 0.16666666666666669;
        double r72341 = r72340 / r72337;
        double r72342 = r72339 - r72341;
        double r72343 = log(r72337);
        double r72344 = 1.0;
        double r72345 = r72343 * r72344;
        double r72346 = fma(r72338, r72342, r72345);
        return r72346;
}

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

    \[\leadsto \color{blue}{0.5 \cdot \frac{1}{n} - \left(1 \cdot \log \left(\frac{1}{n}\right) + 0.1666666666666666851703837437526090070605 \cdot \frac{1}{{n}^{2}}\right)}\]

  4. Simplified0

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{n}, 0.5 - \frac{0.1666666666666666851703837437526090070605}{n}, \log n \cdot 1\right)}\]

  5. Final simplification0

    \[\leadsto \mathsf{fma}\left(\frac{1}{n}, 0.5 - \frac{0.1666666666666666851703837437526090070605}{n}, \log n \cdot 1\right)\]

Reproduce

herbie shell --seed 2020001 +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))