Average Error: 63.0 → 0
Time: 27.1s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\frac{0.5}{n} - \mathsf{fma}\left(-\log n, 1, \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\frac{0.5}{n} - \mathsf{fma}\left(-\log n, 1, \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right)
double f(double n) {
        double r94822 = n;
        double r94823 = 1.0;
        double r94824 = r94822 + r94823;
        double r94825 = log(r94824);
        double r94826 = r94824 * r94825;
        double r94827 = log(r94822);
        double r94828 = r94822 * r94827;
        double r94829 = r94826 - r94828;
        double r94830 = r94829 - r94823;
        return r94830;
}
↓
double f(double n) {
        double r94831 = 0.5;
        double r94832 = n;
        double r94833 = r94831 / r94832;
        double r94834 = log(r94832);
        double r94835 = -r94834;
        double r94836 = 1.0;
        double r94837 = 0.16666666666666669;
        double r94838 = r94832 * r94832;
        double r94839 = r94837 / r94838;
        double r94840 = fma(r94835, r94836, r94839);
        double r94841 = r94833 - r94840;
        return r94841;
}

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

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

  4. Simplified0

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

  5. Final simplification0

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

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(FPCore (n)
  :name "logs (example 3.8)"
  :precision binary64
  :pre (> n 6.8e15)

  :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))