Average Error: 63.0 → 0
Time: 26.7s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\frac{0.5}{n} - \mathsf{fma}\left(-\log n, 1, \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right)\]
\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 r83181 = n;
        double r83182 = 1.0;
        double r83183 = r83181 + r83182;
        double r83184 = log(r83183);
        double r83185 = r83183 * r83184;
        double r83186 = log(r83181);
        double r83187 = r83181 * r83186;
        double r83188 = r83185 - r83187;
        double r83189 = r83188 - r83182;
        return r83189;
}
↓
double f(double n) {
        double r83190 = 0.5;
        double r83191 = n;
        double r83192 = r83190 / r83191;
        double r83193 = log(r83191);
        double r83194 = -r83193;
        double r83195 = 1.0;
        double r83196 = 0.16666666666666669;
        double r83197 = r83191 * r83191;
        double r83198 = r83196 / r83197;
        double r83199 = fma(r83194, r83195, r83198);
        double r83200 = r83192 - r83199;
        return r83200;
}

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 2019325 +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))