Average Error: 63.0 → 0
Time: 26.6s
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 r71321 = n;
        double r71322 = 1.0;
        double r71323 = r71321 + r71322;
        double r71324 = log(r71323);
        double r71325 = r71323 * r71324;
        double r71326 = log(r71321);
        double r71327 = r71321 * r71326;
        double r71328 = r71325 - r71327;
        double r71329 = r71328 - r71322;
        return r71329;
}
↓
double f(double n) {
        double r71330 = 0.5;
        double r71331 = n;
        double r71332 = r71330 / r71331;
        double r71333 = log(r71331);
        double r71334 = -r71333;
        double r71335 = 1.0;
        double r71336 = 0.16666666666666669;
        double r71337 = r71331 * r71331;
        double r71338 = r71336 / r71337;
        double r71339 = fma(r71334, r71335, r71338);
        double r71340 = r71332 - r71339;
        return r71340;
}

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 2019323 +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))