Average Error: 63.0 → 0
Time: 16.5s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\mathsf{fma}\left(\frac{1}{n}, \frac{1}{2} - \frac{\frac{1}{6}}{n}, \log n\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\mathsf{fma}\left(\frac{1}{n}, \frac{1}{2} - \frac{\frac{1}{6}}{n}, \log n\right)
double f(double n) {
        double r1180063 = n;
        double r1180064 = 1.0;
        double r1180065 = r1180063 + r1180064;
        double r1180066 = log(r1180065);
        double r1180067 = r1180065 * r1180066;
        double r1180068 = log(r1180063);
        double r1180069 = r1180063 * r1180068;
        double r1180070 = r1180067 - r1180069;
        double r1180071 = r1180070 - r1180064;
        return r1180071;
}
↓
double f(double n) {
        double r1180072 = 1.0;
        double r1180073 = n;
        double r1180074 = r1180072 / r1180073;
        double r1180075 = 0.5;
        double r1180076 = 0.16666666666666666;
        double r1180077 = r1180076 / r1180073;
        double r1180078 = r1180075 - r1180077;
        double r1180079 = log(r1180073);
        double r1180080 = fma(r1180074, r1180078, r1180079);
        return r1180080;
}

Error

Bits error versus n

Target

Original63.0
Target0.0
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. Simplified61.8

    \[\leadsto \color{blue}{\mathsf{fma}\left(n, \mathsf{log1p}\left(n\right), \mathsf{log1p}\left(n\right) - \mathsf{fma}\left(n, \log n, 1\right)\right)}\]

  3. Taylor expanded around inf 0.0

    \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{1}{n} - \left(\frac{1}{6} \cdot \frac{1}{{n}^{2}} + \log \left(\frac{1}{n}\right)\right)}\]

  4. Simplified0

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{n}, \frac{1}{2} - \frac{\frac{1}{6}}{n}, \log n\right)}\]

  5. Final simplification0

    \[\leadsto \mathsf{fma}\left(\frac{1}{n}, \frac{1}{2} - \frac{\frac{1}{6}}{n}, \log n\right)\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (n)
  :name "logs (example 3.8)"
  :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))