Average Error: 63.0 → 0
Time: 1.0m
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\mathsf{fma}\left(\frac{-1}{6}, \frac{\frac{1}{n}}{n}, \mathsf{fma}\left(\frac{1}{n}, \frac{1}{2}, \log n\right)\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\mathsf{fma}\left(\frac{-1}{6}, \frac{\frac{1}{n}}{n}, \mathsf{fma}\left(\frac{1}{n}, \frac{1}{2}, \log n\right)\right)
double f(double n) {
        double r2291803 = n;
        double r2291804 = 1.0;
        double r2291805 = r2291803 + r2291804;
        double r2291806 = log(r2291805);
        double r2291807 = r2291805 * r2291806;
        double r2291808 = log(r2291803);
        double r2291809 = r2291803 * r2291808;
        double r2291810 = r2291807 - r2291809;
        double r2291811 = r2291810 - r2291804;
        return r2291811;
}
↓
double f(double n) {
        double r2291812 = -0.16666666666666666;
        double r2291813 = 1.0;
        double r2291814 = n;
        double r2291815 = r2291813 / r2291814;
        double r2291816 = r2291815 / r2291814;
        double r2291817 = 0.5;
        double r2291818 = log(r2291814);
        double r2291819 = fma(r2291815, r2291817, r2291818);
        double r2291820 = fma(r2291812, r2291816, r2291819);
        return r2291820;
}

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. Simplified44.2

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

  3. Taylor expanded around -inf 62.0

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

  4. Simplified0.0

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

  5. Taylor expanded around 0 0

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

  6. Simplified0

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

  7. Final simplification0

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

Reproduce

herbie shell --seed 2019151 +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))