Average Error: 63.0 → 0
Time: 18.9s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\frac{0.5}{n} - \mathsf{fma}\left(1, -\log n, \frac{0.16666666666666669}{n \cdot n}\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\frac{0.5}{n} - \mathsf{fma}\left(1, -\log n, \frac{0.16666666666666669}{n \cdot n}\right)
double f(double n) {
        double r67802 = n;
        double r67803 = 1.0;
        double r67804 = r67802 + r67803;
        double r67805 = log(r67804);
        double r67806 = r67804 * r67805;
        double r67807 = log(r67802);
        double r67808 = r67802 * r67807;
        double r67809 = r67806 - r67808;
        double r67810 = r67809 - r67803;
        return r67810;
}
↓
double f(double n) {
        double r67811 = 0.5;
        double r67812 = n;
        double r67813 = r67811 / r67812;
        double r67814 = 1.0;
        double r67815 = log(r67812);
        double r67816 = -r67815;
        double r67817 = 0.16666666666666669;
        double r67818 = r67812 * r67812;
        double r67819 = r67817 / r67818;
        double r67820 = fma(r67814, r67816, r67819);
        double r67821 = r67813 - r67820;
        return r67821;
}

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

    \[\leadsto \color{blue}{0.5 \cdot \frac{1}{n} - \left(1 \cdot \log \left(\frac{1}{n}\right) + 0.16666666666666669 \cdot \frac{1}{{n}^{2}}\right)}\]

  4. Simplified0

    \[\leadsto \color{blue}{\frac{0.5}{n} - \mathsf{fma}\left(1, -\log n, \frac{0.16666666666666669}{n \cdot n}\right)}\]

  5. Final simplification0

    \[\leadsto \frac{0.5}{n} - \mathsf{fma}\left(1, -\log n, \frac{0.16666666666666669}{n \cdot n}\right)\]

Reproduce

herbie shell --seed 2020045 +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))