Average Error: 63.0 → 0.0
Time: 3.5s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\mathsf{fma}\left(\frac{1}{n}, 0.5 - \frac{0.16666666666666669}{n}, \log n \cdot 1\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\mathsf{fma}\left(\frac{1}{n}, 0.5 - \frac{0.16666666666666669}{n}, \log n \cdot 1\right)
double f(double n) {
        double r65958 = n;
        double r65959 = 1.0;
        double r65960 = r65958 + r65959;
        double r65961 = log(r65960);
        double r65962 = r65960 * r65961;
        double r65963 = log(r65958);
        double r65964 = r65958 * r65963;
        double r65965 = r65962 - r65964;
        double r65966 = r65965 - r65959;
        return r65966;
}
↓
double f(double n) {
        double r65967 = 1.0;
        double r65968 = n;
        double r65969 = r65967 / r65968;
        double r65970 = 0.5;
        double r65971 = 0.16666666666666669;
        double r65972 = r65971 / r65968;
        double r65973 = r65970 - r65972;
        double r65974 = log(r65968);
        double r65975 = 1.0;
        double r65976 = r65974 * r65975;
        double r65977 = fma(r65969, r65973, r65976);
        return r65977;
}

Error

Bits error versus n

Target

Original63.0
Target0.0
Herbie0.0
\[\log \left(n + 1\right) - \left(\frac{1}{2 \cdot n} - \left(\frac{1}{3 \cdot \left(n \cdot n\right)} - \frac{4}{{n}^{3}}\right)\right)\]

Derivation

  1. Initial program 63.0

    \[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]

  2. Simplified61.9

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

  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. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020034 +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))