Average Error: 63.0 → 0
Time: 16.7s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\frac{-0.1666666666666666851703837437526090070605}{n \cdot n} + \mathsf{fma}\left(1, \log n, \frac{0.5}{n}\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\frac{-0.1666666666666666851703837437526090070605}{n \cdot n} + \mathsf{fma}\left(1, \log n, \frac{0.5}{n}\right)
double f(double n) {
        double r87104 = n;
        double r87105 = 1.0;
        double r87106 = r87104 + r87105;
        double r87107 = log(r87106);
        double r87108 = r87106 * r87107;
        double r87109 = log(r87104);
        double r87110 = r87104 * r87109;
        double r87111 = r87108 - r87110;
        double r87112 = r87111 - r87105;
        return r87112;
}
↓
double f(double n) {
        double r87113 = 0.16666666666666669;
        double r87114 = -r87113;
        double r87115 = n;
        double r87116 = r87115 * r87115;
        double r87117 = r87114 / r87116;
        double r87118 = 1.0;
        double r87119 = log(r87115);
        double r87120 = 0.5;
        double r87121 = r87120 / r87115;
        double r87122 = fma(r87118, r87119, r87121);
        double r87123 = r87117 + r87122;
        return r87123;
}

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. Simplified61.9

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

  3. Taylor expanded around inf 0.0

    \[\leadsto \color{blue}{0.5 \cdot \frac{1}{n} - \left(0.1666666666666666851703837437526090070605 \cdot \frac{1}{{n}^{2}} + 1 \cdot \log \left(\frac{1}{n}\right)\right)}\]

  4. Simplified0

    \[\leadsto \color{blue}{\frac{-0.1666666666666666851703837437526090070605}{n \cdot n} + \mathsf{fma}\left(1, \log n, \frac{0.5}{n}\right)}\]

  5. Final simplification0

    \[\leadsto \frac{-0.1666666666666666851703837437526090070605}{n \cdot n} + \mathsf{fma}\left(1, \log n, \frac{0.5}{n}\right)\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (n)
  :name "logs (example 3.8)"
  :pre (> n 6.8e+15)

  :herbie-target
  (- (log (+ n 1.0)) (- (/ 1.0 (* 2.0 n)) (- (/ 1.0 (* 3.0 (* n n))) (/ 4.0 (pow n 3.0)))))

  (- (- (* (+ n 1.0) (log (+ n 1.0))) (* n (log n))) 1.0))