Average Error: 63.0 → 0
Time: 17.8s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\frac{0.5}{n} - \mathsf{fma}\left(1, -\log n, \frac{\frac{0.1666666666666666851703837437526090070605}{n}}{n}\right)\]
\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{\frac{0.1666666666666666851703837437526090070605}{n}}{n}\right)
double f(double n) {
        double r3421168 = n;
        double r3421169 = 1.0;
        double r3421170 = r3421168 + r3421169;
        double r3421171 = log(r3421170);
        double r3421172 = r3421170 * r3421171;
        double r3421173 = log(r3421168);
        double r3421174 = r3421168 * r3421173;
        double r3421175 = r3421172 - r3421174;
        double r3421176 = r3421175 - r3421169;
        return r3421176;
}
↓
double f(double n) {
        double r3421177 = 0.5;
        double r3421178 = n;
        double r3421179 = r3421177 / r3421178;
        double r3421180 = 1.0;
        double r3421181 = log(r3421178);
        double r3421182 = -r3421181;
        double r3421183 = 0.16666666666666669;
        double r3421184 = r3421183 / r3421178;
        double r3421185 = r3421184 / r3421178;
        double r3421186 = fma(r3421180, r3421182, r3421185);
        double r3421187 = r3421179 - r3421186;
        return r3421187;
}

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. Simplified62.0

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

  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

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

  5. Final simplification0

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

Reproduce

herbie shell --seed 2019179 +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))