Average Error: 63.0 → 0
Time: 21.0s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\log n + \frac{\frac{-1}{6}}{n \cdot n}\right) + \frac{\frac{1}{2}}{n}\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\log n + \frac{\frac{-1}{6}}{n \cdot n}\right) + \frac{\frac{1}{2}}{n}
double f(double n) {
        double r1608172 = n;
        double r1608173 = 1.0;
        double r1608174 = r1608172 + r1608173;
        double r1608175 = log(r1608174);
        double r1608176 = r1608174 * r1608175;
        double r1608177 = log(r1608172);
        double r1608178 = r1608172 * r1608177;
        double r1608179 = r1608176 - r1608178;
        double r1608180 = r1608179 - r1608173;
        return r1608180;
}
↓
double f(double n) {
        double r1608181 = n;
        double r1608182 = log(r1608181);
        double r1608183 = -0.16666666666666666;
        double r1608184 = r1608181 * r1608181;
        double r1608185 = r1608183 / r1608184;
        double r1608186 = r1608182 + r1608185;
        double r1608187 = 0.5;
        double r1608188 = r1608187 / r1608181;
        double r1608189 = r1608186 + r1608188;
        return r1608189;
}

Error

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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 0.0

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

  4. Simplified0

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

  5. Final simplification0

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

Reproduce

herbie shell --seed 2019149 +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))