Average Error: 63.0 → 0.0
Time: 29.9s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[-1 + \left(\left(\left(1 + \log n\right) + \frac{\frac{1}{2}}{n}\right) + \frac{\frac{-1}{6}}{n \cdot n}\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
-1 + \left(\left(\left(1 + \log n\right) + \frac{\frac{1}{2}}{n}\right) + \frac{\frac{-1}{6}}{n \cdot n}\right)
double f(double n) {
        double r1510204 = n;
        double r1510205 = 1.0;
        double r1510206 = r1510204 + r1510205;
        double r1510207 = log(r1510206);
        double r1510208 = r1510206 * r1510207;
        double r1510209 = log(r1510204);
        double r1510210 = r1510204 * r1510209;
        double r1510211 = r1510208 - r1510210;
        double r1510212 = r1510211 - r1510205;
        return r1510212;
}
↓
double f(double n) {
        double r1510213 = -1.0;
        double r1510214 = 1.0;
        double r1510215 = n;
        double r1510216 = log(r1510215);
        double r1510217 = r1510214 + r1510216;
        double r1510218 = 0.5;
        double r1510219 = r1510218 / r1510215;
        double r1510220 = r1510217 + r1510219;
        double r1510221 = -0.16666666666666666;
        double r1510222 = r1510215 * r1510215;
        double r1510223 = r1510221 / r1510222;
        double r1510224 = r1510220 + r1510223;
        double r1510225 = r1510213 + r1510224;
        return r1510225;
}

Error

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original63.0
Target0
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. 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

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019142 +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))