Average Error: 63.0 → 0
Time: 27.1s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1.0\right) \cdot \log \left(n + 1.0\right) - n \cdot \log n\right) - 1.0\]
↓
\[\left(\frac{0.5}{n} - \frac{0.16666666666666669}{n \cdot n}\right) + 1.0 \cdot \log n\]
\left(\left(n + 1.0\right) \cdot \log \left(n + 1.0\right) - n \cdot \log n\right) - 1.0
↓
\left(\frac{0.5}{n} - \frac{0.16666666666666669}{n \cdot n}\right) + 1.0 \cdot \log n
double f(double n) {
        double r3375327 = n;
        double r3375328 = 1.0;
        double r3375329 = r3375327 + r3375328;
        double r3375330 = log(r3375329);
        double r3375331 = r3375329 * r3375330;
        double r3375332 = log(r3375327);
        double r3375333 = r3375327 * r3375332;
        double r3375334 = r3375331 - r3375333;
        double r3375335 = r3375334 - r3375328;
        return r3375335;
}
↓
double f(double n) {
        double r3375336 = 0.5;
        double r3375337 = n;
        double r3375338 = r3375336 / r3375337;
        double r3375339 = 0.16666666666666669;
        double r3375340 = r3375337 * r3375337;
        double r3375341 = r3375339 / r3375340;
        double r3375342 = r3375338 - r3375341;
        double r3375343 = 1.0;
        double r3375344 = log(r3375337);
        double r3375345 = r3375343 * r3375344;
        double r3375346 = r3375342 + r3375345;
        return r3375346;
}

Error

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original63.0
Target0
Herbie0
\[\log \left(n + 1.0\right) - \left(\frac{1.0}{2.0 \cdot n} - \left(\frac{1.0}{3.0 \cdot \left(n \cdot n\right)} - \frac{4.0}{{n}^{3.0}}\right)\right)\]

Derivation

  1. Initial program 63.0

    \[\left(\left(n + 1.0\right) \cdot \log \left(n + 1.0\right) - n \cdot \log n\right) - 1.0\]

  2. Simplified62.0

    \[\leadsto \color{blue}{\log \left(1.0 + n\right) \cdot \left(1.0 + n\right) - \mathsf{fma}\left(n, \log n, 1.0\right)}\]

  3. Taylor expanded around inf 0.0

    \[\leadsto \color{blue}{0.5 \cdot \frac{1}{n} - \left(0.16666666666666669 \cdot \frac{1}{{n}^{2}} + 1.0 \cdot \log \left(\frac{1}{n}\right)\right)}\]

  4. Simplified0

    \[\leadsto \color{blue}{\left(\frac{0.5}{n} - \frac{0.16666666666666669}{n \cdot n}\right) + 1.0 \cdot \log n}\]

  5. Final simplification0

    \[\leadsto \left(\frac{0.5}{n} - \frac{0.16666666666666669}{n \cdot n}\right) + 1.0 \cdot \log n\]

Reproduce

herbie shell --seed 2019165 +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))