Average Error: 63.0 → 0
Time: 24.8s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\log n \cdot 1 - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) + \frac{0.5}{n}\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\log n \cdot 1 - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) + \frac{0.5}{n}
double f(double n) {
        double r3254360 = n;
        double r3254361 = 1.0;
        double r3254362 = r3254360 + r3254361;
        double r3254363 = log(r3254362);
        double r3254364 = r3254362 * r3254363;
        double r3254365 = log(r3254360);
        double r3254366 = r3254360 * r3254365;
        double r3254367 = r3254364 - r3254366;
        double r3254368 = r3254367 - r3254361;
        return r3254368;
}
↓
double f(double n) {
        double r3254369 = n;
        double r3254370 = log(r3254369);
        double r3254371 = 1.0;
        double r3254372 = r3254370 * r3254371;
        double r3254373 = 0.16666666666666669;
        double r3254374 = r3254369 * r3254369;
        double r3254375 = r3254373 / r3254374;
        double r3254376 = r3254372 - r3254375;
        double r3254377 = 0.5;
        double r3254378 = r3254377 / r3254369;
        double r3254379 = r3254376 + r3254378;
        return r3254379;
}

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. Simplified62.0

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

  3. Taylor expanded around inf 0.0

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

  4. Simplified0

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

  5. Final simplification0

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

Reproduce

herbie shell --seed 2019200 +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))