Average Error: 63.0 → 0
Time: 15.4s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\frac{0.5}{n} + 1\right) + \left(\left(1 \cdot \log n - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) - 1\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\frac{0.5}{n} + 1\right) + \left(\left(1 \cdot \log n - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) - 1\right)
double f(double n) {
        double r63324 = n;
        double r63325 = 1.0;
        double r63326 = r63324 + r63325;
        double r63327 = log(r63326);
        double r63328 = r63326 * r63327;
        double r63329 = log(r63324);
        double r63330 = r63324 * r63329;
        double r63331 = r63328 - r63330;
        double r63332 = r63331 - r63325;
        return r63332;
}
↓
double f(double n) {
        double r63333 = 0.5;
        double r63334 = n;
        double r63335 = r63333 / r63334;
        double r63336 = 1.0;
        double r63337 = r63335 + r63336;
        double r63338 = log(r63334);
        double r63339 = r63336 * r63338;
        double r63340 = 0.16666666666666669;
        double r63341 = r63334 * r63334;
        double r63342 = r63340 / r63341;
        double r63343 = r63339 - r63342;
        double r63344 = r63343 - r63336;
        double r63345 = r63337 + r63344;
        return r63345;
}

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. Taylor expanded around inf 0.0

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

  3. Simplified0.0

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

  5. Applied associate--l+0.0

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

  6. Applied associate--l+0

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

  7. Final simplification0

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

Reproduce

herbie shell --seed 2019235 
(FPCore (n)
  :name "logs (example 3.8)"
  :precision binary64
  :pre (> n 6.8e15)

  :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))