Average Error: 63.0 → 0
Time: 10.4s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\frac{0.5}{n} + \log n \cdot 1\right) - \frac{\frac{0.1666666666666666851703837437526090070605}{n}}{n}\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\frac{0.5}{n} + \log n \cdot 1\right) - \frac{\frac{0.1666666666666666851703837437526090070605}{n}}{n}
double f(double n) {
        double r73321 = n;
        double r73322 = 1.0;
        double r73323 = r73321 + r73322;
        double r73324 = log(r73323);
        double r73325 = r73323 * r73324;
        double r73326 = log(r73321);
        double r73327 = r73321 * r73326;
        double r73328 = r73325 - r73327;
        double r73329 = r73328 - r73322;
        return r73329;
}
↓
double f(double n) {
        double r73330 = 0.5;
        double r73331 = n;
        double r73332 = r73330 / r73331;
        double r73333 = log(r73331);
        double r73334 = 1.0;
        double r73335 = r73333 * r73334;
        double r73336 = r73332 + r73335;
        double r73337 = 0.16666666666666669;
        double r73338 = r73337 / r73331;
        double r73339 = r73338 / r73331;
        double r73340 = r73336 - r73339;
        return r73340;
}

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

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

  4. Final simplification0

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

Reproduce

herbie shell --seed 2019298 
(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))