Average Error: 63.0 → 0.0
Time: 13.1s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(\left(1 + \frac{0.5}{n}\right) - \frac{0.16666666666666669}{n \cdot n}\right) + \log n \cdot 1\right) - 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\left(\left(1 + \frac{0.5}{n}\right) - \frac{0.16666666666666669}{n \cdot n}\right) + \log n \cdot 1\right) - 1
double f(double n) {
        double r75301 = n;
        double r75302 = 1.0;
        double r75303 = r75301 + r75302;
        double r75304 = log(r75303);
        double r75305 = r75303 * r75304;
        double r75306 = log(r75301);
        double r75307 = r75301 * r75306;
        double r75308 = r75305 - r75307;
        double r75309 = r75308 - r75302;
        return r75309;
}
↓
double f(double n) {
        double r75310 = 1.0;
        double r75311 = 0.5;
        double r75312 = n;
        double r75313 = r75311 / r75312;
        double r75314 = r75310 + r75313;
        double r75315 = 0.16666666666666669;
        double r75316 = r75312 * r75312;
        double r75317 = r75315 / r75316;
        double r75318 = r75314 - r75317;
        double r75319 = log(r75312);
        double r75320 = r75319 * r75310;
        double r75321 = r75318 + r75320;
        double r75322 = r75321 - r75310;
        return r75322;
}

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. Taylor expanded around inf 0.0

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

  3. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \left(\left(\left(1 + \frac{0.5}{n}\right) - \frac{0.16666666666666669}{n \cdot n}\right) + \log n \cdot 1\right) - 1\]

Reproduce

herbie shell --seed 2020042 
(FPCore (n)
  :name "logs (example 3.8)"
  :precision binary64
  :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))