Average Error: 63.0 → 0.0
Time: 17.9s
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 r48234 = n;
        double r48235 = 1.0;
        double r48236 = r48234 + r48235;
        double r48237 = log(r48236);
        double r48238 = r48236 * r48237;
        double r48239 = log(r48234);
        double r48240 = r48234 * r48239;
        double r48241 = r48238 - r48240;
        double r48242 = r48241 - r48235;
        return r48242;
}
↓
double f(double n) {
        double r48243 = 1.0;
        double r48244 = 0.5;
        double r48245 = n;
        double r48246 = r48244 / r48245;
        double r48247 = r48243 + r48246;
        double r48248 = 0.16666666666666669;
        double r48249 = r48245 * r48245;
        double r48250 = r48248 / r48249;
        double r48251 = r48247 - r48250;
        double r48252 = log(r48245);
        double r48253 = r48252 * r48243;
        double r48254 = r48251 + r48253;
        double r48255 = r48254 - r48243;
        return r48255;
}

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 2020046 
(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))