Average Error: 63.0 → 0.0
Time: 12.5s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(\left(\frac{0.5}{n} + 1\right) - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) + \log n \cdot 1\right) - 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\left(\left(\frac{0.5}{n} + 1\right) - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) + \log n \cdot 1\right) - 1
double f(double n) {
        double r4693211 = n;
        double r4693212 = 1.0;
        double r4693213 = r4693211 + r4693212;
        double r4693214 = log(r4693213);
        double r4693215 = r4693213 * r4693214;
        double r4693216 = log(r4693211);
        double r4693217 = r4693211 * r4693216;
        double r4693218 = r4693215 - r4693217;
        double r4693219 = r4693218 - r4693212;
        return r4693219;
}
↓
double f(double n) {
        double r4693220 = 0.5;
        double r4693221 = n;
        double r4693222 = r4693220 / r4693221;
        double r4693223 = 1.0;
        double r4693224 = r4693222 + r4693223;
        double r4693225 = 0.16666666666666669;
        double r4693226 = r4693221 * r4693221;
        double r4693227 = r4693225 / r4693226;
        double r4693228 = r4693224 - r4693227;
        double r4693229 = log(r4693221);
        double r4693230 = r4693229 * r4693223;
        double r4693231 = r4693228 + r4693230;
        double r4693232 = r4693231 - r4693223;
        return r4693232;
}

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

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019174 
(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))