Average Error: 63.0 → 0.0
Time: 20.7s
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 r38228 = n;
        double r38229 = 1.0;
        double r38230 = r38228 + r38229;
        double r38231 = log(r38230);
        double r38232 = r38230 * r38231;
        double r38233 = log(r38228);
        double r38234 = r38228 * r38233;
        double r38235 = r38232 - r38234;
        double r38236 = r38235 - r38229;
        return r38236;
}
↓
double f(double n) {
        double r38237 = 0.5;
        double r38238 = n;
        double r38239 = r38237 / r38238;
        double r38240 = 1.0;
        double r38241 = r38239 + r38240;
        double r38242 = 0.16666666666666669;
        double r38243 = r38238 * r38238;
        double r38244 = r38242 / r38243;
        double r38245 = r38241 - r38244;
        double r38246 = log(r38238);
        double r38247 = r38246 * r38240;
        double r38248 = r38245 + r38247;
        double r38249 = r38248 - r38240;
        return r38249;
}

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

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

  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 2019303 
(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))