Average Error: 63.0 → 0.0
Time: 4.7s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(0.5 \cdot \frac{1}{n} - \frac{0.1666666666666666851703837437526090070605}{{n}^{2}}\right) - 1 \cdot \log \left(\frac{1}{n}\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(0.5 \cdot \frac{1}{n} - \frac{0.1666666666666666851703837437526090070605}{{n}^{2}}\right) - 1 \cdot \log \left(\frac{1}{n}\right)
double f(double n) {
        double r76886 = n;
        double r76887 = 1.0;
        double r76888 = r76886 + r76887;
        double r76889 = log(r76888);
        double r76890 = r76888 * r76889;
        double r76891 = log(r76886);
        double r76892 = r76886 * r76891;
        double r76893 = r76890 - r76892;
        double r76894 = r76893 - r76887;
        return r76894;
}
↓
double f(double n) {
        double r76895 = 0.5;
        double r76896 = 1.0;
        double r76897 = n;
        double r76898 = r76896 / r76897;
        double r76899 = r76895 * r76898;
        double r76900 = 0.16666666666666669;
        double r76901 = 2.0;
        double r76902 = pow(r76897, r76901);
        double r76903 = r76900 / r76902;
        double r76904 = r76899 - r76903;
        double r76905 = 1.0;
        double r76906 = log(r76898);
        double r76907 = r76905 * r76906;
        double r76908 = r76904 - r76907;
        return r76908;
}

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

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

  4. Taylor expanded around inf 0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019362 
(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))