Average Error: 63.0 → 0
Time: 19.5s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\frac{0.5}{n} - \left(\frac{0.16666666666666669}{{n}^{2}} - 1 \cdot \log n\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\frac{0.5}{n} - \left(\frac{0.16666666666666669}{{n}^{2}} - 1 \cdot \log n\right)
double f(double n) {
        double r117902 = n;
        double r117903 = 1.0;
        double r117904 = r117902 + r117903;
        double r117905 = log(r117904);
        double r117906 = r117904 * r117905;
        double r117907 = log(r117902);
        double r117908 = r117902 * r117907;
        double r117909 = r117906 - r117908;
        double r117910 = r117909 - r117903;
        return r117910;
}
↓
double f(double n) {
        double r117911 = 0.5;
        double r117912 = n;
        double r117913 = r117911 / r117912;
        double r117914 = 0.16666666666666669;
        double r117915 = 2.0;
        double r117916 = pow(r117912, r117915);
        double r117917 = r117914 / r117916;
        double r117918 = 1.0;
        double r117919 = log(r117912);
        double r117920 = r117918 * r117919;
        double r117921 = r117917 - r117920;
        double r117922 = r117913 - r117921;
        return r117922;
}

Error

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original63.0
Target0
Herbie0
\[\log \left(n + 1\right) - \left(\frac{1}{2 \cdot n} - \left(\frac{1}{3 \cdot \left(n \cdot n\right)} - \frac{4}{{n}^{3}}\right)\right)\]

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

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

  5. Applied associate-+l-0.0

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

  6. Applied associate--l-0.0

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

  7. Simplified0

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

  8. Final simplification0

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

Reproduce

herbie shell --seed 2020045 
(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))