Average Error: 63.0 → 0.0
Time: 20.5s
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 r27934 = n;
        double r27935 = 1.0;
        double r27936 = r27934 + r27935;
        double r27937 = log(r27936);
        double r27938 = r27936 * r27937;
        double r27939 = log(r27934);
        double r27940 = r27934 * r27939;
        double r27941 = r27938 - r27940;
        double r27942 = r27941 - r27935;
        return r27942;
}
↓
double f(double n) {
        double r27943 = 0.5;
        double r27944 = n;
        double r27945 = r27943 / r27944;
        double r27946 = 1.0;
        double r27947 = r27945 + r27946;
        double r27948 = 0.16666666666666669;
        double r27949 = r27944 * r27944;
        double r27950 = r27948 / r27949;
        double r27951 = r27947 - r27950;
        double r27952 = log(r27944);
        double r27953 = r27952 * r27946;
        double r27954 = r27951 + r27953;
        double r27955 = r27954 - r27946;
        return r27955;
}

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 2019326 
(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))