Average Error: 63.0 → 0
Time: 25.4s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[1 \cdot \log n + \left(\frac{0.5}{n} - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
1 \cdot \log n + \left(\frac{0.5}{n} - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right)
double f(double n) {
        double r9026830 = n;
        double r9026831 = 1.0;
        double r9026832 = r9026830 + r9026831;
        double r9026833 = log(r9026832);
        double r9026834 = r9026832 * r9026833;
        double r9026835 = log(r9026830);
        double r9026836 = r9026830 * r9026835;
        double r9026837 = r9026834 - r9026836;
        double r9026838 = r9026837 - r9026831;
        return r9026838;
}
↓
double f(double n) {
        double r9026839 = 1.0;
        double r9026840 = n;
        double r9026841 = log(r9026840);
        double r9026842 = r9026839 * r9026841;
        double r9026843 = 0.5;
        double r9026844 = r9026843 / r9026840;
        double r9026845 = 0.16666666666666669;
        double r9026846 = r9026840 * r9026840;
        double r9026847 = r9026845 / r9026846;
        double r9026848 = r9026844 - r9026847;
        double r9026849 = r9026842 + r9026848;
        return r9026849;
}

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

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

  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. Taylor expanded around inf 0.0

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

  5. Simplified0

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

  6. Final simplification0

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

Reproduce

herbie shell --seed 2019173 
(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))