Average Error: 63.0 → 0
Time: 20.4s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(1 + \frac{0.5}{n}\right) - \frac{0.16666666666666669}{n \cdot n}\right) + \left(\log n \cdot 1 - 1\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\left(1 + \frac{0.5}{n}\right) - \frac{0.16666666666666669}{n \cdot n}\right) + \left(\log n \cdot 1 - 1\right)
double f(double n) {
        double r54808 = n;
        double r54809 = 1.0;
        double r54810 = r54808 + r54809;
        double r54811 = log(r54810);
        double r54812 = r54810 * r54811;
        double r54813 = log(r54808);
        double r54814 = r54808 * r54813;
        double r54815 = r54812 - r54814;
        double r54816 = r54815 - r54809;
        return r54816;
}
↓
double f(double n) {
        double r54817 = 1.0;
        double r54818 = 0.5;
        double r54819 = n;
        double r54820 = r54818 / r54819;
        double r54821 = r54817 + r54820;
        double r54822 = 0.16666666666666669;
        double r54823 = r54819 * r54819;
        double r54824 = r54822 / r54823;
        double r54825 = r54821 - r54824;
        double r54826 = log(r54819);
        double r54827 = r54826 * r54817;
        double r54828 = r54827 - r54817;
        double r54829 = r54825 + r54828;
        return r54829;
}

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

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

  5. Applied associate--l+0

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

  6. Final simplification0

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

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))