Average Error: 63.0 → 0.0
Time: 17.0s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(\left(1 + \frac{\frac{-1}{6}}{n \cdot n}\right) + \frac{\frac{1}{2}}{n}\right) + \log n\right) - 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\left(\left(1 + \frac{\frac{-1}{6}}{n \cdot n}\right) + \frac{\frac{1}{2}}{n}\right) + \log n\right) - 1
double f(double n) {
        double r2998064 = n;
        double r2998065 = 1.0;
        double r2998066 = r2998064 + r2998065;
        double r2998067 = log(r2998066);
        double r2998068 = r2998066 * r2998067;
        double r2998069 = log(r2998064);
        double r2998070 = r2998064 * r2998069;
        double r2998071 = r2998068 - r2998070;
        double r2998072 = r2998071 - r2998065;
        return r2998072;
}
↓
double f(double n) {
        double r2998073 = 1.0;
        double r2998074 = -0.16666666666666666;
        double r2998075 = n;
        double r2998076 = r2998075 * r2998075;
        double r2998077 = r2998074 / r2998076;
        double r2998078 = r2998073 + r2998077;
        double r2998079 = 0.5;
        double r2998080 = r2998079 / r2998075;
        double r2998081 = r2998078 + r2998080;
        double r2998082 = log(r2998075);
        double r2998083 = r2998081 + r2998082;
        double r2998084 = r2998083 - r2998073;
        return r2998084;
}

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

    \[\leadsto \color{blue}{\left(\left(1 + \frac{1}{2} \cdot \frac{1}{n}\right) - \left(\frac{1}{6} \cdot \frac{1}{{n}^{2}} + \log \left(\frac{1}{n}\right)\right)\right)} - 1\]

  3. Simplified0.0

    \[\leadsto \color{blue}{\left(\left(\frac{\frac{1}{2}}{n} + \left(1 + \frac{\frac{-1}{6}}{n \cdot n}\right)\right) + \log n\right)} - 1\]

  4. Final simplification0.0

    \[\leadsto \left(\left(\left(1 + \frac{\frac{-1}{6}}{n \cdot n}\right) + \frac{\frac{1}{2}}{n}\right) + \log n\right) - 1\]

Reproduce

herbie shell --seed 2019158 
(FPCore (n)
  :name "logs (example 3.8)"
  :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))