Average Error: 63.0 → 0.0
Time: 16.4s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(\left(1 + \frac{\frac{-1}{6}}{n \cdot n}\right) + \frac{\frac{1}{2}}{n}\right) + \log n\right) - 1\]
\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 r1449139 = n;
        double r1449140 = 1.0;
        double r1449141 = r1449139 + r1449140;
        double r1449142 = log(r1449141);
        double r1449143 = r1449141 * r1449142;
        double r1449144 = log(r1449139);
        double r1449145 = r1449139 * r1449144;
        double r1449146 = r1449143 - r1449145;
        double r1449147 = r1449146 - r1449140;
        return r1449147;
}
↓
double f(double n) {
        double r1449148 = 1.0;
        double r1449149 = -0.16666666666666666;
        double r1449150 = n;
        double r1449151 = r1449150 * r1449150;
        double r1449152 = r1449149 / r1449151;
        double r1449153 = r1449148 + r1449152;
        double r1449154 = 0.5;
        double r1449155 = r1449154 / r1449150;
        double r1449156 = r1449153 + r1449155;
        double r1449157 = log(r1449150);
        double r1449158 = r1449156 + r1449157;
        double r1449159 = r1449158 - r1449148;
        return r1449159;
}

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 2019163 
(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))