Average Error: 63.0 → 0.0
Time: 11.9s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(\frac{\frac{1}{2}}{n} + 1\right) - \left(\frac{\frac{1}{6}}{n \cdot n} - \log n\right)\right) - 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\left(\frac{\frac{1}{2}}{n} + 1\right) - \left(\frac{\frac{1}{6}}{n \cdot n} - \log n\right)\right) - 1
double f(double n) {
        double r1473135 = n;
        double r1473136 = 1.0;
        double r1473137 = r1473135 + r1473136;
        double r1473138 = log(r1473137);
        double r1473139 = r1473137 * r1473138;
        double r1473140 = log(r1473135);
        double r1473141 = r1473135 * r1473140;
        double r1473142 = r1473139 - r1473141;
        double r1473143 = r1473142 - r1473136;
        return r1473143;
}
↓
double f(double n) {
        double r1473144 = 0.5;
        double r1473145 = n;
        double r1473146 = r1473144 / r1473145;
        double r1473147 = 1.0;
        double r1473148 = r1473146 + r1473147;
        double r1473149 = 0.16666666666666666;
        double r1473150 = r1473145 * r1473145;
        double r1473151 = r1473149 / r1473150;
        double r1473152 = log(r1473145);
        double r1473153 = r1473151 - r1473152;
        double r1473154 = r1473148 - r1473153;
        double r1473155 = r1473154 - r1473147;
        return r1473155;
}

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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019153 
(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))