Average Error: 63.0 → 0.0
Time: 19.2s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(1 + \left(\frac{\frac{-1}{6}}{n \cdot n} + \log n\right)\right) + \frac{\frac{1}{2}}{n}\right) - 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\left(1 + \left(\frac{\frac{-1}{6}}{n \cdot n} + \log n\right)\right) + \frac{\frac{1}{2}}{n}\right) - 1
double f(double n) {
        double r5122176 = n;
        double r5122177 = 1.0;
        double r5122178 = r5122176 + r5122177;
        double r5122179 = log(r5122178);
        double r5122180 = r5122178 * r5122179;
        double r5122181 = log(r5122176);
        double r5122182 = r5122176 * r5122181;
        double r5122183 = r5122180 - r5122182;
        double r5122184 = r5122183 - r5122177;
        return r5122184;
}
↓
double f(double n) {
        double r5122185 = 1.0;
        double r5122186 = -0.16666666666666666;
        double r5122187 = n;
        double r5122188 = r5122187 * r5122187;
        double r5122189 = r5122186 / r5122188;
        double r5122190 = log(r5122187);
        double r5122191 = r5122189 + r5122190;
        double r5122192 = r5122185 + r5122191;
        double r5122193 = 0.5;
        double r5122194 = r5122193 / r5122187;
        double r5122195 = r5122192 + r5122194;
        double r5122196 = r5122195 - r5122185;
        return r5122196;
}

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 62.0

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019125 
(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))