Average Error: 63.0 → 0.0
Time: 18.5s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(1 + \frac{\frac{-1}{6}}{n \cdot n}\right) + \left(\frac{\frac{1}{2}}{n} + \log n\right)\right) - 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\left(1 + \frac{\frac{-1}{6}}{n \cdot n}\right) + \left(\frac{\frac{1}{2}}{n} + \log n\right)\right) - 1
double f(double n) {
        double r901012 = n;
        double r901013 = 1.0;
        double r901014 = r901012 + r901013;
        double r901015 = log(r901014);
        double r901016 = r901014 * r901015;
        double r901017 = log(r901012);
        double r901018 = r901012 * r901017;
        double r901019 = r901016 - r901018;
        double r901020 = r901019 - r901013;
        return r901020;
}
↓
double f(double n) {
        double r901021 = 1.0;
        double r901022 = -0.16666666666666666;
        double r901023 = n;
        double r901024 = r901023 * r901023;
        double r901025 = r901022 / r901024;
        double r901026 = r901021 + r901025;
        double r901027 = 0.5;
        double r901028 = r901027 / r901023;
        double r901029 = log(r901023);
        double r901030 = r901028 + r901029;
        double r901031 = r901026 + r901030;
        double r901032 = r901031 - r901021;
        return r901032;
}

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. Simplified44.3

    \[\leadsto \color{blue}{\mathsf{fma}\left(n, \mathsf{log1p}\left(n\right) - \log n, \mathsf{log1p}\left(n\right)\right) - 1}\]

  3. Taylor expanded around inf 0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019135 +o rules:numerics
(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))