Average Error: 63.0 → 0
Time: 37.0s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\log n + \frac{\frac{\frac{-1}{6}}{n}}{n}\right) + \frac{\frac{1}{2}}{n}\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\log n + \frac{\frac{\frac{-1}{6}}{n}}{n}\right) + \frac{\frac{1}{2}}{n}
double f(double n) {
        double r1409957 = n;
        double r1409958 = 1.0;
        double r1409959 = r1409957 + r1409958;
        double r1409960 = log(r1409959);
        double r1409961 = r1409959 * r1409960;
        double r1409962 = log(r1409957);
        double r1409963 = r1409957 * r1409962;
        double r1409964 = r1409961 - r1409963;
        double r1409965 = r1409964 - r1409958;
        return r1409965;
}
↓
double f(double n) {
        double r1409966 = n;
        double r1409967 = log(r1409966);
        double r1409968 = -0.16666666666666666;
        double r1409969 = r1409968 / r1409966;
        double r1409970 = r1409969 / r1409966;
        double r1409971 = r1409967 + r1409970;
        double r1409972 = 0.5;
        double r1409973 = r1409972 / r1409966;
        double r1409974 = r1409971 + r1409973;
        return r1409974;
}

Error

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original63.0
Target0
Herbie0
\[\log \left(n + 1\right) - \left(\frac{1}{2 \cdot n} - \left(\frac{1}{3 \cdot \left(n \cdot n\right)} - \frac{4}{{n}^{3}}\right)\right)\]

Derivation

  1. Initial program 63.0

    \[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]

  2. Simplified44.2

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

  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

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

  5. Taylor expanded around 0 0

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

  6. Simplified0

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

  7. Final simplification0

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

Reproduce

herbie shell --seed 2019152 +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))