Average Error: 63.0 → 0
Time: 22.0s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\log n + \left(\frac{\frac{-1}{6}}{n \cdot n} + \frac{\frac{1}{2}}{n}\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\log n + \left(\frac{\frac{-1}{6}}{n \cdot n} + \frac{\frac{1}{2}}{n}\right)
double f(double n) {
        double r2645977 = n;
        double r2645978 = 1.0;
        double r2645979 = r2645977 + r2645978;
        double r2645980 = log(r2645979);
        double r2645981 = r2645979 * r2645980;
        double r2645982 = log(r2645977);
        double r2645983 = r2645977 * r2645982;
        double r2645984 = r2645981 - r2645983;
        double r2645985 = r2645984 - r2645978;
        return r2645985;
}
↓
double f(double n) {
        double r2645986 = n;
        double r2645987 = log(r2645986);
        double r2645988 = -0.16666666666666666;
        double r2645989 = r2645986 * r2645986;
        double r2645990 = r2645988 / r2645989;
        double r2645991 = 0.5;
        double r2645992 = r2645991 / r2645986;
        double r2645993 = r2645990 + r2645992;
        double r2645994 = r2645987 + r2645993;
        return r2645994;
}

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. 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. Taylor expanded around 0 0

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

  5. Simplified0

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

  6. Final simplification0

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

Reproduce

herbie shell --seed 2019164 
(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))