Average Error: 63.0 → 0
Time: 1.3m
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\frac{\frac{1}{2}}{n} - \frac{\frac{3002399751580331}{18014398509481984}}{{n}^{2}}\right) + \log n \cdot 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\frac{\frac{1}{2}}{n} - \frac{\frac{3002399751580331}{18014398509481984}}{{n}^{2}}\right) + \log n \cdot 1
double f(double n) {
        double r55703 = n;
        double r55704 = 1.0;
        double r55705 = r55703 + r55704;
        double r55706 = log(r55705);
        double r55707 = r55705 * r55706;
        double r55708 = log(r55703);
        double r55709 = r55703 * r55708;
        double r55710 = r55707 - r55709;
        double r55711 = r55710 - r55704;
        return r55711;
}
↓
double f(double n) {
        double r55712 = 1.0;
        double r55713 = 2.0;
        double r55714 = r55712 / r55713;
        double r55715 = n;
        double r55716 = r55714 / r55715;
        double r55717 = 3002399751580331.0;
        double r55718 = 18014398509481984.0;
        double r55719 = r55717 / r55718;
        double r55720 = 2.0;
        double r55721 = pow(r55715, r55720);
        double r55722 = r55719 / r55721;
        double r55723 = r55716 - r55722;
        double r55724 = log(r55715);
        double r55725 = r55724 * r55712;
        double r55726 = r55723 + r55725;
        return r55726;
}

Error

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original63.0
Target0.0
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

    \[\leadsto \color{blue}{\left(\left(0.5 \cdot \frac{1}{n} + 1\right) - \left(1 \cdot \log \left(\frac{1}{n}\right) + 0.1666666666666666851703837437526090070605 \cdot \frac{1}{{n}^{2}}\right)\right)} - 1\]

  3. Simplified0.0

    \[\leadsto \color{blue}{\left(1 + \left(\left(\frac{\frac{1}{2}}{n} + \frac{-\frac{3002399751580331}{18014398509481984}}{{n}^{2}}\right) + \log n \cdot 1\right)\right)} - 1\]
  4. Using strategy rm

  5. Applied associate--l+0

    \[\leadsto \color{blue}{1 + \left(\left(\left(\frac{\frac{1}{2}}{n} + \frac{-\frac{3002399751580331}{18014398509481984}}{{n}^{2}}\right) + \log n \cdot 1\right) - 1\right)}\]

  6. Simplified0

    \[\leadsto 1 + \color{blue}{\left(\left(\log n \cdot 1 - 1\right) + \left(\frac{\frac{1}{2}}{n} - \frac{\frac{3002399751580331}{18014398509481984}}{{n}^{2}}\right)\right)}\]

  7. Final simplification0

    \[\leadsto \left(\frac{\frac{1}{2}}{n} - \frac{\frac{3002399751580331}{18014398509481984}}{{n}^{2}}\right) + \log n \cdot 1\]

Reproduce

herbie shell --seed 2019304 
(FPCore (n)
  :name "logs (example 3.8)"
  :precision binary64
  :pre (> n 6.8e15)

  :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))