Average Error: 63.0 → 0
Time: 18.2s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\frac{0.5}{n} + 1\right) - \left(\left(\frac{0.1666666666666666851703837437526090070605}{n \cdot n} - \log n \cdot 1\right) + 1\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\frac{0.5}{n} + 1\right) - \left(\left(\frac{0.1666666666666666851703837437526090070605}{n \cdot n} - \log n \cdot 1\right) + 1\right)
double f(double n) {
        double r63755 = n;
        double r63756 = 1.0;
        double r63757 = r63755 + r63756;
        double r63758 = log(r63757);
        double r63759 = r63757 * r63758;
        double r63760 = log(r63755);
        double r63761 = r63755 * r63760;
        double r63762 = r63759 - r63761;
        double r63763 = r63762 - r63756;
        return r63763;
}
↓
double f(double n) {
        double r63764 = 0.5;
        double r63765 = n;
        double r63766 = r63764 / r63765;
        double r63767 = 1.0;
        double r63768 = r63766 + r63767;
        double r63769 = 0.16666666666666669;
        double r63770 = r63765 * r63765;
        double r63771 = r63769 / r63770;
        double r63772 = log(r63765);
        double r63773 = r63772 * r63767;
        double r63774 = r63771 - r63773;
        double r63775 = r63774 + r63767;
        double r63776 = r63768 - r63775;
        return r63776;
}

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

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

  3. Simplified0.0

    \[\leadsto \color{blue}{\left(\left(\left(\frac{0.5}{n} + 1\right) - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) + \log n \cdot 1\right)} - 1\]
  4. Using strategy rm

  5. Applied associate-+l-0.0

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

  6. Applied associate--l-0

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

  7. Final simplification0

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

Reproduce

herbie shell --seed 2019350 
(FPCore (n)
  :name "logs (example 3.8)"
  :precision binary64
  :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))