Average Error: 63.0 → 0
Time: 28.6s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\frac{0.5}{n} - \left(\frac{0.1666666666666666851703837437526090070605}{n \cdot n} - 1 \cdot \log n\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\frac{0.5}{n} - \left(\frac{0.1666666666666666851703837437526090070605}{n \cdot n} - 1 \cdot \log n\right)
double f(double n) {
        double r63768 = n;
        double r63769 = 1.0;
        double r63770 = r63768 + r63769;
        double r63771 = log(r63770);
        double r63772 = r63770 * r63771;
        double r63773 = log(r63768);
        double r63774 = r63768 * r63773;
        double r63775 = r63772 - r63774;
        double r63776 = r63775 - r63769;
        return r63776;
}
↓
double f(double n) {
        double r63777 = 0.5;
        double r63778 = n;
        double r63779 = r63777 / r63778;
        double r63780 = 0.16666666666666669;
        double r63781 = r63778 * r63778;
        double r63782 = r63780 / r63781;
        double r63783 = 1.0;
        double r63784 = log(r63778);
        double r63785 = r63783 * r63784;
        double r63786 = r63782 - r63785;
        double r63787 = r63779 - r63786;
        return r63787;
}

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. Taylor expanded around 0 0

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

  5. Simplified0

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

  6. Final simplification0

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

Reproduce

herbie shell --seed 2019323 
(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))