Average Error: 63.0 → 0.0
Time: 3.9s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(1 - \left(1 \cdot \left(0 - \log n\right) + 0.1666666666666666851703837437526090070605 \cdot \frac{1}{{n}^{2}}\right)\right) + \frac{0.5}{n}\right) - 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\left(1 - \left(1 \cdot \left(0 - \log n\right) + 0.1666666666666666851703837437526090070605 \cdot \frac{1}{{n}^{2}}\right)\right) + \frac{0.5}{n}\right) - 1
double f(double n) {
        double r64755 = n;
        double r64756 = 1.0;
        double r64757 = r64755 + r64756;
        double r64758 = log(r64757);
        double r64759 = r64757 * r64758;
        double r64760 = log(r64755);
        double r64761 = r64755 * r64760;
        double r64762 = r64759 - r64761;
        double r64763 = r64762 - r64756;
        return r64763;
}
↓
double f(double n) {
        double r64764 = 1.0;
        double r64765 = 0.0;
        double r64766 = n;
        double r64767 = log(r64766);
        double r64768 = r64765 - r64767;
        double r64769 = r64764 * r64768;
        double r64770 = 0.16666666666666669;
        double r64771 = 1.0;
        double r64772 = 2.0;
        double r64773 = pow(r64766, r64772);
        double r64774 = r64771 / r64773;
        double r64775 = r64770 * r64774;
        double r64776 = r64769 + r64775;
        double r64777 = r64764 - r64776;
        double r64778 = 0.5;
        double r64779 = r64778 / r64766;
        double r64780 = r64777 + r64779;
        double r64781 = r64780 - r64764;
        return r64781;
}

Error

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

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

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

  5. Applied log-div0.0

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

  6. Simplified0.0

    \[\leadsto \left(\left(1 - \left(1 \cdot \left(\color{blue}{0} - \log n\right) + 0.1666666666666666851703837437526090070605 \cdot \frac{1}{{n}^{2}}\right)\right) + \frac{0.5}{n}\right) - 1\]

  7. Final simplification0.0

    \[\leadsto \left(\left(1 - \left(1 \cdot \left(0 - \log n\right) + 0.1666666666666666851703837437526090070605 \cdot \frac{1}{{n}^{2}}\right)\right) + \frac{0.5}{n}\right) - 1\]

Reproduce

herbie shell --seed 2019352 
(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))