Average Error: 63.0 → 0
Time: 18.5s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\frac{0.5}{n} + \left(1 \cdot \log n - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\frac{0.5}{n} + \left(1 \cdot \log n - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right)
double f(double n) {
        double r2674740 = n;
        double r2674741 = 1.0;
        double r2674742 = r2674740 + r2674741;
        double r2674743 = log(r2674742);
        double r2674744 = r2674742 * r2674743;
        double r2674745 = log(r2674740);
        double r2674746 = r2674740 * r2674745;
        double r2674747 = r2674744 - r2674746;
        double r2674748 = r2674747 - r2674741;
        return r2674748;
}
↓
double f(double n) {
        double r2674749 = 0.5;
        double r2674750 = n;
        double r2674751 = r2674749 / r2674750;
        double r2674752 = 1.0;
        double r2674753 = log(r2674750);
        double r2674754 = r2674752 * r2674753;
        double r2674755 = 0.16666666666666669;
        double r2674756 = r2674750 * r2674750;
        double r2674757 = r2674755 / r2674756;
        double r2674758 = r2674754 - r2674757;
        double r2674759 = r2674751 + r2674758;
        return r2674759;
}

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

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

  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

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

  5. Simplified0

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

  6. Final simplification0

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

Reproduce

herbie shell --seed 2019169 
(FPCore (n)
  :name "logs (example 3.8)"
  :pre (> n 6.8e+15)

  :herbie-target
  (- (log (+ n 1.0)) (- (/ 1.0 (* 2.0 n)) (- (/ 1.0 (* 3.0 (* n n))) (/ 4.0 (pow n 3.0)))))

  (- (- (* (+ n 1.0) (log (+ n 1.0))) (* n (log n))) 1.0))