Average Error: 63.0 → 0
Time: 19.5s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\frac{0.5}{n} - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) + 1 \cdot \log n\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\frac{0.5}{n} - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) + 1 \cdot \log n
double f(double n) {
        double r5723671 = n;
        double r5723672 = 1.0;
        double r5723673 = r5723671 + r5723672;
        double r5723674 = log(r5723673);
        double r5723675 = r5723673 * r5723674;
        double r5723676 = log(r5723671);
        double r5723677 = r5723671 * r5723676;
        double r5723678 = r5723675 - r5723677;
        double r5723679 = r5723678 - r5723672;
        return r5723679;
}
↓
double f(double n) {
        double r5723680 = 0.5;
        double r5723681 = n;
        double r5723682 = r5723680 / r5723681;
        double r5723683 = 0.16666666666666669;
        double r5723684 = r5723681 * r5723681;
        double r5723685 = r5723683 / r5723684;
        double r5723686 = r5723682 - r5723685;
        double r5723687 = 1.0;
        double r5723688 = log(r5723681);
        double r5723689 = r5723687 * r5723688;
        double r5723690 = r5723686 + r5723689;
        return r5723690;
}

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. Simplified62.0

    \[\leadsto \color{blue}{\left(n + 1\right) \cdot \log \left(n + 1\right) - \mathsf{fma}\left(n, \log n, 1\right)}\]

  3. Taylor expanded around inf 0.0

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

  4. Simplified0

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

  5. Final simplification0

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

Reproduce

herbie shell --seed 2019173 +o rules:numerics
(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))