Average Error: 63.0 → 0.0
Time: 13.7s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(1 \cdot \log n + \left(\left(1 + \frac{0.5}{n}\right) - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right)\right) - 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(1 \cdot \log n + \left(\left(1 + \frac{0.5}{n}\right) - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right)\right) - 1
double f(double n) {
        double r5092508 = n;
        double r5092509 = 1.0;
        double r5092510 = r5092508 + r5092509;
        double r5092511 = log(r5092510);
        double r5092512 = r5092510 * r5092511;
        double r5092513 = log(r5092508);
        double r5092514 = r5092508 * r5092513;
        double r5092515 = r5092512 - r5092514;
        double r5092516 = r5092515 - r5092509;
        return r5092516;
}
↓
double f(double n) {
        double r5092517 = 1.0;
        double r5092518 = n;
        double r5092519 = log(r5092518);
        double r5092520 = r5092517 * r5092519;
        double r5092521 = 0.5;
        double r5092522 = r5092521 / r5092518;
        double r5092523 = r5092517 + r5092522;
        double r5092524 = 0.16666666666666669;
        double r5092525 = r5092518 * r5092518;
        double r5092526 = r5092524 / r5092525;
        double r5092527 = r5092523 - r5092526;
        double r5092528 = r5092520 + r5092527;
        double r5092529 = r5092528 - r5092517;
        return r5092529;
}

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

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

  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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019172 
(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))