Average Error: 63.0 → 0.0
Time: 14.5s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(1 \cdot \log n + \left(\left(1 + \frac{0.5}{n}\right) - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right)\right) - 1\]
\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 r2803493 = n;
        double r2803494 = 1.0;
        double r2803495 = r2803493 + r2803494;
        double r2803496 = log(r2803495);
        double r2803497 = r2803495 * r2803496;
        double r2803498 = log(r2803493);
        double r2803499 = r2803493 * r2803498;
        double r2803500 = r2803497 - r2803499;
        double r2803501 = r2803500 - r2803494;
        return r2803501;
}
↓
double f(double n) {
        double r2803502 = 1.0;
        double r2803503 = n;
        double r2803504 = log(r2803503);
        double r2803505 = r2803502 * r2803504;
        double r2803506 = 0.5;
        double r2803507 = r2803506 / r2803503;
        double r2803508 = r2803502 + r2803507;
        double r2803509 = 0.16666666666666669;
        double r2803510 = r2803503 * r2803503;
        double r2803511 = r2803509 / r2803510;
        double r2803512 = r2803508 - r2803511;
        double r2803513 = r2803505 + r2803512;
        double r2803514 = r2803513 - r2803502;
        return r2803514;
}

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 2019171 
(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))