Average Error: 63.0 → 0.0
Time: 15.5s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\log n \cdot 1 - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) + \frac{0.5}{n}\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\log n \cdot 1 - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) + \frac{0.5}{n}
double f(double n) {
        double r4983371 = n;
        double r4983372 = 1.0;
        double r4983373 = r4983371 + r4983372;
        double r4983374 = log(r4983373);
        double r4983375 = r4983373 * r4983374;
        double r4983376 = log(r4983371);
        double r4983377 = r4983371 * r4983376;
        double r4983378 = r4983375 - r4983377;
        double r4983379 = r4983378 - r4983372;
        return r4983379;
}
↓
double f(double n) {
        double r4983380 = n;
        double r4983381 = log(r4983380);
        double r4983382 = 1.0;
        double r4983383 = r4983381 * r4983382;
        double r4983384 = 0.16666666666666669;
        double r4983385 = r4983380 * r4983380;
        double r4983386 = r4983384 / r4983385;
        double r4983387 = r4983383 - r4983386;
        double r4983388 = 0.5;
        double r4983389 = r4983388 / r4983380;
        double r4983390 = r4983387 + r4983389;
        return r4983390;
}

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

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

  4. Taylor expanded around 0 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. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019192 
(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))