Average Error: 63.0 → 0
Time: 14.7s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\log n - \frac{\frac{1}{6}}{n \cdot n}\right) + \frac{\frac{1}{2}}{n}\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\log n - \frac{\frac{1}{6}}{n \cdot n}\right) + \frac{\frac{1}{2}}{n}
double f(double n) {
        double r1837559 = n;
        double r1837560 = 1.0;
        double r1837561 = r1837559 + r1837560;
        double r1837562 = log(r1837561);
        double r1837563 = r1837561 * r1837562;
        double r1837564 = log(r1837559);
        double r1837565 = r1837559 * r1837564;
        double r1837566 = r1837563 - r1837565;
        double r1837567 = r1837566 - r1837560;
        return r1837567;
}
↓
double f(double n) {
        double r1837568 = n;
        double r1837569 = log(r1837568);
        double r1837570 = 0.16666666666666666;
        double r1837571 = r1837568 * r1837568;
        double r1837572 = r1837570 / r1837571;
        double r1837573 = r1837569 - r1837572;
        double r1837574 = 0.5;
        double r1837575 = r1837574 / r1837568;
        double r1837576 = r1837573 + r1837575;
        return r1837576;
}

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

    \[\leadsto \color{blue}{\left(\left(1 + \frac{1}{2} \cdot \frac{1}{n}\right) - \left(\frac{1}{6} \cdot \frac{1}{{n}^{2}} + \log \left(\frac{1}{n}\right)\right)\right)} - 1\]

  3. Simplified0.0

    \[\leadsto \color{blue}{\left(\left(1 + \frac{\frac{1}{2}}{n}\right) - \left(\frac{\frac{1}{6}}{n \cdot n} - \log n\right)\right)} - 1\]

  4. Taylor expanded around 0 0

    \[\leadsto \color{blue}{\left(\log n + \frac{1}{2} \cdot \frac{1}{n}\right) - \frac{1}{6} \cdot \frac{1}{{n}^{2}}}\]

  5. Simplified0

    \[\leadsto \color{blue}{\frac{\frac{1}{2}}{n} + \left(\log n - \frac{\frac{1}{6}}{n \cdot n}\right)}\]

  6. Final simplification0

    \[\leadsto \left(\log n - \frac{\frac{1}{6}}{n \cdot n}\right) + \frac{\frac{1}{2}}{n}\]

Reproduce

herbie shell --seed 2019152 
(FPCore (n)
  :name "logs (example 3.8)"
  :pre (> n 6.8e+15)

  :herbie-target
  (- (log (+ n 1)) (- (/ 1 (* 2 n)) (- (/ 1 (* 3 (* n n))) (/ 4 (pow n 3)))))

  (- (- (* (+ n 1) (log (+ n 1))) (* n (log n))) 1))