Average Error: 63.0 → 0.0
Time: 10.8s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(\frac{\frac{1}{2}}{n} + 1\right) - \left(\frac{\frac{1}{6}}{n \cdot n} - \log n\right)\right) - 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\left(\frac{\frac{1}{2}}{n} + 1\right) - \left(\frac{\frac{1}{6}}{n \cdot n} - \log n\right)\right) - 1
double f(double n) {
        double r1099702 = n;
        double r1099703 = 1.0;
        double r1099704 = r1099702 + r1099703;
        double r1099705 = log(r1099704);
        double r1099706 = r1099704 * r1099705;
        double r1099707 = log(r1099702);
        double r1099708 = r1099702 * r1099707;
        double r1099709 = r1099706 - r1099708;
        double r1099710 = r1099709 - r1099703;
        return r1099710;
}
↓
double f(double n) {
        double r1099711 = 0.5;
        double r1099712 = n;
        double r1099713 = r1099711 / r1099712;
        double r1099714 = 1.0;
        double r1099715 = r1099713 + r1099714;
        double r1099716 = 0.16666666666666666;
        double r1099717 = r1099712 * r1099712;
        double r1099718 = r1099716 / r1099717;
        double r1099719 = log(r1099712);
        double r1099720 = r1099718 - r1099719;
        double r1099721 = r1099715 - r1099720;
        double r1099722 = r1099721 - r1099714;
        return r1099722;
}

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

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

  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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019153 
(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))