Average Error: 63.0 → 0
Time: 31.6s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\frac{0.5}{n} - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) + 1 \cdot \log n\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\frac{0.5}{n} - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) + 1 \cdot \log n
double f(double n) {
        double r4151540 = n;
        double r4151541 = 1.0;
        double r4151542 = r4151540 + r4151541;
        double r4151543 = log(r4151542);
        double r4151544 = r4151542 * r4151543;
        double r4151545 = log(r4151540);
        double r4151546 = r4151540 * r4151545;
        double r4151547 = r4151544 - r4151546;
        double r4151548 = r4151547 - r4151541;
        return r4151548;
}
↓
double f(double n) {
        double r4151549 = 0.5;
        double r4151550 = n;
        double r4151551 = r4151549 / r4151550;
        double r4151552 = 0.16666666666666669;
        double r4151553 = r4151550 * r4151550;
        double r4151554 = r4151552 / r4151553;
        double r4151555 = r4151551 - r4151554;
        double r4151556 = 1.0;
        double r4151557 = log(r4151550);
        double r4151558 = r4151556 * r4151557;
        double r4151559 = r4151555 + r4151558;
        return r4151559;
}

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

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

  3. Simplified0.0

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

  4. Taylor expanded around 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

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

  6. Final simplification0

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

Reproduce

herbie shell --seed 2019168 
(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))