Average Error: 63.0 → 0.0
Time: 20.9s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(\left(\frac{0.5}{n} + 1\right) - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) + \log n \cdot 1\right) - 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\left(\left(\frac{0.5}{n} + 1\right) - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) + \log n \cdot 1\right) - 1
double f(double n) {
        double r45553 = n;
        double r45554 = 1.0;
        double r45555 = r45553 + r45554;
        double r45556 = log(r45555);
        double r45557 = r45555 * r45556;
        double r45558 = log(r45553);
        double r45559 = r45553 * r45558;
        double r45560 = r45557 - r45559;
        double r45561 = r45560 - r45554;
        return r45561;
}
↓
double f(double n) {
        double r45562 = 0.5;
        double r45563 = n;
        double r45564 = r45562 / r45563;
        double r45565 = 1.0;
        double r45566 = r45564 + r45565;
        double r45567 = 0.16666666666666669;
        double r45568 = r45563 * r45563;
        double r45569 = r45567 / r45568;
        double r45570 = r45566 - r45569;
        double r45571 = log(r45563);
        double r45572 = r45571 * r45565;
        double r45573 = r45570 + r45572;
        double r45574 = r45573 - r45565;
        return r45574;
}

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

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

  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

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

Reproduce

herbie shell --seed 2019325 
(FPCore (n)
  :name "logs (example 3.8)"
  :precision binary64
  :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))