Average Error: 63.0 → 0.0
Time: 5.6s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[1 - \left(\left(1 + \left(\frac{0.1666666666666666851703837437526090070605}{{n}^{2}} + 1 \cdot \log \left(\frac{1}{n}\right)\right)\right) - \frac{0.5}{n}\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
1 - \left(\left(1 + \left(\frac{0.1666666666666666851703837437526090070605}{{n}^{2}} + 1 \cdot \log \left(\frac{1}{n}\right)\right)\right) - \frac{0.5}{n}\right)
double f(double n) {
        double r69487 = n;
        double r69488 = 1.0;
        double r69489 = r69487 + r69488;
        double r69490 = log(r69489);
        double r69491 = r69489 * r69490;
        double r69492 = log(r69487);
        double r69493 = r69487 * r69492;
        double r69494 = r69491 - r69493;
        double r69495 = r69494 - r69488;
        return r69495;
}
↓
double f(double n) {
        double r69496 = 1.0;
        double r69497 = 0.16666666666666669;
        double r69498 = n;
        double r69499 = 2.0;
        double r69500 = pow(r69498, r69499);
        double r69501 = r69497 / r69500;
        double r69502 = 1.0;
        double r69503 = r69502 / r69498;
        double r69504 = log(r69503);
        double r69505 = r69496 * r69504;
        double r69506 = r69501 + r69505;
        double r69507 = r69496 + r69506;
        double r69508 = 0.5;
        double r69509 = r69508 / r69498;
        double r69510 = r69507 - r69509;
        double r69511 = r69496 - r69510;
        return r69511;
}

Error

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original63.0
Target0.0
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

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

  5. Applied associate-+l-0.0

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

  6. Applied associate--l-0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019353 
(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))