Average Error: 63.0 → 0
Time: 20.1s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(1 \cdot \log n - \frac{\frac{0.1666666666666666851703837437526090070605}{n}}{n}\right) + \frac{0.5}{n}\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(1 \cdot \log n - \frac{\frac{0.1666666666666666851703837437526090070605}{n}}{n}\right) + \frac{0.5}{n}
double f(double n) {
        double r3680488 = n;
        double r3680489 = 1.0;
        double r3680490 = r3680488 + r3680489;
        double r3680491 = log(r3680490);
        double r3680492 = r3680490 * r3680491;
        double r3680493 = log(r3680488);
        double r3680494 = r3680488 * r3680493;
        double r3680495 = r3680492 - r3680494;
        double r3680496 = r3680495 - r3680489;
        return r3680496;
}
↓
double f(double n) {
        double r3680497 = 1.0;
        double r3680498 = n;
        double r3680499 = log(r3680498);
        double r3680500 = r3680497 * r3680499;
        double r3680501 = 0.16666666666666669;
        double r3680502 = r3680501 / r3680498;
        double r3680503 = r3680502 / r3680498;
        double r3680504 = r3680500 - r3680503;
        double r3680505 = 0.5;
        double r3680506 = r3680505 / r3680498;
        double r3680507 = r3680504 + r3680506;
        return r3680507;
}

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

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

  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

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

  6. Final simplification0

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

Reproduce

herbie shell --seed 2019170 
(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))