Average Error: 63.0 → 0
Time: 43.2s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\log n \cdot 1 - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) + \frac{0.5}{n}\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\log n \cdot 1 - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) + \frac{0.5}{n}
double f(double n) {
        double r5143490 = n;
        double r5143491 = 1.0;
        double r5143492 = r5143490 + r5143491;
        double r5143493 = log(r5143492);
        double r5143494 = r5143492 * r5143493;
        double r5143495 = log(r5143490);
        double r5143496 = r5143490 * r5143495;
        double r5143497 = r5143494 - r5143496;
        double r5143498 = r5143497 - r5143491;
        return r5143498;
}
↓
double f(double n) {
        double r5143499 = n;
        double r5143500 = log(r5143499);
        double r5143501 = 1.0;
        double r5143502 = r5143500 * r5143501;
        double r5143503 = 0.16666666666666669;
        double r5143504 = r5143499 * r5143499;
        double r5143505 = r5143503 / r5143504;
        double r5143506 = r5143502 - r5143505;
        double r5143507 = 0.5;
        double r5143508 = r5143507 / r5143499;
        double r5143509 = r5143506 + r5143508;
        return r5143509;
}

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. Simplified62.0

    \[\leadsto \color{blue}{\log \left(n + 1\right) \cdot \left(n + 1\right) - \mathsf{fma}\left(n, \log n, 1\right)}\]

  3. Taylor expanded around inf 0.0

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

  4. Simplified0

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

  5. Final simplification0

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(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))