Average Error: 63.0 → 0
Time: 17.6s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\frac{0.5}{n} - \mathsf{fma}\left(1, -\log n, \frac{\frac{0.1666666666666666851703837437526090070605}{n}}{n}\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\frac{0.5}{n} - \mathsf{fma}\left(1, -\log n, \frac{\frac{0.1666666666666666851703837437526090070605}{n}}{n}\right)
double f(double n) {
        double r3133411 = n;
        double r3133412 = 1.0;
        double r3133413 = r3133411 + r3133412;
        double r3133414 = log(r3133413);
        double r3133415 = r3133413 * r3133414;
        double r3133416 = log(r3133411);
        double r3133417 = r3133411 * r3133416;
        double r3133418 = r3133415 - r3133417;
        double r3133419 = r3133418 - r3133412;
        return r3133419;
}
↓
double f(double n) {
        double r3133420 = 0.5;
        double r3133421 = n;
        double r3133422 = r3133420 / r3133421;
        double r3133423 = 1.0;
        double r3133424 = log(r3133421);
        double r3133425 = -r3133424;
        double r3133426 = 0.16666666666666669;
        double r3133427 = r3133426 / r3133421;
        double r3133428 = r3133427 / r3133421;
        double r3133429 = fma(r3133423, r3133425, r3133428);
        double r3133430 = r3133422 - r3133429;
        return r3133430;
}

Error

Bits error versus n

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

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

  3. Taylor expanded around inf 0.0

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

  4. Simplified0

    \[\leadsto \color{blue}{\frac{0.5}{n} - \mathsf{fma}\left(1, -\log n, \frac{\frac{0.1666666666666666851703837437526090070605}{n}}{n}\right)}\]

  5. Final simplification0

    \[\leadsto \frac{0.5}{n} - \mathsf{fma}\left(1, -\log n, \frac{\frac{0.1666666666666666851703837437526090070605}{n}}{n}\right)\]

Reproduce

herbie shell --seed 2019172 +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))