Data.Approximate.Numerics:blog from approximate-0.2.2.1

Average Error: 0.2 → 0.1

Time: 3.8s

Precision: 64

Math

\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]

↓

\[\frac{6}{\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{x - 1}}\]

C

double f(double x) {
        double r851511 = 6.0;
        double r851512 = x;
        double r851513 = 1.0;
        double r851514 = r851512 - r851513;
        double r851515 = r851511 * r851514;
        double r851516 = r851512 + r851513;
        double r851517 = 4.0;
        double r851518 = sqrt(r851512);
        double r851519 = r851517 * r851518;
        double r851520 = r851516 + r851519;
        double r851521 = r851515 / r851520;
        return r851521;
}

↓

double f(double x) {
        double r851522 = 6.0;
        double r851523 = x;
        double r851524 = 1.0;
        double r851525 = r851523 + r851524;
        double r851526 = 4.0;
        double r851527 = sqrt(r851523);
        double r851528 = r851526 * r851527;
        double r851529 = r851525 + r851528;
        double r851530 = r851523 - r851524;
        double r851531 = r851529 / r851530;
        double r851532 = r851522 / r851531;
        return r851532;
}

Error

Bits error versus x

Target

Original	0.2
Target	0.1
Herbie	0.1

\[\frac{6}{\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{x - 1}}\]

Derivation

1. Initial program 0.2

   \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
2. Using strategy rm

3. Applied associate-/l* 0.1

   \[\leadsto \color{blue}{\frac{6}{\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{x - 1}}}\]

4. Final simplification 0.1

   \[\leadsto \frac{6}{\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{x - 1}}\]

Reproduce

herbie shell --seed 2020060 
(FPCore (x)
  :name "Data.Approximate.Numerics:blog from approximate-0.2.2.1"
  :precision binary64

  :herbie-target
  (/ 6 (/ (+ (+ x 1) (* 4 (sqrt x))) (- x 1)))

  (/ (* 6 (- x 1)) (+ (+ x 1) (* 4 (sqrt x)))))