Data.Approximate.Numerics:blog from approximate-0.2.2.1

Average Error: 0.2 → 0.1

Time: 3.5s

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 r882584 = 6.0;
        double r882585 = x;
        double r882586 = 1.0;
        double r882587 = r882585 - r882586;
        double r882588 = r882584 * r882587;
        double r882589 = r882585 + r882586;
        double r882590 = 4.0;
        double r882591 = sqrt(r882585);
        double r882592 = r882590 * r882591;
        double r882593 = r882589 + r882592;
        double r882594 = r882588 / r882593;
        return r882594;
}

↓

double f(double x) {
        double r882595 = 6.0;
        double r882596 = x;
        double r882597 = 1.0;
        double r882598 = r882596 + r882597;
        double r882599 = 4.0;
        double r882600 = sqrt(r882596);
        double r882601 = r882599 * r882600;
        double r882602 = r882598 + r882601;
        double r882603 = r882596 - r882597;
        double r882604 = r882602 / r882603;
        double r882605 = r882595 / r882604;
        return r882605;
}

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 2019354 
(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)))))