Data.Approximate.Numerics:blog from approximate-0.2.2.1

Average Error: 0.2 → 0.1

Time: 8.7s

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 r534351 = 6.0;
        double r534352 = x;
        double r534353 = 1.0;
        double r534354 = r534352 - r534353;
        double r534355 = r534351 * r534354;
        double r534356 = r534352 + r534353;
        double r534357 = 4.0;
        double r534358 = sqrt(r534352);
        double r534359 = r534357 * r534358;
        double r534360 = r534356 + r534359;
        double r534361 = r534355 / r534360;
        return r534361;
}

↓

double f(double x) {
        double r534362 = 6.0;
        double r534363 = x;
        double r534364 = 1.0;
        double r534365 = r534363 + r534364;
        double r534366 = 4.0;
        double r534367 = sqrt(r534363);
        double r534368 = r534366 * r534367;
        double r534369 = r534365 + r534368;
        double r534370 = r534363 - r534364;
        double r534371 = r534369 / r534370;
        double r534372 = r534362 / r534371;
        return r534372;
}

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 2019322 
(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)))))