Data.Approximate.Numerics:blog from approximate-0.2.2.1

Average Error: 0.2 → 0.1

Time: 3.9s

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 r981258 = 6.0;
        double r981259 = x;
        double r981260 = 1.0;
        double r981261 = r981259 - r981260;
        double r981262 = r981258 * r981261;
        double r981263 = r981259 + r981260;
        double r981264 = 4.0;
        double r981265 = sqrt(r981259);
        double r981266 = r981264 * r981265;
        double r981267 = r981263 + r981266;
        double r981268 = r981262 / r981267;
        return r981268;
}

↓

double f(double x) {
        double r981269 = 6.0;
        double r981270 = x;
        double r981271 = 1.0;
        double r981272 = r981270 + r981271;
        double r981273 = 4.0;
        double r981274 = sqrt(r981270);
        double r981275 = r981273 * r981274;
        double r981276 = r981272 + r981275;
        double r981277 = r981270 - r981271;
        double r981278 = r981276 / r981277;
        double r981279 = r981269 / r981278;
        return r981279;
}

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)))))