Data.Approximate.Numerics:blog from approximate-0.2.2.1

Average Error: 0.2 → 0.1

Time: 6.1s

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 r868911 = 6.0;
        double r868912 = x;
        double r868913 = 1.0;
        double r868914 = r868912 - r868913;
        double r868915 = r868911 * r868914;
        double r868916 = r868912 + r868913;
        double r868917 = 4.0;
        double r868918 = sqrt(r868912);
        double r868919 = r868917 * r868918;
        double r868920 = r868916 + r868919;
        double r868921 = r868915 / r868920;
        return r868921;
}

↓

double f(double x) {
        double r868922 = 6.0;
        double r868923 = x;
        double r868924 = 1.0;
        double r868925 = r868923 + r868924;
        double r868926 = 4.0;
        double r868927 = sqrt(r868923);
        double r868928 = r868926 * r868927;
        double r868929 = r868925 + r868928;
        double r868930 = r868923 - r868924;
        double r868931 = r868929 / r868930;
        double r868932 = r868922 / r868931;
        return r868932;
}

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 2020056 
(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)))))