Data.Approximate.Numerics:blog from approximate-0.2.2.1

Average Error: 0.2 → 0.1

Time: 5.1s

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 r856071 = 6.0;
        double r856072 = x;
        double r856073 = 1.0;
        double r856074 = r856072 - r856073;
        double r856075 = r856071 * r856074;
        double r856076 = r856072 + r856073;
        double r856077 = 4.0;
        double r856078 = sqrt(r856072);
        double r856079 = r856077 * r856078;
        double r856080 = r856076 + r856079;
        double r856081 = r856075 / r856080;
        return r856081;
}

↓

double f(double x) {
        double r856082 = 6.0;
        double r856083 = x;
        double r856084 = 1.0;
        double r856085 = r856083 + r856084;
        double r856086 = 4.0;
        double r856087 = sqrt(r856083);
        double r856088 = r856086 * r856087;
        double r856089 = r856085 + r856088;
        double r856090 = r856083 - r856084;
        double r856091 = r856089 / r856090;
        double r856092 = r856082 / r856091;
        return r856092;
}

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 2020049 
(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)))))