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 r775166 = 6.0;
        double r775167 = x;
        double r775168 = 1.0;
        double r775169 = r775167 - r775168;
        double r775170 = r775166 * r775169;
        double r775171 = r775167 + r775168;
        double r775172 = 4.0;
        double r775173 = sqrt(r775167);
        double r775174 = r775172 * r775173;
        double r775175 = r775171 + r775174;
        double r775176 = r775170 / r775175;
        return r775176;
}

↓

double f(double x) {
        double r775177 = 6.0;
        double r775178 = x;
        double r775179 = 1.0;
        double r775180 = r775178 + r775179;
        double r775181 = 4.0;
        double r775182 = sqrt(r775178);
        double r775183 = r775181 * r775182;
        double r775184 = r775180 + r775183;
        double r775185 = r775178 - r775179;
        double r775186 = r775184 / r775185;
        double r775187 = r775177 / r775186;
        return r775187;
}

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)))))