Data.Approximate.Numerics:blog from approximate-0.2.2.1

Average Error: 0.2 → 0.1

Time: 18.7s

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 r1367034 = 6.0;
        double r1367035 = x;
        double r1367036 = 1.0;
        double r1367037 = r1367035 - r1367036;
        double r1367038 = r1367034 * r1367037;
        double r1367039 = r1367035 + r1367036;
        double r1367040 = 4.0;
        double r1367041 = sqrt(r1367035);
        double r1367042 = r1367040 * r1367041;
        double r1367043 = r1367039 + r1367042;
        double r1367044 = r1367038 / r1367043;
        return r1367044;
}

↓

double f(double x) {
        double r1367045 = 6.0;
        double r1367046 = x;
        double r1367047 = 1.0;
        double r1367048 = r1367046 + r1367047;
        double r1367049 = 4.0;
        double r1367050 = sqrt(r1367046);
        double r1367051 = r1367049 * r1367050;
        double r1367052 = r1367048 + r1367051;
        double r1367053 = r1367046 - r1367047;
        double r1367054 = r1367052 / r1367053;
        double r1367055 = r1367045 / r1367054;
        return r1367055;
}

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 2019199 
(FPCore (x)
  :name "Data.Approximate.Numerics:blog from approximate-0.2.2.1"

  :herbie-target
  (/ 6.0 (/ (+ (+ x 1.0) (* 4.0 (sqrt x))) (- x 1.0)))

  (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))