Data.Approximate.Numerics:blog from approximate-0.2.2.1

Average Error: 0.2 → 0.1

Time: 4.5s

Precision: 64

Math

\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]

↓

\[\frac{x - 1}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}}\]

C

double f(double x) {
        double r1192227 = 6.0;
        double r1192228 = x;
        double r1192229 = 1.0;
        double r1192230 = r1192228 - r1192229;
        double r1192231 = r1192227 * r1192230;
        double r1192232 = r1192228 + r1192229;
        double r1192233 = 4.0;
        double r1192234 = sqrt(r1192228);
        double r1192235 = r1192233 * r1192234;
        double r1192236 = r1192232 + r1192235;
        double r1192237 = r1192231 / r1192236;
        return r1192237;
}

↓

double f(double x) {
        double r1192238 = x;
        double r1192239 = 1.0;
        double r1192240 = r1192238 - r1192239;
        double r1192241 = sqrt(r1192238);
        double r1192242 = 4.0;
        double r1192243 = r1192238 + r1192239;
        double r1192244 = fma(r1192241, r1192242, r1192243);
        double r1192245 = 6.0;
        double r1192246 = r1192244 / r1192245;
        double r1192247 = r1192240 / r1192246;
        return r1192247;
}

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. Simplified 0.1

   \[\leadsto \color{blue}{\frac{x - 1}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}}}\]

3. Final simplification 0.1

   \[\leadsto \frac{x - 1}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}}\]

Reproduce

herbie shell --seed 2020034 +o rules:numerics
(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)))))