Data.Approximate.Numerics:blog from approximate-0.2.2.1

Average Error: 0.2 → 0.1

Time: 14.8s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r486590 = 6.0;
        double r486591 = x;
        double r486592 = 1.0;
        double r486593 = r486591 - r486592;
        double r486594 = r486590 * r486593;
        double r486595 = r486591 + r486592;
        double r486596 = 4.0;
        double r486597 = sqrt(r486591);
        double r486598 = r486596 * r486597;
        double r486599 = r486595 + r486598;
        double r486600 = r486594 / r486599;
        return r486600;
}

↓

double f(double x) {
        double r486601 = 1.0;
        double r486602 = x;
        double r486603 = sqrt(r486602);
        double r486604 = 4.0;
        double r486605 = 1.0;
        double r486606 = r486602 + r486605;
        double r486607 = fma(r486603, r486604, r486606);
        double r486608 = r486602 - r486605;
        double r486609 = r486607 / r486608;
        double r486610 = 6.0;
        double r486611 = r486609 / r486610;
        double r486612 = r486601 / r486611;
        return r486612;
}

Error

Bits error versus x

Target

Original	0.2
Target	0.0
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 0.0

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

4. Applied clear-num 0.1

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

5. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019212 +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)))))