Data.Approximate.Numerics:blog from approximate-0.2.2.1

Average Error: 0.2 → 0.0

Time: 15.3s

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 r2397582 = 6.0;
        double r2397583 = x;
        double r2397584 = 1.0;
        double r2397585 = r2397583 - r2397584;
        double r2397586 = r2397582 * r2397585;
        double r2397587 = r2397583 + r2397584;
        double r2397588 = 4.0;
        double r2397589 = sqrt(r2397583);
        double r2397590 = r2397588 * r2397589;
        double r2397591 = r2397587 + r2397590;
        double r2397592 = r2397586 / r2397591;
        return r2397592;
}

↓

double f(double x) {
        double r2397593 = x;
        double r2397594 = 1.0;
        double r2397595 = r2397593 - r2397594;
        double r2397596 = sqrt(r2397593);
        double r2397597 = 4.0;
        double r2397598 = r2397593 + r2397594;
        double r2397599 = fma(r2397596, r2397597, r2397598);
        double r2397600 = 6.0;
        double r2397601 = r2397599 / r2397600;
        double r2397602 = r2397595 / r2397601;
        return r2397602;
}

Error

Bits error versus x

Target

Original	0.2
Target	0.1
Herbie	0.0

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

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

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

3. Final simplification 0.0

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

Reproduce

herbie shell --seed 2020018 +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)))))