Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, B

Average Error: 0.1 → 0.1

Time: 16.7s

Precision: 64

Math

\[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)\]

↓

\[\frac{1}{2} \cdot \mathsf{fma}\left(\sqrt{z}, y, x\right)\]

C

double f(double x, double y, double z) {
        double r198340 = 1.0;
        double r198341 = 2.0;
        double r198342 = r198340 / r198341;
        double r198343 = x;
        double r198344 = y;
        double r198345 = z;
        double r198346 = sqrt(r198345);
        double r198347 = r198344 * r198346;
        double r198348 = r198343 + r198347;
        double r198349 = r198342 * r198348;
        return r198349;
}

↓

double f(double x, double y, double z) {
        double r198350 = 1.0;
        double r198351 = 2.0;
        double r198352 = r198350 / r198351;
        double r198353 = z;
        double r198354 = sqrt(r198353);
        double r198355 = y;
        double r198356 = x;
        double r198357 = fma(r198354, r198355, r198356);
        double r198358 = r198352 * r198357;
        return r198358;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

1. Initial program 0.1

   \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)\]

2. Simplified 0.1

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \mathsf{fma}\left(\sqrt{z}, y, x\right)}\]

3. Final simplification 0.1

   \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(\sqrt{z}, y, x\right)\]

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, B"
  :precision binary64
  (* (/ 1 2) (+ x (* y (sqrt z)))))