Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B

Average Error: 5.9 → 3.1

Time: 14.3s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot z \le 1.090511535234420829903305418329171804458 \cdot 10^{305}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r543424 = x;
        double r543425 = r543424 * r543424;
        double r543426 = y;
        double r543427 = 4.0;
        double r543428 = r543426 * r543427;
        double r543429 = z;
        double r543430 = r543429 * r543429;
        double r543431 = t;
        double r543432 = r543430 - r543431;
        double r543433 = r543428 * r543432;
        double r543434 = r543425 - r543433;
        return r543434;
}

↓

double f(double x, double y, double z, double t) {
        double r543435 = z;
        double r543436 = r543435 * r543435;
        double r543437 = 1.0905115352344208e+305;
        bool r543438 = r543436 <= r543437;
        double r543439 = x;
        double r543440 = r543439 * r543439;
        double r543441 = y;
        double r543442 = 4.0;
        double r543443 = r543441 * r543442;
        double r543444 = t;
        double r543445 = r543436 - r543444;
        double r543446 = r543443 * r543445;
        double r543447 = r543440 - r543446;
        double r543448 = sqrt(r543444);
        double r543449 = r543435 + r543448;
        double r543450 = r543443 * r543449;
        double r543451 = r543435 - r543448;
        double r543452 = r543450 * r543451;
        double r543453 = r543440 - r543452;
        double r543454 = r543438 ? r543447 : r543453;
        return r543454;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	5.9
Target	5.9
Herbie	3.1

\[x \cdot x - 4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right)\]

Derivation

1. Split input into 2 regimes

2. if (* z z) < 1.0905115352344208e+305

   1. Initial program 0.1

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]

   if 1.0905115352344208e+305 < (* z z)

   1. Initial program 62.8

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 63.4

      \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - \color{blue}{\sqrt{t} \cdot \sqrt{t}}\right)\]

   4. Applied difference-of-squares 63.4

      \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot \color{blue}{\left(\left(z + \sqrt{t}\right) \cdot \left(z - \sqrt{t}\right)\right)}\]

   5. Applied associate-*r* 32.3

      \[\leadsto x \cdot x - \color{blue}{\left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 3.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \le 1.090511535234420829903305418329171804458 \cdot 10^{305}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

  :herbie-target
  (- (* x x) (* 4 (* y (- (* z z) t))))

  (- (* x x) (* (* y 4) (- (* z z) t))))