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

Average Error: 6.3 → 4.1

Time: 13.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot z \le 4.052108237063258146574851851160968341006 \cdot 10^{262}:\\ \;\;\;\;x \cdot x - \left(4 \cdot y\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(\left(4 \cdot y\right) \cdot \left(\sqrt{t} + z\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r543487 = x;
        double r543488 = r543487 * r543487;
        double r543489 = y;
        double r543490 = 4.0;
        double r543491 = r543489 * r543490;
        double r543492 = z;
        double r543493 = r543492 * r543492;
        double r543494 = t;
        double r543495 = r543493 - r543494;
        double r543496 = r543491 * r543495;
        double r543497 = r543488 - r543496;
        return r543497;
}

↓

double f(double x, double y, double z, double t) {
        double r543498 = z;
        double r543499 = r543498 * r543498;
        double r543500 = 4.052108237063258e+262;
        bool r543501 = r543499 <= r543500;
        double r543502 = x;
        double r543503 = r543502 * r543502;
        double r543504 = 4.0;
        double r543505 = y;
        double r543506 = r543504 * r543505;
        double r543507 = t;
        double r543508 = r543499 - r543507;
        double r543509 = r543506 * r543508;
        double r543510 = r543503 - r543509;
        double r543511 = sqrt(r543507);
        double r543512 = r543511 + r543498;
        double r543513 = r543506 * r543512;
        double r543514 = r543498 - r543511;
        double r543515 = r543513 * r543514;
        double r543516 = r543503 - r543515;
        double r543517 = r543501 ? r543510 : r543516;
        return r543517;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.3
Target	6.3
Herbie	4.1

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

Derivation

1. Split input into 2 regimes

2. if (* z z) < 4.052108237063258e+262

   1. Initial program 0.1

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

   3. Applied pow1 0.1

      \[\leadsto \color{blue}{{\left(x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\right)}^{1}}\]

   if 4.052108237063258e+262 < (* z z)

   1. Initial program 49.1

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

   3. Applied add-sqr-sqrt 56.5

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

   4. Applied difference-of-squares 56.5

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

   5. Applied associate-*r* 31.3

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

   6. Simplified 31.3

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

4. Final simplification 4.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \le 4.052108237063258146574851851160968341006 \cdot 10^{262}:\\ \;\;\;\;x \cdot x - \left(4 \cdot y\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(\left(4 \cdot y\right) \cdot \left(\sqrt{t} + z\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"

  :herbie-target
  (- (* x x) (* 4.0 (* y (- (* z z) t))))

  (- (* x x) (* (* y 4.0) (- (* z z) t))))