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

Average Error: 5.8 → 3.3

Time: 11.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot z \le 2.131141250228609434937483959890669684472 \cdot 10^{293}:\\ \;\;\;\;x \cdot x - y \cdot \left(4 \cdot \left(z \cdot z - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r536730 = x;
        double r536731 = r536730 * r536730;
        double r536732 = y;
        double r536733 = 4.0;
        double r536734 = r536732 * r536733;
        double r536735 = z;
        double r536736 = r536735 * r536735;
        double r536737 = t;
        double r536738 = r536736 - r536737;
        double r536739 = r536734 * r536738;
        double r536740 = r536731 - r536739;
        return r536740;
}

↓

double f(double x, double y, double z, double t) {
        double r536741 = z;
        double r536742 = r536741 * r536741;
        double r536743 = 2.1311412502286094e+293;
        bool r536744 = r536742 <= r536743;
        double r536745 = x;
        double r536746 = r536745 * r536745;
        double r536747 = y;
        double r536748 = 4.0;
        double r536749 = t;
        double r536750 = r536742 - r536749;
        double r536751 = r536748 * r536750;
        double r536752 = r536747 * r536751;
        double r536753 = r536746 - r536752;
        double r536754 = r536747 * r536748;
        double r536755 = sqrt(r536749);
        double r536756 = r536741 + r536755;
        double r536757 = r536754 * r536756;
        double r536758 = r536741 - r536755;
        double r536759 = r536757 * r536758;
        double r536760 = r536746 - r536759;
        double r536761 = r536744 ? r536753 : r536760;
        return r536761;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	5.8
Target	5.8
Herbie	3.3

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

Derivation

1. Split input into 2 regimes

2. if (* z z) < 2.1311412502286094e+293

   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 associate-*l* 0.1

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

   if 2.1311412502286094e+293 < (* z z)

   1. Initial program 58.1

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

   3. Applied add-sqr-sqrt 61.2

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

   4. Applied difference-of-squares 61.2

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

   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 3.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \le 2.131141250228609434937483959890669684472 \cdot 10^{293}:\\ \;\;\;\;x \cdot x - y \cdot \left(4 \cdot \left(z \cdot z - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019294 
(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))))