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

Average Error: 5.7 → 3.0

Time: 21.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot z \le 4.316651475360915328553043429326191138193 \cdot 10^{304}:\\ \;\;\;\;x \cdot x - \left(4 \cdot y\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(\left(\sqrt{t} + z\right) \cdot \left(4 \cdot y\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r30016175 = x;
        double r30016176 = r30016175 * r30016175;
        double r30016177 = y;
        double r30016178 = 4.0;
        double r30016179 = r30016177 * r30016178;
        double r30016180 = z;
        double r30016181 = r30016180 * r30016180;
        double r30016182 = t;
        double r30016183 = r30016181 - r30016182;
        double r30016184 = r30016179 * r30016183;
        double r30016185 = r30016176 - r30016184;
        return r30016185;
}

↓

double f(double x, double y, double z, double t) {
        double r30016186 = z;
        double r30016187 = r30016186 * r30016186;
        double r30016188 = 4.3166514753609153e+304;
        bool r30016189 = r30016187 <= r30016188;
        double r30016190 = x;
        double r30016191 = r30016190 * r30016190;
        double r30016192 = 4.0;
        double r30016193 = y;
        double r30016194 = r30016192 * r30016193;
        double r30016195 = t;
        double r30016196 = r30016187 - r30016195;
        double r30016197 = r30016194 * r30016196;
        double r30016198 = r30016191 - r30016197;
        double r30016199 = sqrt(r30016195);
        double r30016200 = r30016199 + r30016186;
        double r30016201 = r30016200 * r30016194;
        double r30016202 = r30016186 - r30016199;
        double r30016203 = r30016201 * r30016202;
        double r30016204 = r30016191 - r30016203;
        double r30016205 = r30016189 ? r30016198 : r30016204;
        return r30016205;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	5.7
Target	5.7
Herbie	3.0

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

Derivation

1. Split input into 2 regimes

2. if (* z z) < 4.3166514753609153e+304

   1. Initial program 0.1

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

   if 4.3166514753609153e+304 < (* z z)

   1. Initial program 62.6

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

   3. Applied add-sqr-sqrt 63.3

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

   4. Applied difference-of-squares 63.3

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

   5. Applied associate-*r* 32.5

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

4. Final simplification 3.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \le 4.316651475360915328553043429326191138193 \cdot 10^{304}:\\ \;\;\;\;x \cdot x - \left(4 \cdot y\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(\left(\sqrt{t} + z\right) \cdot \left(4 \cdot y\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(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))))