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

Average Error: 5.7 → 2.9

Time: 1.1m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot z \le 4.858358169893005910299857370632243557955 \cdot 10^{306}:\\ \;\;\;\;x \cdot x - \left(z \cdot z - t\right) \cdot \left(4 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(z - \sqrt{t}\right) \cdot \left(\left(\sqrt{t} + z\right) \cdot \left(4 \cdot y\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r31081279 = x;
        double r31081280 = r31081279 * r31081279;
        double r31081281 = y;
        double r31081282 = 4.0;
        double r31081283 = r31081281 * r31081282;
        double r31081284 = z;
        double r31081285 = r31081284 * r31081284;
        double r31081286 = t;
        double r31081287 = r31081285 - r31081286;
        double r31081288 = r31081283 * r31081287;
        double r31081289 = r31081280 - r31081288;
        return r31081289;
}

↓

double f(double x, double y, double z, double t) {
        double r31081290 = z;
        double r31081291 = r31081290 * r31081290;
        double r31081292 = 4.858358169893006e+306;
        bool r31081293 = r31081291 <= r31081292;
        double r31081294 = x;
        double r31081295 = r31081294 * r31081294;
        double r31081296 = t;
        double r31081297 = r31081291 - r31081296;
        double r31081298 = 4.0;
        double r31081299 = y;
        double r31081300 = r31081298 * r31081299;
        double r31081301 = r31081297 * r31081300;
        double r31081302 = r31081295 - r31081301;
        double r31081303 = sqrt(r31081296);
        double r31081304 = r31081290 - r31081303;
        double r31081305 = r31081303 + r31081290;
        double r31081306 = r31081305 * r31081300;
        double r31081307 = r31081304 * r31081306;
        double r31081308 = r31081295 - r31081307;
        double r31081309 = r31081293 ? r31081302 : r31081308;
        return r31081309;
}

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	2.9

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

Derivation

1. Split input into 2 regimes

2. if (* z z) < 4.858358169893006e+306

   1. Initial program 0.1

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

   if 4.858358169893006e+306 < (* z z)

   1. Initial program 63.1

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

   3. Applied add-sqr-sqrt 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* 31.3

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

4. Final simplification 2.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \le 4.858358169893005910299857370632243557955 \cdot 10^{306}:\\ \;\;\;\;x \cdot x - \left(z \cdot z - t\right) \cdot \left(4 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(z - \sqrt{t}\right) \cdot \left(\left(\sqrt{t} + z\right) \cdot \left(4 \cdot y\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(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))))