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

Average Error: 5.7 → 3.0

Time: 21.5s

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 r29721920 = x;
        double r29721921 = r29721920 * r29721920;
        double r29721922 = y;
        double r29721923 = 4.0;
        double r29721924 = r29721922 * r29721923;
        double r29721925 = z;
        double r29721926 = r29721925 * r29721925;
        double r29721927 = t;
        double r29721928 = r29721926 - r29721927;
        double r29721929 = r29721924 * r29721928;
        double r29721930 = r29721921 - r29721929;
        return r29721930;
}

↓

double f(double x, double y, double z, double t) {
        double r29721931 = z;
        double r29721932 = r29721931 * r29721931;
        double r29721933 = 4.3166514753609153e+304;
        bool r29721934 = r29721932 <= r29721933;
        double r29721935 = x;
        double r29721936 = r29721935 * r29721935;
        double r29721937 = 4.0;
        double r29721938 = y;
        double r29721939 = r29721937 * r29721938;
        double r29721940 = t;
        double r29721941 = r29721932 - r29721940;
        double r29721942 = r29721939 * r29721941;
        double r29721943 = r29721936 - r29721942;
        double r29721944 = sqrt(r29721940);
        double r29721945 = r29721944 + r29721931;
        double r29721946 = r29721945 * r29721939;
        double r29721947 = r29721931 - r29721944;
        double r29721948 = r29721946 * r29721947;
        double r29721949 = r29721936 - r29721948;
        double r29721950 = r29721934 ? r29721943 : r29721949;
        return r29721950;
}

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 
(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))))