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

Average Error: 6.1 → 3.2

Time: 18.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot z \le 2.007625814536843067127930964300867649199 \cdot 10^{304}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \mathsf{fma}\left(z, z, -t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r632331 = x;
        double r632332 = r632331 * r632331;
        double r632333 = y;
        double r632334 = 4.0;
        double r632335 = r632333 * r632334;
        double r632336 = z;
        double r632337 = r632336 * r632336;
        double r632338 = t;
        double r632339 = r632337 - r632338;
        double r632340 = r632335 * r632339;
        double r632341 = r632332 - r632340;
        return r632341;
}

↓

double f(double x, double y, double z, double t) {
        double r632342 = z;
        double r632343 = r632342 * r632342;
        double r632344 = 2.007625814536843e+304;
        bool r632345 = r632343 <= r632344;
        double r632346 = x;
        double r632347 = r632346 * r632346;
        double r632348 = y;
        double r632349 = 4.0;
        double r632350 = r632348 * r632349;
        double r632351 = t;
        double r632352 = -r632351;
        double r632353 = fma(r632342, r632342, r632352);
        double r632354 = r632350 * r632353;
        double r632355 = r632347 - r632354;
        double r632356 = sqrt(r632351);
        double r632357 = r632342 + r632356;
        double r632358 = r632350 * r632357;
        double r632359 = r632342 - r632356;
        double r632360 = r632358 * r632359;
        double r632361 = r632347 - r632360;
        double r632362 = r632345 ? r632355 : r632361;
        return r632362;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.1
Target	6.1
Herbie	3.2

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

Derivation

1. Split input into 2 regimes

2. if (* z z) < 2.007625814536843e+304

   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 fma-neg 0.1

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

   if 2.007625814536843e+304 < (* z z)

   1. Initial program 63.2

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

   3. Applied add-sqr-sqrt 63.6

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

   4. Applied difference-of-squares 63.6

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

   5. Applied associate-*r* 32.8

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

4. Final simplification 3.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \le 2.007625814536843067127930964300867649199 \cdot 10^{304}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \mathsf{fma}\left(z, z, -t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(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))))