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

Average Error: 6.1 → 3.1

Time: 6.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot z \le 2.94628321762693042 \cdot 10^{297}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(y \cdot 4\right) \cdot \left(\sqrt{t} + z\right)\right) \cdot \left(\sqrt{t} - z\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r743456 = x;
        double r743457 = r743456 * r743456;
        double r743458 = y;
        double r743459 = 4.0;
        double r743460 = r743458 * r743459;
        double r743461 = z;
        double r743462 = r743461 * r743461;
        double r743463 = t;
        double r743464 = r743462 - r743463;
        double r743465 = r743460 * r743464;
        double r743466 = r743457 - r743465;
        return r743466;
}

↓

double f(double x, double y, double z, double t) {
        double r743467 = z;
        double r743468 = r743467 * r743467;
        double r743469 = 2.9462832176269304e+297;
        bool r743470 = r743468 <= r743469;
        double r743471 = x;
        double r743472 = y;
        double r743473 = 4.0;
        double r743474 = r743472 * r743473;
        double r743475 = t;
        double r743476 = r743475 - r743468;
        double r743477 = r743474 * r743476;
        double r743478 = fma(r743471, r743471, r743477);
        double r743479 = sqrt(r743475);
        double r743480 = r743479 + r743467;
        double r743481 = r743474 * r743480;
        double r743482 = r743479 - r743467;
        double r743483 = r743481 * r743482;
        double r743484 = fma(r743471, r743471, r743483);
        double r743485 = r743470 ? r743478 : r743484;
        return r743485;
}

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	3.1

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

Derivation

1. Split input into 2 regimes

2. if (* z z) < 2.9462832176269304e+297

   1. Initial program 0.1

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

   2. Simplified 0.1

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

   if 2.9462832176269304e+297 < (* z z)

   1. Initial program 59.8

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

   2. Simplified 59.8

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

   4. Applied add-sqr-sqrt 62.0

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

   5. Applied difference-of-squares 62.0

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

   6. Applied associate-*r* 29.8

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

4. Final simplification 3.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \le 2.94628321762693042 \cdot 10^{297}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(y \cdot 4\right) \cdot \left(\sqrt{t} + z\right)\right) \cdot \left(\sqrt{t} - z\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 +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))))