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

Average Error: 5.9 → 3.1

Time: 5.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot z \le 8.0569891955755611 \cdot 10^{300}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right) + \left(y \cdot 4\right) \cdot \mathsf{fma}\left(-z, z, z \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(y \cdot 4\right) \cdot \left(\sqrt{t} + z\right)\right) \cdot \left(\sqrt{t} - z\right)\right)\\ \end{array}\]

C

double code(double x, double y, double z, double t) {
	return ((x * x) - ((y * 4.0) * ((z * z) - t)));
}

↓

double code(double x, double y, double z, double t) {
	double VAR;
	if (((z * z) <= 8.056989195575561e+300)) {
		VAR = fma(x, x, (((y * 4.0) * (t - (z * z))) + ((y * 4.0) * fma(-z, z, (z * z)))));
	} else {
		VAR = fma(x, x, (((y * 4.0) * (sqrt(t) + z)) * (sqrt(t) - z)));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	5.9
Target	5.9
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) < 8.056989195575561e+300

   1. Initial program 0.1

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

   2. Simplified 0.1

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

   4. Applied add-sqr-sqrt 31.7

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

   5. Applied prod-diff 31.7

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

   6. Applied distribute-lft-in 31.7

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

   7. Simplified 0.1

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

   if 8.056989195575561e+300 < (* z z)

   1. Initial program 61.4

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

   2. Simplified 61.4

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

   4. Applied add-sqr-sqrt 62.9

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

   5. Applied difference-of-squares 62.9

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

   6. Applied associate-*r* 32.0

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

4. Final simplification 3.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \le 8.0569891955755611 \cdot 10^{300}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right) + \left(y \cdot 4\right) \cdot \mathsf{fma}\left(-z, z, z \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(y \cdot 4\right) \cdot \left(\sqrt{t} + z\right)\right) \cdot \left(\sqrt{t} - z\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020091 +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))))