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

Average Error: 6.0 → 3.2

Time: 4.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot z \le 1.8646532487958418 \cdot 10^{304}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 4, t - {z}^{2}, {x}^{2}\right) + y \cdot \left(\mathsf{fma}\left(-z, z, z \cdot z\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(y \cdot 4\right) \cdot \left(\sqrt{t} + z\right)\right) \cdot \left(\sqrt{t} - z\right)\right)\\ \end{array}\]

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 temp;
	if (((z * z) <= 1.8646532487958418e+304)) {
		temp = (fma((y * 4.0), (t - pow(z, 2.0)), pow(x, 2.0)) + (y * (fma(-z, z, (z * z)) * 4.0)));
	} else {
		temp = fma(x, x, (((y * 4.0) * (sqrt(t) + z)) * (sqrt(t) - z)));
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.0
Target	5.9
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) < 1.8646532487958418e+304

   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 associate-*l* 0.1

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

   6. Applied fma-udef 0.1

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

   7. Simplified 0.1

      \[\leadsto \color{blue}{{x}^{2}} + y \cdot \left(4 \cdot \left(t - z \cdot z\right)\right)\]
   8. Using strategy rm

   9. Applied add-sqr-sqrt 32.3

      \[\leadsto {x}^{2} + y \cdot \left(4 \cdot \left(\color{blue}{\sqrt{t} \cdot \sqrt{t}} - z \cdot z\right)\right)\]

   10. Applied prod-diff 32.3

       \[\leadsto {x}^{2} + y \cdot \left(4 \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt{t}, \sqrt{t}, -z \cdot z\right) + \mathsf{fma}\left(-z, z, z \cdot z\right)\right)}\right)\]

   11. Applied distribute-rgt-in 32.3

       \[\leadsto {x}^{2} + y \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt{t}, \sqrt{t}, -z \cdot z\right) \cdot 4 + \mathsf{fma}\left(-z, z, z \cdot z\right) \cdot 4\right)}\]

   12. Applied distribute-lft-in 32.3

       \[\leadsto {x}^{2} + \color{blue}{\left(y \cdot \left(\mathsf{fma}\left(\sqrt{t}, \sqrt{t}, -z \cdot z\right) \cdot 4\right) + y \cdot \left(\mathsf{fma}\left(-z, z, z \cdot z\right) \cdot 4\right)\right)}\]

   13. Applied associate-+r+ 32.3

       \[\leadsto \color{blue}{\left({x}^{2} + y \cdot \left(\mathsf{fma}\left(\sqrt{t}, \sqrt{t}, -z \cdot z\right) \cdot 4\right)\right) + y \cdot \left(\mathsf{fma}\left(-z, z, z \cdot z\right) \cdot 4\right)}\]

   14. Simplified 0.1

       \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot 4, t - {z}^{2}, {x}^{2}\right)} + y \cdot \left(\mathsf{fma}\left(-z, z, z \cdot z\right) \cdot 4\right)\]

   if 1.8646532487958418e+304 < (* z z)

   1. Initial program 62.3

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

   2. Simplified 62.3

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

   4. Applied add-sqr-sqrt 63.3

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

   5. Applied difference-of-squares 63.3

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

   6. Applied associate-*r* 32.8

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

4. Final simplification 3.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \le 1.8646532487958418 \cdot 10^{304}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 4, t - {z}^{2}, {x}^{2}\right) + y \cdot \left(\mathsf{fma}\left(-z, z, z \cdot z\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(y \cdot 4\right) \cdot \left(\sqrt{t} + z\right)\right) \cdot \left(\sqrt{t} - z\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 +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))))