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

Average Error: 5.7 → 0.1

Time: 2.9s

Precision: 64

Math

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

↓

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

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

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	0.1

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

Derivation

1. Initial program 5.7

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

3. Applied *-commutative 5.7

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

4. Applied associate-*l* 5.7

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

6. Applied sub-neg 5.7

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

7. Applied distribute-lft-in 5.7

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

8. Applied distribute-lft-in 5.7

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

9. Applied associate--r+ 5.7

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

10. Simplified 5.7

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

12. Applied associate-*l* 0.1

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

13. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020078 
(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))))