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

Average Error: 5.8 → 3.5

Time: 11.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -1.6331768204397996 \cdot 10^{+153}:\\ \;\;\;\;x \cdot x - \frac{4 \cdot \left(z + \sqrt{t}\right)}{\frac{\frac{1}{z - \sqrt{t}}}{y}}\\ \mathbf{elif}\;z \leq 2.5730812676030823 \cdot 10^{+126}:\\ \;\;\;\;x \cdot x - \left(4 \cdot y\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(z - \sqrt{t}\right) \cdot \left(\left(z + \sqrt{t}\right) \cdot \left(4 \cdot y\right)\right)\\ \end{array}\]

FPCore

(FPCore (x y z t) :precision binary64 (- (* x x) (* (* y 4.0) (- (* z z) t))))

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.6331768204397996e+153)
   (- (* x x) (/ (* 4.0 (+ z (sqrt t))) (/ (/ 1.0 (- z (sqrt t))) y)))
   (if (<= z 2.5730812676030823e+126)
     (- (* x x) (* (* 4.0 y) (- (* z z) t)))
     (- (* x x) (* (- z (sqrt t)) (* (+ z (sqrt t)) (* 4.0 y)))))))

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 tmp;
	if (z <= -1.6331768204397996e+153) {
		tmp = (x * x) - ((4.0 * (z + sqrt(t))) / ((1.0 / (z - sqrt(t))) / y));
	} else if (z <= 2.5730812676030823e+126) {
		tmp = (x * x) - ((4.0 * y) * ((z * z) - t));
	} else {
		tmp = (x * x) - ((z - sqrt(t)) * ((z + sqrt(t)) * (4.0 * y)));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	5.8
Target	5.8
Herbie	3.5

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

Derivation

1. Split input into 3 regimes

2. if z < -1.6331768204397996e153

   1. Initial program 62.3

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

   3. Applied flip--_binary64_17444 64.0

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

   4. Applied associate-*r/_binary64_17411 64.0

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

   5. Simplified 64.0

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

   7. Applied clear-num_binary64_17468 64.0

      \[\leadsto x \cdot x - \color{blue}{\frac{1}{\frac{z \cdot z + t}{\left(y \cdot 4\right) \cdot \left({z}^{4} - t \cdot t\right)}}}\]

   8. Simplified 62.3

      \[\leadsto x \cdot x - \frac{1}{\color{blue}{\frac{\frac{1}{z \cdot z - t}}{4 \cdot y}}}\]
   9. Using strategy rm

   10. Applied add-sqr-sqrt_binary64_17491 63.2

       \[\leadsto x \cdot x - \frac{1}{\frac{\frac{1}{z \cdot z - \color{blue}{\sqrt{t} \cdot \sqrt{t}}}}{4 \cdot y}}\]

   11. Applied difference-of-squares_binary64_17438 63.2

       \[\leadsto x \cdot x - \frac{1}{\frac{\frac{1}{\color{blue}{\left(z + \sqrt{t}\right) \cdot \left(z - \sqrt{t}\right)}}}{4 \cdot y}}\]

   12. Applied add-sqr-sqrt_binary64_17491 63.2

       \[\leadsto x \cdot x - \frac{1}{\frac{\frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{\left(z + \sqrt{t}\right) \cdot \left(z - \sqrt{t}\right)}}{4 \cdot y}}\]

   13. Applied times-frac_binary64_17475 61.4

       \[\leadsto x \cdot x - \frac{1}{\frac{\color{blue}{\frac{\sqrt{1}}{z + \sqrt{t}} \cdot \frac{\sqrt{1}}{z - \sqrt{t}}}}{4 \cdot y}}\]

   14. Applied times-frac_binary64_17475 29.6

       \[\leadsto x \cdot x - \frac{1}{\color{blue}{\frac{\frac{\sqrt{1}}{z + \sqrt{t}}}{4} \cdot \frac{\frac{\sqrt{1}}{z - \sqrt{t}}}{y}}}\]

   15. Applied associate-/r*_binary64_17413 29.5

       \[\leadsto x \cdot x - \color{blue}{\frac{\frac{1}{\frac{\frac{\sqrt{1}}{z + \sqrt{t}}}{4}}}{\frac{\frac{\sqrt{1}}{z - \sqrt{t}}}{y}}}\]

   16. Simplified 29.4

       \[\leadsto x \cdot x - \frac{\color{blue}{4 \cdot \left(z + \sqrt{t}\right)}}{\frac{\frac{\sqrt{1}}{z - \sqrt{t}}}{y}}\]

   if -1.6331768204397996e153 < z < 2.5730812676030823e126

   1. Initial program 0.1

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

   if 2.5730812676030823e126 < z

   1. Initial program 44.3

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

   3. Applied add-sqr-sqrt_binary64_17491 54.3

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

   4. Applied difference-of-squares_binary64_17438 54.3

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

   5. Applied associate-*r*_binary64_17409 31.9

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

4. Final simplification 3.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6331768204397996 \cdot 10^{+153}:\\ \;\;\;\;x \cdot x - \frac{4 \cdot \left(z + \sqrt{t}\right)}{\frac{\frac{1}{z - \sqrt{t}}}{y}}\\ \mathbf{elif}\;z \leq 2.5730812676030823 \cdot 10^{+126}:\\ \;\;\;\;x \cdot x - \left(4 \cdot y\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(z - \sqrt{t}\right) \cdot \left(\left(z + \sqrt{t}\right) \cdot \left(4 \cdot y\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021043 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

  :herbie-target
  (- (* x x) (* 4.0 (* y (- (* z z) t))))

  (- (* x x) (* (* y 4.0) (- (* z z) t))))