Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Average Error: 2.1 → 0.5

Time: 6.9s

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

\[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\]

↓

\[\begin{array}{l} \mathbf{if}\;b \le -2.5642226124843436 \cdot 10^{59}:\\ \;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(\left(a \cdot z\right) \cdot \left(\sqrt[3]{b} \cdot \left(\sqrt[3]{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \sqrt[3]{\sqrt[3]{b}}\right)\right)\right) \cdot \sqrt[3]{b}\\ \mathbf{elif}\;b \le 1.56252021443305207 \cdot 10^{31}:\\ \;\;\;\;a \cdot \left(t + z \cdot b\right) + \left(x + y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(\left(a \cdot z\right) \cdot \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \sqrt[3]{\sqrt[3]{b}}\right)\\ \end{array}\]

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((double) (((double) (((double) (x + ((double) (y * z)))) + ((double) (t * a)))) + ((double) (((double) (a * z)) * b))));
}

↓

double code(double x, double y, double z, double t, double a, double b) {
	double VAR;
	if ((b <= -2.5642226124843436e+59)) {
		VAR = ((double) (((double) (((double) (x + ((double) (y * z)))) + ((double) (t * a)))) + ((double) (((double) (((double) (a * z)) * ((double) (((double) cbrt(b)) * ((double) (((double) cbrt(((double) (((double) cbrt(b)) * ((double) cbrt(b)))))) * ((double) cbrt(((double) cbrt(b)))))))))) * ((double) cbrt(b))))));
	} else {
		double VAR_1;
		if ((b <= 1.562520214433052e+31)) {
			VAR_1 = ((double) (((double) (a * ((double) (t + ((double) (z * b)))))) + ((double) (x + ((double) (y * z))))));
		} else {
			VAR_1 = ((double) (((double) (((double) (x + ((double) (y * z)))) + ((double) (t * a)))) + ((double) (((double) (((double) (a * z)) * ((double) (((double) cbrt(b)) * ((double) cbrt(b)))))) * ((double) (((double) cbrt(((double) (((double) cbrt(b)) * ((double) cbrt(b)))))) * ((double) cbrt(((double) cbrt(b))))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original	2.1
Target	0.4
Herbie	0.5

\[\begin{array}{l} \mathbf{if}\;z \lt -11820553527347888000:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \mathbf{elif}\;z \lt 4.75897431883642871 \cdot 10^{-122}:\\ \;\;\;\;\left(b \cdot z + t\right) \cdot a + \left(z \cdot y + x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if b < -2.5642226124843436e59

   1. Initial program 0.6

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 0.9

      \[\leadsto \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot \color{blue}{\left(\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \sqrt[3]{b}\right)}\]

   4. Applied associate-*r* 0.9

      \[\leadsto \left(\left(x + y \cdot z\right) + t \cdot a\right) + \color{blue}{\left(\left(a \cdot z\right) \cdot \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)\right) \cdot \sqrt[3]{b}}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 0.9

      \[\leadsto \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(\left(a \cdot z\right) \cdot \left(\sqrt[3]{b} \cdot \sqrt[3]{\color{blue}{\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \sqrt[3]{b}}}\right)\right) \cdot \sqrt[3]{b}\]

   7. Applied cbrt-prod 1.0

      \[\leadsto \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(\left(a \cdot z\right) \cdot \left(\sqrt[3]{b} \cdot \color{blue}{\left(\sqrt[3]{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \sqrt[3]{\sqrt[3]{b}}\right)}\right)\right) \cdot \sqrt[3]{b}\]

   if -2.5642226124843436e59 < b < 1.56252021443305207e31

   1. Initial program 3.0

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\]

   2. Simplified 0.2

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

   if 1.56252021443305207e31 < b

   1. Initial program 0.7

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 1.0

      \[\leadsto \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot \color{blue}{\left(\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \sqrt[3]{b}\right)}\]

   4. Applied associate-*r* 1.0

      \[\leadsto \left(\left(x + y \cdot z\right) + t \cdot a\right) + \color{blue}{\left(\left(a \cdot z\right) \cdot \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)\right) \cdot \sqrt[3]{b}}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 1.0

      \[\leadsto \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(\left(a \cdot z\right) \cdot \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)\right) \cdot \sqrt[3]{\color{blue}{\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \sqrt[3]{b}}}\]

   7. Applied cbrt-prod 1.0

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.5642226124843436 \cdot 10^{59}:\\ \;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(\left(a \cdot z\right) \cdot \left(\sqrt[3]{b} \cdot \left(\sqrt[3]{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \sqrt[3]{\sqrt[3]{b}}\right)\right)\right) \cdot \sqrt[3]{b}\\ \mathbf{elif}\;b \le 1.56252021443305207 \cdot 10^{31}:\\ \;\;\;\;a \cdot \left(t + z \cdot b\right) + \left(x + y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(\left(a \cdot z\right) \cdot \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \sqrt[3]{\sqrt[3]{b}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(FPCore (x y z t a b)
  :name "Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1"
  :precision binary64

  :herbie-target
  (if (< z -1.1820553527347888e+19) (+ (* z (+ (* b a) y)) (+ x (* t a))) (if (< z 4.7589743188364287e-122) (+ (* (+ (* b z) t) a) (+ (* z y) x)) (+ (* z (+ (* b a) y)) (+ x (* t a)))))

  (+ (+ (+ x (* y z)) (* t a)) (* (* a z) b)))