Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Average Error: 1.9 → 0.4

Time: 5.8min

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -2.5099893648265318 \cdot 10^{-101} \lor \neg \left(b \le 3.16136548725884987 \cdot 10^{-32}\right):\\ \;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \left(\sqrt[3]{a} \cdot \left(z \cdot b\right)\right)\\ \end{array}\]

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 <= -2.5099893648265318e-101) || !(b <= 3.16136548725885e-32))) {
		VAR = ((double) (((double) (((double) (x + ((double) (y * z)))) + ((double) (t * a)))) + ((double) (((double) (a * z)) * b))));
	} else {
		VAR = ((double) (((double) (((double) (x + ((double) (y * z)))) + ((double) (t * a)))) + ((double) (((double) (((double) cbrt(a)) * ((double) cbrt(a)))) * ((double) (((double) cbrt(a)) * ((double) (z * b))))))));
	}
	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	1.9
Target	0.3
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if b < -2.5099893648265318e-101 or 3.16136548725884987e-32 < b

   1. Initial program 0.6

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

   if -2.5099893648265318e-101 < b < 3.16136548725884987e-32

   1. Initial program 3.5

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

   3. Applied associate-*l* 0.0

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

   5. Applied add-cube-cbrt 0.1

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

   6. Applied associate-*l* 0.1

      \[\leadsto \left(\left(x + y \cdot z\right) + t \cdot a\right) + \color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \left(\sqrt[3]{a} \cdot \left(z \cdot b\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.5099893648265318 \cdot 10^{-101} \lor \neg \left(b \le 3.16136548725884987 \cdot 10^{-32}\right):\\ \;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \left(\sqrt[3]{a} \cdot \left(z \cdot b\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(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)))