Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Average Error: 2.1 → 0.6

Time: 7.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -5.121552918984049 \cdot 10^{+119} \lor \neg \left(b \leq 2.6475876084449683 \cdot 10^{-67}\right):\\ \;\;\;\;\left(x + \left(z \cdot y + a \cdot t\right)\right) + b \cdot \left(z \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + a \cdot t\right) + z \cdot \left(y + b \cdot a\right)\\ \end{array}\]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (+ (+ (+ x (* y z)) (* t a)) (* (* a z) b)))

↓

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= b -5.121552918984049e+119) (not (<= b 2.6475876084449683e-67)))
   (+ (+ x (+ (* z y) (* a t))) (* b (* z a)))
   (+ (+ x (* a t)) (* z (+ y (* b a))))))

C

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

↓

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -5.121552918984049e+119) || !(b <= 2.6475876084449683e-67)) {
		tmp = (x + ((z * y) + (a * t))) + (b * (z * a));
	} else {
		tmp = (x + (a * t)) + (z * (y + (b * a)));
	}
	return tmp;
}

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	0.6

\[\begin{array}{l} \mathbf{if}\;z < -1.1820553527347888 \cdot 10^{+19}:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \mathbf{elif}\;z < 4.7589743188364287 \cdot 10^{-122}:\\ \;\;\;\;\left(b \cdot z + t\right) \cdot a + \left(z \cdot y + x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if b < -5.12155291898404904e119 or 2.64758760844496831e-67 < b

   1. Initial program 0.9

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

   3. Applied associate-+l+_binary64 0.9

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

   4. Simplified 0.9

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

   if -5.12155291898404904e119 < b < 2.64758760844496831e-67

   1. Initial program 2.9

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

   2. Taylor expanded around 0 0.4

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

   3. Simplified 0.4

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

   5. Applied associate-+r+_binary64 0.4

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

   6. Simplified 0.4

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.121552918984049 \cdot 10^{+119} \lor \neg \left(b \leq 2.6475876084449683 \cdot 10^{-67}\right):\\ \;\;\;\;\left(x + \left(z \cdot y + a \cdot t\right)\right) + b \cdot \left(z \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + a \cdot t\right) + z \cdot \left(y + b \cdot a\right)\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021118 
(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)))