Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Average Error: 2.2 → 0.3

Time: 3.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -126080516211816400 \lor \neg \left(a \le 1.92081113332191634 \cdot 10^{-104}\right):\\ \;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + a \cdot \left(z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, b, y\right), z, \mathsf{fma}\left(a, t, x\right)\right)\\ \end{array}\]

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 VAR;
	if (((a <= -1.260805162118164e+17) || !(a <= 1.9208111333219163e-104))) {
		VAR = (((x + (y * z)) + (t * a)) + (a * (z * b)));
	} else {
		VAR = fma(fma(a, b, y), z, fma(a, t, x));
	}
	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	2.2
Target	0.4
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if a < -1.260805162118164e+17 or 1.9208111333219163e-104 < a

   1. Initial program 4.3

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

   3. Applied associate-*l* 0.5

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

   if -1.260805162118164e+17 < a < 1.9208111333219163e-104

   1. Initial program 0.4

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

   2. Simplified 0.0

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -126080516211816400 \lor \neg \left(a \le 1.92081113332191634 \cdot 10^{-104}\right):\\ \;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + a \cdot \left(z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, b, y\right), z, \mathsf{fma}\left(a, t, x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020091 +o rules:numerics
(FPCore (x y z t a b)
  :name "Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1"
  :precision binary64

  :herbie-target
  (if (< z -11820553527347888000) (+ (* 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)))