Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Average Error: 2.1 → 0.5

Time: 46.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \mathsf{fma}\left(a \cdot z, b, \mathsf{fma}\left(a, t, \mathsf{fma}\left(z, y, x\right)\right)\right)\\ \mathbf{if}\;b \leq -9.112644010531083 \cdot 10^{+36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 2.4356118753451166 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(y, z, \mathsf{fma}\left(a, \mathsf{fma}\left(z, b, t\right), x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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
 (let* ((t_1 (fma (* a z) b (fma a t (fma z y x)))))
   (if (<= b -9.112644010531083e+36)
     t_1
     (if (<= b 2.4356118753451166e+116) (fma y z (fma a (fma z b t) x)) t_1))))

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 t_1 = fma((a * z), b, fma(a, t, fma(z, y, x)));
	double tmp;
	if (b <= -9.112644010531083e+36) {
		tmp = t_1;
	} else if (b <= 2.4356118753451166e+116) {
		tmp = fma(y, z, fma(a, fma(z, b, t), x));
	} else {
		tmp = t_1;
	}
	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	0.5

\[\begin{array}{l} \mathbf{if}\;z < -11820553527347888000:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \mathbf{elif}\;z < 4.7589743188364287 \cdot 10^{-122}:\\ \;\;\;\;\left(b \cdot z + t\right) \cdot a + \left(z \cdot y + x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if b < -9.11264401053108332e36 or 2.43561187534511665e116 < 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. Applied egg-rr 0.9

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

   if -9.11264401053108332e36 < b < 2.43561187534511665e116

   1. Initial program 2.7

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

   2. Simplified 0.3

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.112644010531083 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot z, b, \mathsf{fma}\left(a, t, \mathsf{fma}\left(z, y, x\right)\right)\right)\\ \mathbf{elif}\;b \leq 2.4356118753451166 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(y, z, \mathsf{fma}\left(a, \mathsf{fma}\left(z, b, t\right), x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot z, b, \mathsf{fma}\left(a, t, \mathsf{fma}\left(z, y, x\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(FPCore (x y z t a b)
  :name "Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1"
  :precision binary64

  :herbie-target
  (if (< z -11820553527347888000.0) (+ (* 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)))