Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Average Error: 2.1 → 0.4

Time: 4.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1.07422854836868 \cdot 10^{-15} \lor \neg \left(z \le 2.9993001878202427 \cdot 10^{122}\right):\\ \;\;\;\;\mathsf{fma}\left(t \cdot 1, a, \mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot 1, a, a \cdot \left(z \cdot b\right) + \mathsf{fma}\left(z, y, x\right)\right)\\ \end{array}\]

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 (((z <= -1.0742285483686794e-15) || !(z <= 2.9993001878202427e+122))) {
		VAR = ((double) fma(((double) (t * 1.0)), a, ((double) fma(((double) fma(b, a, y)), z, x))));
	} else {
		VAR = ((double) fma(((double) (t * 1.0)), a, ((double) (((double) (a * ((double) (z * b)))) + ((double) fma(z, y, 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	2.1
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 z < -1.0742285483686794e-15 or 2.9993001878202427e+122 < z

   1. Initial program 5.6

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

   3. Applied +-commutative 5.6

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

   4. Applied associate-+l+ 5.6

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

   5. Simplified 5.6

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

   7. Applied *-un-lft-identity 5.6

      \[\leadsto t \cdot \color{blue}{\left(1 \cdot a\right)} + \mathsf{fma}\left(a \cdot z, b, \mathsf{fma}\left(z, y, x\right)\right)\]

   8. Applied associate-*r* 5.6

      \[\leadsto \color{blue}{\left(t \cdot 1\right) \cdot a} + \mathsf{fma}\left(a \cdot z, b, \mathsf{fma}\left(z, y, x\right)\right)\]

   9. Applied fma-def 5.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot 1, a, \mathsf{fma}\left(a \cdot z, b, \mathsf{fma}\left(z, y, x\right)\right)\right)}\]

   10. Taylor expanded around inf 8.2

       \[\leadsto \mathsf{fma}\left(t \cdot 1, a, \color{blue}{a \cdot \left(z \cdot b\right) + \left(x + z \cdot y\right)}\right)\]

   11. Simplified 0.1

       \[\leadsto \mathsf{fma}\left(t \cdot 1, a, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)}\right)\]

   if -1.0742285483686794e-15 < z < 2.9993001878202427e+122

   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 +-commutative 0.7

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

   4. Applied associate-+l+ 0.7

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

   5. Simplified 0.7

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

   7. Applied *-un-lft-identity 0.7

      \[\leadsto t \cdot \color{blue}{\left(1 \cdot a\right)} + \mathsf{fma}\left(a \cdot z, b, \mathsf{fma}\left(z, y, x\right)\right)\]

   8. Applied associate-*r* 0.7

      \[\leadsto \color{blue}{\left(t \cdot 1\right) \cdot a} + \mathsf{fma}\left(a \cdot z, b, \mathsf{fma}\left(z, y, x\right)\right)\]

   9. Applied fma-def 0.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot 1, a, \mathsf{fma}\left(a \cdot z, b, \mathsf{fma}\left(z, y, x\right)\right)\right)}\]
   10. Using strategy rm

   11. Applied fma-udef 0.7

       \[\leadsto \mathsf{fma}\left(t \cdot 1, a, \color{blue}{\left(a \cdot z\right) \cdot b + \mathsf{fma}\left(z, y, x\right)}\right)\]

   12. Simplified 0.6

       \[\leadsto \mathsf{fma}\left(t \cdot 1, a, \color{blue}{a \cdot \left(z \cdot b\right)} + \mathsf{fma}\left(z, y, x\right)\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.07422854836868 \cdot 10^{-15} \lor \neg \left(z \le 2.9993001878202427 \cdot 10^{122}\right):\\ \;\;\;\;\mathsf{fma}\left(t \cdot 1, a, \mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot 1, a, a \cdot \left(z \cdot b\right) + \mathsf{fma}\left(z, y, x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 +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)))