Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Average Error: 2.1 → 0.8

Time: 5.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \mathsf{fma}\left(a, t, \mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), x\right)\right)\\ \mathbf{if}\;z \leq -6.404233806295518 \cdot 10^{+43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.3935108643919645 \cdot 10^{-188}:\\ \;\;\;\;x + \left(z \cdot y + a \cdot \mathsf{fma}\left(z, b, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 t (fma z (fma a b y) x))))
   (if (<= z -6.404233806295518e+43)
     t_1
     (if (<= z 4.3935108643919645e-188)
       (+ x (+ (* z y) (* a (fma z b t))))
       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, t, fma(z, fma(a, b, y), x));
	double tmp;
	if (z <= -6.404233806295518e+43) {
		tmp = t_1;
	} else if (z <= 4.3935108643919645e-188) {
		tmp = x + ((z * y) + (a * fma(z, b, t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(x + Float64(y * z)) + Float64(t * a)) + Float64(Float64(a * z) * b))
end

↓

function code(x, y, z, t, a, b)
	t_1 = fma(a, t, fma(z, fma(a, b, y), x))
	tmp = 0.0
	if (z <= -6.404233806295518e+43)
		tmp = t_1;
	elseif (z <= 4.3935108643919645e-188)
		tmp = Float64(x + Float64(Float64(z * y) + Float64(a * fma(z, b, t))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision] + N[(t * a), $MachinePrecision]), $MachinePrecision] + N[(N[(a * z), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a * t + N[(z * N[(a * b + y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -6.404233806295518e+43], t$95$1, If[LessEqual[z, 4.3935108643919645e-188], N[(x + N[(N[(z * y), $MachinePrecision] + N[(a * N[(z * b + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

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	0.5
Herbie	0.8

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

Derivation

1. Split input into 2 regimes

2. if z < -6.40423380629551797e43 or 4.39351086439196451e-188 < z

   1. Initial program 3.5

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

   2. Taylor expanded in x around 0 5.4

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

   3. Simplified 1.4

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

   if -6.40423380629551797e43 < z < 4.39351086439196451e-188

   1. Initial program 0.5

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

   2. Simplified 0.2

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

   3. Applied egg-rr 0.2

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.404233806295518 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(a, t, \mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), x\right)\right)\\ \mathbf{elif}\;z \leq 4.3935108643919645 \cdot 10^{-188}:\\ \;\;\;\;x + \left(z \cdot y + a \cdot \mathsf{fma}\left(z, b, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, \mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), x\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022150 
(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)))