Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Average Error: 2.0 → 1.5

Time: 5.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq 2.2714426383117235 \cdot 10^{-113}:\\ \;\;\;\;\mathsf{fma}\left(a, z \cdot b, \mathsf{fma}\left(a, t, \mathsf{fma}\left(z, y, x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), \mathsf{fma}\left(a, t, x\right)\right)\\ \end{array} \]

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 (<= z 2.2714426383117235e-113)
   (fma a (* z b) (fma a t (fma z y x)))
   (fma z (fma a b y) (fma a t x))))

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 (z <= 2.2714426383117235e-113) {
		tmp = fma(a, (z * b), fma(a, t, fma(z, y, x)));
	} else {
		tmp = fma(z, fma(a, b, y), fma(a, t, x));
	}
	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)
	tmp = 0.0
	if (z <= 2.2714426383117235e-113)
		tmp = fma(a, Float64(z * b), fma(a, t, fma(z, y, x)));
	else
		tmp = fma(z, fma(a, b, y), fma(a, t, x));
	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_] := If[LessEqual[z, 2.2714426383117235e-113], N[(a * N[(z * b), $MachinePrecision] + N[(a * t + N[(z * y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(a * b + y), $MachinePrecision] + N[(a * t + x), $MachinePrecision]), $MachinePrecision]]

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	2.0
Target	0.3
Herbie	1.5

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

Derivation

1. Split input into 2 regimes

2. if z < 2.27144263831172354e-113

   1. Initial program 1.6

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

   2. Applied egg-rr 1.9

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

   if 2.27144263831172354e-113 < z

   1. Initial program 3.1

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

   2. Simplified 4.6

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

   3. Applied egg-rr 5.7

      \[\leadsto \color{blue}{\sqrt[3]{\mathsf{fma}\left(y, z, \mathsf{fma}\left(a, \mathsf{fma}\left(z, b, t\right), x\right)\right)} \cdot {\left(\sqrt[3]{\mathsf{fma}\left(y, z, \mathsf{fma}\left(a, \mathsf{fma}\left(z, b, t\right), x\right)\right)}\right)}^{2}} \]

   4. Taylor expanded in y around 0 4.6

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

   5. Simplified 0.7

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

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.2714426383117235 \cdot 10^{-113}:\\ \;\;\;\;\mathsf{fma}\left(a, z \cdot b, \mathsf{fma}\left(a, t, \mathsf{fma}\left(z, y, x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), \mathsf{fma}\left(a, t, x\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(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)))