Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Average Error: 3.09% → 1.13%

Time: 12.5s

Precision: binary64

Cost: 19913

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+86} \lor \neg \left(z \leq 5 \cdot 10^{+141}\right):\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), \mathsf{fma}\left(t, a, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z, \mathsf{fma}\left(a, \mathsf{fma}\left(z, b, t\right), x\right)\right)\\ \end{array} \]

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 (or (<= z -1.35e+86) (not (<= z 5e+141)))
   (fma z (fma a b y) (fma t a x))
   (fma y z (fma a (fma z b 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 <= -1.35e+86) || !(z <= 5e+141)) {
		tmp = fma(z, fma(a, b, y), fma(t, a, x));
	} else {
		tmp = fma(y, z, fma(a, fma(z, b, 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 <= -1.35e+86) || !(z <= 5e+141))
		tmp = fma(z, fma(a, b, y), fma(t, a, x));
	else
		tmp = fma(y, z, fma(a, fma(z, b, 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[Or[LessEqual[z, -1.35e+86], N[Not[LessEqual[z, 5e+141]], $MachinePrecision]], N[(z * N[(a * b + y), $MachinePrecision] + N[(t * a + x), $MachinePrecision]), $MachinePrecision], N[(y * z + N[(a * N[(z * b + t), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]

Target

Original	3.09%
Target	0.61%
Herbie	1.13%

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

Derivation

1. Split input into 2 regimes

2. if z < -1.35000000000000009e86 or 5.00000000000000025e141 < z

   1. Initial program 12.06

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

   2. Simplified 0.13

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

      [Start] 12.06	\[ \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
      +-commutative [=>] 12.06	\[ \color{blue}{\left(a \cdot z\right) \cdot b + \left(\left(x + y \cdot z\right) + t \cdot a\right)} \]
      +-commutative [=>] 12.06	\[ \left(a \cdot z\right) \cdot b + \left(\color{blue}{\left(y \cdot z + x\right)} + t \cdot a\right) \]
      associate-+l+ [=>] 12.06	\[ \left(a \cdot z\right) \cdot b + \color{blue}{\left(y \cdot z + \left(x + t \cdot a\right)\right)} \]
      associate-+r+ [=>] 12.06	\[ \color{blue}{\left(\left(a \cdot z\right) \cdot b + y \cdot z\right) + \left(x + t \cdot a\right)} \]
      *-commutative [=>] 12.06	\[ \left(\color{blue}{\left(z \cdot a\right)} \cdot b + y \cdot z\right) + \left(x + t \cdot a\right) \]
      associate-*l* [=>] 0.14	\[ \left(\color{blue}{z \cdot \left(a \cdot b\right)} + y \cdot z\right) + \left(x + t \cdot a\right) \]
      *-commutative [=>] 0.14	\[ \left(z \cdot \left(a \cdot b\right) + \color{blue}{z \cdot y}\right) + \left(x + t \cdot a\right) \]
      distribute-lft-out [=>] 0.14	\[ \color{blue}{z \cdot \left(a \cdot b + y\right)} + \left(x + t \cdot a\right) \]
      fma-def [=>] 0.14	\[ \color{blue}{\mathsf{fma}\left(z, a \cdot b + y, x + t \cdot a\right)} \]
      fma-def [=>] 0.13	\[ \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(a, b, y\right)}, x + t \cdot a\right) \]
      +-commutative [=>] 0.13	\[ \mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), \color{blue}{t \cdot a + x}\right) \]
      fma-def [=>] 0.13	\[ \mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), \color{blue}{\mathsf{fma}\left(t, a, x\right)}\right) \]

   if -1.35000000000000009e86 < z < 5.00000000000000025e141

   1. Initial program 1.03

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

   2. Simplified 1.35

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

      [Start] 1.03	\[ \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
      associate-+l+ [=>] 1.03	\[ \color{blue}{\left(x + y \cdot z\right) + \left(t \cdot a + \left(a \cdot z\right) \cdot b\right)} \]
      +-commutative [=>] 1.03	\[ \color{blue}{\left(y \cdot z + x\right)} + \left(t \cdot a + \left(a \cdot z\right) \cdot b\right) \]
      associate-+l+ [=>] 1.03	\[ \color{blue}{y \cdot z + \left(x + \left(t \cdot a + \left(a \cdot z\right) \cdot b\right)\right)} \]
      fma-def [=>] 1.02	\[ \color{blue}{\mathsf{fma}\left(y, z, x + \left(t \cdot a + \left(a \cdot z\right) \cdot b\right)\right)} \]
      +-commutative [=>] 1.02	\[ \mathsf{fma}\left(y, z, \color{blue}{\left(t \cdot a + \left(a \cdot z\right) \cdot b\right) + x}\right) \]
      *-commutative [=>] 1.02	\[ \mathsf{fma}\left(y, z, \left(\color{blue}{a \cdot t} + \left(a \cdot z\right) \cdot b\right) + x\right) \]
      associate-*l* [=>] 1.36	\[ \mathsf{fma}\left(y, z, \left(a \cdot t + \color{blue}{a \cdot \left(z \cdot b\right)}\right) + x\right) \]
      distribute-lft-out [=>] 1.36	\[ \mathsf{fma}\left(y, z, \color{blue}{a \cdot \left(t + z \cdot b\right)} + x\right) \]
      fma-def [=>] 1.35	\[ \mathsf{fma}\left(y, z, \color{blue}{\mathsf{fma}\left(a, t + z \cdot b, x\right)}\right) \]
      +-commutative [=>] 1.35	\[ \mathsf{fma}\left(y, z, \mathsf{fma}\left(a, \color{blue}{z \cdot b + t}, x\right)\right) \]
      fma-def [=>] 1.35	\[ \mathsf{fma}\left(y, z, \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(z, b, t\right)}, x\right)\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 1.13

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+86} \lor \neg \left(z \leq 5 \cdot 10^{+141}\right):\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), \mathsf{fma}\left(t, a, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z, \mathsf{fma}\left(a, \mathsf{fma}\left(z, b, t\right), x\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	2.14%
Cost	19780

\[\begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-131}:\\ \;\;\;\;\mathsf{fma}\left(y, z, \mathsf{fma}\left(a, \mathsf{fma}\left(z, b, t\right), x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot t + \left(x + z \cdot y\right)\right) + b \cdot \left(z \cdot a\right)\\ \end{array} \]

Alternative 2
Error	2.12%
Cost	13508

\[\begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-128}:\\ \;\;\;\;\mathsf{fma}\left(a, t + z \cdot b, \mathsf{fma}\left(y, z, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot t + \left(x + z \cdot y\right)\right) + b \cdot \left(z \cdot a\right)\\ \end{array} \]

Alternative 3
Error	65.51%
Cost	1644

\[\begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+126}:\\ \;\;\;\;z \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{+72}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1 \cdot 10^{+19}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;z \leq -6.2 \cdot 10^{-53}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;z \leq -5.3 \cdot 10^{-68}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -6.4 \cdot 10^{-111}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;z \leq -1.05 \cdot 10^{-186}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{-289}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-186}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{-122}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+40}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \]

Alternative 4
Error	34.03%
Cost	1376

\[\begin{array}{l} t_1 := x + b \cdot \left(z \cdot a\right)\\ t_2 := x + a \cdot t\\ t_3 := z \cdot y + a \cdot t\\ t_4 := x + z \cdot y\\ \mathbf{if}\;x \leq -4.4 \cdot 10^{+148}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{+123}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -310000000:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-85}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-141}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 10^{-127}:\\ \;\;\;\;x + a \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-53}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 5
Error	34.2%
Cost	1245

\[\begin{array}{l} t_1 := x + a \cdot t\\ t_2 := a \cdot \left(t + z \cdot b\right)\\ t_3 := x + z \cdot y\\ \mathbf{if}\;z \leq -1.15 \cdot 10^{+48}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{-52}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.16 \cdot 10^{-66}:\\ \;\;\;\;x + b \cdot \left(z \cdot a\right)\\ \mathbf{elif}\;z \leq -1.75 \cdot 10^{-93}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{-113}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.22 \cdot 10^{-141} \lor \neg \left(z \leq 1350000\right):\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	35.26%
Cost	1240

\[\begin{array}{l} t_1 := a \cdot \left(t + z \cdot b\right)\\ t_2 := x + z \cdot y\\ \mathbf{if}\;x \leq -2300000000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-69}:\\ \;\;\;\;x + a \cdot t\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-144}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -6.3 \cdot 10^{-245}:\\ \;\;\;\;z \cdot \left(y + a \cdot b\right)\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-54}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	2.12%
Cost	1092

\[\begin{array}{l} t_1 := x + z \cdot y\\ \mathbf{if}\;a \leq -1 \cdot 10^{-128}:\\ \;\;\;\;\left(a \cdot \left(z \cdot b\right) + a \cdot t\right) + t_1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot t + t_1\right) + b \cdot \left(z \cdot a\right)\\ \end{array} \]

Alternative 8
Error	41.75%
Cost	981

\[\begin{array}{l} t_1 := x + a \cdot t\\ \mathbf{if}\;z \leq -1.1 \cdot 10^{+161}:\\ \;\;\;\;z \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;z \leq -1.55 \cdot 10^{+67}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.6 \cdot 10^{+19}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+37} \lor \neg \left(z \leq 1.4 \cdot 10^{+95}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \]

Alternative 9
Error	3.96%
Cost	960

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

Alternative 10
Error	15.39%
Cost	841

\[\begin{array}{l} \mathbf{if}\;b \leq -3.4 \cdot 10^{+180} \lor \neg \left(b \leq 6.5 \cdot 10^{+170}\right):\\ \;\;\;\;x + b \cdot \left(z \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot y + \left(x + a \cdot t\right)\\ \end{array} \]

Alternative 11
Error	12.62%
Cost	841

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{+32} \lor \neg \left(t \leq 8.6 \cdot 10^{-93}\right):\\ \;\;\;\;z \cdot y + \left(x + a \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(y + a \cdot b\right)\\ \end{array} \]

Alternative 12
Error	32.79%
Cost	716

\[\begin{array}{l} t_1 := x + z \cdot y\\ \mathbf{if}\;z \leq -1.15 \cdot 10^{+48}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.6 \cdot 10^{+19}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;z \leq 85000:\\ \;\;\;\;x + a \cdot t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	31.4%
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{+19}:\\ \;\;\;\;z \cdot \left(y + a \cdot b\right)\\ \mathbf{elif}\;z \leq 6500:\\ \;\;\;\;x + a \cdot t\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot y\\ \end{array} \]

Alternative 14
Error	51.95%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -3000000000000:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{-42}:\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 15
Error	62.61%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023103 
(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)))