Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Average Error: 1.9 → 0.2

Time: 12.9s

Precision: binary64

Cost: 19913

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \leq -2000 \lor \neg \left(a \leq 5 \cdot 10^{-83}\right):\\ \;\;\;\;\mathsf{fma}\left(a, t + z \cdot b, \mathsf{fma}\left(y, z, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), \mathsf{fma}\left(t, a, x\right)\right)\\ \end{array} \]

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 (<= a -2000.0) (not (<= a 5e-83)))
   (fma a (+ t (* z b)) (fma y z x))
   (fma z (fma a b y) (fma t a 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 ((a <= -2000.0) || !(a <= 5e-83)) {
		tmp = fma(a, (t + (z * b)), fma(y, z, x));
	} else {
		tmp = fma(z, fma(a, b, y), fma(t, a, 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 ((a <= -2000.0) || !(a <= 5e-83))
		tmp = fma(a, Float64(t + Float64(z * b)), fma(y, z, x));
	else
		tmp = fma(z, fma(a, b, y), fma(t, a, 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[a, -2000.0], N[Not[LessEqual[a, 5e-83]], $MachinePrecision]], N[(a * N[(t + N[(z * b), $MachinePrecision]), $MachinePrecision] + N[(y * z + x), $MachinePrecision]), $MachinePrecision], N[(z * N[(a * b + y), $MachinePrecision] + N[(t * a + x), $MachinePrecision]), $MachinePrecision]]

Target

Original	1.9
Target	0.3
Herbie	0.2

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

Derivation

1. Split input into 2 regimes

2. if a < -2e3 or 5e-83 < a

   1. Initial program 3.8

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

   2. Simplified 0.3

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

      [Start] 3.8	\[ \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
      associate-+l+ [=>] 3.8	\[ \color{blue}{\left(x + y \cdot z\right) + \left(t \cdot a + \left(a \cdot z\right) \cdot b\right)} \]
      +-commutative [=>] 3.8	\[ \color{blue}{\left(t \cdot a + \left(a \cdot z\right) \cdot b\right) + \left(x + y \cdot z\right)} \]
      *-commutative [=>] 3.8	\[ \left(\color{blue}{a \cdot t} + \left(a \cdot z\right) \cdot b\right) + \left(x + y \cdot z\right) \]
      associate-*l* [=>] 0.3	\[ \left(a \cdot t + \color{blue}{a \cdot \left(z \cdot b\right)}\right) + \left(x + y \cdot z\right) \]
      distribute-lft-out [=>] 0.3	\[ \color{blue}{a \cdot \left(t + z \cdot b\right)} + \left(x + y \cdot z\right) \]
      fma-def [=>] 0.3	\[ \color{blue}{\mathsf{fma}\left(a, t + z \cdot b, x + y \cdot z\right)} \]
      +-commutative [=>] 0.3	\[ \mathsf{fma}\left(a, t + z \cdot b, \color{blue}{y \cdot z + x}\right) \]
      fma-def [=>] 0.3	\[ \mathsf{fma}\left(a, t + z \cdot b, \color{blue}{\mathsf{fma}\left(y, z, x\right)}\right) \]

   if -2e3 < a < 5e-83

   1. Initial program 0.3

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

   2. Simplified 0.0

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

      [Start] 0.3	\[ \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
      +-commutative [=>] 0.3	\[ \color{blue}{\left(a \cdot z\right) \cdot b + \left(\left(x + y \cdot z\right) + t \cdot a\right)} \]
      +-commutative [=>] 0.3	\[ \left(a \cdot z\right) \cdot b + \left(\color{blue}{\left(y \cdot z + x\right)} + t \cdot a\right) \]
      associate-+l+ [=>] 0.3	\[ \left(a \cdot z\right) \cdot b + \color{blue}{\left(y \cdot z + \left(x + t \cdot a\right)\right)} \]
      associate-+r+ [=>] 0.3	\[ \color{blue}{\left(\left(a \cdot z\right) \cdot b + y \cdot z\right) + \left(x + t \cdot a\right)} \]
      *-commutative [=>] 0.3	\[ \left(\color{blue}{\left(z \cdot a\right)} \cdot b + y \cdot z\right) + \left(x + t \cdot a\right) \]
      associate-*l* [=>] 0.0	\[ \left(\color{blue}{z \cdot \left(a \cdot b\right)} + y \cdot z\right) + \left(x + t \cdot a\right) \]
      *-commutative [=>] 0.0	\[ \left(z \cdot \left(a \cdot b\right) + \color{blue}{z \cdot y}\right) + \left(x + t \cdot a\right) \]
      distribute-lft-out [=>] 0.0	\[ \color{blue}{z \cdot \left(a \cdot b + y\right)} + \left(x + t \cdot a\right) \]
      fma-def [=>] 0.0	\[ \color{blue}{\mathsf{fma}\left(z, a \cdot b + y, x + t \cdot a\right)} \]
      fma-def [=>] 0.0	\[ \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(a, b, y\right)}, x + t \cdot a\right) \]
      +-commutative [=>] 0.0	\[ \mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), \color{blue}{t \cdot a + x}\right) \]
      fma-def [=>] 0.0	\[ \mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), \color{blue}{\mathsf{fma}\left(t, a, x\right)}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2000 \lor \neg \left(a \leq 5 \cdot 10^{-83}\right):\\ \;\;\;\;\mathsf{fma}\left(a, t + z \cdot b, \mathsf{fma}\left(y, z, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), \mathsf{fma}\left(t, a, x\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	13640

\[\begin{array}{l} t_1 := x + z \cdot y\\ \mathbf{if}\;a \leq -2 \cdot 10^{+109}:\\ \;\;\;\;t_1 + \left(a \cdot \left(z \cdot b\right) + a \cdot t\right)\\ \mathbf{elif}\;a \leq 10^{-86}:\\ \;\;\;\;\left(t_1 + a \cdot t\right) + b \cdot \left(a \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t + z \cdot b, \mathsf{fma}\left(y, z, x\right)\right)\\ \end{array} \]

Alternative 2
Error	0.5
Cost	7496

\[\begin{array}{l} t_1 := a \cdot \left(z \cdot b\right)\\ t_2 := x + z \cdot y\\ \mathbf{if}\;a \leq -1 \cdot 10^{+109}:\\ \;\;\;\;t_2 + \left(t_1 + a \cdot t\right)\\ \mathbf{elif}\;a \leq 10^{-90}:\\ \;\;\;\;\left(t_2 + a \cdot t\right) + b \cdot \left(a \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 + \mathsf{fma}\left(t, a, t_1\right)\\ \end{array} \]

Alternative 3
Error	10.3
Cost	1236

\[\begin{array}{l} t_1 := x + z \cdot y\\ t_2 := t_1 + a \cdot \left(z \cdot b\right)\\ t_3 := x + a \cdot \left(t + z \cdot b\right)\\ t_4 := t_1 + a \cdot t\\ \mathbf{if}\;a \leq -5.5 \cdot 10^{+60}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -1.06 \cdot 10^{-72}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 4.3 \cdot 10^{-215}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;a \leq 2.55 \cdot 10^{-170}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{-102}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 4
Error	0.5
Cost	1225

\[\begin{array}{l} t_1 := x + z \cdot y\\ \mathbf{if}\;a \leq -2 \cdot 10^{+110} \lor \neg \left(a \leq 2 \cdot 10^{-82}\right):\\ \;\;\;\;t_1 + \left(a \cdot \left(z \cdot b\right) + a \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 + a \cdot t\right) + b \cdot \left(a \cdot z\right)\\ \end{array} \]

Alternative 5
Error	33.6
Cost	1116

\[\begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-19}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -9.2 \cdot 10^{-104}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-265}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{-215}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-112}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-41}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+44}:\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	34.2
Cost	1116

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-20}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-105}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-258}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-224}:\\ \;\;\;\;z \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-113}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-41}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{+44}:\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	34.2
Cost	1116

\[\begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-20}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-104}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;x \leq -5.4 \cdot 10^{-261}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-215}:\\ \;\;\;\;b \cdot \left(a \cdot z\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-111}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-41}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+45}:\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	10.5
Cost	1104

\[\begin{array}{l} t_1 := \left(x + z \cdot y\right) + a \cdot t\\ t_2 := x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{if}\;a \leq -1.4 \cdot 10^{+57}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -6000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -2.6 \cdot 10^{-56}:\\ \;\;\;\;x + a \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{-102}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	38.2
Cost	984

\[\begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{+104}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;a \leq 2.7 \cdot 10^{-102}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{-73}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;a \leq 2.6 \cdot 10^{-32}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 85000000000:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{+170}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\\ \end{array} \]

Alternative 10
Error	20.6
Cost	977

\[\begin{array}{l} \mathbf{if}\;a \leq -2.6 \cdot 10^{+58}:\\ \;\;\;\;a \cdot \left(t + z \cdot b\right)\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{-147}:\\ \;\;\;\;x + z \cdot y\\ \mathbf{elif}\;a \leq 9 \cdot 10^{+90} \lor \neg \left(a \leq 9 \cdot 10^{+190}\right):\\ \;\;\;\;x + a \cdot t\\ \mathbf{else}:\\ \;\;\;\;x + a \cdot \left(z \cdot b\right)\\ \end{array} \]

Alternative 11
Error	2.9
Cost	960

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

Alternative 12
Error	33.2
Cost	852

\[\begin{array}{l} \mathbf{if}\;x \leq -5.3 \cdot 10^{-20}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -8.6 \cdot 10^{-105}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-113}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-41}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+43}:\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13
Error	26.4
Cost	849

\[\begin{array}{l} t_1 := x + a \cdot t\\ \mathbf{if}\;x \leq -2.6 \cdot 10^{-20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-63}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-271} \lor \neg \left(x \leq 1.9 \cdot 10^{-215}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(a \cdot z\right)\\ \end{array} \]

Alternative 14
Error	12.5
Cost	841

\[\begin{array}{l} \mathbf{if}\;a \leq -3.2 \cdot 10^{-58} \lor \neg \left(a \leq 7.2 \cdot 10^{-149}\right):\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot y\\ \end{array} \]

Alternative 15
Error	20.5
Cost	585

\[\begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+54} \lor \neg \left(a \leq 2.3 \cdot 10^{-148}\right):\\ \;\;\;\;x + a \cdot t\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot y\\ \end{array} \]

Alternative 16
Error	20.4
Cost	584

\[\begin{array}{l} \mathbf{if}\;a \leq -3.9 \cdot 10^{+56}:\\ \;\;\;\;a \cdot \left(t + z \cdot b\right)\\ \mathbf{elif}\;a \leq 8 \cdot 10^{-149}:\\ \;\;\;\;x + z \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + a \cdot t\\ \end{array} \]

Alternative 17
Error	39.4
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023047 
(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)))