Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Average Error: 2.1 → 0.7

Time: 12.0s

Precision: binary64

Cost: 19913

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-169} \lor \neg \left(a \leq 1.4 \cdot 10^{+69}\right):\\ \;\;\;\;\mathsf{fma}\left(a, t + z \cdot b, \mathsf{fma}\left(y, z, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), \mathsf{fma}\left(t, a, x\right)\right)\\ \end{array} \]

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 -1e-169) (not (<= a 1.4e+69)))
   (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 <= -1e-169) || !(a <= 1.4e+69)) {
		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 <= -1e-169) || !(a <= 1.4e+69))
		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, -1e-169], N[Not[LessEqual[a, 1.4e+69]], $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	2.1
Target	0.3
Herbie	0.7

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

Derivation

1. Split input into 2 regimes

2. if a < -1.00000000000000002e-169 or 1.39999999999999991e69 < a

   1. Initial program 3.9

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

   2. Simplified 1.2

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

      [Start] 3.9	\[ \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
      associate-+l+ [=>] 3.9	\[ \color{blue}{\left(x + y \cdot z\right) + \left(t \cdot a + \left(a \cdot z\right) \cdot b\right)} \]
      +-commutative [=>] 3.9	\[ \color{blue}{\left(t \cdot a + \left(a \cdot z\right) \cdot b\right) + \left(x + y \cdot z\right)} \]
      *-commutative [=>] 3.9	\[ \left(\color{blue}{a \cdot t} + \left(a \cdot z\right) \cdot b\right) + \left(x + y \cdot z\right) \]
      associate-*l* [=>] 1.2	\[ \left(a \cdot t + \color{blue}{a \cdot \left(z \cdot b\right)}\right) + \left(x + y \cdot z\right) \]
      distribute-lft-out [=>] 1.2	\[ \color{blue}{a \cdot \left(t + z \cdot b\right)} + \left(x + y \cdot z\right) \]
      fma-def [=>] 1.2	\[ \color{blue}{\mathsf{fma}\left(a, t + z \cdot b, x + y \cdot z\right)} \]
      +-commutative [=>] 1.2	\[ \mathsf{fma}\left(a, t + z \cdot b, \color{blue}{y \cdot z + x}\right) \]
      fma-def [=>] 1.2	\[ \mathsf{fma}\left(a, t + z \cdot b, \color{blue}{\mathsf{fma}\left(y, z, x\right)}\right) \]

   if -1.00000000000000002e-169 < a < 1.39999999999999991e69

   1. Initial program 0.6

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

   2. Simplified 0.3

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

      [Start] 0.6	\[ \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
      +-commutative [=>] 0.6	\[ \color{blue}{\left(a \cdot z\right) \cdot b + \left(\left(x + y \cdot z\right) + t \cdot a\right)} \]
      +-commutative [=>] 0.6	\[ \left(a \cdot z\right) \cdot b + \left(\color{blue}{\left(y \cdot z + x\right)} + t \cdot a\right) \]
      associate-+l+ [=>] 0.6	\[ \left(a \cdot z\right) \cdot b + \color{blue}{\left(y \cdot z + \left(x + t \cdot a\right)\right)} \]
      associate-+r+ [=>] 0.6	\[ \color{blue}{\left(\left(a \cdot z\right) \cdot b + y \cdot z\right) + \left(x + t \cdot a\right)} \]
      *-commutative [=>] 0.6	\[ \left(\color{blue}{\left(z \cdot a\right)} \cdot b + y \cdot z\right) + \left(x + t \cdot a\right) \]
      associate-*l* [=>] 0.3	\[ \left(\color{blue}{z \cdot \left(a \cdot b\right)} + y \cdot z\right) + \left(x + t \cdot a\right) \]
      *-commutative [=>] 0.3	\[ \left(z \cdot \left(a \cdot b\right) + \color{blue}{z \cdot y}\right) + \left(x + t \cdot a\right) \]
      distribute-lft-out [=>] 0.3	\[ \color{blue}{z \cdot \left(a \cdot b + y\right)} + \left(x + t \cdot a\right) \]
      fma-def [=>] 0.3	\[ \color{blue}{\mathsf{fma}\left(z, a \cdot b + y, x + t \cdot a\right)} \]
      fma-def [=>] 0.3	\[ \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(a, b, y\right)}, x + t \cdot a\right) \]
      +-commutative [=>] 0.3	\[ \mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), \color{blue}{t \cdot a + x}\right) \]
      fma-def [=>] 0.3	\[ \mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), \color{blue}{\mathsf{fma}\left(t, a, x\right)}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-169} \lor \neg \left(a \leq 1.4 \cdot 10^{+69}\right):\\ \;\;\;\;\mathsf{fma}\left(a, t + z \cdot b, \mathsf{fma}\left(y, z, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), \mathsf{fma}\left(t, a, x\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	2.6
Cost	13376

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

Alternative 2
Error	22.5
Cost	1376

\[\begin{array}{l} t_1 := z \cdot \left(y + a \cdot b\right)\\ t_2 := x + a \cdot t\\ t_3 := a \cdot \left(t + z \cdot b\right)\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{+17}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{-29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3.6 \cdot 10^{-72}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -4.4 \cdot 10^{-282}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.14 \cdot 10^{-259}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{-139}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-13}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+21}:\\ \;\;\;\;x + z \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	25.3
Cost	1114

\[\begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+163}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;y \leq -9.2 \cdot 10^{+57} \lor \neg \left(y \leq -6.2 \cdot 10^{+38}\right) \land \left(y \leq 7.6 \cdot 10^{+15} \lor \neg \left(y \leq 2.3 \cdot 10^{+27}\right) \land y \leq 1.4 \cdot 10^{+129}\right):\\ \;\;\;\;x + a \cdot t\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \]

Alternative 4
Error	23.0
Cost	978

\[\begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{+17} \lor \neg \left(x \leq 1.32 \cdot 10^{-291}\right) \land \left(x \leq 1.66 \cdot 10^{-261} \lor \neg \left(x \leq 4.8 \cdot 10^{-134}\right)\right):\\ \;\;\;\;x + a \cdot t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y + a \cdot b\right)\\ \end{array} \]

Alternative 5
Error	7.8
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-24} \lor \neg \left(y \leq 7.5 \cdot 10^{+14}\right):\\ \;\;\;\;z \cdot y + \left(x + a \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot t + \left(x + a \cdot \left(z \cdot b\right)\right)\\ \end{array} \]

Alternative 6
Error	2.6
Cost	960

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

Alternative 7
Error	32.7
Cost	852

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+18}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-277}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-255}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-136}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-13}:\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	20.5
Cost	850

\[\begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+163} \lor \neg \left(y \leq -1.7 \cdot 10^{+59}\right) \land \left(y \leq -2.4 \cdot 10^{-22} \lor \neg \left(y \leq 1.3 \cdot 10^{-15}\right)\right):\\ \;\;\;\;x + z \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + a \cdot t\\ \end{array} \]

Alternative 9
Error	18.7
Cost	848

\[\begin{array}{l} t_1 := x + a \cdot t\\ \mathbf{if}\;x \leq -4.5 \cdot 10^{+22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-78}:\\ \;\;\;\;z \cdot y + a \cdot t\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-13}:\\ \;\;\;\;a \cdot \left(t + z \cdot b\right)\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+21}:\\ \;\;\;\;x + z \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	7.8
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-23} \lor \neg \left(y \leq 7.2 \cdot 10^{+14}\right):\\ \;\;\;\;z \cdot y + \left(x + a \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \end{array} \]

Alternative 11
Error	14.5
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -5.6 \cdot 10^{+38}:\\ \;\;\;\;z \cdot y + a \cdot t\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+16}:\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot y\\ \end{array} \]

Alternative 12
Error	33.5
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{+101}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-13}:\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13
Error	40.0
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023027 
(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)))