Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Average Error: 1.9 → 0.5

Time: 11.9s

Precision: binary64

Cost: 19913

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+118} \lor \neg \left(z \leq 4 \cdot 10^{-77}\right):\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), \mathsf{fma}\left(t, a, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t + z \cdot b, \mathsf{fma}\left(y, z, x\right)\right)\\ \end{array} \]

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 -3e+118) (not (<= z 4e-77)))
   (fma z (fma a b y) (fma t a x))
   (fma a (+ t (* z b)) (fma y z 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 <= -3e+118) || !(z <= 4e-77)) {
		tmp = fma(z, fma(a, b, y), fma(t, a, x));
	} else {
		tmp = fma(a, (t + (z * b)), fma(y, z, 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 <= -3e+118) || !(z <= 4e-77))
		tmp = fma(z, fma(a, b, y), fma(t, a, x));
	else
		tmp = fma(a, Float64(t + Float64(z * b)), fma(y, z, 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, -3e+118], N[Not[LessEqual[z, 4e-77]], $MachinePrecision]], N[(z * N[(a * b + y), $MachinePrecision] + N[(t * a + x), $MachinePrecision]), $MachinePrecision], N[(a * N[(t + N[(z * b), $MachinePrecision]), $MachinePrecision] + N[(y * z + x), $MachinePrecision]), $MachinePrecision]]

Target

Original	1.9
Target	0.3
Herbie	0.5

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

Derivation

1. Split input into 2 regimes

2. if z < -3e118 or 3.9999999999999997e-77 < z

   1. Initial program 4.3

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

   2. Simplified 0.4

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

      [Start] 4.3	\[ \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
      +-commutative [=>] 4.3	\[ \color{blue}{\left(a \cdot z\right) \cdot b + \left(\left(x + y \cdot z\right) + t \cdot a\right)} \]
      +-commutative [=>] 4.3	\[ \left(a \cdot z\right) \cdot b + \left(\color{blue}{\left(y \cdot z + x\right)} + t \cdot a\right) \]
      associate-+l+ [=>] 4.3	\[ \left(a \cdot z\right) \cdot b + \color{blue}{\left(y \cdot z + \left(x + t \cdot a\right)\right)} \]
      associate-+r+ [=>] 4.3	\[ \color{blue}{\left(\left(a \cdot z\right) \cdot b + y \cdot z\right) + \left(x + t \cdot a\right)} \]
      *-commutative [=>] 4.3	\[ \left(\color{blue}{\left(z \cdot a\right)} \cdot b + y \cdot z\right) + \left(x + t \cdot a\right) \]
      associate-*l* [=>] 0.4	\[ \left(\color{blue}{z \cdot \left(a \cdot b\right)} + y \cdot z\right) + \left(x + t \cdot a\right) \]
      *-commutative [=>] 0.4	\[ \left(z \cdot \left(a \cdot b\right) + \color{blue}{z \cdot y}\right) + \left(x + t \cdot a\right) \]
      distribute-lft-out [=>] 0.4	\[ \color{blue}{z \cdot \left(a \cdot b + y\right)} + \left(x + t \cdot a\right) \]
      fma-def [=>] 0.4	\[ \color{blue}{\mathsf{fma}\left(z, a \cdot b + y, x + t \cdot a\right)} \]
      fma-def [=>] 0.4	\[ \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(a, b, y\right)}, x + t \cdot a\right) \]
      +-commutative [=>] 0.4	\[ \mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), \color{blue}{t \cdot a + x}\right) \]
      fma-def [=>] 0.4	\[ \mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), \color{blue}{\mathsf{fma}\left(t, a, x\right)}\right) \]

   if -3e118 < z < 3.9999999999999997e-77

   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 0.5

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

      [Start] 0.6	\[ \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
      associate-+l+ [=>] 0.6	\[ \color{blue}{\left(x + y \cdot z\right) + \left(t \cdot a + \left(a \cdot z\right) \cdot b\right)} \]
      +-commutative [=>] 0.6	\[ \color{blue}{\left(t \cdot a + \left(a \cdot z\right) \cdot b\right) + \left(x + y \cdot z\right)} \]
      *-commutative [=>] 0.6	\[ \left(\color{blue}{a \cdot t} + \left(a \cdot z\right) \cdot b\right) + \left(x + y \cdot z\right) \]
      associate-*l* [=>] 0.5	\[ \left(a \cdot t + \color{blue}{a \cdot \left(z \cdot b\right)}\right) + \left(x + y \cdot z\right) \]
      distribute-lft-out [=>] 0.5	\[ \color{blue}{a \cdot \left(t + z \cdot b\right)} + \left(x + y \cdot z\right) \]
      fma-def [=>] 0.5	\[ \color{blue}{\mathsf{fma}\left(a, t + z \cdot b, x + y \cdot z\right)} \]
      +-commutative [=>] 0.5	\[ \mathsf{fma}\left(a, t + z \cdot b, \color{blue}{y \cdot z + x}\right) \]
      fma-def [=>] 0.5	\[ \mathsf{fma}\left(a, t + z \cdot b, \color{blue}{\mathsf{fma}\left(y, z, x\right)}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+118} \lor \neg \left(z \leq 4 \cdot 10^{-77}\right):\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), \mathsf{fma}\left(t, a, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t + z \cdot b, \mathsf{fma}\left(y, z, x\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	13641

\[\begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+22} \lor \neg \left(b \leq 10^{-78}\right):\\ \;\;\;\;\left(a \cdot t + \left(x + z \cdot y\right)\right) + b \cdot \left(z \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t + z \cdot b, \mathsf{fma}\left(y, z, x\right)\right)\\ \end{array} \]

Alternative 2
Error	20.3
Cost	1504

\[\begin{array}{l} t_1 := a \cdot t + z \cdot y\\ t_2 := x + a \cdot t\\ t_3 := z \cdot \left(y + a \cdot b\right)\\ \mathbf{if}\;x \leq -17500000000:\\ \;\;\;\;x + z \cdot y\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-42}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-90}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -2.06 \cdot 10^{-109}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{-207}:\\ \;\;\;\;a \cdot \left(t + z \cdot b\right)\\ \mathbf{elif}\;x \leq 1.255 \cdot 10^{-290}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-34}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	26.6
Cost	1247

\[\begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+108} \lor \neg \left(z \leq -3.5 \cdot 10^{+36}\right) \land \left(z \leq -1.8 \cdot 10^{-63} \lor \neg \left(z \leq 1.7 \cdot 10^{+34}\right) \land \left(z \leq 5.2 \cdot 10^{+64} \lor \neg \left(z \leq 2 \cdot 10^{+128}\right) \land z \leq 1.8 \cdot 10^{+261}\right)\right):\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + a \cdot t\\ \end{array} \]

Alternative 4
Error	21.3
Cost	1241

\[\begin{array}{l} t_1 := x + a \cdot t\\ t_2 := z \cdot \left(y + a \cdot b\right)\\ \mathbf{if}\;z \leq -2.6 \cdot 10^{+101}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{+35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{-67}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 7.4 \cdot 10^{+28}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+123} \lor \neg \left(z \leq 4.9 \cdot 10^{+249}\right):\\ \;\;\;\;x + z \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	15.4
Cost	1241

\[\begin{array}{l} t_1 := x + a \cdot \left(t + z \cdot b\right)\\ t_2 := z \cdot \left(y + a \cdot b\right)\\ t_3 := x + z \cdot y\\ \mathbf{if}\;z \leq -2.95 \cdot 10^{+107}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{+40}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{-62}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+33}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+123} \lor \neg \left(z \leq 10^{+251}\right):\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	2.9
Cost	1225

\[\begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-85} \lor \neg \left(a \leq 4 \cdot 10^{-171}\right):\\ \;\;\;\;\left(a \cdot \left(z \cdot b\right) + a \cdot t\right) + \left(x + z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + a \cdot t\right) + z \cdot y\\ \end{array} \]

Alternative 7
Error	0.3
Cost	1225

\[\begin{array}{l} t_1 := x + z \cdot y\\ \mathbf{if}\;b \leq -10000000000 \lor \neg \left(b \leq 10^{-78}\right):\\ \;\;\;\;\left(a \cdot t + t_1\right) + b \cdot \left(z \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(z \cdot b\right) + a \cdot t\right) + t_1\\ \end{array} \]

Alternative 8
Error	8.0
Cost	969

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-48} \lor \neg \left(x \leq 3.5 \cdot 10^{-140}\right):\\ \;\;\;\;\left(x + a \cdot t\right) + z \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot t + z \cdot \left(y + a \cdot b\right)\\ \end{array} \]

Alternative 9
Error	32.6
Cost	852

\[\begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{+16}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.02 \cdot 10^{-208}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-293}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-249}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-34}:\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	21.0
Cost	850

\[\begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+116} \lor \neg \left(y \leq -3.1 \cdot 10^{-37}\right) \land \left(y \leq -7.6 \cdot 10^{-125} \lor \neg \left(y \leq 4.1 \cdot 10^{-26}\right)\right):\\ \;\;\;\;x + z \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + a \cdot t\\ \end{array} \]

Alternative 11
Error	8.7
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{-135} \lor \neg \left(y \leq 3.8 \cdot 10^{-160}\right):\\ \;\;\;\;\left(x + a \cdot t\right) + z \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \end{array} \]

Alternative 12
Error	32.6
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+16}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+14}:\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13
Error	39.7
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022349 
(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)))