Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Average Accuracy: 96.5% → 99.8%

Time: 15.9s

Precision: binary64

Cost: 19913

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+27} \lor \neg \left(z \leq 2 \cdot 10^{-56}\right):\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), \mathsf{fma}\left(t, a, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t + z \cdot b, \mathsf{fma}\left(y, z, x\right)\right)\\ \end{array} \]

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 -4e+27) (not (<= z 2e-56)))
   (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 <= -4e+27) || !(z <= 2e-56)) {
		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 <= -4e+27) || !(z <= 2e-56))
		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, -4e+27], N[Not[LessEqual[z, 2e-56]], $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	96.5%
Target	99.4%
Herbie	99.8%

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

Derivation

1. Split input into 2 regimes

2. if z < -4.0000000000000001e27 or 2.0000000000000001e-56 < z

   1. Initial program 92.1%

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

   2. Simplified 99.6%

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

      [Start] 92.1	\[ \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
      +-commutative [=>] 92.1	\[ \color{blue}{\left(a \cdot z\right) \cdot b + \left(\left(x + y \cdot z\right) + t \cdot a\right)} \]
      +-commutative [=>] 92.1	\[ \left(a \cdot z\right) \cdot b + \left(\color{blue}{\left(y \cdot z + x\right)} + t \cdot a\right) \]
      associate-+l+ [=>] 92.1	\[ \left(a \cdot z\right) \cdot b + \color{blue}{\left(y \cdot z + \left(x + t \cdot a\right)\right)} \]
      associate-+r+ [=>] 92.1	\[ \color{blue}{\left(\left(a \cdot z\right) \cdot b + y \cdot z\right) + \left(x + t \cdot a\right)} \]
      *-commutative [=>] 92.1	\[ \left(\color{blue}{\left(z \cdot a\right)} \cdot b + y \cdot z\right) + \left(x + t \cdot a\right) \]
      associate-*l* [=>] 99.6	\[ \left(\color{blue}{z \cdot \left(a \cdot b\right)} + y \cdot z\right) + \left(x + t \cdot a\right) \]
      *-commutative [=>] 99.6	\[ \left(z \cdot \left(a \cdot b\right) + \color{blue}{z \cdot y}\right) + \left(x + t \cdot a\right) \]
      distribute-lft-out [=>] 99.6	\[ \color{blue}{z \cdot \left(a \cdot b + y\right)} + \left(x + t \cdot a\right) \]
      fma-def [=>] 99.6	\[ \color{blue}{\mathsf{fma}\left(z, a \cdot b + y, x + t \cdot a\right)} \]
      fma-def [=>] 99.6	\[ \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(a, b, y\right)}, x + t \cdot a\right) \]
      +-commutative [=>] 99.6	\[ \mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), \color{blue}{t \cdot a + x}\right) \]
      fma-def [=>] 99.6	\[ \mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), \color{blue}{\mathsf{fma}\left(t, a, x\right)}\right) \]

   if -4.0000000000000001e27 < z < 2.0000000000000001e-56

   1. Initial program 99.4%

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

   2. Simplified 99.9%

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

      [Start] 99.4	\[ \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
      associate-+l+ [=>] 99.4	\[ \color{blue}{\left(x + y \cdot z\right) + \left(t \cdot a + \left(a \cdot z\right) \cdot b\right)} \]
      +-commutative [=>] 99.4	\[ \color{blue}{\left(t \cdot a + \left(a \cdot z\right) \cdot b\right) + \left(x + y \cdot z\right)} \]
      *-commutative [=>] 99.4	\[ \left(\color{blue}{a \cdot t} + \left(a \cdot z\right) \cdot b\right) + \left(x + y \cdot z\right) \]
      associate-*l* [=>] 99.9	\[ \left(a \cdot t + \color{blue}{a \cdot \left(z \cdot b\right)}\right) + \left(x + y \cdot z\right) \]
      distribute-lft-out [=>] 99.9	\[ \color{blue}{a \cdot \left(t + z \cdot b\right)} + \left(x + y \cdot z\right) \]
      fma-def [=>] 99.9	\[ \color{blue}{\mathsf{fma}\left(a, t + z \cdot b, x + y \cdot z\right)} \]
      +-commutative [=>] 99.9	\[ \mathsf{fma}\left(a, t + z \cdot b, \color{blue}{y \cdot z + x}\right) \]
      fma-def [=>] 99.9	\[ \mathsf{fma}\left(a, t + z \cdot b, \color{blue}{\mathsf{fma}\left(y, z, x\right)}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+27} \lor \neg \left(z \leq 2 \cdot 10^{-56}\right):\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), \mathsf{fma}\left(t, a, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t + z \cdot b, \mathsf{fma}\left(y, z, x\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.3%
Cost	13508

\[\begin{array}{l} t_1 := x + z \cdot y\\ \mathbf{if}\;a \leq -1 \cdot 10^{+109}:\\ \;\;\;\;\mathsf{fma}\left(a, t + z \cdot b, \mathsf{fma}\left(y, z, x\right)\right)\\ \mathbf{elif}\;a \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\left(t_1 + a \cdot t\right) + b \cdot \left(z \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \mathsf{fma}\left(t, a, a \cdot \left(z \cdot b\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	99.3%
Cost	8904

\[\begin{array}{l} t_1 := x + z \cdot y\\ t_2 := \left(t_1 + a \cdot t\right) + b \cdot \left(z \cdot a\right)\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1 + \left(a \cdot \left(z \cdot b\right) + a \cdot t\right)\\ \mathbf{elif}\;t_2 \leq 4 \cdot 10^{+307}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t + z \cdot b, a, x\right)\\ \end{array} \]

Alternative 3
Accuracy	99.3%
Cost	7496

\[\begin{array}{l} t_1 := a \cdot \left(z \cdot b\right)\\ t_2 := x + z \cdot y\\ \mathbf{if}\;a \leq -7.2 \cdot 10^{+108}:\\ \;\;\;\;t_2 + \left(t_1 + a \cdot t\right)\\ \mathbf{elif}\;a \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\left(t_2 + a \cdot t\right) + b \cdot \left(z \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 + \mathsf{fma}\left(t, a, t_1\right)\\ \end{array} \]

Alternative 4
Accuracy	47.3%
Cost	1244

\[\begin{array}{l} t_1 := a \cdot \left(z \cdot b\right)\\ \mathbf{if}\;x \leq -3000000000:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-275}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-230}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-208}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-53}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2200000:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;x \leq 45000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Accuracy	47.0%
Cost	1244

\[\begin{array}{l} \mathbf{if}\;x \leq -13500000:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-276}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;x \leq 10^{-230}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-207}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-40}:\\ \;\;\;\;z \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;x \leq 45:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+17}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Accuracy	54.0%
Cost	1244

\[\begin{array}{l} t_1 := x + a \cdot t\\ \mathbf{if}\;x \leq -7.8 \cdot 10^{-59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-98}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;x \leq -5.9 \cdot 10^{-139}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{-249}:\\ \;\;\;\;b \cdot \left(z \cdot a\right)\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-276}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-179}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-40}:\\ \;\;\;\;z \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Accuracy	64.6%
Cost	1240

\[\begin{array}{l} t_1 := a \cdot \left(t + z \cdot b\right)\\ t_2 := x + a \cdot t\\ t_3 := z \cdot \left(y + a \cdot b\right)\\ \mathbf{if}\;x \leq -7.2 \cdot 10^{+39}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-11}:\\ \;\;\;\;x + z \cdot y\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-57}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-277}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-183}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6.3 \cdot 10^{-40}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Accuracy	65.1%
Cost	1240

\[\begin{array}{l} t_1 := x + a \cdot t\\ t_2 := z \cdot \left(y + a \cdot b\right)\\ \mathbf{if}\;x \leq -7 \cdot 10^{+39}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-14}:\\ \;\;\;\;x + z \cdot y\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-57}:\\ \;\;\;\;a \cdot \left(t + z \cdot b\right)\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-292}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-246}:\\ \;\;\;\;a \cdot t + z \cdot y\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-39}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Accuracy	99.3%
Cost	1225

\[\begin{array}{l} t_1 := x + z \cdot y\\ \mathbf{if}\;a \leq -2 \cdot 10^{+110} \lor \neg \left(a \leq 5 \cdot 10^{-12}\right):\\ \;\;\;\;t_1 + \left(a \cdot \left(z \cdot b\right) + a \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 + a \cdot t\right) + b \cdot \left(z \cdot a\right)\\ \end{array} \]

Alternative 10
Accuracy	96.1%
Cost	1224

\[\begin{array}{l} t_1 := x + z \cdot \left(y + a \cdot b\right)\\ \mathbf{if}\;z \leq -5.2 \cdot 10^{+46}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+113}:\\ \;\;\;\;\left(x + z \cdot y\right) + \left(a \cdot \left(z \cdot b\right) + a \cdot t\right)\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+288}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\\ \end{array} \]

Alternative 11
Accuracy	65.8%
Cost	1112

\[\begin{array}{l} t_1 := x + a \cdot t\\ t_2 := b \cdot \left(z \cdot a\right)\\ t_3 := x + z \cdot y\\ \mathbf{if}\;y \leq -3.1 \cdot 10^{+17}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -5 \cdot 10^{-142}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -2.9 \cdot 10^{-252}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{-290}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-53}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 12
Accuracy	71.4%
Cost	1104

\[\begin{array}{l} t_1 := z \cdot \left(y + a \cdot b\right)\\ t_2 := x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{-139}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-292}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-244}:\\ \;\;\;\;a \cdot t + z \cdot y\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-40}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 13
Accuracy	64.7%
Cost	977

\[\begin{array}{l} t_1 := x + a \cdot t\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{+37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -9.8 \cdot 10^{-9}:\\ \;\;\;\;x + z \cdot y\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-25} \lor \neg \left(x \leq 4.6 \cdot 10^{-40}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y + a \cdot b\right)\\ \end{array} \]

Alternative 14
Accuracy	84.5%
Cost	972

\[\begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+26}:\\ \;\;\;\;\left(x + z \cdot y\right) + a \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-53}:\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{elif}\;y \leq 108000000000:\\ \;\;\;\;x + z \cdot \left(y + a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + a \cdot t\right) + z \cdot y\\ \end{array} \]

Alternative 15
Accuracy	49.1%
Cost	852

\[\begin{array}{l} \mathbf{if}\;x \leq -47000000:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-275}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{-158}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{-40}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;x \leq 205000000:\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 16
Accuracy	47.4%
Cost	848

\[\begin{array}{l} \mathbf{if}\;x \leq -8200000:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-275}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-178}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-15}:\\ \;\;\;\;b \cdot \left(z \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 17
Accuracy	86.9%
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -9.4 \cdot 10^{+24} \lor \neg \left(y \leq 0.00046\right):\\ \;\;\;\;\left(x + a \cdot t\right) + z \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \end{array} \]

Alternative 18
Accuracy	86.3%
Cost	841

\[\begin{array}{l} \mathbf{if}\;a \leq -5.6 \cdot 10^{-69} \lor \neg \left(a \leq 8.5 \cdot 10^{-101}\right):\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(y + a \cdot b\right)\\ \end{array} \]

Alternative 19
Accuracy	49.0%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -1.16 \cdot 10^{-14}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 50000000:\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 20
Accuracy	37.7%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023122 
(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)))