Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Average Accuracy: 97.1% → 96.9%

Time: 15.0s

Precision: binary64

Cost: 19780

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-248}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), \mathsf{fma}\left(t, a, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot t + \left(x + z \cdot y\right)\right) + b \cdot \left(z \cdot a\right)\\ \end{array} \]

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 (<= z -2e-248)
   (fma z (fma a b y) (fma t a x))
   (+ (+ (* a t) (+ x (* z y))) (* b (* z a)))))

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 <= -2e-248) {
		tmp = fma(z, fma(a, b, y), fma(t, a, x));
	} else {
		tmp = ((a * t) + (x + (z * y))) + (b * (z * a));
	}
	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 <= -2e-248)
		tmp = fma(z, fma(a, b, y), fma(t, a, x));
	else
		tmp = Float64(Float64(Float64(a * t) + Float64(x + Float64(z * y))) + Float64(b * Float64(z * a)));
	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[LessEqual[z, -2e-248], N[(z * N[(a * b + y), $MachinePrecision] + N[(t * a + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * t), $MachinePrecision] + N[(x + N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	97.1%
Target	99.5%
Herbie	96.9%

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

Derivation

1. Split input into 2 regimes

2. if z < -1.99999999999999996e-248

   1. Initial program 96.7%

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

   2. Simplified 96.2%

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

      [Start] 96.7	\[ \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
      +-commutative [=>] 96.7	\[ \color{blue}{\left(a \cdot z\right) \cdot b + \left(\left(x + y \cdot z\right) + t \cdot a\right)} \]
      +-commutative [=>] 96.7	\[ \left(a \cdot z\right) \cdot b + \left(\color{blue}{\left(y \cdot z + x\right)} + t \cdot a\right) \]
      associate-+l+ [=>] 96.7	\[ \left(a \cdot z\right) \cdot b + \color{blue}{\left(y \cdot z + \left(x + t \cdot a\right)\right)} \]
      associate-+r+ [=>] 96.7	\[ \color{blue}{\left(\left(a \cdot z\right) \cdot b + y \cdot z\right) + \left(x + t \cdot a\right)} \]
      *-commutative [=>] 96.7	\[ \left(\color{blue}{\left(z \cdot a\right)} \cdot b + y \cdot z\right) + \left(x + t \cdot a\right) \]
      associate-*l* [=>] 96.2	\[ \left(\color{blue}{z \cdot \left(a \cdot b\right)} + y \cdot z\right) + \left(x + t \cdot a\right) \]
      *-commutative [=>] 96.2	\[ \left(z \cdot \left(a \cdot b\right) + \color{blue}{z \cdot y}\right) + \left(x + t \cdot a\right) \]
      distribute-lft-out [=>] 96.2	\[ \color{blue}{z \cdot \left(a \cdot b + y\right)} + \left(x + t \cdot a\right) \]
      fma-def [=>] 96.2	\[ \color{blue}{\mathsf{fma}\left(z, a \cdot b + y, x + t \cdot a\right)} \]
      fma-def [=>] 96.2	\[ \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(a, b, y\right)}, x + t \cdot a\right) \]
      +-commutative [=>] 96.2	\[ \mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), \color{blue}{t \cdot a + x}\right) \]
      fma-def [=>] 96.2	\[ \mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), \color{blue}{\mathsf{fma}\left(t, a, x\right)}\right) \]

   if -1.99999999999999996e-248 < z

   1. Initial program 97.4%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-248}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), \mathsf{fma}\left(t, a, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot t + \left(x + z \cdot y\right)\right) + b \cdot \left(z \cdot a\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.8%
Cost	14404

\[\begin{array}{l} t_1 := x + z \cdot y\\ t_2 := \left(a \cdot t + t_1\right) + b \cdot \left(z \cdot a\right)\\ \mathbf{if}\;t_2 \leq 2 \cdot 10^{-89}:\\ \;\;\;\;\mathsf{fma}\left(a, t + z \cdot b, \mathsf{fma}\left(y, z, x\right)\right)\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+294}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(z \cdot b\right) + a \cdot t\right) + t_1\\ \end{array} \]

Alternative 2
Accuracy	97.8%
Cost	3017

\[\begin{array}{l} t_1 := x + z \cdot y\\ t_2 := \left(a \cdot t + t_1\right) + b \cdot \left(z \cdot a\right)\\ \mathbf{if}\;t_2 \leq 2 \cdot 10^{-89} \lor \neg \left(t_2 \leq 2 \cdot 10^{+294}\right):\\ \;\;\;\;\left(a \cdot \left(z \cdot b\right) + a \cdot t\right) + t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Accuracy	59.5%
Cost	1246

\[\begin{array}{l} t_1 := z \cdot \left(a \cdot b\right)\\ \mathbf{if}\;z \leq -5.8 \cdot 10^{+255}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;z \leq -6.4 \cdot 10^{+140}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.16 \cdot 10^{+111}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;z \leq -5.9 \cdot 10^{+69} \lor \neg \left(z \leq -7.6 \cdot 10^{+53}\right) \land \left(z \leq 3.6 \cdot 10^{+92} \lor \neg \left(z \leq 1.08 \cdot 10^{+201}\right)\right):\\ \;\;\;\;x + a \cdot t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	43.2%
Cost	1244

\[\begin{array}{l} t_1 := a \cdot \left(z \cdot b\right)\\ \mathbf{if}\;x \leq -3.3 \cdot 10^{-76}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{-298}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-189}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-90}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{+78}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+101}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+134}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Accuracy	64.8%
Cost	1240

\[\begin{array}{l} t_1 := x + a \cdot t\\ t_2 := x + z \cdot y\\ t_3 := x + b \cdot \left(z \cdot a\right)\\ \mathbf{if}\;b \leq -4.2 \cdot 10^{+145}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq -2.2 \cdot 10^{-72}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -4.3 \cdot 10^{-172}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{-183}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-100}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{+63}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 6
Accuracy	96.1%
Cost	1225

\[\begin{array}{l} \mathbf{if}\;z \leq 4.8 \cdot 10^{+89} \lor \neg \left(z \leq 4.3 \cdot 10^{+198}\right):\\ \;\;\;\;\left(a \cdot \left(z \cdot b\right) + a \cdot t\right) + \left(x + z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(y + a \cdot b\right)\\ \end{array} \]

Alternative 7
Accuracy	96.1%
Cost	1225

\[\begin{array}{l} t_1 := x + z \cdot y\\ \mathbf{if}\;z \leq 4.5 \cdot 10^{+89} \lor \neg \left(z \leq 1.05 \cdot 10^{+200}\right):\\ \;\;\;\;\left(a \cdot \left(z \cdot b\right) + a \cdot t\right) + t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{\frac{1}{b}}{a}}{z}} + t_1\\ \end{array} \]

Alternative 8
Accuracy	86.3%
Cost	1100

\[\begin{array}{l} t_1 := x + a \cdot t\\ t_2 := x + z \cdot \left(y + a \cdot b\right)\\ \mathbf{if}\;z \leq -3.8 \cdot 10^{+53}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -3.3 \cdot 10^{-78}:\\ \;\;\;\;t_1 + z \cdot y\\ \mathbf{elif}\;z \leq 11.6:\\ \;\;\;\;a \cdot \left(z \cdot b\right) + t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Accuracy	44.1%
Cost	980

\[\begin{array}{l} t_1 := z \cdot \left(a \cdot b\right)\\ \mathbf{if}\;x \leq -2.22 \cdot 10^{-76}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -4.4 \cdot 10^{-175}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-300}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-188}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{+20}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Accuracy	68.2%
Cost	849

\[\begin{array}{l} t_1 := x + z \cdot y\\ \mathbf{if}\;z \leq -6.4 \cdot 10^{+216}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{+140}:\\ \;\;\;\;z \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;z \leq -5.2 \cdot 10^{-79} \lor \neg \left(z \leq 85\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + a \cdot t\\ \end{array} \]

Alternative 11
Accuracy	69.1%
Cost	844

\[\begin{array}{l} t_1 := z \cdot \left(y + a \cdot b\right)\\ \mathbf{if}\;z \leq -4 \cdot 10^{+53}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.8:\\ \;\;\;\;x + a \cdot t\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{+92}:\\ \;\;\;\;x + z \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Accuracy	83.7%
Cost	841

\[\begin{array}{l} \mathbf{if}\;b \leq -5.6 \cdot 10^{+166} \lor \neg \left(b \leq 2.6 \cdot 10^{+67}\right):\\ \;\;\;\;x + b \cdot \left(z \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + a \cdot t\right) + z \cdot y\\ \end{array} \]

Alternative 13
Accuracy	88.0%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{+53} \lor \neg \left(z \leq 7.5 \cdot 10^{+35}\right):\\ \;\;\;\;x + z \cdot \left(y + a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + a \cdot t\right) + z \cdot y\\ \end{array} \]

Alternative 14
Accuracy	48.2%
Cost	720

\[\begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-76}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-281}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-189}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+32}:\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 15
Accuracy	48.1%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -2.22 \cdot 10^{-76}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+29}:\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 16
Accuracy	37.9%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023137 
(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)))