Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Percentage Accurate: 92.2% → 96.5%

Time: 10.2s

Precision: binary64

Cost: 1988

• 32.5% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} t_1 := \left(a \cdot t + \left(x + y \cdot z\right)\right) + b \cdot \left(z \cdot a\right)\\ \mathbf{if}\;t_1 \leq \infty:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(t + z \cdot b\right)\\ \end{array} \]

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
 (let* ((t_1 (+ (+ (* a t) (+ x (* y z))) (* b (* z a)))))
   (if (<= t_1 INFINITY) t_1 (* a (+ t (* z b))))))

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 t_1 = ((a * t) + (x + (y * z))) + (b * (z * a));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = a * (t + (z * b));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return ((x + (y * z)) + (t * a)) + ((a * z) * b);
}

↓

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((a * t) + (x + (y * z))) + (b * (z * a));
	double tmp;
	if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = a * (t + (z * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	return ((x + (y * z)) + (t * a)) + ((a * z) * b)

↓

def code(x, y, z, t, a, b):
	t_1 = ((a * t) + (x + (y * z))) + (b * (z * a))
	tmp = 0
	if t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = a * (t + (z * b))
	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)
	t_1 = Float64(Float64(Float64(a * t) + Float64(x + Float64(y * z))) + Float64(b * Float64(z * a)))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(a * Float64(t + Float64(z * b)));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((x + (y * z)) + (t * a)) + ((a * z) * b);
end

↓

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = ((a * t) + (x + (y * z))) + (b * (z * a));
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = a * (t + (z * b));
	end
	tmp_2 = 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_] := Block[{t$95$1 = N[(N[(N[(a * t), $MachinePrecision] + N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(a * N[(t + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	92.2%
Target	97.4%
Herbie	96.5%

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 x (*.f64 y z)) (*.f64 t a)) (*.f64 (*.f64 a z) b)) < +inf.0

   1. Initial program 98.1%

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

   if +inf.0 < (+.f64 (+.f64 (+.f64 x (*.f64 y z)) (*.f64 t a)) (*.f64 (*.f64 a z) b))

   1. Initial program 0.0%

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

   2. Simplified 88.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, z, \mathsf{fma}\left(a, \mathsf{fma}\left(z, b, t\right), x\right)\right)} \]
      Step-by-step derivation

      [Start] 0.0%	\[ \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
      associate-+l+ [=>] 0.0%	\[ \color{blue}{\left(x + y \cdot z\right) + \left(t \cdot a + \left(a \cdot z\right) \cdot b\right)} \]
      +-commutative [=>] 0.0%	\[ \color{blue}{\left(y \cdot z + x\right)} + \left(t \cdot a + \left(a \cdot z\right) \cdot b\right) \]
      associate-+l+ [=>] 0.0%	\[ \color{blue}{y \cdot z + \left(x + \left(t \cdot a + \left(a \cdot z\right) \cdot b\right)\right)} \]
      fma-def [=>] 11.1%	\[ \color{blue}{\mathsf{fma}\left(y, z, x + \left(t \cdot a + \left(a \cdot z\right) \cdot b\right)\right)} \]
      +-commutative [=>] 11.1%	\[ \mathsf{fma}\left(y, z, \color{blue}{\left(t \cdot a + \left(a \cdot z\right) \cdot b\right) + x}\right) \]
      *-commutative [=>] 11.1%	\[ \mathsf{fma}\left(y, z, \left(\color{blue}{a \cdot t} + \left(a \cdot z\right) \cdot b\right) + x\right) \]
      associate-*l* [=>] 11.1%	\[ \mathsf{fma}\left(y, z, \left(a \cdot t + \color{blue}{a \cdot \left(z \cdot b\right)}\right) + x\right) \]
      distribute-lft-out [=>] 88.9%	\[ \mathsf{fma}\left(y, z, \color{blue}{a \cdot \left(t + z \cdot b\right)} + x\right) \]
      fma-def [=>] 88.9%	\[ \mathsf{fma}\left(y, z, \color{blue}{\mathsf{fma}\left(a, t + z \cdot b, x\right)}\right) \]
      +-commutative [=>] 88.9%	\[ \mathsf{fma}\left(y, z, \mathsf{fma}\left(a, \color{blue}{z \cdot b + t}, x\right)\right) \]
      fma-def [=>] 88.9%	\[ \mathsf{fma}\left(y, z, \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(z, b, t\right)}, x\right)\right) \]

   3. Taylor expanded in a around inf 88.9%

      \[\leadsto \color{blue}{a \cdot \left(z \cdot b + t\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a \cdot t + \left(x + y \cdot z\right)\right) + b \cdot \left(z \cdot a\right) \leq \infty:\\ \;\;\;\;\left(a \cdot t + \left(x + y \cdot z\right)\right) + b \cdot \left(z \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(t + z \cdot b\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	96.5%
Cost	1988

\[\begin{array}{l} t_1 := \left(a \cdot t + \left(x + y \cdot z\right)\right) + b \cdot \left(z \cdot a\right)\\ \mathbf{if}\;t_1 \leq \infty:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(t + z \cdot b\right)\\ \end{array} \]

Alternative 2
Accuracy	95.8%
Cost	19648

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

Alternative 3
Accuracy	94.8%
Cost	13376

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

Alternative 4
Accuracy	81.5%
Cost	1104

\[\begin{array}{l} t_1 := y \cdot z + \left(x + a \cdot t\right)\\ t_2 := a \cdot \left(t + z \cdot b\right)\\ \mathbf{if}\;a \leq -7.5 \cdot 10^{+163}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{+31}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.5 \cdot 10^{+110}:\\ \;\;\;\;x + z \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{+160}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Accuracy	93.9%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;z \leq 6.2 \cdot 10^{+149}:\\ \;\;\;\;\left(a \cdot \left(z \cdot b\right) + a \cdot t\right) + \left(x + y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(y + a \cdot b\right)\\ \end{array} \]

Alternative 6
Accuracy	40.8%
Cost	980

\[\begin{array}{l} \mathbf{if}\;a \leq -2.6 \cdot 10^{-48}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;a \leq -1.5 \cdot 10^{-209}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;a \leq 1.42 \cdot 10^{-157}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 2 \cdot 10^{-49}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;a \leq 9.8 \cdot 10^{+30}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \end{array} \]

Alternative 7
Accuracy	39.8%
Cost	852

\[\begin{array}{l} \mathbf{if}\;a \leq -4.4 \cdot 10^{-48}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;a \leq -2.3 \cdot 10^{-209}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;a \leq 2.1 \cdot 10^{-157}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{-55}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{+56}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\\ \end{array} \]

Alternative 8
Accuracy	60.8%
Cost	849

\[\begin{array}{l} t_1 := a \cdot \left(z \cdot b\right)\\ \mathbf{if}\;z \leq -2.6 \cdot 10^{+82}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 0.008:\\ \;\;\;\;x + a \cdot t\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+182} \lor \neg \left(z \leq 4.6 \cdot 10^{+236}\right):\\ \;\;\;\;x + y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Accuracy	73.7%
Cost	844

\[\begin{array}{l} t_1 := a \cdot \left(t + z \cdot b\right)\\ \mathbf{if}\;a \leq -1.45 \cdot 10^{-49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{-73}:\\ \;\;\;\;x + y \cdot z\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{+64}:\\ \;\;\;\;x + z \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Accuracy	86.8%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+82} \lor \neg \left(z \leq 42000000000000\right):\\ \;\;\;\;x + z \cdot \left(y + a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z + \left(x + a \cdot t\right)\\ \end{array} \]

Alternative 11
Accuracy	74.7%
Cost	713

\[\begin{array}{l} \mathbf{if}\;a \leq -2.65 \cdot 10^{-48} \lor \neg \left(a \leq 8.1 \cdot 10^{+24}\right):\\ \;\;\;\;a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \]

Alternative 12
Accuracy	58.0%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{+83} \lor \neg \left(z \leq 6.4 \cdot 10^{+161}\right):\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + a \cdot t\\ \end{array} \]

Alternative 13
Accuracy	39.5%
Cost	456

\[\begin{array}{l} \mathbf{if}\;a \leq -9.5 \cdot 10^{-67}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;a \leq 1.75 \cdot 10^{+62}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\\ \end{array} \]

Alternative 14
Accuracy	26.6%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023243 
(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)))