Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Percentage Accurate: 92.6% → 98.0%

Time: 14.1s

Precision: binary64

Cost: 19912

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \leq -1.65 \cdot 10^{-104}:\\ \;\;\;\;\mathsf{fma}\left(y, z, \mathsf{fma}\left(a, \mathsf{fma}\left(z, b, t\right), x\right)\right)\\ \mathbf{elif}\;a \leq 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), \mathsf{fma}\left(t, a, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t + z \cdot b, \mathsf{fma}\left(y, z, x\right)\right)\\ \end{array} \]

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 (<= a -1.65e-104)
   (fma y z (fma a (fma z b t) x))
   (if (<= a 1e-16)
     (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 (a <= -1.65e-104) {
		tmp = fma(y, z, fma(a, fma(z, b, t), x));
	} else if (a <= 1e-16) {
		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 (a <= -1.65e-104)
		tmp = fma(y, z, fma(a, fma(z, b, t), x));
	elseif (a <= 1e-16)
		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[LessEqual[a, -1.65e-104], N[(y * z + N[(a * N[(z * b + t), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1e-16], 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	92.6%
Target	97.4%
Herbie	98.0%

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

Derivation

1. Split input into 3 regimes

2. if a < -1.65000000000000001e-104

   1. Initial program 90.2%

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

   2. Simplified 100.0%

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

      [Start] 90.2	\[ \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
      associate-+l+ [=>] 90.2	\[ \color{blue}{\left(x + y \cdot z\right) + \left(t \cdot a + \left(a \cdot z\right) \cdot b\right)} \]
      +-commutative [=>] 90.2	\[ \color{blue}{\left(y \cdot z + x\right)} + \left(t \cdot a + \left(a \cdot z\right) \cdot b\right) \]
      associate-+l+ [=>] 90.2	\[ \color{blue}{y \cdot z + \left(x + \left(t \cdot a + \left(a \cdot z\right) \cdot b\right)\right)} \]
      fma-def [=>] 91.6	\[ \color{blue}{\mathsf{fma}\left(y, z, x + \left(t \cdot a + \left(a \cdot z\right) \cdot b\right)\right)} \]
      +-commutative [=>] 91.6	\[ \mathsf{fma}\left(y, z, \color{blue}{\left(t \cdot a + \left(a \cdot z\right) \cdot b\right) + x}\right) \]
      *-commutative [=>] 91.6	\[ \mathsf{fma}\left(y, z, \left(\color{blue}{a \cdot t} + \left(a \cdot z\right) \cdot b\right) + x\right) \]
      associate-*l* [=>] 93.0	\[ \mathsf{fma}\left(y, z, \left(a \cdot t + \color{blue}{a \cdot \left(z \cdot b\right)}\right) + x\right) \]
      distribute-lft-out [=>] 100.0	\[ \mathsf{fma}\left(y, z, \color{blue}{a \cdot \left(t + z \cdot b\right)} + x\right) \]
      fma-def [=>] 100.0	\[ \mathsf{fma}\left(y, z, \color{blue}{\mathsf{fma}\left(a, t + z \cdot b, x\right)}\right) \]
      +-commutative [=>] 100.0	\[ \mathsf{fma}\left(y, z, \mathsf{fma}\left(a, \color{blue}{z \cdot b + t}, x\right)\right) \]
      fma-def [=>] 100.0	\[ \mathsf{fma}\left(y, z, \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(z, b, t\right)}, x\right)\right) \]

   if -1.65000000000000001e-104 < a < 9.9999999999999998e-17

   1. Initial program 96.5%

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

   2. Simplified 100.0%

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

      [Start] 96.5	\[ \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
      +-commutative [=>] 96.5	\[ \color{blue}{\left(a \cdot z\right) \cdot b + \left(\left(x + y \cdot z\right) + t \cdot a\right)} \]
      +-commutative [=>] 96.5	\[ \left(a \cdot z\right) \cdot b + \left(\color{blue}{\left(y \cdot z + x\right)} + t \cdot a\right) \]
      associate-+l+ [=>] 96.5	\[ \left(a \cdot z\right) \cdot b + \color{blue}{\left(y \cdot z + \left(x + t \cdot a\right)\right)} \]
      associate-+r+ [=>] 96.5	\[ \color{blue}{\left(\left(a \cdot z\right) \cdot b + y \cdot z\right) + \left(x + t \cdot a\right)} \]
      *-commutative [=>] 96.5	\[ \left(\color{blue}{\left(z \cdot a\right)} \cdot b + y \cdot z\right) + \left(x + t \cdot a\right) \]
      associate-*l* [=>] 97.3	\[ \left(\color{blue}{z \cdot \left(a \cdot b\right)} + y \cdot z\right) + \left(x + t \cdot a\right) \]
      *-commutative [=>] 97.3	\[ \left(z \cdot \left(a \cdot b\right) + \color{blue}{z \cdot y}\right) + \left(x + t \cdot a\right) \]
      distribute-lft-out [=>] 100.0	\[ \color{blue}{z \cdot \left(a \cdot b + y\right)} + \left(x + t \cdot a\right) \]
      fma-def [=>] 100.0	\[ \color{blue}{\mathsf{fma}\left(z, a \cdot b + y, x + t \cdot a\right)} \]
      fma-def [=>] 100.0	\[ \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(a, b, y\right)}, x + t \cdot a\right) \]
      +-commutative [<=] 100.0	\[ \mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), \color{blue}{t \cdot a + x}\right) \]
      fma-def [=>] 100.0	\[ \mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), \color{blue}{\mathsf{fma}\left(t, a, x\right)}\right) \]

   if 9.9999999999999998e-17 < a

   1. Initial program 91.7%

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

   2. Simplified 99.9%

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

      [Start] 91.7	\[ \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
      associate-+l+ [=>] 91.7	\[ \color{blue}{\left(x + y \cdot z\right) + \left(t \cdot a + \left(a \cdot z\right) \cdot b\right)} \]
      +-commutative [=>] 91.7	\[ \color{blue}{\left(t \cdot a + \left(a \cdot z\right) \cdot b\right) + \left(x + y \cdot z\right)} \]
      *-commutative [=>] 91.7	\[ \left(\color{blue}{a \cdot t} + \left(a \cdot z\right) \cdot b\right) + \left(x + y \cdot z\right) \]
      associate-*l* [=>] 97.1	\[ \left(a \cdot t + \color{blue}{a \cdot \left(z \cdot b\right)}\right) + \left(x + y \cdot z\right) \]
      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 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.65 \cdot 10^{-104}:\\ \;\;\;\;\mathsf{fma}\left(y, z, \mathsf{fma}\left(a, \mathsf{fma}\left(z, b, t\right), x\right)\right)\\ \mathbf{elif}\;a \leq 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), \mathsf{fma}\left(t, a, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t + z \cdot b, \mathsf{fma}\left(y, z, x\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.1%
Cost	19913

\[\begin{array}{l} \mathbf{if}\;a \leq -3.2 \cdot 10^{-99} \lor \neg \left(a \leq 3.45 \cdot 10^{-146}\right):\\ \;\;\;\;\mathsf{fma}\left(y, z, \mathsf{fma}\left(a, \mathsf{fma}\left(z, b, t\right), x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot t + \left(x + y \cdot z\right)\right) + b \cdot \left(a \cdot z\right)\\ \end{array} \]

Alternative 2
Accuracy	95.5%
Cost	13640

\[\begin{array}{l} t_1 := t + z \cdot b\\ \mathbf{if}\;a \leq -8.2 \cdot 10^{+202}:\\ \;\;\;\;\mathsf{fma}\left(t_1, a, x\right)\\ \mathbf{elif}\;a \leq 5 \cdot 10^{-167}:\\ \;\;\;\;\left(a \cdot t + \left(x + y \cdot z\right)\right) + b \cdot \left(a \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t_1, \mathsf{fma}\left(y, z, x\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	96.8%
Cost	1988

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

Alternative 4
Accuracy	39.4%
Cost	1772

\[\begin{array}{l} t_1 := b \cdot \left(a \cdot z\right)\\ \mathbf{if}\;z \leq -9.5 \cdot 10^{+275}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.35 \cdot 10^{+178}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -2.6 \cdot 10^{+25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4 \cdot 10^{-39}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -6.3 \cdot 10^{-139}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{-178}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-202}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-146}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{-113}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-70}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+16}:\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	37.0%
Cost	1640

\[\begin{array}{l} t_1 := a \cdot \left(z \cdot b\right)\\ \mathbf{if}\;z \leq -7.5 \cdot 10^{+24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.9 \cdot 10^{-36}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -1.9 \cdot 10^{-72}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.15 \cdot 10^{-138}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-178}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 6.9 \cdot 10^{-203}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-145}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-110}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{-71}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+68}:\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Accuracy	72.6%
Cost	1240

\[\begin{array}{l} t_1 := x + y \cdot z\\ t_2 := a \cdot \left(t + z \cdot b\right)\\ \mathbf{if}\;a \leq -2.3 \cdot 10^{-77}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{-82}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 4 \cdot 10^{-38}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 7.8 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 3.05 \cdot 10^{+35}:\\ \;\;\;\;a \cdot t + y \cdot z\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{+76}:\\ \;\;\;\;x + a \cdot \left(z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Accuracy	85.3%
Cost	1105

\[\begin{array}{l} t_1 := x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{if}\;b \leq -2.4 \cdot 10^{+81}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -2.3 \cdot 10^{+14}:\\ \;\;\;\;\left(a \cdot t + y \cdot z\right) + b \cdot \left(a \cdot z\right)\\ \mathbf{elif}\;b \leq -0.01 \lor \neg \left(b \leq 2.75 \cdot 10^{+106}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot t + \left(x + y \cdot z\right)\\ \end{array} \]

Alternative 8
Accuracy	92.2%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;a \leq -7.8 \cdot 10^{+230}:\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(z \cdot b\right) + a \cdot t\right) + \left(x + y \cdot z\right)\\ \end{array} \]

Alternative 9
Accuracy	60.0%
Cost	980

\[\begin{array}{l} t_1 := x + y \cdot z\\ t_2 := b \cdot \left(a \cdot z\right)\\ \mathbf{if}\;z \leq -3.4 \cdot 10^{+276}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{+122}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -3.9 \cdot 10^{+25}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.75 \cdot 10^{+68}:\\ \;\;\;\;x + a \cdot t\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 10
Accuracy	39.3%
Cost	852

\[\begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{-40}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-267}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-182}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-86}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+122}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\\ \end{array} \]

Alternative 11
Accuracy	58.6%
Cost	849

\[\begin{array}{l} t_1 := b \cdot \left(a \cdot z\right)\\ \mathbf{if}\;z \leq -6.6 \cdot 10^{+276}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -5.2 \cdot 10^{+178}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -9 \cdot 10^{+23} \lor \neg \left(z \leq 4.6 \cdot 10^{+68}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + a \cdot t\\ \end{array} \]

Alternative 12
Accuracy	82.1%
Cost	841

\[\begin{array}{l} \mathbf{if}\;a \leq -3.5 \cdot 10^{-144} \lor \neg \left(a \leq 1.8 \cdot 10^{-114}\right):\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \]

Alternative 13
Accuracy	85.6%
Cost	841

\[\begin{array}{l} \mathbf{if}\;b \leq -0.0125 \lor \neg \left(b \leq 7.5 \cdot 10^{+104}\right):\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot t + \left(x + y \cdot z\right)\\ \end{array} \]

Alternative 14
Accuracy	73.3%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1.06 \cdot 10^{-14} \lor \neg \left(z \leq 4.25 \cdot 10^{+68}\right):\\ \;\;\;\;z \cdot \left(y + a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + a \cdot t\\ \end{array} \]

Alternative 15
Accuracy	39.3%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+49}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+23}:\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 16
Accuracy	25.9%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023160 
(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)))