Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Average Error: 2.2 → 2.2

Time: 9.2s

Precision: binary64

Cost: 960

Math

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

↓

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

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
 (+ (+ (* t a) (+ x (* y z))) (* (* z a) 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) {
	return ((t * a) + (x + (y * z))) + ((z * a) * b);
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((x + (y * z)) + (t * a)) + ((a * z) * b)
end function

↓

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((t * a) + (x + (y * z))) + ((z * a) * b)
end function

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) {
	return ((t * a) + (x + (y * z))) + ((z * a) * b);
}

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

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)
	return Float64(Float64(Float64(t * a) + Float64(x + Float64(y * z))) + Float64(Float64(z * a) * b))
end

MATLAB

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

↓

function tmp = code(x, y, z, t, a, b)
	tmp = ((t * a) + (x + (y * z))) + ((z * a) * b);
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_] := N[(N[(N[(t * a), $MachinePrecision] + N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(z * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]

Target

Original	2.2
Target	0.4
Herbie	2.2

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

Derivation

1. Initial program 2.2

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

2. Final simplification 2.2

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

Alternatives

Alternative 1
Error	23.0
Cost	2032

\[\begin{array}{l} t_1 := x + y \cdot z\\ t_2 := a \cdot \left(t + z \cdot b\right)\\ t_3 := t \cdot a + y \cdot z\\ t_4 := x + t \cdot a\\ \mathbf{if}\;t \leq -7.4 \cdot 10^{+50}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -5.2 \cdot 10^{-16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -5 \cdot 10^{-28}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -5.6 \cdot 10^{-84}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq -2.5 \cdot 10^{-112}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.1 \cdot 10^{-123}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.4 \cdot 10^{-172}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.45 \cdot 10^{-225}:\\ \;\;\;\;z \cdot \left(y + a \cdot b\right)\\ \mathbf{elif}\;t \leq -1.25 \cdot 10^{-241}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-114}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{+23}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq 6.1 \cdot 10^{+63}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 2
Error	39.4
Cost	1248

\[\begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{-39}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{-234}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -2.5 \cdot 10^{-288}:\\ \;\;\;\;t \cdot a\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-192}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-64}:\\ \;\;\;\;t \cdot a\\ \mathbf{elif}\;z \leq 8500:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+40}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 10^{+145}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 3
Error	21.7
Cost	1112

\[\begin{array}{l} t_1 := x + t \cdot a\\ t_2 := z \cdot \left(y + a \cdot b\right)\\ \mathbf{if}\;z \leq -1.35 \cdot 10^{-17}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-143}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-63}:\\ \;\;\;\;a \cdot \left(t + z \cdot b\right)\\ \mathbf{elif}\;z \leq 21000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+40}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+65}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \]

Alternative 4
Error	13.7
Cost	1104

\[\begin{array}{l} t_1 := \left(x + t \cdot a\right) + y \cdot z\\ \mathbf{if}\;x \leq -3.6 \cdot 10^{-103}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{-175}:\\ \;\;\;\;z \cdot \left(y + a \cdot b\right)\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-200}:\\ \;\;\;\;t \cdot a + y \cdot z\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-214}:\\ \;\;\;\;a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	13.7
Cost	1104

\[\begin{array}{l} t_1 := \left(x + t \cdot a\right) + y \cdot z\\ \mathbf{if}\;x \leq -3.6 \cdot 10^{-103}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-175}:\\ \;\;\;\;a \cdot \left(z \cdot b\right) + y \cdot z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-200}:\\ \;\;\;\;t \cdot a + y \cdot z\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-214}:\\ \;\;\;\;a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	8.9
Cost	968

\[\begin{array}{l} t_1 := x + t \cdot a\\ t_2 := t_1 + y \cdot z\\ \mathbf{if}\;y \leq -4.2 \cdot 10^{-138}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+122}:\\ \;\;\;\;\left(z \cdot a\right) \cdot b + t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	7.3
Cost	968

\[\begin{array}{l} t_1 := \left(x + t \cdot a\right) + y \cdot z\\ \mathbf{if}\;t \leq -1.16 \cdot 10^{-48}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.15 \cdot 10^{-102}:\\ \;\;\;\;a \cdot \left(z \cdot b\right) + \left(x + y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	3.0
Cost	960

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

Alternative 9
Error	24.9
Cost	848

\[\begin{array}{l} t_1 := x + t \cdot a\\ \mathbf{if}\;z \leq -2.8 \cdot 10^{+19}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+40}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+158}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 10
Error	20.2
Cost	848

\[\begin{array}{l} t_1 := x + t \cdot a\\ t_2 := z \cdot \left(y + a \cdot b\right)\\ \mathbf{if}\;z \leq -2.45 \cdot 10^{-11}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-107}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-99}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \]

Alternative 11
Error	19.7
Cost	584

\[\begin{array}{l} t_1 := x + t \cdot a\\ \mathbf{if}\;a \leq -9500000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 6.6 \cdot 10^{+21}:\\ \;\;\;\;x + y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	32.2
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00055:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+34}:\\ \;\;\;\;t \cdot a\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13
Error	39.1
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022325 
(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)))