Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Average Error: 2.0 → 0.3

Time: 10.9s

Precision: binary64

Cost: 1356

Math

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

↓

\[\begin{array}{l} t_1 := z \cdot \left(a \cdot b + y\right) + \left(a \cdot t + x\right)\\ t_2 := x + z \cdot y\\ \mathbf{if}\;z \leq -5 \cdot 10^{+82}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -5 \cdot 10^{-229}:\\ \;\;\;\;\left(a \cdot t + t_2\right) + b \cdot \left(z \cdot a\right)\\ \mathbf{elif}\;z \leq 10^{-76}:\\ \;\;\;\;\left(a \cdot t + a \cdot \left(z \cdot b\right)\right) + t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (+ (* z (+ (* a b) y)) (+ (* a t) x))) (t_2 (+ x (* z y))))
   (if (<= z -5e+82)
     t_1
     (if (<= z -5e-229)
       (+ (+ (* a t) t_2) (* b (* z a)))
       (if (<= z 1e-76) (+ (+ (* a t) (* a (* z b))) t_2) t_1)))))

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 = (z * ((a * b) + y)) + ((a * t) + x);
	double t_2 = x + (z * y);
	double tmp;
	if (z <= -5e+82) {
		tmp = t_1;
	} else if (z <= -5e-229) {
		tmp = ((a * t) + t_2) + (b * (z * a));
	} else if (z <= 1e-76) {
		tmp = ((a * t) + (a * (z * b))) + t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (z * ((a * b) + y)) + ((a * t) + x)
    t_2 = x + (z * y)
    if (z <= (-5d+82)) then
        tmp = t_1
    else if (z <= (-5d-229)) then
        tmp = ((a * t) + t_2) + (b * (z * a))
    else if (z <= 1d-76) then
        tmp = ((a * t) + (a * (z * b))) + t_2
    else
        tmp = t_1
    end if
    code = tmp
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) {
	double t_1 = (z * ((a * b) + y)) + ((a * t) + x);
	double t_2 = x + (z * y);
	double tmp;
	if (z <= -5e+82) {
		tmp = t_1;
	} else if (z <= -5e-229) {
		tmp = ((a * t) + t_2) + (b * (z * a));
	} else if (z <= 1e-76) {
		tmp = ((a * t) + (a * (z * b))) + t_2;
	} else {
		tmp = t_1;
	}
	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 = (z * ((a * b) + y)) + ((a * t) + x)
	t_2 = x + (z * y)
	tmp = 0
	if z <= -5e+82:
		tmp = t_1
	elif z <= -5e-229:
		tmp = ((a * t) + t_2) + (b * (z * a))
	elif z <= 1e-76:
		tmp = ((a * t) + (a * (z * b))) + t_2
	else:
		tmp = t_1
	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(z * Float64(Float64(a * b) + y)) + Float64(Float64(a * t) + x))
	t_2 = Float64(x + Float64(z * y))
	tmp = 0.0
	if (z <= -5e+82)
		tmp = t_1;
	elseif (z <= -5e-229)
		tmp = Float64(Float64(Float64(a * t) + t_2) + Float64(b * Float64(z * a)));
	elseif (z <= 1e-76)
		tmp = Float64(Float64(Float64(a * t) + Float64(a * Float64(z * b))) + t_2);
	else
		tmp = t_1;
	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 = (z * ((a * b) + y)) + ((a * t) + x);
	t_2 = x + (z * y);
	tmp = 0.0;
	if (z <= -5e+82)
		tmp = t_1;
	elseif (z <= -5e-229)
		tmp = ((a * t) + t_2) + (b * (z * a));
	elseif (z <= 1e-76)
		tmp = ((a * t) + (a * (z * b))) + t_2;
	else
		tmp = t_1;
	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[(z * N[(N[(a * b), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + N[(N[(a * t), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(z * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5e+82], t$95$1, If[LessEqual[z, -5e-229], N[(N[(N[(a * t), $MachinePrecision] + t$95$2), $MachinePrecision] + N[(b * N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1e-76], N[(N[(N[(a * t), $MachinePrecision] + N[(a * N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision], t$95$1]]]]]

Target

Original	2.0
Target	0.4
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if z < -5.00000000000000015e82 or 9.99999999999999927e-77 < z

   1. Initial program 4.4

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

   2. Simplified 7.0

      \[\leadsto \color{blue}{\left(x + y \cdot z\right) + \left(t \cdot a + a \cdot \left(z \cdot b\right)\right)} \]
      Proof
      (+.f64 (+.f64 x (*.f64 y z)) (+.f64 (*.f64 t a) (*.f64 a (*.f64 z b)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 x (*.f64 y z)) (+.f64 (*.f64 t a) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 a z) b)))): 14 points increase in error, 21 points decrease in error
      (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 x (*.f64 y z)) (*.f64 t a)) (*.f64 (*.f64 a z) b))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in z around 0 0.3

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

   if -5.00000000000000015e82 < z < -5.00000000000000016e-229

   1. Initial program 0.5

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

   if -5.00000000000000016e-229 < z < 9.99999999999999927e-77

   1. Initial program 0.7

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

   2. Simplified 0.0

      \[\leadsto \color{blue}{\left(x + y \cdot z\right) + \left(t \cdot a + a \cdot \left(z \cdot b\right)\right)} \]
      Proof
      (+.f64 (+.f64 x (*.f64 y z)) (+.f64 (*.f64 t a) (*.f64 a (*.f64 z b)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 x (*.f64 y z)) (+.f64 (*.f64 t a) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 a z) b)))): 14 points increase in error, 21 points decrease in error
      (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 x (*.f64 y z)) (*.f64 t a)) (*.f64 (*.f64 a z) b))): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+82}:\\ \;\;\;\;z \cdot \left(a \cdot b + y\right) + \left(a \cdot t + x\right)\\ \mathbf{elif}\;z \leq -5 \cdot 10^{-229}:\\ \;\;\;\;\left(a \cdot t + \left(x + z \cdot y\right)\right) + b \cdot \left(z \cdot a\right)\\ \mathbf{elif}\;z \leq 10^{-76}:\\ \;\;\;\;\left(a \cdot t + a \cdot \left(z \cdot b\right)\right) + \left(x + z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(a \cdot b + y\right) + \left(a \cdot t + x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	22.5
Cost	1504

\[\begin{array}{l} t_1 := x + z \cdot y\\ t_2 := a \cdot \left(t + z \cdot b\right)\\ \mathbf{if}\;a \leq -2.45 \cdot 10^{+64}:\\ \;\;\;\;a \cdot t + x\\ \mathbf{elif}\;a \leq -1.25 \cdot 10^{-27}:\\ \;\;\;\;z \cdot \left(a \cdot b + y\right)\\ \mathbf{elif}\;a \leq -2.7 \cdot 10^{-36}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 10^{-166}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2 \cdot 10^{-133}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 420000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{+147}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 2.5 \cdot 10^{+161}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	22.0
Cost	1372

\[\begin{array}{l} t_1 := a \cdot t + x\\ t_2 := x + z \cdot y\\ \mathbf{if}\;y \leq -2.8 \cdot 10^{+175}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -2.65 \cdot 10^{+121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -255000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -6.2 \cdot 10^{-304}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{-236}:\\ \;\;\;\;x + a \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{+143}:\\ \;\;\;\;z \cdot \left(a \cdot b + y\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	33.8
Cost	1248

\[\begin{array}{l} t_1 := z \cdot \left(a \cdot b\right)\\ \mathbf{if}\;x \leq -1.26 \cdot 10^{+36}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-240}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-265}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{-296}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-166}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-127}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;x \leq 1.58 \cdot 10^{-115}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	26.4
Cost	1112

\[\begin{array}{l} t_1 := a \cdot t + x\\ \mathbf{if}\;y \leq -3.2 \cdot 10^{+176}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{+120}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -3.5 \cdot 10^{+21}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{+33}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+62}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+155}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \]

Alternative 5
Error	21.6
Cost	1108

\[\begin{array}{l} t_1 := a \cdot t + x\\ t_2 := x + z \cdot y\\ \mathbf{if}\;y \leq -2.8 \cdot 10^{+175}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -2.65 \cdot 10^{+121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1260000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+143}:\\ \;\;\;\;z \cdot \left(a \cdot b + y\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	23.2
Cost	1108

\[\begin{array}{l} t_1 := a \cdot t + x\\ \mathbf{if}\;y \leq -0.4:\\ \;\;\;\;a \cdot t + z \cdot y\\ \mathbf{elif}\;y \leq -6.2 \cdot 10^{-304}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-232}:\\ \;\;\;\;x + a \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+143}:\\ \;\;\;\;z \cdot \left(a \cdot b + y\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot y\\ \end{array} \]

Alternative 7
Error	23.2
Cost	1108

\[\begin{array}{l} t_1 := a \cdot t + x\\ \mathbf{if}\;y \leq -56:\\ \;\;\;\;a \cdot t + z \cdot y\\ \mathbf{elif}\;y \leq -3.7 \cdot 10^{-284}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-183}:\\ \;\;\;\;x + z \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+143}:\\ \;\;\;\;z \cdot \left(a \cdot b + y\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot y\\ \end{array} \]

Alternative 8
Error	15.6
Cost	1104

\[\begin{array}{l} t_1 := x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{if}\;y \leq -3.5 \cdot 10^{+21}:\\ \;\;\;\;a \cdot t + z \cdot y\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+63}:\\ \;\;\;\;z \cdot \left(a \cdot b + y\right)\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+143}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot y\\ \end{array} \]

Alternative 9
Error	9.3
Cost	1104

\[\begin{array}{l} t_1 := x + a \cdot \left(t + z \cdot b\right)\\ t_2 := \left(a \cdot t + x\right) + z \cdot y\\ \mathbf{if}\;y \leq -265000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+33}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+90}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+143}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 10
Error	8.5
Cost	1104

\[\begin{array}{l} t_1 := \left(a \cdot t + x\right) + z \cdot y\\ t_2 := z \cdot \left(a \cdot b + y\right) + x\\ \mathbf{if}\;z \leq -4.5 \cdot 10^{+30}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 2200:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.16 \cdot 10^{+103}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{+149}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 11
Error	2.9
Cost	1096

\[\begin{array}{l} t_1 := z \cdot \left(a \cdot b + y\right) + \left(a \cdot t + x\right)\\ \mathbf{if}\;t \leq 7.6 \cdot 10^{-231}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-151}:\\ \;\;\;\;a \cdot \left(z \cdot b\right) + \left(x + z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	1.6
Cost	1092

\[\begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+94}:\\ \;\;\;\;\left(a \cdot t + a \cdot \left(z \cdot b\right)\right) + \left(x + z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(a \cdot b + y\right) + \left(a \cdot t + x\right)\\ \end{array} \]

Alternative 13
Error	38.0
Cost	984

\[\begin{array}{l} \mathbf{if}\;y \leq -2.45 \cdot 10^{+180}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;y \leq -2.35 \cdot 10^{+121}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;y \leq -9.2 \cdot 10^{+20}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+43}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+96}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+155}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \]

Alternative 14
Error	20.3
Cost	848

\[\begin{array}{l} t_1 := a \cdot t + x\\ t_2 := x + z \cdot y\\ \mathbf{if}\;y \leq -5.9 \cdot 10^{+175}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -2.65 \cdot 10^{+121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -41000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+33}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 15
Error	33.0
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+38}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 18000000000:\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 16
Error	39.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022337 
(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)))