Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Percentage Accurate: 92.3% → 97.4%

Time: 9.7s

Alternatives: 9

Speedup: 0.5×

• 32.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (+ (+ (+ x (* y z)) (* t a)) (* (* a 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);
}

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

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);
}

Python

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

MATLAB

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

Initial Program: 92.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (+ (+ (+ x (* y z)) (* t a)) (* (* a 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);
}

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

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);
}

Python

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

MATLAB

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

Alternative 1: 97.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(z \cdot a\right) \cdot b\\ \mathbf{if}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(\frac{z}{t}, \mathsf{fma}\left(a, b, y\right), a\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ (+ x (* y z)) (* t a)) (* (* z a) b))))
   (if (<= t_1 INFINITY) t_1 (fma t (fma (/ z t) (fma a b y) a) x))))

C

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

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(x + Float64(y * z)) + Float64(t * a)) + Float64(Float64(z * a) * b))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = fma(t, fma(Float64(z / t), fma(a, b, y), a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision] + N[(t * a), $MachinePrecision]), $MachinePrecision] + N[(N[(z * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(t * N[(N[(z / t), $MachinePrecision] * N[(a * b + y), $MachinePrecision] + a), $MachinePrecision] + x), $MachinePrecision]]]

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 97.1%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   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. Add Preprocessing

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 55.0%

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

   6. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

         \[\leadsto t \cdot \left(a + \left(\frac{a \cdot \left(b \cdot z\right)}{t} + \frac{y \cdot z}{t}\right)\right) + x \cdot \color{blue}{1} \]

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 N/A

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

   8. Applied rewrites 90.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(\frac{z}{t}, \mathsf{fma}\left(a, b, y\right), a\right), x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(z \cdot a\right) \cdot b \leq \infty:\\ \;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(z \cdot a\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(\frac{z}{t}, \mathsf{fma}\left(a, b, y\right), a\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(a, t, \mathsf{fma}\left(z, y, x\right)\right)\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-150}:\\ \;\;\;\;\left(z \cdot a\right) \cdot b + \mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(\frac{z}{t}, \mathsf{fma}\left(a, b, y\right), a\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -7.5e-7)
   (fma a t (fma z y x))
   (if (<= t 7.5e-150)
     (+ (* (* z a) b) (fma z y x))
     (fma t (fma (/ z t) (fma a b y) a) x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -7.5e-7) {
		tmp = fma(a, t, fma(z, y, x));
	} else if (t <= 7.5e-150) {
		tmp = ((z * a) * b) + fma(z, y, x);
	} else {
		tmp = fma(t, fma((z / t), fma(a, b, y), a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -7.5e-7)
		tmp = fma(a, t, fma(z, y, x));
	elseif (t <= 7.5e-150)
		tmp = Float64(Float64(Float64(z * a) * b) + fma(z, y, x));
	else
		tmp = fma(t, fma(Float64(z / t), fma(a, b, y), a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -7.5e-7], N[(a * t + N[(z * y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7.5e-150], N[(N[(N[(z * a), $MachinePrecision] * b), $MachinePrecision] + N[(z * y + x), $MachinePrecision]), $MachinePrecision], N[(t * N[(N[(z / t), $MachinePrecision] * N[(a * b + y), $MachinePrecision] + a), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.5000000000000002e-7

   1. Initial program 91.5%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 85.0%

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

   if -7.5000000000000002e-7 < t < 7.5000000000000004e-150

   1. Initial program 93.6%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 87.8

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

   5. Applied rewrites 87.8%

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

   if 7.5000000000000004e-150 < t

   1. Initial program 91.7%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 81.2%

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

   6. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

         \[\leadsto t \cdot \left(a + \left(\frac{a \cdot \left(b \cdot z\right)}{t} + \frac{y \cdot z}{t}\right)\right) + x \cdot \color{blue}{1} \]

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 N/A

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

   8. Applied rewrites 91.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(\frac{z}{t}, \mathsf{fma}\left(a, b, y\right), a\right), x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(a, t, \mathsf{fma}\left(z, y, x\right)\right)\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-150}:\\ \;\;\;\;\left(z \cdot a\right) \cdot b + \mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(\frac{z}{t}, \mathsf{fma}\left(a, b, y\right), a\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 97.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \mathbf{if}\;z < -11820553527347888000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z < 4.7589743188364287 \cdot 10^{-122}:\\ \;\;\;\;\left(b \cdot z + t\right) \cdot a + \left(z \cdot y + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* z (+ (* b a) y)) (+ x (* t a)))))
   (if (< z -11820553527347888000.0)
     t_1
     (if (< z 4.7589743188364287e-122)
       (+ (* (+ (* b z) t) a) (+ (* z y) x))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (z * ((b * a) + y)) + (x + (t * a));
	double tmp;
	if (z < -11820553527347888000.0) {
		tmp = t_1;
	} else if (z < 4.7589743188364287e-122) {
		tmp = (((b * z) + t) * a) + ((z * y) + x);
	} 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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * ((b * a) + y)) + (x + (t * a))
    if (z < (-11820553527347888000.0d0)) then
        tmp = t_1
    else if (z < 4.7589743188364287d-122) then
        tmp = (((b * z) + t) * a) + ((z * y) + x)
    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) {
	double t_1 = (z * ((b * a) + y)) + (x + (t * a));
	double tmp;
	if (z < -11820553527347888000.0) {
		tmp = t_1;
	} else if (z < 4.7589743188364287e-122) {
		tmp = (((b * z) + t) * a) + ((z * y) + x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (z * ((b * a) + y)) + (x + (t * a))
	tmp = 0
	if z < -11820553527347888000.0:
		tmp = t_1
	elif z < 4.7589743188364287e-122:
		tmp = (((b * z) + t) * a) + ((z * y) + x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(z * Float64(Float64(b * a) + y)) + Float64(x + Float64(t * a)))
	tmp = 0.0
	if (z < -11820553527347888000.0)
		tmp = t_1;
	elseif (z < 4.7589743188364287e-122)
		tmp = Float64(Float64(Float64(Float64(b * z) + t) * a) + Float64(Float64(z * y) + x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (z * ((b * a) + y)) + (x + (t * a));
	tmp = 0.0;
	if (z < -11820553527347888000.0)
		tmp = t_1;
	elseif (z < 4.7589743188364287e-122)
		tmp = (((b * z) + t) * a) + ((z * y) + x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(z * N[(N[(b * a), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + N[(x + N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -11820553527347888000.0], t$95$1, If[Less[z, 4.7589743188364287e-122], N[(N[(N[(N[(b * z), $MachinePrecision] + t), $MachinePrecision] * a), $MachinePrecision] + N[(N[(z * y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024223 
(FPCore (x y z t a b)
  :name "Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -11820553527347888000) (+ (* z (+ (* b a) y)) (+ x (* t a))) (if (< z 47589743188364287/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (* (+ (* b z) t) a) (+ (* z y) x)) (+ (* z (+ (* b a) y)) (+ x (* t a))))))

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