Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Percentage Accurate: 92.3% → 97.4%

Time: 8.2s

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 84.1%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 90.3%

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

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

   1. Initial program 95.3%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 89.5

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

   5. Applied rewrites 89.5%

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

   if 7.5000000000000004e-150 < t

   1. Initial program 89.2%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 77.5%

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

   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 93.5%

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

4. Final simplification 91.2%

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

Alternative 3: 87.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, \mathsf{fma}\left(b, z, t\right), x\right)\\ \mathbf{if}\;a \leq -2.3 \cdot 10^{-67}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3.8 \cdot 10^{-35}:\\ \;\;\;\;\mathsf{fma}\left(a, t, \mathsf{fma}\left(z, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma a (fma b z t) x)))
   (if (<= a -2.3e-67) t_1 (if (<= a 3.8e-35) (fma a t (fma z y x)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(a, fma(b, z, t), x);
	double tmp;
	if (a <= -2.3e-67) {
		tmp = t_1;
	} else if (a <= 3.8e-35) {
		tmp = fma(a, t, fma(z, y, x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(a, fma(b, z, t), x)
	tmp = 0.0
	if (a <= -2.3e-67)
		tmp = t_1;
	elseif (a <= 3.8e-35)
		tmp = fma(a, t, fma(z, y, x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a * N[(b * z + t), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -2.3e-67], t$95$1, If[LessEqual[a, 3.8e-35], N[(a * t + N[(z * y + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.3e-67 or 3.8000000000000001e-35 < a

   1. Initial program 81.3%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 88.4

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

   5. Applied rewrites 88.4%

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

   if -2.3e-67 < a < 3.8000000000000001e-35

   1. Initial program 99.9%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 90.3%

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

Alternative 4: 83.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \mathsf{fma}\left(b, z, t\right)\\ \mathbf{if}\;a \leq -4.6 \cdot 10^{+157}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.5 \cdot 10^{+76}:\\ \;\;\;\;\mathsf{fma}\left(a, t, \mathsf{fma}\left(z, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* a (fma b z t))))
   (if (<= a -4.6e+157) t_1 (if (<= a 1.5e+76) (fma a t (fma z y x)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a * fma(b, z, t);
	double tmp;
	if (a <= -4.6e+157) {
		tmp = t_1;
	} else if (a <= 1.5e+76) {
		tmp = fma(a, t, fma(z, y, x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a * fma(b, z, t))
	tmp = 0.0
	if (a <= -4.6e+157)
		tmp = t_1;
	elseif (a <= 1.5e+76)
		tmp = fma(a, t, fma(z, y, x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a * N[(b * z + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -4.6e+157], t$95$1, If[LessEqual[a, 1.5e+76], N[(a * t + N[(z * y + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.60000000000000008e157 or 1.4999999999999999e76 < a

   1. Initial program 74.3%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 88.4

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

   5. Applied rewrites 88.4%

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

   if -4.60000000000000008e157 < a < 1.4999999999999999e76

   1. Initial program 96.0%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 85.2%

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

Alternative 5: 74.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \mathsf{fma}\left(b, z, t\right)\\ \mathbf{if}\;a \leq -4 \cdot 10^{-40}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 4.1 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* a (fma b z t))))
   (if (<= a -4e-40) t_1 (if (<= a 4.1e+20) (fma z y x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a * fma(b, z, t);
	double tmp;
	if (a <= -4e-40) {
		tmp = t_1;
	} else if (a <= 4.1e+20) {
		tmp = fma(z, y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a * fma(b, z, t))
	tmp = 0.0
	if (a <= -4e-40)
		tmp = t_1;
	elseif (a <= 4.1e+20)
		tmp = fma(z, y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a * N[(b * z + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -4e-40], t$95$1, If[LessEqual[a, 4.1e+20], N[(z * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.9999999999999997e-40 or 4.1e20 < a

   1. Initial program 80.3%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 77.2

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

   5. Applied rewrites 77.2%

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

   if -3.9999999999999997e-40 < a < 4.1e20

   1. Initial program 99.1%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + y \cdot z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot z + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{z \cdot y} + x \]

      3. lower-fma.f64 74.7

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

   5. Applied rewrites 74.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, y, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 62.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.02 \cdot 10^{-108}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{elif}\;a \leq 9.2 \cdot 10^{-73}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -1.02e-108)
   (fma a t x)
   (if (<= a 9.2e-73) (fma z y x) (fma a t x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -1.02e-108) {
		tmp = fma(a, t, x);
	} else if (a <= 9.2e-73) {
		tmp = fma(z, y, x);
	} else {
		tmp = fma(a, t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -1.02e-108)
		tmp = fma(a, t, x);
	elseif (a <= 9.2e-73)
		tmp = fma(z, y, x);
	else
		tmp = fma(a, t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -1.02e-108], N[(a * t + x), $MachinePrecision], If[LessEqual[a, 9.2e-73], N[(z * y + x), $MachinePrecision], N[(a * t + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.02000000000000008e-108 or 9.19999999999999953e-73 < a

   1. Initial program 83.3%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + a \cdot t} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot t + x} \]

      2. lower-fma.f64 57.2

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

   5. Applied rewrites 57.2%

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

   if -1.02000000000000008e-108 < a < 9.19999999999999953e-73

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + y \cdot z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot z + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{z \cdot y} + x \]

      3. lower-fma.f64 79.9

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

   5. Applied rewrites 79.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, y, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 59.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.36 \cdot 10^{+163}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{+136}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -1.36e+163) (* y z) (if (<= y 2.25e+136) (fma a t x) (* y z))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.36e+163) {
		tmp = y * z;
	} else if (y <= 2.25e+136) {
		tmp = fma(a, t, x);
	} else {
		tmp = y * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -1.36e+163)
		tmp = Float64(y * z);
	elseif (y <= 2.25e+136)
		tmp = fma(a, t, x);
	else
		tmp = Float64(y * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -1.36e+163], N[(y * z), $MachinePrecision], If[LessEqual[y, 2.25e+136], N[(a * t + x), $MachinePrecision], N[(y * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.36000000000000001e163 or 2.25e136 < y

   1. Initial program 85.9%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot z} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{z \cdot y} \]

      2. lower-*.f64 62.5

         \[\leadsto \color{blue}{z \cdot y} \]

   5. Applied rewrites 62.5%

      \[\leadsto \color{blue}{z \cdot y} \]

   if -1.36000000000000001e163 < y < 2.25e136

   1. Initial program 90.7%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + a \cdot t} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot t + x} \]

      2. lower-fma.f64 61.6

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

   5. Applied rewrites 61.6%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.36 \cdot 10^{+163}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{+136}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 37.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.02 \cdot 10^{-108}:\\ \;\;\;\;t \cdot a\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{-73}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -1.02e-108) (* t a) (if (<= a 2.3e-73) (* y z) (* t a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -1.02e-108) {
		tmp = t * a;
	} else if (a <= 2.3e-73) {
		tmp = y * z;
	} else {
		tmp = t * a;
	}
	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) :: tmp
    if (a <= (-1.02d-108)) then
        tmp = t * a
    else if (a <= 2.3d-73) then
        tmp = y * z
    else
        tmp = t * a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -1.02e-108) {
		tmp = t * a;
	} else if (a <= 2.3e-73) {
		tmp = y * z;
	} else {
		tmp = t * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -1.02e-108:
		tmp = t * a
	elif a <= 2.3e-73:
		tmp = y * z
	else:
		tmp = t * a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -1.02e-108)
		tmp = Float64(t * a);
	elseif (a <= 2.3e-73)
		tmp = Float64(y * z);
	else
		tmp = Float64(t * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -1.02e-108)
		tmp = t * a;
	elseif (a <= 2.3e-73)
		tmp = y * z;
	else
		tmp = t * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -1.02e-108], N[(t * a), $MachinePrecision], If[LessEqual[a, 2.3e-73], N[(y * z), $MachinePrecision], N[(t * a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.02000000000000008e-108 or 2.29999999999999988e-73 < a

   1. Initial program 83.3%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{a \cdot t} \]
   4. Step-by-step derivation

      1. lower-*.f64 41.3

         \[\leadsto \color{blue}{a \cdot t} \]

   5. Applied rewrites 41.3%

      \[\leadsto \color{blue}{a \cdot t} \]

   if -1.02000000000000008e-108 < a < 2.29999999999999988e-73

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot z} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{z \cdot y} \]

      2. lower-*.f64 45.9

         \[\leadsto \color{blue}{z \cdot y} \]

   5. Applied rewrites 45.9%

      \[\leadsto \color{blue}{z \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.02 \cdot 10^{-108}:\\ \;\;\;\;t \cdot a\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{-73}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 28.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ t \cdot a \end{array} \]

FPCore

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

C

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

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 = t * a
end function

Java

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

Python

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

Julia

function code(x, y, z, t, a, b)
	return Float64(t * a)
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(t * a), $MachinePrecision]

Derivation

1. Initial program 89.5%

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

3. Taylor expanded in t around inf

   \[\leadsto \color{blue}{a \cdot t} \]
4. Step-by-step derivation

   1. lower-*.f64 30.4

      \[\leadsto \color{blue}{a \cdot t} \]

5. Applied rewrites 30.4%

   \[\leadsto \color{blue}{a \cdot t} \]

6. Final simplification 30.4%

   \[\leadsto t \cdot a \]
7. 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)))