Data.Histogram.Bin.BinF:$cfromIndex from histogram-fill-0.8.4.1

Percentage Accurate: 100.0% → 100.0%

Time: 4.1s

Alternatives: 6

Speedup: 2.3×

Specification

Math

\[\begin{array}{l} \\ \left(\frac{x}{2} + y \cdot x\right) + z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (+ (/ x 2.0) (* y x)) z))

C

double code(double x, double y, double z) {
	return ((x / 2.0) + (y * x)) + z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((x / 2.0d0) + (y * x)) + z
end function

Java

public static double code(double x, double y, double z) {
	return ((x / 2.0) + (y * x)) + z;
}

Python

def code(x, y, z):
	return ((x / 2.0) + (y * x)) + z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x / 2.0) + Float64(y * x)) + z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x / 2.0) + (y * x)) + z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x / 2.0), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{x}{2} + y \cdot x\right) + z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (+ (/ x 2.0) (* y x)) z))

C

double code(double x, double y, double z) {
	return ((x / 2.0) + (y * x)) + z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((x / 2.0d0) + (y * x)) + z
end function

Java

public static double code(double x, double y, double z) {
	return ((x / 2.0) + (y * x)) + z;
}

Python

def code(x, y, z):
	return ((x / 2.0) + (y * x)) + z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x / 2.0) + Float64(y * x)) + z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x / 2.0) + (y * x)) + z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x / 2.0), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, x, \mathsf{fma}\left(0.5, x, z\right)\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma y x (fma 0.5 x z)))

C

double code(double x, double y, double z) {
	return fma(y, x, fma(0.5, x, z));
}

Julia

function code(x, y, z)
	return fma(y, x, fma(0.5, x, z))
end

Wolfram

code[x_, y_, z_] := N[(y * x + N[(0.5 * x + z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{x}{2} + y \cdot x\right) + z \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{x}{2} + y \cdot x\right) + z} \]

   2. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{x}{2} + y \cdot x\right)} + z \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left(y \cdot x + \frac{x}{2}\right)} + z \]

   4. associate-+l+ N/A

      \[\leadsto \color{blue}{y \cdot x + \left(\frac{x}{2} + z\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto \color{blue}{y \cdot x} + \left(\frac{x}{2} + z\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \frac{x}{2} + z\right)} \]

   7. lift-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{\frac{x}{2}} + z\right) \]

   8. clear-num N/A

      \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{\frac{1}{\frac{2}{x}}} + z\right) \]

   9. associate-/r/ N/A

      \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{\frac{1}{2} \cdot x} + z\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, x, z\right)}\right) \]

   11. metadata-eval 100.0

       \[\leadsto \mathsf{fma}\left(y, x, \mathsf{fma}\left(\color{blue}{0.5}, x, z\right)\right) \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \mathsf{fma}\left(0.5, x, z\right)\right)} \]
5. Add Preprocessing

Alternative 2: 84.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot y + \frac{x}{2}\\ t_1 := \left(y - -0.5\right) \cdot x\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+166}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+161}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (* x y) (/ x 2.0))) (t_1 (* (- y -0.5) x)))
   (if (<= t_0 -4e+166) t_1 (if (<= t_0 5e+161) (fma 0.5 x z) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = (x * y) + (x / 2.0);
	double t_1 = (y - -0.5) * x;
	double tmp;
	if (t_0 <= -4e+166) {
		tmp = t_1;
	} else if (t_0 <= 5e+161) {
		tmp = fma(0.5, x, z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * y) + Float64(x / 2.0))
	t_1 = Float64(Float64(y - -0.5) * x)
	tmp = 0.0
	if (t_0 <= -4e+166)
		tmp = t_1;
	elseif (t_0 <= 5e+161)
		tmp = fma(0.5, x, z);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * y), $MachinePrecision] + N[(x / 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y - -0.5), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+166], t$95$1, If[LessEqual[t$95$0, 5e+161], N[(0.5 * x + z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 x #s(literal 2 binary64)) (*.f64 y x)) < -3.99999999999999976e166 or 4.9999999999999997e161 < (+.f64 (/.f64 x #s(literal 2 binary64)) (*.f64 y x))

   1. Initial program 100.0%

      \[\left(\frac{x}{2} + y \cdot x\right) + z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{z + \frac{1}{2} \cdot x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot x + z} \]

      2. lower-fma.f64 26.5

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x, z\right)} \]

   5. Applied rewrites 26.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x, z\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + y\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} + y\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} + y\right) \cdot x} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(y + \frac{1}{2}\right)} \cdot x \]

      4. metadata-eval N/A

         \[\leadsto \left(y + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)}\right) \cdot x \]

      5. sub-neg N/A

         \[\leadsto \color{blue}{\left(y - \frac{-1}{2}\right)} \cdot x \]

      6. lower--.f64 95.2

         \[\leadsto \color{blue}{\left(y - -0.5\right)} \cdot x \]

   8. Applied rewrites 95.2%

      \[\leadsto \color{blue}{\left(y - -0.5\right) \cdot x} \]

   if -3.99999999999999976e166 < (+.f64 (/.f64 x #s(literal 2 binary64)) (*.f64 y x)) < 4.9999999999999997e161

   1. Initial program 100.0%

      \[\left(\frac{x}{2} + y \cdot x\right) + z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{z + \frac{1}{2} \cdot x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot x + z} \]

      2. lower-fma.f64 82.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x, z\right)} \]

   5. Applied rewrites 82.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x, z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y + \frac{x}{2} \leq -4 \cdot 10^{+166}:\\ \;\;\;\;\left(y - -0.5\right) \cdot x\\ \mathbf{elif}\;x \cdot y + \frac{x}{2} \leq 5 \cdot 10^{+161}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - -0.5\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.4 \cdot 10^{+19}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -6.4e+19) (* x y) (if (<= y 1.4e+16) (fma 0.5 x z) (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -6.4e+19) {
		tmp = x * y;
	} else if (y <= 1.4e+16) {
		tmp = fma(0.5, x, z);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -6.4e+19)
		tmp = Float64(x * y);
	elseif (y <= 1.4e+16)
		tmp = fma(0.5, x, z);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -6.4e+19], N[(x * y), $MachinePrecision], If[LessEqual[y, 1.4e+16], N[(0.5 * x + z), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.4e19 or 1.4e16 < y

   1. Initial program 100.0%

      \[\left(\frac{x}{2} + y \cdot x\right) + z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{z + \frac{1}{2} \cdot x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot x + z} \]

      2. lower-fma.f64 26.8

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x, z\right)} \]

   5. Applied rewrites 26.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x, z\right)} \]

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 74.7

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

   8. Applied rewrites 74.7%

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

   if -6.4e19 < y < 1.4e16

   1. Initial program 100.0%

      \[\left(\frac{x}{2} + y \cdot x\right) + z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{z + \frac{1}{2} \cdot x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot x + z} \]

      2. lower-fma.f64 97.3

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x, z\right)} \]

   5. Applied rewrites 97.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x, z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.4 \cdot 10^{+19}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.8 \cdot 10^{-8}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 0.5:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -9.8e-8) (* x y) (if (<= y 0.5) (* 0.5 x) (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -9.8e-8) {
		tmp = x * y;
	} else if (y <= 0.5) {
		tmp = 0.5 * x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-9.8d-8)) then
        tmp = x * y
    else if (y <= 0.5d0) then
        tmp = 0.5d0 * x
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -9.8e-8) {
		tmp = x * y;
	} else if (y <= 0.5) {
		tmp = 0.5 * x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -9.8e-8:
		tmp = x * y
	elif y <= 0.5:
		tmp = 0.5 * x
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -9.8e-8)
		tmp = Float64(x * y);
	elseif (y <= 0.5)
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -9.8e-8)
		tmp = x * y;
	elseif (y <= 0.5)
		tmp = 0.5 * x;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -9.8e-8], N[(x * y), $MachinePrecision], If[LessEqual[y, 0.5], N[(0.5 * x), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -9.8000000000000004e-8 or 0.5 < y

   1. Initial program 100.0%

      \[\left(\frac{x}{2} + y \cdot x\right) + z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{z + \frac{1}{2} \cdot x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot x + z} \]

      2. lower-fma.f64 30.6

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x, z\right)} \]

   5. Applied rewrites 30.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x, z\right)} \]

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 70.3

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

   8. Applied rewrites 70.3%

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

   if -9.8000000000000004e-8 < y < 0.5

   1. Initial program 100.0%

      \[\left(\frac{x}{2} + y \cdot x\right) + z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{z + \frac{1}{2} \cdot x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot x + z} \]

      2. lower-fma.f64 99.4

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x, z\right)} \]

   5. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x, z\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{1}{2} \cdot \color{blue}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.4%

      \[\leadsto 0.5 \cdot \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.8 \cdot 10^{-8}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 0.5:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 100.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.5 + y, x, z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (+ 0.5 y) x z))

C

double code(double x, double y, double z) {
	return fma((0.5 + y), x, z);
}

Julia

function code(x, y, z)
	return fma(Float64(0.5 + y), x, z)
end

Wolfram

code[x_, y_, z_] := N[(N[(0.5 + y), $MachinePrecision] * x + z), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{x}{2} + y \cdot x\right) + z \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{x}{2} + y \cdot x\right) + z} \]

   2. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{x}{2} + y \cdot x\right)} + z \]

   3. lift-/.f64 N/A

      \[\leadsto \left(\color{blue}{\frac{x}{2}} + y \cdot x\right) + z \]

   4. div-inv N/A

      \[\leadsto \left(\color{blue}{x \cdot \frac{1}{2}} + y \cdot x\right) + z \]

   5. lift-*.f64 N/A

      \[\leadsto \left(x \cdot \frac{1}{2} + \color{blue}{y \cdot x}\right) + z \]

   6. *-commutative N/A

      \[\leadsto \left(x \cdot \frac{1}{2} + \color{blue}{x \cdot y}\right) + z \]

   7. distribute-lft-out N/A

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + y\right)} + z \]

   8. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} + y\right) \cdot x} + z \]

   9. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + y, x, z\right)} \]

   10. lower-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{2} + y}, x, z\right) \]

   11. metadata-eval 100.0

       \[\leadsto \mathsf{fma}\left(\color{blue}{0.5} + y, x, z\right) \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 + y, x, z\right)} \]
5. Add Preprocessing

Alternative 6: 25.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* 0.5 x))

C

double code(double x, double y, double z) {
	return 0.5 * x;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.5d0 * x
end function

Java

public static double code(double x, double y, double z) {
	return 0.5 * x;
}

Python

def code(x, y, z):
	return 0.5 * x

Julia

function code(x, y, z)
	return Float64(0.5 * x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.5 * x;
end

Wolfram

code[x_, y_, z_] := N[(0.5 * x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{x}{2} + y \cdot x\right) + z \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{z + \frac{1}{2} \cdot x} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x + z} \]

   2. lower-fma.f64 62.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x, z\right)} \]

5. Applied rewrites 62.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x, z\right)} \]

6. Taylor expanded in x around inf

   \[\leadsto \frac{1}{2} \cdot \color{blue}{x} \]
7. Step-by-step derivation

8. Applied rewrites 23.2%

   \[\leadsto 0.5 \cdot \color{blue}{x} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024295 
(FPCore (x y z)
  :name "Data.Histogram.Bin.BinF:$cfromIndex from histogram-fill-0.8.4.1"
  :precision binary64
  (+ (+ (/ x 2.0) (* y x)) z))