Optimisation.CirclePacking:place from circle-packing-0.1.0.4, G

Percentage Accurate: 100.0% → 100.0%

Time: 4.8s

Alternatives: 7

Speedup: 1.0×

• 20.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x + y\right) \cdot \left(z + 1\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* (+ x y) (+ z 1.0)))

C

double code(double x, double y, double z) {
	return (x + y) * (z + 1.0);
}

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 + y) * (z + 1.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return (x + y) * (z + 1.0);
}

Python

def code(x, y, z):
	return (x + y) * (z + 1.0)

Julia

function code(x, y, z)
	return Float64(Float64(x + y) * Float64(z + 1.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x + y) * (z + 1.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(x + y), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x + y\right) \cdot \left(z + 1\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* (+ x y) (+ z 1.0)))

C

double code(double x, double y, double z) {
	return (x + y) * (z + 1.0);
}

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 + y) * (z + 1.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return (x + y) * (z + 1.0);
}

Python

def code(x, y, z):
	return (x + y) * (z + 1.0)

Julia

function code(x, y, z)
	return Float64(Float64(x + y) * Float64(z + 1.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x + y) * (z + 1.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(x + y), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.9× speedup

Math

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

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (+ (fma z (+ x y) y) x))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return fma(z, (x + y), y) + x;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(fma(z, Float64(x + y), y) + x)
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(N[(z * N[(x + y), $MachinePrecision] + y), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(x + y\right) \cdot \left(z + 1\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. associate-+r+ N/A

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

   8. lower-+.f64 N/A

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

   9. lower-fma.f64 100.0

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 75.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;1 + z \leq -4 \cdot 10^{+93}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;1 + z \leq -400000:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;1 + z \leq 100:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (+ 1.0 z) -4e+93)
   (* x z)
   (if (<= (+ 1.0 z) -400000.0)
     (* y z)
     (if (<= (+ 1.0 z) 100.0) (+ x y) (* y z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((1.0 + z) <= -4e+93) {
		tmp = x * z;
	} else if ((1.0 + z) <= -400000.0) {
		tmp = y * z;
	} else if ((1.0 + z) <= 100.0) {
		tmp = x + y;
	} else {
		tmp = y * z;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
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 ((1.0d0 + z) <= (-4d+93)) then
        tmp = x * z
    else if ((1.0d0 + z) <= (-400000.0d0)) then
        tmp = y * z
    else if ((1.0d0 + z) <= 100.0d0) then
        tmp = x + y
    else
        tmp = y * z
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if ((1.0 + z) <= -4e+93) {
		tmp = x * z;
	} else if ((1.0 + z) <= -400000.0) {
		tmp = y * z;
	} else if ((1.0 + z) <= 100.0) {
		tmp = x + y;
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if (1.0 + z) <= -4e+93:
		tmp = x * z
	elif (1.0 + z) <= -400000.0:
		tmp = y * z
	elif (1.0 + z) <= 100.0:
		tmp = x + y
	else:
		tmp = y * z
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(1.0 + z) <= -4e+93)
		tmp = Float64(x * z);
	elseif (Float64(1.0 + z) <= -400000.0)
		tmp = Float64(y * z);
	elseif (Float64(1.0 + z) <= 100.0)
		tmp = Float64(x + y);
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((1.0 + z) <= -4e+93)
		tmp = x * z;
	elseif ((1.0 + z) <= -400000.0)
		tmp = y * z;
	elseif ((1.0 + z) <= 100.0)
		tmp = x + y;
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(1.0 + z), $MachinePrecision], -4e+93], N[(x * z), $MachinePrecision], If[LessEqual[N[(1.0 + z), $MachinePrecision], -400000.0], N[(y * z), $MachinePrecision], If[LessEqual[N[(1.0 + z), $MachinePrecision], 100.0], N[(x + y), $MachinePrecision], N[(y * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 z #s(literal 1 binary64)) < -4.00000000000000017e93

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + z\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 50.0

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

   5. Applied rewrites 50.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 50.0%

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

   if -4.00000000000000017e93 < (+.f64 z #s(literal 1 binary64)) < -4e5 or 100 < (+.f64 z #s(literal 1 binary64))

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(1 + z\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{z \cdot y + 1 \cdot y} \]

      3. *-lft-identity N/A

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

      4. lower-fma.f64 54.6

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

   5. Applied rewrites 54.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 52.8%

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

   if -4e5 < (+.f64 z #s(literal 1 binary64)) < 100

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 96.3

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

   5. Applied rewrites 96.3%

      \[\leadsto \color{blue}{y + x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + z \leq -4 \cdot 10^{+93}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;1 + z \leq -400000:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;1 + z \leq 100:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;1 + z \leq -400000:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;1 + z \leq 5 \cdot 10^{+33}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (+ 1.0 z) -400000.0)
   (* x z)
   (if (<= (+ 1.0 z) 5e+33) (+ x y) (* x z))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((1.0 + z) <= -400000.0) {
		tmp = x * z;
	} else if ((1.0 + z) <= 5e+33) {
		tmp = x + y;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
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 ((1.0d0 + z) <= (-400000.0d0)) then
        tmp = x * z
    else if ((1.0d0 + z) <= 5d+33) then
        tmp = x + y
    else
        tmp = x * z
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if ((1.0 + z) <= -400000.0) {
		tmp = x * z;
	} else if ((1.0 + z) <= 5e+33) {
		tmp = x + y;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if (1.0 + z) <= -400000.0:
		tmp = x * z
	elif (1.0 + z) <= 5e+33:
		tmp = x + y
	else:
		tmp = x * z
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(1.0 + z) <= -400000.0)
		tmp = Float64(x * z);
	elseif (Float64(1.0 + z) <= 5e+33)
		tmp = Float64(x + y);
	else
		tmp = Float64(x * z);
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((1.0 + z) <= -400000.0)
		tmp = x * z;
	elseif ((1.0 + z) <= 5e+33)
		tmp = x + y;
	else
		tmp = x * z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(1.0 + z), $MachinePrecision], -400000.0], N[(x * z), $MachinePrecision], If[LessEqual[N[(1.0 + z), $MachinePrecision], 5e+33], N[(x + y), $MachinePrecision], N[(x * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 z #s(literal 1 binary64)) < -4e5 or 4.99999999999999973e33 < (+.f64 z #s(literal 1 binary64))

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + z\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 49.3

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

   5. Applied rewrites 49.3%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 49.1%

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

   if -4e5 < (+.f64 z #s(literal 1 binary64)) < 4.99999999999999973e33

   1. Initial program 99.9%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 90.7

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

   5. Applied rewrites 90.7%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + z \leq -400000:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;1 + z \leq 5 \cdot 10^{+33}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-273}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (+ x y) -2e-273) (fma z x x) (fma z y y)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((x + y) <= -2e-273) {
		tmp = fma(z, x, x);
	} else {
		tmp = fma(z, y, y);
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(x + y) <= -2e-273)
		tmp = fma(z, x, x);
	else
		tmp = fma(z, y, y);
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(x + y), $MachinePrecision], -2e-273], N[(z * x + x), $MachinePrecision], N[(z * y + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -2e-273

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + z\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 48.9

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

   5. Applied rewrites 48.9%

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

   if -2e-273 < (+.f64 x y)

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(1 + z\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{z \cdot y + 1 \cdot y} \]

      3. *-lft-identity N/A

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

      4. lower-fma.f64 60.1

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

   5. Applied rewrites 60.1%

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

Alternative 5: 73.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-273}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (+ x y) -2e-273) (fma z x x) (* y z)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((x + y) <= -2e-273) {
		tmp = fma(z, x, x);
	} else {
		tmp = y * z;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(x + y) <= -2e-273)
		tmp = fma(z, x, x);
	else
		tmp = Float64(y * z);
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(x + y), $MachinePrecision], -2e-273], N[(z * x + x), $MachinePrecision], N[(y * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -2e-273

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + z\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 48.9

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

   5. Applied rewrites 48.9%

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

   if -2e-273 < (+.f64 x y)

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(1 + z\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{z \cdot y + 1 \cdot y} \]

      3. *-lft-identity N/A

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

      4. lower-fma.f64 60.1

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

   5. Applied rewrites 60.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 34.1%

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

4. Final simplification 41.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-273}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \left(1 + z\right) \cdot \left(x + y\right) \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (* (+ 1.0 z) (+ x y)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return (1.0 + z) * (x + y);
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (1.0d0 + z) * (x + y)
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	return (1.0 + z) * (x + y);
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return (1.0 + z) * (x + y)

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(Float64(1.0 + z) * Float64(x + y))
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = (1.0 + z) * (x + y);
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(N[(1.0 + z), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(x + y\right) \cdot \left(z + 1\right) \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto \left(1 + z\right) \cdot \left(x + y\right) \]
4. Add Preprocessing

Alternative 7: 51.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ x + y \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (+ x y))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return x + y;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + y
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	return x + y;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return x + y

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(x + y)
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = x + y;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(x + y), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(x + y\right) \cdot \left(z + 1\right) \]
2. Add Preprocessing

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 46.0

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

5. Applied rewrites 46.0%

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

6. Final simplification 46.0%

   \[\leadsto x + y \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024268 
(FPCore (x y z)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, G"
  :precision binary64
  (* (+ x y) (+ z 1.0)))