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

Percentage Accurate: 100.0% → 100.0%

Time: 4.4s

Alternatives: 8

Speedup: 1.0×

• 21.5% 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

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

FPCore

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

C

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

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(fma(z, Float64(x + y), 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[(z * N[(x + y), $MachinePrecision] + x), $MachinePrecision] + y), $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. associate-+r+ N/A

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

   7. lower-+.f64 N/A

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

   8. lower-fma.f64 100.0

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 74.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;1 + z \leq -2 \cdot 10^{+217}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;1 + z \leq -5 \cdot 10^{+154}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;1 + z \leq -200:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;1 + z \leq 2:\\ \;\;\;\;x + y\\ \mathbf{elif}\;1 + z \leq 2 \cdot 10^{+65}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

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) -2e+217)
   (* y z)
   (if (<= (+ 1.0 z) -5e+154)
     (* x z)
     (if (<= (+ 1.0 z) -200.0)
       (* y z)
       (if (<= (+ 1.0 z) 2.0)
         (+ x y)
         (if (<= (+ 1.0 z) 2e+65) (* y z) (* x z)))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((1.0 + z) <= -2e+217) {
		tmp = y * z;
	} else if ((1.0 + z) <= -5e+154) {
		tmp = x * z;
	} else if ((1.0 + z) <= -200.0) {
		tmp = y * z;
	} else if ((1.0 + z) <= 2.0) {
		tmp = x + y;
	} else if ((1.0 + z) <= 2e+65) {
		tmp = y * z;
	} 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) <= (-2d+217)) then
        tmp = y * z
    else if ((1.0d0 + z) <= (-5d+154)) then
        tmp = x * z
    else if ((1.0d0 + z) <= (-200.0d0)) then
        tmp = y * z
    else if ((1.0d0 + z) <= 2.0d0) then
        tmp = x + y
    else if ((1.0d0 + z) <= 2d+65) then
        tmp = y * z
    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) <= -2e+217) {
		tmp = y * z;
	} else if ((1.0 + z) <= -5e+154) {
		tmp = x * z;
	} else if ((1.0 + z) <= -200.0) {
		tmp = y * z;
	} else if ((1.0 + z) <= 2.0) {
		tmp = x + y;
	} else if ((1.0 + z) <= 2e+65) {
		tmp = y * z;
	} 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) <= -2e+217:
		tmp = y * z
	elif (1.0 + z) <= -5e+154:
		tmp = x * z
	elif (1.0 + z) <= -200.0:
		tmp = y * z
	elif (1.0 + z) <= 2.0:
		tmp = x + y
	elif (1.0 + z) <= 2e+65:
		tmp = y * z
	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) <= -2e+217)
		tmp = Float64(y * z);
	elseif (Float64(1.0 + z) <= -5e+154)
		tmp = Float64(x * z);
	elseif (Float64(1.0 + z) <= -200.0)
		tmp = Float64(y * z);
	elseif (Float64(1.0 + z) <= 2.0)
		tmp = Float64(x + y);
	elseif (Float64(1.0 + z) <= 2e+65)
		tmp = Float64(y * z);
	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) <= -2e+217)
		tmp = y * z;
	elseif ((1.0 + z) <= -5e+154)
		tmp = x * z;
	elseif ((1.0 + z) <= -200.0)
		tmp = y * z;
	elseif ((1.0 + z) <= 2.0)
		tmp = x + y;
	elseif ((1.0 + z) <= 2e+65)
		tmp = y * z;
	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], -2e+217], N[(y * z), $MachinePrecision], If[LessEqual[N[(1.0 + z), $MachinePrecision], -5e+154], N[(x * z), $MachinePrecision], If[LessEqual[N[(1.0 + z), $MachinePrecision], -200.0], N[(y * z), $MachinePrecision], If[LessEqual[N[(1.0 + z), $MachinePrecision], 2.0], N[(x + y), $MachinePrecision], If[LessEqual[N[(1.0 + z), $MachinePrecision], 2e+65], N[(y * z), $MachinePrecision], N[(x * z), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 z #s(literal 1 binary64)) < -1.99999999999999992e217 or -5.00000000000000004e154 < (+.f64 z #s(literal 1 binary64)) < -200 or 2 < (+.f64 z #s(literal 1 binary64)) < 2e65

   1. Initial program 99.9%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]
   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 54.9

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

   5. Applied rewrites 54.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 53.3%

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

   if -1.99999999999999992e217 < (+.f64 z #s(literal 1 binary64)) < -5.00000000000000004e154 or 2e65 < (+.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 53.3

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

   5. Applied rewrites 53.3%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 53.3%

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

   if -200 < (+.f64 z #s(literal 1 binary64)) < 2

   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 98.4

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

   5. Applied rewrites 98.4%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + z \leq -2 \cdot 10^{+217}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;1 + z \leq -5 \cdot 10^{+154}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;1 + z \leq -200:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;1 + z \leq 2:\\ \;\;\;\;x + y\\ \mathbf{elif}\;1 + z \leq 2 \cdot 10^{+65}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-258}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{elif}\;x + y \leq 2 \cdot 10^{+32}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (+ x y) -1e-258) (fma z x x) (if (<= (+ x y) 2e+32) (+ x y) (* y z))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((x + y) <= -1e-258) {
		tmp = fma(z, x, x);
	} else if ((x + y) <= 2e+32) {
		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(x + y) <= -1e-258)
		tmp = fma(z, x, x);
	elseif (Float64(x + y) <= 2e+32)
		tmp = Float64(x + y);
	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], -1e-258], N[(z * x + x), $MachinePrecision], If[LessEqual[N[(x + y), $MachinePrecision], 2e+32], N[(x + y), $MachinePrecision], N[(y * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x y) < -9.99999999999999954e-259

   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 48.7

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

   5. Applied rewrites 48.7%

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

   if -9.99999999999999954e-259 < (+.f64 x y) < 2.00000000000000011e32

   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 64.9

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

   5. Applied rewrites 64.9%

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

   if 2.00000000000000011e32 < (+.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 56.0

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

   5. Applied rewrites 56.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 38.3%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-258}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{elif}\;x + y \leq 2 \cdot 10^{+32}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;1 + z \leq -200:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;1 + z \leq 500000000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

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) -200.0)
   (* x z)
   (if (<= (+ 1.0 z) 500000000.0) (+ x y) (* x z))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((1.0 + z) <= -200.0) {
		tmp = x * z;
	} else if ((1.0 + z) <= 500000000.0) {
		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) <= (-200.0d0)) then
        tmp = x * z
    else if ((1.0d0 + z) <= 500000000.0d0) 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) <= -200.0) {
		tmp = x * z;
	} else if ((1.0 + z) <= 500000000.0) {
		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) <= -200.0:
		tmp = x * z
	elif (1.0 + z) <= 500000000.0:
		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) <= -200.0)
		tmp = Float64(x * z);
	elseif (Float64(1.0 + z) <= 500000000.0)
		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) <= -200.0)
		tmp = x * z;
	elseif ((1.0 + z) <= 500000000.0)
		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], -200.0], N[(x * z), $MachinePrecision], If[LessEqual[N[(1.0 + z), $MachinePrecision], 500000000.0], N[(x + y), $MachinePrecision], N[(x * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 z #s(literal 1 binary64)) < -200 or 5e8 < (+.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 51.1

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

   5. Applied rewrites 51.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 50.4%

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

   if -200 < (+.f64 z #s(literal 1 binary64)) < 5e8

   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 96.9

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

   5. Applied rewrites 96.9%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + z \leq -200:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;1 + z \leq 500000000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-258}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + z\right) \cdot y\\ \end{array} \end{array} \]

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) -1e-258) (fma z x x) (* (+ 1.0 z) y)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((x + y) <= -1e-258) {
		tmp = fma(z, x, x);
	} else {
		tmp = (1.0 + z) * y;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(x + y) <= -1e-258)
		tmp = fma(z, x, x);
	else
		tmp = Float64(Float64(1.0 + z) * 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], -1e-258], N[(z * x + x), $MachinePrecision], N[(N[(1.0 + z), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -9.99999999999999954e-259

   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 48.7

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

   5. Applied rewrites 48.7%

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

   if -9.99999999999999954e-259 < (+.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 49.3

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

   5. Applied rewrites 49.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, y, y\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 49.3%

      \[\leadsto \left(1 + z\right) \cdot \color{blue}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-258}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, y\right)\\ \end{array} \end{array} \]

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) -1e-258) (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) <= -1e-258) {
		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) <= -1e-258)
		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], -1e-258], N[(z * x + x), $MachinePrecision], N[(z * y + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -9.99999999999999954e-259

   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 48.7

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

   5. Applied rewrites 48.7%

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

   if -9.99999999999999954e-259 < (+.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 49.3

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

   5. Applied rewrites 49.3%

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

Alternative 7: 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 8: 50.5% 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 45.7

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

5. Applied rewrites 45.7%

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

6. Final simplification 45.7%

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

Reproduce

herbie shell --seed 2024277 
(FPCore (x y z)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, G"
  :precision binary64
  (* (+ x y) (+ z 1.0)))