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

Percentage Accurate: 100.0% → 100.0%

Time: 5.3s

Alternatives: 8

Speedup: 1.0×

• 20.9% 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

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

FPCore

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

C

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

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(fma(z, Float64(y + x), 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[(y + x), $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. Add Preprocessing

Alternative 2: 74.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z + 1 \leq -5000000 \lor \neg \left(z + 1 \leq 200\right):\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= (+ z 1.0) -5000000.0) (not (<= (+ z 1.0) 200.0)))
   (* z y)
   (+ y x)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (((z + 1.0) <= -5000000.0) || !((z + 1.0) <= 200.0)) {
		tmp = z * y;
	} else {
		tmp = y + x;
	}
	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 (((z + 1.0d0) <= (-5000000.0d0)) .or. (.not. ((z + 1.0d0) <= 200.0d0))) then
        tmp = z * y
    else
        tmp = y + x
    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 (((z + 1.0) <= -5000000.0) || !((z + 1.0) <= 200.0)) {
		tmp = z * y;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if ((z + 1.0) <= -5000000.0) or not ((z + 1.0) <= 200.0):
		tmp = z * y
	else:
		tmp = y + x
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if ((Float64(z + 1.0) <= -5000000.0) || !(Float64(z + 1.0) <= 200.0))
		tmp = Float64(z * y);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((z + 1.0) <= -5000000.0) || ~(((z + 1.0) <= 200.0)))
		tmp = z * y;
	else
		tmp = y + x;
	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[Or[LessEqual[N[(z + 1.0), $MachinePrecision], -5000000.0], N[Not[LessEqual[N[(z + 1.0), $MachinePrecision], 200.0]], $MachinePrecision]], N[(z * y), $MachinePrecision], N[(y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 z #s(literal 1 binary64)) < -5e6 or 200 < (+.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 z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 54.3%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 49.0%

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

   if -5e6 < (+.f64 z #s(literal 1 binary64)) < 200

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 3.6

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

   5. Applied rewrites 3.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 3.4%

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

   9. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. lower-+.f64 97.5

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

   11. Applied rewrites 97.5%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z + 1 \leq -5000000 \lor \neg \left(z + 1 \leq 200\right):\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z + 1 \leq -5000000 \lor \neg \left(z + 1 \leq 10^{+39}\right):\\ \;\;\;\;x \cdot z\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= (+ z 1.0) -5000000.0) (not (<= (+ z 1.0) 1e+39)))
   (* x z)
   (+ y x)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (((z + 1.0) <= -5000000.0) || !((z + 1.0) <= 1e+39)) {
		tmp = x * z;
	} else {
		tmp = y + x;
	}
	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 (((z + 1.0d0) <= (-5000000.0d0)) .or. (.not. ((z + 1.0d0) <= 1d+39))) then
        tmp = x * z
    else
        tmp = y + x
    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 (((z + 1.0) <= -5000000.0) || !((z + 1.0) <= 1e+39)) {
		tmp = x * z;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if ((z + 1.0) <= -5000000.0) or not ((z + 1.0) <= 1e+39):
		tmp = x * z
	else:
		tmp = y + x
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if ((Float64(z + 1.0) <= -5000000.0) || !(Float64(z + 1.0) <= 1e+39))
		tmp = Float64(x * z);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((z + 1.0) <= -5000000.0) || ~(((z + 1.0) <= 1e+39)))
		tmp = x * z;
	else
		tmp = y + x;
	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[Or[LessEqual[N[(z + 1.0), $MachinePrecision], -5000000.0], N[Not[LessEqual[N[(z + 1.0), $MachinePrecision], 1e+39]], $MachinePrecision]], N[(x * z), $MachinePrecision], N[(y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 z #s(literal 1 binary64)) < -5e6 or 9.9999999999999994e38 < (+.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 z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 53.3%

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

   if -5e6 < (+.f64 z #s(literal 1 binary64)) < 9.9999999999999994e38

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 7.4

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

   5. Applied rewrites 7.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 6.5%

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

   9. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. lower-+.f64 93.6

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

   11. Applied rewrites 93.6%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z + 1 \leq -5000000 \lor \neg \left(z + 1 \leq 10^{+39}\right):\\ \;\;\;\;x \cdot z\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-220}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{elif}\;x + y \leq 2 \cdot 10^{+76}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \end{array} \]

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) -2e-220) (fma z x x) (if (<= (+ x y) 2e+76) (+ y x) (* z y))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((x + y) <= -2e-220) {
		tmp = fma(z, x, x);
	} else if ((x + y) <= 2e+76) {
		tmp = y + x;
	} else {
		tmp = z * y;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(x + y) <= -2e-220)
		tmp = fma(z, x, x);
	elseif (Float64(x + y) <= 2e+76)
		tmp = Float64(y + x);
	else
		tmp = Float64(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], -2e-220], N[(z * x + x), $MachinePrecision], If[LessEqual[N[(x + y), $MachinePrecision], 2e+76], N[(y + x), $MachinePrecision], N[(z * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x y) < -1.99999999999999998e-220

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + z\right)} \]
   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 54.9

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

   5. Applied rewrites 54.9%

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

   if -1.99999999999999998e-220 < (+.f64 x y) < 2.0000000000000001e76

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 43.1

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

   5. Applied rewrites 43.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 25.9%

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

   9. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. lower-+.f64 57.9

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

   11. Applied rewrites 57.9%

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

   if 2.0000000000000001e76 < (+.f64 x y)

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 57.4

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

   5. Applied rewrites 57.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 30.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 30.9%

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

Alternative 5: 97.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-220}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + z\right) \cdot y\\ \end{array} \end{array} \]

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) -2e-220) (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) <= -2e-220) {
		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) <= -2e-220)
		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], -2e-220], 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) < -1.99999999999999998e-220

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + z\right)} \]
   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 54.9

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

   5. Applied rewrites 54.9%

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

   if -1.99999999999999998e-220 < (+.f64 x y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y \cdot \left(1 + z\right)} \]
   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 51.8

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

   5. Applied rewrites 51.8%

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

   7. Applied rewrites 51.8%

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

Alternative 6: 97.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-220}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, y\right)\\ \end{array} \end{array} \]

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-220) (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-220) {
		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-220)
		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-220], N[(z * x + x), $MachinePrecision], N[(z * y + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -1.99999999999999998e-220

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + z\right)} \]
   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 54.9

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

   5. Applied rewrites 54.9%

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

   if -1.99999999999999998e-220 < (+.f64 x y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y \cdot \left(1 + z\right)} \]
   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 51.8

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

   5. Applied rewrites 51.8%

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

Alternative 7: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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) * (z + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 8: 51.3% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

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 = y + x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 50.3

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

5. Applied rewrites 50.3%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 28.3%

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

9. Taylor expanded in z around 0

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

    1. +-commutative N/A

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

    2. lower-+.f64 51.5

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

11. Applied rewrites 51.5%

    \[\leadsto \color{blue}{y + x} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024326 
(FPCore (x y z)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, G"
  :precision binary64
  (* (+ x y) (+ z 1.0)))