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

Percentage Accurate: 100.0% → 100.0%

Time: 5.2s

Alternatives: 6

Speedup: 1.0×

• 20.5% 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, 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]

Derivation

1. Initial program 100.0%

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

Alternative 2: 74.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z + 1 \leq -2 \cdot 10^{+133}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z + 1 \leq -5000:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{elif}\;z + 1 \leq 20000:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z + 1 \leq 4 \cdot 10^{+154}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ z 1.0) -2e+133)
   (* y z)
   (if (<= (+ z 1.0) -5000.0)
     (fma z x x)
     (if (<= (+ z 1.0) 20000.0)
       (+ y x)
       (if (<= (+ z 1.0) 4e+154) (* y z) (fma z x x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z + 1.0) <= -2e+133) {
		tmp = y * z;
	} else if ((z + 1.0) <= -5000.0) {
		tmp = fma(z, x, x);
	} else if ((z + 1.0) <= 20000.0) {
		tmp = y + x;
	} else if ((z + 1.0) <= 4e+154) {
		tmp = y * z;
	} else {
		tmp = fma(z, x, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z + 1.0) <= -2e+133)
		tmp = Float64(y * z);
	elseif (Float64(z + 1.0) <= -5000.0)
		tmp = fma(z, x, x);
	elseif (Float64(z + 1.0) <= 20000.0)
		tmp = Float64(y + x);
	elseif (Float64(z + 1.0) <= 4e+154)
		tmp = Float64(y * z);
	else
		tmp = fma(z, x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z + 1.0), $MachinePrecision], -2e+133], N[(y * z), $MachinePrecision], If[LessEqual[N[(z + 1.0), $MachinePrecision], -5000.0], N[(z * x + x), $MachinePrecision], If[LessEqual[N[(z + 1.0), $MachinePrecision], 20000.0], N[(y + x), $MachinePrecision], If[LessEqual[N[(z + 1.0), $MachinePrecision], 4e+154], N[(y * z), $MachinePrecision], N[(z * x + x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 z #s(literal 1 binary64)) < -2e133 or 2e4 < (+.f64 z #s(literal 1 binary64)) < 4.00000000000000015e154

   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 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 60.1%

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

   if -2e133 < (+.f64 z #s(literal 1 binary64)) < -5e3 or 4.00000000000000015e154 < (+.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 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 44.2

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

   5. Applied rewrites 44.2%

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

   if -5e3 < (+.f64 z #s(literal 1 binary64)) < 2e4

   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 3.9

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

   5. Applied rewrites 3.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 3.6%

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

   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 97.3

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

   11. Applied rewrites 97.3%

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

Alternative 3: 74.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z + 1 \leq -2 \cdot 10^{+133}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z + 1 \leq -5000:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z + 1 \leq 20000:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z + 1 \leq 4 \cdot 10^{+154}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ z 1.0) -2e+133)
   (* y z)
   (if (<= (+ z 1.0) -5000.0)
     (* x z)
     (if (<= (+ z 1.0) 20000.0)
       (+ y x)
       (if (<= (+ z 1.0) 4e+154) (* y z) (* x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z + 1.0) <= -2e+133) {
		tmp = y * z;
	} else if ((z + 1.0) <= -5000.0) {
		tmp = x * z;
	} else if ((z + 1.0) <= 20000.0) {
		tmp = y + x;
	} else if ((z + 1.0) <= 4e+154) {
		tmp = y * z;
	} else {
		tmp = x * z;
	}
	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 ((z + 1.0d0) <= (-2d+133)) then
        tmp = y * z
    else if ((z + 1.0d0) <= (-5000.0d0)) then
        tmp = x * z
    else if ((z + 1.0d0) <= 20000.0d0) then
        tmp = y + x
    else if ((z + 1.0d0) <= 4d+154) then
        tmp = y * z
    else
        tmp = x * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z + 1.0) <= -2e+133) {
		tmp = y * z;
	} else if ((z + 1.0) <= -5000.0) {
		tmp = x * z;
	} else if ((z + 1.0) <= 20000.0) {
		tmp = y + x;
	} else if ((z + 1.0) <= 4e+154) {
		tmp = y * z;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z + 1.0) <= -2e+133:
		tmp = y * z
	elif (z + 1.0) <= -5000.0:
		tmp = x * z
	elif (z + 1.0) <= 20000.0:
		tmp = y + x
	elif (z + 1.0) <= 4e+154:
		tmp = y * z
	else:
		tmp = x * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z + 1.0) <= -2e+133)
		tmp = Float64(y * z);
	elseif (Float64(z + 1.0) <= -5000.0)
		tmp = Float64(x * z);
	elseif (Float64(z + 1.0) <= 20000.0)
		tmp = Float64(y + x);
	elseif (Float64(z + 1.0) <= 4e+154)
		tmp = Float64(y * z);
	else
		tmp = Float64(x * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z + 1.0) <= -2e+133)
		tmp = y * z;
	elseif ((z + 1.0) <= -5000.0)
		tmp = x * z;
	elseif ((z + 1.0) <= 20000.0)
		tmp = y + x;
	elseif ((z + 1.0) <= 4e+154)
		tmp = y * z;
	else
		tmp = x * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z + 1.0), $MachinePrecision], -2e+133], N[(y * z), $MachinePrecision], If[LessEqual[N[(z + 1.0), $MachinePrecision], -5000.0], N[(x * z), $MachinePrecision], If[LessEqual[N[(z + 1.0), $MachinePrecision], 20000.0], N[(y + x), $MachinePrecision], If[LessEqual[N[(z + 1.0), $MachinePrecision], 4e+154], N[(y * z), $MachinePrecision], N[(x * z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 z #s(literal 1 binary64)) < -2e133 or 2e4 < (+.f64 z #s(literal 1 binary64)) < 4.00000000000000015e154

   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 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 60.1%

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

   if -2e133 < (+.f64 z #s(literal 1 binary64)) < -5e3 or 4.00000000000000015e154 < (+.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 97.3

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

   5. Applied rewrites 97.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 42.8%

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

   if -5e3 < (+.f64 z #s(literal 1 binary64)) < 2e4

   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 3.9

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

   5. Applied rewrites 3.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 3.6%

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

   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 97.3

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

   11. Applied rewrites 97.3%

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

Alternative 4: 74.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z + 1 \leq -5000 \lor \neg \left(z + 1 \leq 2\right):\\ \;\;\;\;x \cdot z\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (+ z 1.0) -5000.0) (not (<= (+ z 1.0) 2.0))) (* x z) (+ y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((z + 1.0) <= -5000.0) || !((z + 1.0) <= 2.0)) {
		tmp = x * z;
	} else {
		tmp = y + x;
	}
	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 (((z + 1.0d0) <= (-5000.0d0)) .or. (.not. ((z + 1.0d0) <= 2.0d0))) then
        tmp = x * z
    else
        tmp = y + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (((z + 1.0) <= -5000.0) || !((z + 1.0) <= 2.0)) {
		tmp = x * z;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((z + 1.0) <= -5000.0) or not ((z + 1.0) <= 2.0):
		tmp = x * z
	else:
		tmp = y + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(z + 1.0) <= -5000.0) || !(Float64(z + 1.0) <= 2.0))
		tmp = Float64(x * z);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((z + 1.0) <= -5000.0) || ~(((z + 1.0) <= 2.0)))
		tmp = x * z;
	else
		tmp = y + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(z + 1.0), $MachinePrecision], -5000.0], N[Not[LessEqual[N[(z + 1.0), $MachinePrecision], 2.0]], $MachinePrecision]], N[(x * z), $MachinePrecision], N[(y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 z #s(literal 1 binary64)) < -5e3 or 2 < (+.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 97.2

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 41.7%

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

   if -5e3 < (+.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 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 3.5

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

   5. Applied rewrites 3.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 3.3%

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

   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 98.5

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

   11. Applied rewrites 98.5%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z + 1 \leq -5000 \lor \neg \left(z + 1 \leq 2\right):\\ \;\;\;\;x \cdot z\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-208}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ x y) -1e-208) (fma z x x) (fma z y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x + y) <= -1e-208) {
		tmp = fma(z, x, x);
	} else {
		tmp = fma(z, y, y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x + y) <= -1e-208)
		tmp = fma(z, x, x);
	else
		tmp = fma(z, y, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x + y), $MachinePrecision], -1e-208], N[(z * x + x), $MachinePrecision], N[(z * y + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -1.0000000000000001e-208

   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 45.7

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

   5. Applied rewrites 45.7%

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

   if -1.0000000000000001e-208 < (+.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 54.2

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

   5. Applied rewrites 54.2%

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

Alternative 6: 50.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ y + x \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return y + 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 = y + x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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 48.9

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

5. Applied rewrites 48.9%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 21.9%

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

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 52.9

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

11. Applied rewrites 52.9%

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

Reproduce

herbie shell --seed 2024318 
(FPCore (x y z)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, G"
  :precision binary64
  (* (+ x y) (+ z 1.0)))