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

Percentage Accurate: 100.0% → 100.0%

Time: 5.9s

Alternatives: 8

Speedup: 1.0×

• 20.4% 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

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (z + 1.0d0) * (x + y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(z + 1.0), $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%

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

Alternative 2: 75.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z + 1 \leq 0.999998:\\ \;\;\;\;\mathsf{fma}\left(y, z, y\right)\\ \mathbf{elif}\;z + 1 \leq 1.00001:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z + 1 \leq 4 \cdot 10^{+89}:\\ \;\;\;\;\mathsf{fma}\left(y, z, y\right)\\ \mathbf{elif}\;z + 1 \leq 10^{+245}:\\ \;\;\;\;x \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ z 1.0) 0.999998)
   (fma y z y)
   (if (<= (+ z 1.0) 1.00001)
     (+ x y)
     (if (<= (+ z 1.0) 4e+89)
       (fma y z y)
       (if (<= (+ z 1.0) 1e+245) (* x z) (fma y z y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z + 1.0) <= 0.999998) {
		tmp = fma(y, z, y);
	} else if ((z + 1.0) <= 1.00001) {
		tmp = x + y;
	} else if ((z + 1.0) <= 4e+89) {
		tmp = fma(y, z, y);
	} else if ((z + 1.0) <= 1e+245) {
		tmp = x * z;
	} else {
		tmp = fma(y, z, y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z + 1.0) <= 0.999998)
		tmp = fma(y, z, y);
	elseif (Float64(z + 1.0) <= 1.00001)
		tmp = Float64(x + y);
	elseif (Float64(z + 1.0) <= 4e+89)
		tmp = fma(y, z, y);
	elseif (Float64(z + 1.0) <= 1e+245)
		tmp = Float64(x * z);
	else
		tmp = fma(y, z, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z + 1.0), $MachinePrecision], 0.999998], N[(y * z + y), $MachinePrecision], If[LessEqual[N[(z + 1.0), $MachinePrecision], 1.00001], N[(x + y), $MachinePrecision], If[LessEqual[N[(z + 1.0), $MachinePrecision], 4e+89], N[(y * z + y), $MachinePrecision], If[LessEqual[N[(z + 1.0), $MachinePrecision], 1e+245], N[(x * z), $MachinePrecision], N[(y * z + y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 z #s(literal 1 binary64)) < 0.999998000000000054 or 1.0000100000000001 < (+.f64 z #s(literal 1 binary64)) < 3.99999999999999998e89 or 1.00000000000000004e245 < (+.f64 z #s(literal 1 binary64))

   1. Initial program 99.9%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]
   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-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. accelerator-lowering-fma.f64 51.3

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

   5. Simplified 51.3%

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

   if 0.999998000000000054 < (+.f64 z #s(literal 1 binary64)) < 1.0000100000000001

   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. +-lowering-+.f64 99.3

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

   5. Simplified 99.3%

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

   if 3.99999999999999998e89 < (+.f64 z #s(literal 1 binary64)) < 1.00000000000000004e245

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 70.3%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z + 1 \leq 0.999998:\\ \;\;\;\;\mathsf{fma}\left(y, z, y\right)\\ \mathbf{elif}\;z + 1 \leq 1.00001:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z + 1 \leq 4 \cdot 10^{+89}:\\ \;\;\;\;\mathsf{fma}\left(y, z, y\right)\\ \mathbf{elif}\;z + 1 \leq 10^{+245}:\\ \;\;\;\;x \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z + 1 \leq -50000000:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z + 1 \leq 40000000000000:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z + 1 \leq 10^{+245}:\\ \;\;\;\;x \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ z 1.0) -50000000.0)
   (* y z)
   (if (<= (+ z 1.0) 40000000000000.0)
     (+ x y)
     (if (<= (+ z 1.0) 1e+245) (* x z) (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z + 1.0) <= -50000000.0) {
		tmp = y * z;
	} else if ((z + 1.0) <= 40000000000000.0) {
		tmp = x + y;
	} else if ((z + 1.0) <= 1e+245) {
		tmp = x * z;
	} else {
		tmp = y * 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) <= (-50000000.0d0)) then
        tmp = y * z
    else if ((z + 1.0d0) <= 40000000000000.0d0) then
        tmp = x + y
    else if ((z + 1.0d0) <= 1d+245) then
        tmp = x * z
    else
        tmp = y * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z + 1.0) <= -50000000.0) {
		tmp = y * z;
	} else if ((z + 1.0) <= 40000000000000.0) {
		tmp = x + y;
	} else if ((z + 1.0) <= 1e+245) {
		tmp = x * z;
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z + 1.0) <= -50000000.0:
		tmp = y * z
	elif (z + 1.0) <= 40000000000000.0:
		tmp = x + y
	elif (z + 1.0) <= 1e+245:
		tmp = x * z
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z + 1.0) <= -50000000.0)
		tmp = Float64(y * z);
	elseif (Float64(z + 1.0) <= 40000000000000.0)
		tmp = Float64(x + y);
	elseif (Float64(z + 1.0) <= 1e+245)
		tmp = Float64(x * z);
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z + 1.0) <= -50000000.0)
		tmp = y * z;
	elseif ((z + 1.0) <= 40000000000000.0)
		tmp = x + y;
	elseif ((z + 1.0) <= 1e+245)
		tmp = x * z;
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z + 1.0), $MachinePrecision], -50000000.0], N[(y * z), $MachinePrecision], If[LessEqual[N[(z + 1.0), $MachinePrecision], 40000000000000.0], N[(x + y), $MachinePrecision], If[LessEqual[N[(z + 1.0), $MachinePrecision], 1e+245], N[(x * z), $MachinePrecision], N[(y * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 z #s(literal 1 binary64)) < -5e7 or 1.00000000000000004e245 < (+.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

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

   5. Simplified 98.6%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 54.4%

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

   if -5e7 < (+.f64 z #s(literal 1 binary64)) < 4e13

   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. +-lowering-+.f64 96.7

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

   5. Simplified 96.7%

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

   if 4e13 < (+.f64 z #s(literal 1 binary64)) < 1.00000000000000004e245

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 68.6%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z + 1 \leq -50000000:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z + 1 \leq 40000000000000:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z + 1 \leq 10^{+245}:\\ \;\;\;\;x \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z + 1 \leq -50000000:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z + 1 \leq 40000000000000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ z 1.0) -50000000.0)
   (* x z)
   (if (<= (+ z 1.0) 40000000000000.0) (+ x y) (* x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z + 1.0) <= -50000000.0) {
		tmp = x * z;
	} else if ((z + 1.0) <= 40000000000000.0) {
		tmp = x + y;
	} 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) <= (-50000000.0d0)) then
        tmp = x * z
    else if ((z + 1.0d0) <= 40000000000000.0d0) then
        tmp = x + y
    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) <= -50000000.0) {
		tmp = x * z;
	} else if ((z + 1.0) <= 40000000000000.0) {
		tmp = x + y;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z + 1.0) <= -50000000.0:
		tmp = x * z
	elif (z + 1.0) <= 40000000000000.0:
		tmp = x + y
	else:
		tmp = x * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z + 1.0) <= -50000000.0)
		tmp = Float64(x * z);
	elseif (Float64(z + 1.0) <= 40000000000000.0)
		tmp = Float64(x + y);
	else
		tmp = Float64(x * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z + 1.0) <= -50000000.0)
		tmp = x * z;
	elseif ((z + 1.0) <= 40000000000000.0)
		tmp = x + y;
	else
		tmp = x * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z + 1.0), $MachinePrecision], -50000000.0], N[(x * z), $MachinePrecision], If[LessEqual[N[(z + 1.0), $MachinePrecision], 40000000000000.0], N[(x + y), $MachinePrecision], N[(x * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 z #s(literal 1 binary64)) < -5e7 or 4e13 < (+.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

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

   5. Simplified 99.1%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 56.9%

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

   if -5e7 < (+.f64 z #s(literal 1 binary64)) < 4e13

   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. +-lowering-+.f64 96.7

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

   5. Simplified 96.7%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z + 1 \leq -50000000:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z + 1 \leq 40000000000000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 26.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(x + y\right) \leq -1 \cdot 10^{-269}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* (+ z 1.0) (+ x y)) -1e-269) x y))

C

double code(double x, double y, double z) {
	double tmp;
	if (((z + 1.0) * (x + y)) <= -1e-269) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((z + 1.0d0) * (x + y)) <= (-1d-269)) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (((z + 1.0) * (x + y)) <= -1e-269) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((z + 1.0) * (x + y)) <= -1e-269:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(z + 1.0) * Float64(x + y)) <= -1e-269)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((z + 1.0) * (x + y)) <= -1e-269)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(z + 1.0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision], -1e-269], x, y]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (+.f64 x y) (+.f64 z #s(literal 1 binary64))) < -9.9999999999999996e-270

   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. +-lowering-+.f64 53.7

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

   5. Simplified 53.7%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 26.3%

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

   if -9.9999999999999996e-270 < (*.f64 (+.f64 x y) (+.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 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-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. accelerator-lowering-fma.f64 44.6

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

   5. Simplified 44.6%

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

   6. Taylor expanded in z around 0

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

   8. Simplified 20.7%

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

4. Final simplification 23.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(x + y\right) \leq -1 \cdot 10^{-269}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-269}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ x y) -1e-269) (fma z x x) (fma y z y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x + y) <= -1e-269) {
		tmp = fma(z, x, x);
	} else {
		tmp = fma(y, z, y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x + y) <= -1e-269)
		tmp = fma(z, x, x);
	else
		tmp = fma(y, z, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x + y), $MachinePrecision], -1e-269], N[(z * x + x), $MachinePrecision], N[(y * z + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -9.9999999999999996e-270

   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. accelerator-lowering-fma.f64 53.3

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

   5. Simplified 53.3%

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

   if -9.9999999999999996e-270 < (+.f64 x y)

   1. Initial program 99.9%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]
   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-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. accelerator-lowering-fma.f64 42.2

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

   5. Simplified 42.2%

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

Alternative 7: 51.3% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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. +-lowering-+.f64 53.3

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

5. Simplified 53.3%

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

6. Final simplification 53.3%

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

Alternative 8: 26.7% accurate, 12.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

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. +-lowering-+.f64 53.3

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

5. Simplified 53.3%

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

6. Taylor expanded in y around 0

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

8. Simplified 30.2%

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

Reproduce

herbie shell --seed 2024195 
(FPCore (x y z)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, G"
  :precision binary64
  (* (+ x y) (+ z 1.0)))