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

Percentage Accurate: 100.0% → 100.0%

Time: 4.7s

Alternatives: 8

Speedup: 1.0×

• 21.1% 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: 73.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z + 1 \leq -1 \cdot 10^{+161}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z + 1 \leq -1 \cdot 10^{+80}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z + 1 \leq -2 \cdot 10^{+19}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z + 1 \leq 100000:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z + 1 \leq 5 \cdot 10^{+261}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ z 1.0) -1e+161)
   (* y z)
   (if (<= (+ z 1.0) -1e+80)
     (* x z)
     (if (<= (+ z 1.0) -2e+19)
       (* y z)
       (if (<= (+ z 1.0) 100000.0)
         (+ x y)
         (if (<= (+ z 1.0) 5e+261) (* y z) (* x z)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z + 1.0) <= -1e+161) {
		tmp = y * z;
	} else if ((z + 1.0) <= -1e+80) {
		tmp = x * z;
	} else if ((z + 1.0) <= -2e+19) {
		tmp = y * z;
	} else if ((z + 1.0) <= 100000.0) {
		tmp = x + y;
	} else if ((z + 1.0) <= 5e+261) {
		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) <= (-1d+161)) then
        tmp = y * z
    else if ((z + 1.0d0) <= (-1d+80)) then
        tmp = x * z
    else if ((z + 1.0d0) <= (-2d+19)) then
        tmp = y * z
    else if ((z + 1.0d0) <= 100000.0d0) then
        tmp = x + y
    else if ((z + 1.0d0) <= 5d+261) 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) <= -1e+161) {
		tmp = y * z;
	} else if ((z + 1.0) <= -1e+80) {
		tmp = x * z;
	} else if ((z + 1.0) <= -2e+19) {
		tmp = y * z;
	} else if ((z + 1.0) <= 100000.0) {
		tmp = x + y;
	} else if ((z + 1.0) <= 5e+261) {
		tmp = y * z;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z + 1.0) <= -1e+161:
		tmp = y * z
	elif (z + 1.0) <= -1e+80:
		tmp = x * z
	elif (z + 1.0) <= -2e+19:
		tmp = y * z
	elif (z + 1.0) <= 100000.0:
		tmp = x + y
	elif (z + 1.0) <= 5e+261:
		tmp = y * z
	else:
		tmp = x * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z + 1.0) <= -1e+161)
		tmp = Float64(y * z);
	elseif (Float64(z + 1.0) <= -1e+80)
		tmp = Float64(x * z);
	elseif (Float64(z + 1.0) <= -2e+19)
		tmp = Float64(y * z);
	elseif (Float64(z + 1.0) <= 100000.0)
		tmp = Float64(x + y);
	elseif (Float64(z + 1.0) <= 5e+261)
		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) <= -1e+161)
		tmp = y * z;
	elseif ((z + 1.0) <= -1e+80)
		tmp = x * z;
	elseif ((z + 1.0) <= -2e+19)
		tmp = y * z;
	elseif ((z + 1.0) <= 100000.0)
		tmp = x + y;
	elseif ((z + 1.0) <= 5e+261)
		tmp = y * z;
	else
		tmp = x * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z + 1.0), $MachinePrecision], -1e+161], N[(y * z), $MachinePrecision], If[LessEqual[N[(z + 1.0), $MachinePrecision], -1e+80], N[(x * z), $MachinePrecision], If[LessEqual[N[(z + 1.0), $MachinePrecision], -2e+19], N[(y * z), $MachinePrecision], If[LessEqual[N[(z + 1.0), $MachinePrecision], 100000.0], N[(x + y), $MachinePrecision], If[LessEqual[N[(z + 1.0), $MachinePrecision], 5e+261], N[(y * z), $MachinePrecision], N[(x * z), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 z #s(literal 1 binary64)) < -1e161 or -1e80 < (+.f64 z #s(literal 1 binary64)) < -2e19 or 1e5 < (+.f64 z #s(literal 1 binary64)) < 5.0000000000000001e261

   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 99.5%

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

   6. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 54.7

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

   8. Simplified 54.7%

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

   if -1e161 < (+.f64 z #s(literal 1 binary64)) < -1e80 or 5.0000000000000001e261 < (+.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 100.0%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 65.2

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

   8. Simplified 65.2%

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

   if -2e19 < (+.f64 z #s(literal 1 binary64)) < 1e5

   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 98.1

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

   5. Simplified 98.1%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z + 1 \leq -1 \cdot 10^{+161}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z + 1 \leq -1 \cdot 10^{+80}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z + 1 \leq -2 \cdot 10^{+19}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z + 1 \leq 100000:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z + 1 \leq 5 \cdot 10^{+261}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 42.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-5}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x + y \leq -2 \cdot 10^{-228}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ x y) -2e-5) (* x z) (if (<= (+ x y) -2e-228) (+ x y) (fma y z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x + y) <= -2e-5) {
		tmp = x * z;
	} else if ((x + y) <= -2e-228) {
		tmp = x + y;
	} else {
		tmp = fma(y, z, y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x + y) <= -2e-5)
		tmp = Float64(x * z);
	elseif (Float64(x + y) <= -2e-228)
		tmp = Float64(x + y);
	else
		tmp = fma(y, z, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x + y), $MachinePrecision], -2e-5], N[(x * z), $MachinePrecision], If[LessEqual[N[(x + y), $MachinePrecision], -2e-228], N[(x + y), $MachinePrecision], N[(y * z + y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x y) < -2.00000000000000016e-5

   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 54.8%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 30.8

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

   8. Simplified 30.8%

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

   if -2.00000000000000016e-5 < (+.f64 x y) < -2.00000000000000007e-228

   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. +-lowering-+.f64 76.7

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

   5. Simplified 76.7%

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

   if -2.00000000000000007e-228 < (+.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-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 52.5

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

   5. Simplified 52.5%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-5}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x + y \leq -2 \cdot 10^{-228}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z + 1 \leq -2 \cdot 10^{+19}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z + 1 \leq 100000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ z 1.0) -2e+19)
   (* y z)
   (if (<= (+ z 1.0) 100000.0) (+ x y) (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z + 1.0) <= -2e+19) {
		tmp = y * z;
	} else if ((z + 1.0) <= 100000.0) {
		tmp = x + y;
	} 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) <= (-2d+19)) then
        tmp = y * z
    else if ((z + 1.0d0) <= 100000.0d0) then
        tmp = x + y
    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) <= -2e+19) {
		tmp = y * z;
	} else if ((z + 1.0) <= 100000.0) {
		tmp = x + y;
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z + 1.0) <= -2e+19:
		tmp = y * z
	elif (z + 1.0) <= 100000.0:
		tmp = x + y
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z + 1.0) <= -2e+19)
		tmp = Float64(y * z);
	elseif (Float64(z + 1.0) <= 100000.0)
		tmp = Float64(x + y);
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z + 1.0) <= -2e+19)
		tmp = y * z;
	elseif ((z + 1.0) <= 100000.0)
		tmp = x + y;
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z + 1.0), $MachinePrecision], -2e+19], N[(y * z), $MachinePrecision], If[LessEqual[N[(z + 1.0), $MachinePrecision], 100000.0], N[(x + y), $MachinePrecision], N[(y * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 z #s(literal 1 binary64)) < -2e19 or 1e5 < (+.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 99.6%

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

   6. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 52.1

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

   8. Simplified 52.1%

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

   if -2e19 < (+.f64 z #s(literal 1 binary64)) < 1e5

   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 98.1

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

   5. Simplified 98.1%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z + 1 \leq -2 \cdot 10^{+19}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z + 1 \leq 100000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 26.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(x + y\right) \leq -2 \cdot 10^{-228}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* (+ z 1.0) (+ x y)) -2e-228) x y))

C

double code(double x, double y, double z) {
	double tmp;
	if (((z + 1.0) * (x + y)) <= -2e-228) {
		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)) <= (-2d-228)) 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)) <= -2e-228) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((z + 1.0) * (x + y)) <= -2e-228:
		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)) <= -2e-228)
		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)) <= -2e-228)
		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], -2e-228], x, y]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (+.f64 x y) (+.f64 z #s(literal 1 binary64))) < -2.00000000000000007e-228

   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 50.7

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

   5. Simplified 50.7%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 19.4%

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

   if -2.00000000000000007e-228 < (*.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 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 50.6

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

   5. Simplified 50.6%

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

   6. Taylor expanded in y around inf

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

   8. Simplified 28.3%

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

4. Final simplification 23.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(x + y\right) \leq -2 \cdot 10^{-228}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-228}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ x y) -2e-228) (fma z x x) (fma y z y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x + y) <= -2e-228) {
		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) <= -2e-228)
		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], -2e-228], N[(z * x + x), $MachinePrecision], N[(y * z + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -2.00000000000000007e-228

   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 45.2

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

   5. Simplified 45.2%

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

   if -2.00000000000000007e-228 < (+.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-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 52.5

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

   5. Simplified 52.5%

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

Alternative 7: 50.5% 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 50.7

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

5. Simplified 50.7%

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

6. Final simplification 50.7%

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

Alternative 8: 25.5% 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 50.7

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

5. Simplified 50.7%

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

6. Taylor expanded in y around 0

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

8. Simplified 21.3%

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

Reproduce

herbie shell --seed 2024198 
(FPCore (x y z)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, G"
  :precision binary64
  (* (+ x y) (+ z 1.0)))