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

Percentage Accurate: 100.0% → 100.0%

Time: 5.4s

Alternatives: 9

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, 0.8× speedup

Math

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

FPCore

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

C

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

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 + ((x + y) * z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(x + N[(y + N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $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. distribute-rgt-in N/A

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

   2. *-lft-identity N/A

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

   3. +-commutative N/A

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

   4. associate-+r+ N/A

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

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(z \cdot \left(x + y\right) + y\right), \color{blue}{x}\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(z \cdot \left(x + y\right)\right), y\right), x\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\left(x + y\right) \cdot z\right), y\right), x\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + y\right), z\right), y\right), x\right) \]

   9. +-lowering-+.f64 100.0%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), z\right), y\right), x\right) \]

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 50.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq 4.9 \cdot 10^{-172}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0) (* x z) (if (<= z 4.9e-172) y (if (<= z 1.0) x (* x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = x * z;
	} else if (z <= 4.9e-172) {
		tmp = y;
	} else if (z <= 1.0) {
		tmp = x;
	} 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)) then
        tmp = x * z
    else if (z <= 4.9d-172) then
        tmp = y
    else if (z <= 1.0d0) then
        tmp = x
    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) {
		tmp = x * z;
	} else if (z <= 4.9e-172) {
		tmp = y;
	} else if (z <= 1.0) {
		tmp = x;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = x * z
	elif z <= 4.9e-172:
		tmp = y
	elif z <= 1.0:
		tmp = x
	else:
		tmp = x * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(x * z);
	elseif (z <= 4.9e-172)
		tmp = y;
	elseif (z <= 1.0)
		tmp = x;
	else
		tmp = Float64(x * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.0)
		tmp = x * z;
	elseif (z <= 4.9e-172)
		tmp = y;
	elseif (z <= 1.0)
		tmp = x;
	else
		tmp = x * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.0], N[(x * z), $MachinePrecision], If[LessEqual[z, 4.9e-172], y, If[LessEqual[z, 1.0], x, N[(x * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1 or 1 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \color{blue}{z}\right) \]
   4. Step-by-step derivation

   5. Simplified 98.7%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 52.6%

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{x}\right) \]

   8. Simplified 52.6%

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

   if -1 < z < 4.9000000000000001e-172

   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

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

      2. +-lowering-+.f64 99.4%

         \[\leadsto \mathsf{+.f64}\left(y, \color{blue}{x}\right) \]

   5. Simplified 99.4%

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

   6. Taylor expanded in y around inf

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

   8. Simplified 52.3%

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

   if 4.9000000000000001e-172 < z < 1

   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

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

      2. +-lowering-+.f64 99.1%

         \[\leadsto \mathsf{+.f64}\left(y, \color{blue}{x}\right) \]

   5. Simplified 99.1%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 42.7%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq 4.9 \cdot 10^{-172}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -29500:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-172}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 510000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -29500.0)
   (* y z)
   (if (<= z 6e-172) y (if (<= z 510000000.0) x (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -29500.0) {
		tmp = y * z;
	} else if (z <= 6e-172) {
		tmp = y;
	} else if (z <= 510000000.0) {
		tmp = x;
	} 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 <= (-29500.0d0)) then
        tmp = y * z
    else if (z <= 6d-172) then
        tmp = y
    else if (z <= 510000000.0d0) then
        tmp = x
    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 <= -29500.0) {
		tmp = y * z;
	} else if (z <= 6e-172) {
		tmp = y;
	} else if (z <= 510000000.0) {
		tmp = x;
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -29500.0:
		tmp = y * z
	elif z <= 6e-172:
		tmp = y
	elif z <= 510000000.0:
		tmp = x
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -29500.0)
		tmp = Float64(y * z);
	elseif (z <= 6e-172)
		tmp = y;
	elseif (z <= 510000000.0)
		tmp = x;
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -29500.0)
		tmp = y * z;
	elseif (z <= 6e-172)
		tmp = y;
	elseif (z <= 510000000.0)
		tmp = x;
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -29500.0], N[(y * z), $MachinePrecision], If[LessEqual[z, 6e-172], y, If[LessEqual[z, 510000000.0], x, N[(y * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -29500 or 5.1e8 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \color{blue}{z}\right) \]
   4. Step-by-step derivation

   5. Simplified 99.4%

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

   6. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 51.2%

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{z}\right) \]

   8. Simplified 51.2%

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

   if -29500 < z < 5.99999999999999967e-172

   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

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

      2. +-lowering-+.f64 98.3%

         \[\leadsto \mathsf{+.f64}\left(y, \color{blue}{x}\right) \]

   5. Simplified 98.3%

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

   6. Taylor expanded in y around inf

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

   8. Simplified 51.8%

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

   if 5.99999999999999967e-172 < z < 5.1e8

   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

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

      2. +-lowering-+.f64 96.4%

         \[\leadsto \mathsf{+.f64}\left(y, \color{blue}{x}\right) \]

   5. Simplified 96.4%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 41.7%

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

Alternative 4: 74.8% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ z 1.0) -20000.0) (* x z) (if (<= (+ z 1.0) 2.0) (+ x y) (* x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z + 1.0) <= -20000.0) {
		tmp = x * z;
	} else if ((z + 1.0) <= 2.0) {
		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) <= (-20000.0d0)) then
        tmp = x * z
    else if ((z + 1.0d0) <= 2.0d0) 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) <= -20000.0) {
		tmp = x * z;
	} else if ((z + 1.0) <= 2.0) {
		tmp = x + y;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z + 1.0) <= -20000.0:
		tmp = x * z
	elif (z + 1.0) <= 2.0:
		tmp = x + y
	else:
		tmp = x * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z + 1.0) <= -20000.0)
		tmp = Float64(x * z);
	elseif (Float64(z + 1.0) <= 2.0)
		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) <= -20000.0)
		tmp = x * z;
	elseif ((z + 1.0) <= 2.0)
		tmp = x + y;
	else
		tmp = x * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z + 1.0), $MachinePrecision], -20000.0], N[(x * z), $MachinePrecision], If[LessEqual[N[(z + 1.0), $MachinePrecision], 2.0], N[(x + y), $MachinePrecision], N[(x * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \color{blue}{z}\right) \]
   4. Step-by-step derivation

   5. Simplified 98.7%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 52.6%

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{x}\right) \]

   8. Simplified 52.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 99.3%

         \[\leadsto \mathsf{+.f64}\left(y, \color{blue}{x}\right) \]

   5. Simplified 99.3%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z + 1 \leq -20000:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z + 1 \leq 2:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -5 \cdot 10^{-235}:\\ \;\;\;\;x \cdot \left(z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ x y) -5e-235) (* x (+ z 1.0)) (+ y (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x + y) <= -5e-235) {
		tmp = x * (z + 1.0);
	} else {
		tmp = y + (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 ((x + y) <= (-5d-235)) then
        tmp = x * (z + 1.0d0)
    else
        tmp = y + (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x + y) <= -5e-235) {
		tmp = x * (z + 1.0);
	} else {
		tmp = y + (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x + y) <= -5e-235:
		tmp = x * (z + 1.0)
	else:
		tmp = y + (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x + y) <= -5e-235)
		tmp = Float64(x * Float64(z + 1.0));
	else
		tmp = Float64(y + Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x + y) <= -5e-235)
		tmp = x * (z + 1.0);
	else
		tmp = y + (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x + y), $MachinePrecision], -5e-235], N[(x * N[(z + 1.0), $MachinePrecision]), $MachinePrecision], N[(y + N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -4.9999999999999998e-235

   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

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + z\right), \color{blue}{x}\right) \]

      3. +-lowering-+.f64 50.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, z\right), x\right) \]

   5. Simplified 50.8%

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

   if -4.9999999999999998e-235 < (+.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

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{y}, \mathsf{+.f64}\left(z, 1\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 52.8%

      \[\leadsto \color{blue}{y} \cdot \left(z + 1\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. distribute-lft1-in N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(z \cdot y\right), \color{blue}{y}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(y \cdot z\right), y\right) \]

      5. *-lowering-*.f64 52.8%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, z\right), y\right) \]

   7. Applied egg-rr 52.8%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -5 \cdot 10^{-235}:\\ \;\;\;\;x \cdot \left(z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -5 \cdot 10^{-235}:\\ \;\;\;\;x \cdot \left(z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ x y) -5e-235) (* x (+ z 1.0)) (* y (+ z 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x + y) <= -5e-235) {
		tmp = x * (z + 1.0);
	} else {
		tmp = y * (z + 1.0);
	}
	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 ((x + y) <= (-5d-235)) then
        tmp = x * (z + 1.0d0)
    else
        tmp = y * (z + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x + y) <= -5e-235) {
		tmp = x * (z + 1.0);
	} else {
		tmp = y * (z + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x + y) <= -5e-235:
		tmp = x * (z + 1.0)
	else:
		tmp = y * (z + 1.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x + y) <= -5e-235)
		tmp = Float64(x * Float64(z + 1.0));
	else
		tmp = Float64(y * Float64(z + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x + y) <= -5e-235)
		tmp = x * (z + 1.0);
	else
		tmp = y * (z + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x + y), $MachinePrecision], -5e-235], N[(x * N[(z + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -4.9999999999999998e-235

   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

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + z\right), \color{blue}{x}\right) \]

      3. +-lowering-+.f64 50.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, z\right), x\right) \]

   5. Simplified 50.8%

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

   if -4.9999999999999998e-235 < (+.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

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{y}, \mathsf{+.f64}\left(z, 1\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 52.8%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -5 \cdot 10^{-235}:\\ \;\;\;\;x \cdot \left(z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 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 8: 31.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-88}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= x -2.5e-88) x y))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.5e-88) {
		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 (x <= (-2.5d-88)) 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 (x <= -2.5e-88) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.5e-88:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.5e-88)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.5e-88)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.5e-88], x, y]

Derivation

1. Split input into 2 regimes

2. if x < -2.50000000000000004e-88

   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

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

      2. +-lowering-+.f64 39.0%

         \[\leadsto \mathsf{+.f64}\left(y, \color{blue}{x}\right) \]

   5. Simplified 39.0%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 20.8%

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

   if -2.50000000000000004e-88 < x

   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

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

      2. +-lowering-+.f64 51.2%

         \[\leadsto \mathsf{+.f64}\left(y, \color{blue}{x}\right) \]

   5. Simplified 51.2%

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

   6. Taylor expanded in y around inf

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

   8. Simplified 28.7%

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

Alternative 9: 26.4% accurate, 7.0× 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

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

   2. +-lowering-+.f64 47.4%

      \[\leadsto \mathsf{+.f64}\left(y, \color{blue}{x}\right) \]

5. Simplified 47.4%

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

6. Taylor expanded in y around 0

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

8. Simplified 23.4%

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

Reproduce

herbie shell --seed 2024155 
(FPCore (x y z)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, G"
  :precision binary64
  (* (+ x y) (+ z 1.0)))