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

Percentage Accurate: 100.0% → 100.0%

Time: 5.2s

Alternatives: 8

Speedup: 1.0×

• 21.2% 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: 50.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+25}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -1:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-230}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-83}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.3e+25)
   (* y z)
   (if (<= z -1.0)
     (* x z)
     (if (<= z 2.7e-230) y (if (<= z 7e-83) x (if (<= z 1.0) y (* y z)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.3e+25) {
		tmp = y * z;
	} else if (z <= -1.0) {
		tmp = x * z;
	} else if (z <= 2.7e-230) {
		tmp = y;
	} else if (z <= 7e-83) {
		tmp = x;
	} else if (z <= 1.0) {
		tmp = 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 <= (-3.3d+25)) then
        tmp = y * z
    else if (z <= (-1.0d0)) then
        tmp = x * z
    else if (z <= 2.7d-230) then
        tmp = y
    else if (z <= 7d-83) then
        tmp = x
    else if (z <= 1.0d0) then
        tmp = 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 <= -3.3e+25) {
		tmp = y * z;
	} else if (z <= -1.0) {
		tmp = x * z;
	} else if (z <= 2.7e-230) {
		tmp = y;
	} else if (z <= 7e-83) {
		tmp = x;
	} else if (z <= 1.0) {
		tmp = y;
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3.3e+25:
		tmp = y * z
	elif z <= -1.0:
		tmp = x * z
	elif z <= 2.7e-230:
		tmp = y
	elif z <= 7e-83:
		tmp = x
	elif z <= 1.0:
		tmp = y
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.3e+25)
		tmp = Float64(y * z);
	elseif (z <= -1.0)
		tmp = Float64(x * z);
	elseif (z <= 2.7e-230)
		tmp = y;
	elseif (z <= 7e-83)
		tmp = x;
	elseif (z <= 1.0)
		tmp = y;
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -3.3e+25)
		tmp = y * z;
	elseif (z <= -1.0)
		tmp = x * z;
	elseif (z <= 2.7e-230)
		tmp = y;
	elseif (z <= 7e-83)
		tmp = x;
	elseif (z <= 1.0)
		tmp = y;
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.3e+25], N[(y * z), $MachinePrecision], If[LessEqual[z, -1.0], N[(x * z), $MachinePrecision], If[LessEqual[z, 2.7e-230], y, If[LessEqual[z, 7e-83], x, If[LessEqual[z, 1.0], y, N[(y * z), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -3.3000000000000001e25 or 1 < z

   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 54.1%

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

   6. Taylor expanded in z around inf

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

   8. Simplified 53.4%

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

   if -3.3000000000000001e25 < z < -1

   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 22.1%

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

   5. Simplified 22.1%

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

   6. Taylor expanded in z around inf

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

   8. Simplified 12.1%

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

   if -1 < z < 2.70000000000000011e-230 or 7.00000000000000061e-83 < 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 98.1%

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

   5. Simplified 98.1%

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

   6. Taylor expanded in y around inf

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

   8. Simplified 51.5%

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

   if 2.70000000000000011e-230 < z < 7.00000000000000061e-83

   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 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 52.4%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+25}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -1:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-230}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-83}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z + 1 \leq -5 \cdot 10^{+25}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z + 1 \leq -1000000:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z + 1 \leq 1:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ z 1.0) -5e+25)
   (* y z)
   (if (<= (+ z 1.0) -1000000.0)
     (* x z)
     (if (<= (+ z 1.0) 1.0) (+ x y) (* y (+ z 1.0))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z + 1.0) <= -5e+25) {
		tmp = y * z;
	} else if ((z + 1.0) <= -1000000.0) {
		tmp = x * z;
	} else if ((z + 1.0) <= 1.0) {
		tmp = x + y;
	} 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 ((z + 1.0d0) <= (-5d+25)) then
        tmp = y * z
    else if ((z + 1.0d0) <= (-1000000.0d0)) then
        tmp = x * z
    else if ((z + 1.0d0) <= 1.0d0) then
        tmp = x + y
    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 ((z + 1.0) <= -5e+25) {
		tmp = y * z;
	} else if ((z + 1.0) <= -1000000.0) {
		tmp = x * z;
	} else if ((z + 1.0) <= 1.0) {
		tmp = x + y;
	} else {
		tmp = y * (z + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z + 1.0) <= -5e+25:
		tmp = y * z
	elif (z + 1.0) <= -1000000.0:
		tmp = x * z
	elif (z + 1.0) <= 1.0:
		tmp = x + y
	else:
		tmp = y * (z + 1.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z + 1.0) <= -5e+25)
		tmp = Float64(y * z);
	elseif (Float64(z + 1.0) <= -1000000.0)
		tmp = Float64(x * z);
	elseif (Float64(z + 1.0) <= 1.0)
		tmp = Float64(x + y);
	else
		tmp = Float64(y * Float64(z + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z + 1.0) <= -5e+25)
		tmp = y * z;
	elseif ((z + 1.0) <= -1000000.0)
		tmp = x * z;
	elseif ((z + 1.0) <= 1.0)
		tmp = x + y;
	else
		tmp = y * (z + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z + 1.0), $MachinePrecision], -5e+25], N[(y * z), $MachinePrecision], If[LessEqual[N[(z + 1.0), $MachinePrecision], -1000000.0], N[(x * z), $MachinePrecision], If[LessEqual[N[(z + 1.0), $MachinePrecision], 1.0], N[(x + y), $MachinePrecision], N[(y * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 z #s(literal 1 binary64)) < -5.00000000000000024e25

   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 56.2%

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

   6. Taylor expanded in z around inf

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

   8. Simplified 56.2%

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

   if -5.00000000000000024e25 < (+.f64 z #s(literal 1 binary64)) < -1e6

   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 22.1%

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

   5. Simplified 22.1%

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

   6. Taylor expanded in z around inf

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

   8. Simplified 12.1%

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

   if -1e6 < (+.f64 z #s(literal 1 binary64)) < 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 99.6%

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

   5. Simplified 99.6%

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

   if 1 < (+.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

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

   5. Simplified 52.9%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z + 1 \leq -5 \cdot 10^{+25}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z + 1 \leq -1000000:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z + 1 \leq 1:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 50.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-230}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-83}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0)
   (* y z)
   (if (<= z 6.8e-230) y (if (<= z 6.2e-83) x (if (<= z 1.0) y (* y z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = y * z;
	} else if (z <= 6.8e-230) {
		tmp = y;
	} else if (z <= 6.2e-83) {
		tmp = x;
	} else if (z <= 1.0) {
		tmp = 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)) then
        tmp = y * z
    else if (z <= 6.8d-230) then
        tmp = y
    else if (z <= 6.2d-83) then
        tmp = x
    else if (z <= 1.0d0) then
        tmp = 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) {
		tmp = y * z;
	} else if (z <= 6.8e-230) {
		tmp = y;
	} else if (z <= 6.2e-83) {
		tmp = x;
	} else if (z <= 1.0) {
		tmp = y;
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = y * z
	elif z <= 6.8e-230:
		tmp = y
	elif z <= 6.2e-83:
		tmp = x
	elif z <= 1.0:
		tmp = y
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(y * z);
	elseif (z <= 6.8e-230)
		tmp = y;
	elseif (z <= 6.2e-83)
		tmp = x;
	elseif (z <= 1.0)
		tmp = 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)
		tmp = y * z;
	elseif (z <= 6.8e-230)
		tmp = y;
	elseif (z <= 6.2e-83)
		tmp = x;
	elseif (z <= 1.0)
		tmp = y;
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.0], N[(y * z), $MachinePrecision], If[LessEqual[z, 6.8e-230], y, If[LessEqual[z, 6.2e-83], x, If[LessEqual[z, 1.0], y, N[(y * 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 x around 0

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

   5. Simplified 55.2%

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

   6. Taylor expanded in z around inf

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

   8. Simplified 54.4%

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

   if -1 < z < 6.8e-230 or 6.19999999999999985e-83 < 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 98.1%

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

   5. Simplified 98.1%

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

   6. Taylor expanded in y around inf

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

   8. Simplified 51.5%

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

   if 6.8e-230 < z < 6.19999999999999985e-83

   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 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 52.4%

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

Alternative 5: 74.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+24}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -1:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq 40000000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -6e+24)
   (* y z)
   (if (<= z -1.0) (* x z) (if (<= z 40000000.0) (+ x y) (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -6e+24) {
		tmp = y * z;
	} else if (z <= -1.0) {
		tmp = x * z;
	} else if (z <= 40000000.0) {
		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 <= (-6d+24)) then
        tmp = y * z
    else if (z <= (-1.0d0)) then
        tmp = x * z
    else if (z <= 40000000.0d0) 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 <= -6e+24) {
		tmp = y * z;
	} else if (z <= -1.0) {
		tmp = x * z;
	} else if (z <= 40000000.0) {
		tmp = x + y;
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -6e+24:
		tmp = y * z
	elif z <= -1.0:
		tmp = x * z
	elif z <= 40000000.0:
		tmp = x + y
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -6e+24)
		tmp = Float64(y * z);
	elseif (z <= -1.0)
		tmp = Float64(x * z);
	elseif (z <= 40000000.0)
		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 <= -6e+24)
		tmp = y * z;
	elseif (z <= -1.0)
		tmp = x * z;
	elseif (z <= 40000000.0)
		tmp = x + y;
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -6e+24], N[(y * z), $MachinePrecision], If[LessEqual[z, -1.0], N[(x * z), $MachinePrecision], If[LessEqual[z, 40000000.0], N[(x + y), $MachinePrecision], N[(y * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.9999999999999999e24 or 4e7 < z

   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 55.0%

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

   6. Taylor expanded in z around inf

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

   8. Simplified 54.6%

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

   if -5.9999999999999999e24 < z < -1

   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 22.1%

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

   5. Simplified 22.1%

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

   6. Taylor expanded in z around inf

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

   8. Simplified 12.1%

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

   if -1 < z < 4e7

   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 95.9%

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

   5. Simplified 95.9%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+24}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -1:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq 40000000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-226}:\\ \;\;\;\;x \cdot \left(z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ x y) -1e-226) (* x (+ z 1.0)) (* y (+ z 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x + y) <= -1e-226) {
		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) <= (-1d-226)) 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) <= -1e-226) {
		tmp = x * (z + 1.0);
	} else {
		tmp = y * (z + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x + y) <= -1e-226:
		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) <= -1e-226)
		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) <= -1e-226)
		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], -1e-226], 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) < -9.99999999999999921e-227

   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 46.1%

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

   5. Simplified 46.1%

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

   if -9.99999999999999921e-227 < (+.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 51.3%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-226}:\\ \;\;\;\;x \cdot \left(z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 32.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-51}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= x -4e-51) x y))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4e-51) {
		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 <= (-4d-51)) 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 <= -4e-51) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -4e-51:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4e-51)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -4e-51)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4e-51], x, y]

Derivation

1. Split input into 2 regimes

2. if x < -4e-51

   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 58.9%

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

   5. Simplified 58.9%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 45.1%

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

   if -4e-51 < 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 46.6%

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

   5. Simplified 46.6%

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

   6. Taylor expanded in y around inf

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

   8. Simplified 29.6%

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

Alternative 8: 26.0% 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 49.6%

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

5. Simplified 49.6%

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

6. Taylor expanded in y around 0

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

8. Simplified 25.6%

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

Reproduce

herbie shell --seed 2024158 
(FPCore (x y z)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, G"
  :precision binary64
  (* (+ x y) (+ z 1.0)))