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

Percentage Accurate: 100.0% → 100.0%

Time: 4.2s

Alternatives: 9

Speedup: 1.0×

• 20.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(x + y\right) \cdot \left(z + 1\right) \]

2. Final simplification 100.0%

   \[\leadsto \left(x + y\right) \cdot \left(z + 1\right) \]

Alternative 2: 49.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{-29}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq -1.95 \cdot 10^{-140}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -2.75 \cdot 10^{-178}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-43}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 80000:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+87}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{+153}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -8.2e-29)
   (* x z)
   (if (<= z -1.95e-140)
     x
     (if (<= z -2.75e-178)
       y
       (if (<= z 2.3e-43)
         x
         (if (<= z 80000.0)
           y
           (if (<= z 3.3e+87)
             (* x z)
             (if (<= z 8.6e+153) (* y z) (* x z)))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -8.2e-29) {
		tmp = x * z;
	} else if (z <= -1.95e-140) {
		tmp = x;
	} else if (z <= -2.75e-178) {
		tmp = y;
	} else if (z <= 2.3e-43) {
		tmp = x;
	} else if (z <= 80000.0) {
		tmp = y;
	} else if (z <= 3.3e+87) {
		tmp = x * z;
	} else if (z <= 8.6e+153) {
		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 <= (-8.2d-29)) then
        tmp = x * z
    else if (z <= (-1.95d-140)) then
        tmp = x
    else if (z <= (-2.75d-178)) then
        tmp = y
    else if (z <= 2.3d-43) then
        tmp = x
    else if (z <= 80000.0d0) then
        tmp = y
    else if (z <= 3.3d+87) then
        tmp = x * z
    else if (z <= 8.6d+153) 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 <= -8.2e-29) {
		tmp = x * z;
	} else if (z <= -1.95e-140) {
		tmp = x;
	} else if (z <= -2.75e-178) {
		tmp = y;
	} else if (z <= 2.3e-43) {
		tmp = x;
	} else if (z <= 80000.0) {
		tmp = y;
	} else if (z <= 3.3e+87) {
		tmp = x * z;
	} else if (z <= 8.6e+153) {
		tmp = y * z;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -8.2e-29:
		tmp = x * z
	elif z <= -1.95e-140:
		tmp = x
	elif z <= -2.75e-178:
		tmp = y
	elif z <= 2.3e-43:
		tmp = x
	elif z <= 80000.0:
		tmp = y
	elif z <= 3.3e+87:
		tmp = x * z
	elif z <= 8.6e+153:
		tmp = y * z
	else:
		tmp = x * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -8.2e-29)
		tmp = Float64(x * z);
	elseif (z <= -1.95e-140)
		tmp = x;
	elseif (z <= -2.75e-178)
		tmp = y;
	elseif (z <= 2.3e-43)
		tmp = x;
	elseif (z <= 80000.0)
		tmp = y;
	elseif (z <= 3.3e+87)
		tmp = Float64(x * z);
	elseif (z <= 8.6e+153)
		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 <= -8.2e-29)
		tmp = x * z;
	elseif (z <= -1.95e-140)
		tmp = x;
	elseif (z <= -2.75e-178)
		tmp = y;
	elseif (z <= 2.3e-43)
		tmp = x;
	elseif (z <= 80000.0)
		tmp = y;
	elseif (z <= 3.3e+87)
		tmp = x * z;
	elseif (z <= 8.6e+153)
		tmp = y * z;
	else
		tmp = x * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -8.2e-29], N[(x * z), $MachinePrecision], If[LessEqual[z, -1.95e-140], x, If[LessEqual[z, -2.75e-178], y, If[LessEqual[z, 2.3e-43], x, If[LessEqual[z, 80000.0], y, If[LessEqual[z, 3.3e+87], N[(x * z), $MachinePrecision], If[LessEqual[z, 8.6e+153], N[(y * z), $MachinePrecision], N[(x * z), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -8.1999999999999996e-29 or 8e4 < z < 3.3000000000000001e87 or 8.5999999999999995e153 < z

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. distribute-lft-in 99.9%

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

      3. *-rgt-identity 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. associate-+l+ 99.9%

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

      2. flip-+ 21.8%

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

      3. +-commutative 21.8%

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

      4. fma-def 21.8%

         \[\leadsto \frac{x \cdot x - \color{blue}{\mathsf{fma}\left(x + y, z, y\right)} \cdot \left(y + \left(x + y\right) \cdot z\right)}{x - \left(y + \left(x + y\right) \cdot z\right)} \]

      5. +-commutative 21.8%

         \[\leadsto \frac{x \cdot x - \mathsf{fma}\left(x + y, z, y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot z + y\right)}}{x - \left(y + \left(x + y\right) \cdot z\right)} \]

      6. fma-def 21.8%

         \[\leadsto \frac{x \cdot x - \mathsf{fma}\left(x + y, z, y\right) \cdot \color{blue}{\mathsf{fma}\left(x + y, z, y\right)}}{x - \left(y + \left(x + y\right) \cdot z\right)} \]

      7. +-commutative 21.8%

         \[\leadsto \frac{x \cdot x - \mathsf{fma}\left(x + y, z, y\right) \cdot \mathsf{fma}\left(x + y, z, y\right)}{x - \color{blue}{\left(\left(x + y\right) \cdot z + y\right)}} \]

      8. fma-def 21.8%

         \[\leadsto \frac{x \cdot x - \mathsf{fma}\left(x + y, z, y\right) \cdot \mathsf{fma}\left(x + y, z, y\right)}{x - \color{blue}{\mathsf{fma}\left(x + y, z, y\right)}} \]

   5. Applied egg-rr 21.8%

      \[\leadsto \color{blue}{\frac{x \cdot x - \mathsf{fma}\left(x + y, z, y\right) \cdot \mathsf{fma}\left(x + y, z, y\right)}{x - \mathsf{fma}\left(x + y, z, y\right)}} \]
   6. Step-by-step derivation

      1. difference-of-squares 22.6%

         \[\leadsto \frac{\color{blue}{\left(x + \mathsf{fma}\left(x + y, z, y\right)\right) \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      2. +-commutative 22.6%

         \[\leadsto \frac{\color{blue}{\left(\mathsf{fma}\left(x + y, z, y\right) + x\right)} \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      3. fma-udef 22.6%

         \[\leadsto \frac{\left(\color{blue}{\left(\left(x + y\right) \cdot z + y\right)} + x\right) \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      4. associate-+r+ 22.6%

         \[\leadsto \frac{\color{blue}{\left(\left(x + y\right) \cdot z + \left(y + x\right)\right)} \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      5. +-commutative 22.6%

         \[\leadsto \frac{\left(\left(x + y\right) \cdot z + \color{blue}{\left(x + y\right)}\right) \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      6. *-rgt-identity 22.6%

         \[\leadsto \frac{\left(\left(x + y\right) \cdot z + \color{blue}{\left(x + y\right) \cdot 1}\right) \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      7. distribute-lft-in 22.6%

         \[\leadsto \frac{\color{blue}{\left(\left(x + y\right) \cdot \left(z + 1\right)\right)} \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      8. associate-/l* 54.2%

         \[\leadsto \color{blue}{\frac{\left(x + y\right) \cdot \left(z + 1\right)}{\frac{x - \mathsf{fma}\left(x + y, z, y\right)}{x - \mathsf{fma}\left(x + y, z, y\right)}}} \]

      9. *-inverses 100.0%

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

      10. associate-/l* 99.7%

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

      11. +-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in z around inf 95.2%

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

   9. Taylor expanded in y around 0 50.0%

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

   if -8.1999999999999996e-29 < z < -1.9500000000000001e-140 or -2.75000000000000014e-178 < z < 2.2999999999999999e-43

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]

   2. Taylor expanded in x around inf 49.9%

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

   3. Taylor expanded in z around 0 49.9%

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

   if -1.9500000000000001e-140 < z < -2.75000000000000014e-178 or 2.2999999999999999e-43 < z < 8e4

   1. Initial program 99.9%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]

   2. Taylor expanded in x around 0 40.2%

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

   3. Taylor expanded in z around 0 40.2%

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

   if 3.3000000000000001e87 < z < 8.5999999999999995e153

   1. Initial program 99.9%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]
   2. Step-by-step derivation

      1. +-commutative 99.9%

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

      2. distribute-lft-in 99.9%

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

      3. *-rgt-identity 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around 0 52.9%

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

   5. Taylor expanded in z around inf 52.9%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{-29}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq -1.95 \cdot 10^{-140}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -2.75 \cdot 10^{-178}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-43}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 80000:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+87}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{+153}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]

Alternative 3: 50.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -3.6 \cdot 10^{-140}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -2.3 \cdot 10^{-178}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-43}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0)
   (* y z)
   (if (<= z -3.6e-140)
     x
     (if (<= z -2.3e-178) y (if (<= z 6.8e-43) 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 <= -3.6e-140) {
		tmp = x;
	} else if (z <= -2.3e-178) {
		tmp = y;
	} else if (z <= 6.8e-43) {
		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 <= (-3.6d-140)) then
        tmp = x
    else if (z <= (-2.3d-178)) then
        tmp = y
    else if (z <= 6.8d-43) 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 <= -3.6e-140) {
		tmp = x;
	} else if (z <= -2.3e-178) {
		tmp = y;
	} else if (z <= 6.8e-43) {
		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 <= -3.6e-140:
		tmp = x
	elif z <= -2.3e-178:
		tmp = y
	elif z <= 6.8e-43:
		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 <= -3.6e-140)
		tmp = x;
	elseif (z <= -2.3e-178)
		tmp = y;
	elseif (z <= 6.8e-43)
		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 <= -3.6e-140)
		tmp = x;
	elseif (z <= -2.3e-178)
		tmp = y;
	elseif (z <= 6.8e-43)
		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, -3.6e-140], x, If[LessEqual[z, -2.3e-178], y, If[LessEqual[z, 6.8e-43], 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. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. distribute-lft-in 99.9%

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

      3. *-rgt-identity 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around 0 52.2%

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

   5. Taylor expanded in z around inf 51.3%

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

   if -1 < z < -3.6e-140 or -2.29999999999999994e-178 < z < 6.8000000000000001e-43

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]

   2. Taylor expanded in x around inf 49.9%

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

   3. Taylor expanded in z around 0 49.9%

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

   if -3.6e-140 < z < -2.29999999999999994e-178 or 6.8000000000000001e-43 < z < 1

   1. Initial program 99.9%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]

   2. Taylor expanded in x around 0 43.6%

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

   3. Taylor expanded in z around 0 43.6%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -3.6 \cdot 10^{-140}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -2.3 \cdot 10^{-178}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-43}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 4: 75.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.1 \cdot 10^{+63}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-6}:\\ \;\;\;\;y \cdot \left(z + 1\right)\\ \mathbf{elif}\;z \leq 80000:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+90}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+149}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -6.1e+63)
   (* x z)
   (if (<= z -1.7e-6)
     (* y (+ z 1.0))
     (if (<= z 80000.0)
       (+ x y)
       (if (<= z 3.6e+90) (* x z) (if (<= z 1.25e+149) (* y z) (* x z)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -6.1e+63) {
		tmp = x * z;
	} else if (z <= -1.7e-6) {
		tmp = y * (z + 1.0);
	} else if (z <= 80000.0) {
		tmp = x + y;
	} else if (z <= 3.6e+90) {
		tmp = x * z;
	} else if (z <= 1.25e+149) {
		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 <= (-6.1d+63)) then
        tmp = x * z
    else if (z <= (-1.7d-6)) then
        tmp = y * (z + 1.0d0)
    else if (z <= 80000.0d0) then
        tmp = x + y
    else if (z <= 3.6d+90) then
        tmp = x * z
    else if (z <= 1.25d+149) 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 <= -6.1e+63) {
		tmp = x * z;
	} else if (z <= -1.7e-6) {
		tmp = y * (z + 1.0);
	} else if (z <= 80000.0) {
		tmp = x + y;
	} else if (z <= 3.6e+90) {
		tmp = x * z;
	} else if (z <= 1.25e+149) {
		tmp = y * z;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -6.1e+63:
		tmp = x * z
	elif z <= -1.7e-6:
		tmp = y * (z + 1.0)
	elif z <= 80000.0:
		tmp = x + y
	elif z <= 3.6e+90:
		tmp = x * z
	elif z <= 1.25e+149:
		tmp = y * z
	else:
		tmp = x * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -6.1e+63)
		tmp = Float64(x * z);
	elseif (z <= -1.7e-6)
		tmp = Float64(y * Float64(z + 1.0));
	elseif (z <= 80000.0)
		tmp = Float64(x + y);
	elseif (z <= 3.6e+90)
		tmp = Float64(x * z);
	elseif (z <= 1.25e+149)
		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 <= -6.1e+63)
		tmp = x * z;
	elseif (z <= -1.7e-6)
		tmp = y * (z + 1.0);
	elseif (z <= 80000.0)
		tmp = x + y;
	elseif (z <= 3.6e+90)
		tmp = x * z;
	elseif (z <= 1.25e+149)
		tmp = y * z;
	else
		tmp = x * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -6.1e+63], N[(x * z), $MachinePrecision], If[LessEqual[z, -1.7e-6], N[(y * N[(z + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 80000.0], N[(x + y), $MachinePrecision], If[LessEqual[z, 3.6e+90], N[(x * z), $MachinePrecision], If[LessEqual[z, 1.25e+149], N[(y * z), $MachinePrecision], N[(x * z), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -6.09999999999999968e63 or 8e4 < z < 3.6e90 or 1.24999999999999998e149 < z

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. *-rgt-identity 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. associate-+l+ 100.0%

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

      2. flip-+ 18.0%

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

      3. +-commutative 18.0%

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

      4. fma-def 18.0%

         \[\leadsto \frac{x \cdot x - \color{blue}{\mathsf{fma}\left(x + y, z, y\right)} \cdot \left(y + \left(x + y\right) \cdot z\right)}{x - \left(y + \left(x + y\right) \cdot z\right)} \]

      5. +-commutative 18.0%

         \[\leadsto \frac{x \cdot x - \mathsf{fma}\left(x + y, z, y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot z + y\right)}}{x - \left(y + \left(x + y\right) \cdot z\right)} \]

      6. fma-def 18.0%

         \[\leadsto \frac{x \cdot x - \mathsf{fma}\left(x + y, z, y\right) \cdot \color{blue}{\mathsf{fma}\left(x + y, z, y\right)}}{x - \left(y + \left(x + y\right) \cdot z\right)} \]

      7. +-commutative 18.0%

         \[\leadsto \frac{x \cdot x - \mathsf{fma}\left(x + y, z, y\right) \cdot \mathsf{fma}\left(x + y, z, y\right)}{x - \color{blue}{\left(\left(x + y\right) \cdot z + y\right)}} \]

      8. fma-def 18.0%

         \[\leadsto \frac{x \cdot x - \mathsf{fma}\left(x + y, z, y\right) \cdot \mathsf{fma}\left(x + y, z, y\right)}{x - \color{blue}{\mathsf{fma}\left(x + y, z, y\right)}} \]

   5. Applied egg-rr 18.0%

      \[\leadsto \color{blue}{\frac{x \cdot x - \mathsf{fma}\left(x + y, z, y\right) \cdot \mathsf{fma}\left(x + y, z, y\right)}{x - \mathsf{fma}\left(x + y, z, y\right)}} \]
   6. Step-by-step derivation

      1. difference-of-squares 18.4%

         \[\leadsto \frac{\color{blue}{\left(x + \mathsf{fma}\left(x + y, z, y\right)\right) \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      2. +-commutative 18.4%

         \[\leadsto \frac{\color{blue}{\left(\mathsf{fma}\left(x + y, z, y\right) + x\right)} \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      3. fma-udef 18.4%

         \[\leadsto \frac{\left(\color{blue}{\left(\left(x + y\right) \cdot z + y\right)} + x\right) \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      4. associate-+r+ 18.4%

         \[\leadsto \frac{\color{blue}{\left(\left(x + y\right) \cdot z + \left(y + x\right)\right)} \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      5. +-commutative 18.4%

         \[\leadsto \frac{\left(\left(x + y\right) \cdot z + \color{blue}{\left(x + y\right)}\right) \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      6. *-rgt-identity 18.4%

         \[\leadsto \frac{\left(\left(x + y\right) \cdot z + \color{blue}{\left(x + y\right) \cdot 1}\right) \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      7. distribute-lft-in 18.4%

         \[\leadsto \frac{\color{blue}{\left(\left(x + y\right) \cdot \left(z + 1\right)\right)} \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      8. associate-/l* 45.3%

         \[\leadsto \color{blue}{\frac{\left(x + y\right) \cdot \left(z + 1\right)}{\frac{x - \mathsf{fma}\left(x + y, z, y\right)}{x - \mathsf{fma}\left(x + y, z, y\right)}}} \]

      9. *-inverses 100.0%

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

      10. associate-/l* 99.8%

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

      11. +-commutative 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in z around inf 99.5%

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

   9. Taylor expanded in y around 0 50.3%

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

   if -6.09999999999999968e63 < z < -1.70000000000000003e-6

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]

   2. Taylor expanded in x around 0 41.1%

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

   if -1.70000000000000003e-6 < z < 8e4

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]

   2. Taylor expanded in z around 0 97.8%

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

   if 3.6e90 < z < 1.24999999999999998e149

   1. Initial program 99.9%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]
   2. Step-by-step derivation

      1. +-commutative 99.9%

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

      2. distribute-lft-in 99.9%

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

      3. *-rgt-identity 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around 0 52.9%

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

   5. Taylor expanded in z around inf 52.9%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.1 \cdot 10^{+63}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-6}:\\ \;\;\;\;y \cdot \left(z + 1\right)\\ \mathbf{elif}\;z \leq 80000:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+90}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+149}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]

Alternative 5: 74.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq 80000:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+94}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{+149}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0)
   (* x z)
   (if (<= z 80000.0)
     (+ x y)
     (if (<= z 1.4e+94) (* x z) (if (<= z 4.4e+149) (* y z) (* x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = x * z;
	} else if (z <= 80000.0) {
		tmp = x + y;
	} else if (z <= 1.4e+94) {
		tmp = x * z;
	} else if (z <= 4.4e+149) {
		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)) then
        tmp = x * z
    else if (z <= 80000.0d0) then
        tmp = x + y
    else if (z <= 1.4d+94) then
        tmp = x * z
    else if (z <= 4.4d+149) 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) {
		tmp = x * z;
	} else if (z <= 80000.0) {
		tmp = x + y;
	} else if (z <= 1.4e+94) {
		tmp = x * z;
	} else if (z <= 4.4e+149) {
		tmp = y * z;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = x * z
	elif z <= 80000.0:
		tmp = x + y
	elif z <= 1.4e+94:
		tmp = x * z
	elif z <= 4.4e+149:
		tmp = y * z
	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 <= 80000.0)
		tmp = Float64(x + y);
	elseif (z <= 1.4e+94)
		tmp = Float64(x * z);
	elseif (z <= 4.4e+149)
		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)
		tmp = x * z;
	elseif (z <= 80000.0)
		tmp = x + y;
	elseif (z <= 1.4e+94)
		tmp = x * z;
	elseif (z <= 4.4e+149)
		tmp = y * z;
	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, 80000.0], N[(x + y), $MachinePrecision], If[LessEqual[z, 1.4e+94], N[(x * z), $MachinePrecision], If[LessEqual[z, 4.4e+149], N[(y * z), $MachinePrecision], N[(x * z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1 or 8e4 < z < 1.39999999999999999e94 or 4.4e149 < z

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. distribute-lft-in 99.9%

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

      3. *-rgt-identity 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. associate-+l+ 99.9%

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

      2. flip-+ 21.3%

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

      3. +-commutative 21.3%

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

      4. fma-def 21.3%

         \[\leadsto \frac{x \cdot x - \color{blue}{\mathsf{fma}\left(x + y, z, y\right)} \cdot \left(y + \left(x + y\right) \cdot z\right)}{x - \left(y + \left(x + y\right) \cdot z\right)} \]

      5. +-commutative 21.3%

         \[\leadsto \frac{x \cdot x - \mathsf{fma}\left(x + y, z, y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot z + y\right)}}{x - \left(y + \left(x + y\right) \cdot z\right)} \]

      6. fma-def 21.3%

         \[\leadsto \frac{x \cdot x - \mathsf{fma}\left(x + y, z, y\right) \cdot \color{blue}{\mathsf{fma}\left(x + y, z, y\right)}}{x - \left(y + \left(x + y\right) \cdot z\right)} \]

      7. +-commutative 21.3%

         \[\leadsto \frac{x \cdot x - \mathsf{fma}\left(x + y, z, y\right) \cdot \mathsf{fma}\left(x + y, z, y\right)}{x - \color{blue}{\left(\left(x + y\right) \cdot z + y\right)}} \]

      8. fma-def 21.3%

         \[\leadsto \frac{x \cdot x - \mathsf{fma}\left(x + y, z, y\right) \cdot \mathsf{fma}\left(x + y, z, y\right)}{x - \color{blue}{\mathsf{fma}\left(x + y, z, y\right)}} \]

   5. Applied egg-rr 21.3%

      \[\leadsto \color{blue}{\frac{x \cdot x - \mathsf{fma}\left(x + y, z, y\right) \cdot \mathsf{fma}\left(x + y, z, y\right)}{x - \mathsf{fma}\left(x + y, z, y\right)}} \]
   6. Step-by-step derivation

      1. difference-of-squares 22.1%

         \[\leadsto \frac{\color{blue}{\left(x + \mathsf{fma}\left(x + y, z, y\right)\right) \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      2. +-commutative 22.1%

         \[\leadsto \frac{\color{blue}{\left(\mathsf{fma}\left(x + y, z, y\right) + x\right)} \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      3. fma-udef 22.1%

         \[\leadsto \frac{\left(\color{blue}{\left(\left(x + y\right) \cdot z + y\right)} + x\right) \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      4. associate-+r+ 22.1%

         \[\leadsto \frac{\color{blue}{\left(\left(x + y\right) \cdot z + \left(y + x\right)\right)} \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      5. +-commutative 22.1%

         \[\leadsto \frac{\left(\left(x + y\right) \cdot z + \color{blue}{\left(x + y\right)}\right) \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      6. *-rgt-identity 22.1%

         \[\leadsto \frac{\left(\left(x + y\right) \cdot z + \color{blue}{\left(x + y\right) \cdot 1}\right) \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      7. distribute-lft-in 22.1%

         \[\leadsto \frac{\color{blue}{\left(\left(x + y\right) \cdot \left(z + 1\right)\right)} \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      8. associate-/l* 53.4%

         \[\leadsto \color{blue}{\frac{\left(x + y\right) \cdot \left(z + 1\right)}{\frac{x - \mathsf{fma}\left(x + y, z, y\right)}{x - \mathsf{fma}\left(x + y, z, y\right)}}} \]

      9. *-inverses 100.0%

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

      10. associate-/l* 99.7%

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

      11. +-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in z around inf 96.8%

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

   9. Taylor expanded in y around 0 50.8%

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

   if -1 < z < 8e4

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]

   2. Taylor expanded in z around 0 97.4%

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

   if 1.39999999999999999e94 < z < 4.4e149

   1. Initial program 99.9%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]
   2. Step-by-step derivation

      1. +-commutative 99.9%

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

      2. distribute-lft-in 99.9%

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

      3. *-rgt-identity 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around 0 52.9%

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

   5. Taylor expanded in z around inf 52.9%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq 80000:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+94}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{+149}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]

Alternative 6: 98.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]

   2. Taylor expanded in z around inf 96.4%

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

   if -1 < z < 1

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]

   2. Taylor expanded in z around 0 98.8%

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

4. Final simplification 97.5%

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

Alternative 7: 63.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-54}:\\ \;\;\;\;x \cdot \left(z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.1e-54) (* x (+ z 1.0)) (* y (+ z 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.1e-54) {
		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 <= (-3.1d-54)) 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 <= -3.1e-54) {
		tmp = x * (z + 1.0);
	} else {
		tmp = y * (z + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.1e-54:
		tmp = x * (z + 1.0)
	else:
		tmp = y * (z + 1.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.1e-54)
		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 <= -3.1e-54)
		tmp = x * (z + 1.0);
	else
		tmp = y * (z + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.1e-54], 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 x < -3.10000000000000004e-54

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]

   2. Taylor expanded in x around inf 78.2%

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

   if -3.10000000000000004e-54 < x

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]

   2. Taylor expanded in x around 0 62.2%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-54}:\\ \;\;\;\;x \cdot \left(z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z + 1\right)\\ \end{array} \]

Alternative 8: 32.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-126}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= x -1.3e-126) x y))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.3e-126) {
		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 <= (-1.3d-126)) 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 <= -1.3e-126) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.3e-126:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.3e-126)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.3e-126)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.3e-126], x, y]

Derivation

1. Split input into 2 regimes

2. if x < -1.3e-126

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]

   2. Taylor expanded in x around inf 72.9%

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

   3. Taylor expanded in z around 0 35.5%

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

   if -1.3e-126 < x

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]

   2. Taylor expanded in x around 0 62.5%

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

   3. Taylor expanded in z around 0 34.1%

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

4. Final simplification 34.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-126}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 9: 26.2% 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. Taylor expanded in x around inf 51.9%

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

3. Taylor expanded in z around 0 25.1%

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

4. Final simplification 25.1%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023200 
(FPCore (x y z)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, G"
  :precision binary64
  (* (+ x y) (+ z 1.0)))