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

Percentage Accurate: 100.0% → 100.0%

Time: 7.2s

Alternatives: 9

Speedup: 1.0×

• 21.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x + y) * (1.0 - 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) * (1.0d0 - z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x + y) * (1.0 - 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) * (1.0d0 - z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (1.0 - z) * (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 = (1.0d0 - z) * (x + y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 74.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(-z\right)\\ t_1 := x \cdot \left(-z\right)\\ \mathbf{if}\;z \leq -1.1 \cdot 10^{+118}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -1.9 \cdot 10^{+70}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -2.05 \cdot 10^{+14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+140}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- z))) (t_1 (* x (- z))))
   (if (<= z -1.1e+118)
     t_0
     (if (<= z -1.9e+70)
       t_1
       (if (<= z -2.05e+14)
         t_0
         (if (<= z 1.0) (+ x y) (if (<= z 1.1e+140) t_0 t_1)))))))

C

double code(double x, double y, double z) {
	double t_0 = y * -z;
	double t_1 = x * -z;
	double tmp;
	if (z <= -1.1e+118) {
		tmp = t_0;
	} else if (z <= -1.9e+70) {
		tmp = t_1;
	} else if (z <= -2.05e+14) {
		tmp = t_0;
	} else if (z <= 1.0) {
		tmp = x + y;
	} else if (z <= 1.1e+140) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y * -z
    t_1 = x * -z
    if (z <= (-1.1d+118)) then
        tmp = t_0
    else if (z <= (-1.9d+70)) then
        tmp = t_1
    else if (z <= (-2.05d+14)) then
        tmp = t_0
    else if (z <= 1.0d0) then
        tmp = x + y
    else if (z <= 1.1d+140) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * -z;
	double t_1 = x * -z;
	double tmp;
	if (z <= -1.1e+118) {
		tmp = t_0;
	} else if (z <= -1.9e+70) {
		tmp = t_1;
	} else if (z <= -2.05e+14) {
		tmp = t_0;
	} else if (z <= 1.0) {
		tmp = x + y;
	} else if (z <= 1.1e+140) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * -z
	t_1 = x * -z
	tmp = 0
	if z <= -1.1e+118:
		tmp = t_0
	elif z <= -1.9e+70:
		tmp = t_1
	elif z <= -2.05e+14:
		tmp = t_0
	elif z <= 1.0:
		tmp = x + y
	elif z <= 1.1e+140:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(-z))
	t_1 = Float64(x * Float64(-z))
	tmp = 0.0
	if (z <= -1.1e+118)
		tmp = t_0;
	elseif (z <= -1.9e+70)
		tmp = t_1;
	elseif (z <= -2.05e+14)
		tmp = t_0;
	elseif (z <= 1.0)
		tmp = Float64(x + y);
	elseif (z <= 1.1e+140)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * -z;
	t_1 = x * -z;
	tmp = 0.0;
	if (z <= -1.1e+118)
		tmp = t_0;
	elseif (z <= -1.9e+70)
		tmp = t_1;
	elseif (z <= -2.05e+14)
		tmp = t_0;
	elseif (z <= 1.0)
		tmp = x + y;
	elseif (z <= 1.1e+140)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * (-z)), $MachinePrecision]}, Block[{t$95$1 = N[(x * (-z)), $MachinePrecision]}, If[LessEqual[z, -1.1e+118], t$95$0, If[LessEqual[z, -1.9e+70], t$95$1, If[LessEqual[z, -2.05e+14], t$95$0, If[LessEqual[z, 1.0], N[(x + y), $MachinePrecision], If[LessEqual[z, 1.1e+140], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.09999999999999993e118 or -1.8999999999999999e70 < z < -2.05e14 or 1 < z < 1.0999999999999999e140

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 98.8%

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

      1. neg-mul-1 52.9%

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

   5. Simplified 98.8%

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

   6. Taylor expanded in x around 0 51.7%

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

      1. associate-*r* 51.7%

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

      2. mul-1-neg 51.7%

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

   8. Simplified 51.7%

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

   if -1.09999999999999993e118 < z < -1.8999999999999999e70 or 1.0999999999999999e140 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 52.6%

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

      1. *-commutative 52.6%

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

   5. Simplified 52.6%

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

   6. Taylor expanded in z around inf 52.6%

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

      1. neg-mul-1 52.6%

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

   8. Simplified 52.6%

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

   if -2.05e14 < z < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 96.9%

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

      1. +-commutative 96.9%

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

   5. Simplified 96.9%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+118}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq -1.9 \cdot 10^{+70}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq -2.05 \cdot 10^{+14}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+140}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - z \leq -2 \cdot 10^{+145}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{elif}\;1 - z \leq 1:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \mathbf{elif}\;1 - z \leq 4 \cdot 10^{+14}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (- 1.0 z) -2e+145)
   (* x (- z))
   (if (<= (- 1.0 z) 1.0)
     (* y (- 1.0 z))
     (if (<= (- 1.0 z) 4e+14) (+ x y) (* y (- z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((1.0 - z) <= -2e+145) {
		tmp = x * -z;
	} else if ((1.0 - z) <= 1.0) {
		tmp = y * (1.0 - z);
	} else if ((1.0 - z) <= 4e+14) {
		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 ((1.0d0 - z) <= (-2d+145)) then
        tmp = x * -z
    else if ((1.0d0 - z) <= 1.0d0) then
        tmp = y * (1.0d0 - z)
    else if ((1.0d0 - z) <= 4d+14) 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 ((1.0 - z) <= -2e+145) {
		tmp = x * -z;
	} else if ((1.0 - z) <= 1.0) {
		tmp = y * (1.0 - z);
	} else if ((1.0 - z) <= 4e+14) {
		tmp = x + y;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (1.0 - z) <= -2e+145:
		tmp = x * -z
	elif (1.0 - z) <= 1.0:
		tmp = y * (1.0 - z)
	elif (1.0 - z) <= 4e+14:
		tmp = x + y
	else:
		tmp = y * -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(1.0 - z) <= -2e+145)
		tmp = Float64(x * Float64(-z));
	elseif (Float64(1.0 - z) <= 1.0)
		tmp = Float64(y * Float64(1.0 - z));
	elseif (Float64(1.0 - z) <= 4e+14)
		tmp = Float64(x + y);
	else
		tmp = Float64(y * Float64(-z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((1.0 - z) <= -2e+145)
		tmp = x * -z;
	elseif ((1.0 - z) <= 1.0)
		tmp = y * (1.0 - z);
	elseif ((1.0 - z) <= 4e+14)
		tmp = x + y;
	else
		tmp = y * -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(1.0 - z), $MachinePrecision], -2e+145], N[(x * (-z)), $MachinePrecision], If[LessEqual[N[(1.0 - z), $MachinePrecision], 1.0], N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 - z), $MachinePrecision], 4e+14], N[(x + y), $MachinePrecision], N[(y * (-z)), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 #s(literal 1 binary64) z) < -2e145

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 51.8%

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

      1. *-commutative 51.8%

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

   5. Simplified 51.8%

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

   6. Taylor expanded in z around inf 51.8%

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

      1. neg-mul-1 51.8%

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

   8. Simplified 51.8%

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

   if -2e145 < (-.f64 #s(literal 1 binary64) z) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 46.1%

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

   if 1 < (-.f64 #s(literal 1 binary64) z) < 4e14

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 11.2%

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

      1. +-commutative 11.2%

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

   5. Simplified 11.2%

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

   if 4e14 < (-.f64 #s(literal 1 binary64) z)

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 100.0%

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

      1. neg-mul-1 54.9%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 51.5%

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

      1. associate-*r* 51.5%

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

      2. mul-1-neg 51.5%

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

   8. Simplified 51.5%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - z \leq -2 \cdot 10^{+145}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{elif}\;1 - z \leq 1:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \mathbf{elif}\;1 - z \leq 4 \cdot 10^{+14}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.05 \cdot 10^{+14} \lor \neg \left(z \leq 1\right):\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.05e+14) (not (<= z 1.0))) (* y (- z)) (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.05e+14) || !(z <= 1.0)) {
		tmp = y * -z;
	} 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 <= (-2.05d+14)) .or. (.not. (z <= 1.0d0))) then
        tmp = y * -z
    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 <= -2.05e+14) || !(z <= 1.0)) {
		tmp = y * -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.05e+14) or not (z <= 1.0):
		tmp = y * -z
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.05e+14) || !(z <= 1.0))
		tmp = Float64(y * Float64(-z));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2.05e+14) || ~((z <= 1.0)))
		tmp = y * -z;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.05e+14], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(y * (-z)), $MachinePrecision], N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.05e14 or 1 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 99.3%

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

      1. neg-mul-1 52.8%

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

   5. Simplified 99.3%

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

   6. Taylor expanded in x around 0 52.0%

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

      1. associate-*r* 52.0%

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

      2. mul-1-neg 52.0%

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

   8. Simplified 52.0%

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

   if -2.05e14 < z < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 96.9%

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

      1. +-commutative 96.9%

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

   5. Simplified 96.9%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.05 \cdot 10^{+14} \lor \neg \left(z \leq 1\right):\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -4 \cdot 10^{-224}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ x y) -4e-224) (* x (- 1.0 z)) (- y (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x + y) <= -4e-224) {
		tmp = x * (1.0 - z);
	} 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) <= (-4d-224)) then
        tmp = x * (1.0d0 - z)
    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) <= -4e-224) {
		tmp = x * (1.0 - z);
	} else {
		tmp = y - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x + y) <= -4e-224:
		tmp = x * (1.0 - z)
	else:
		tmp = y - (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x + y) <= -4e-224)
		tmp = Float64(x * Float64(1.0 - z));
	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) <= -4e-224)
		tmp = x * (1.0 - z);
	else
		tmp = y - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x + y), $MachinePrecision], -4e-224], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(y - N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -4.0000000000000001e-224

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 55.5%

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

      1. *-commutative 55.5%

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

   5. Simplified 55.5%

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

   if -4.0000000000000001e-224 < (+.f64 x y)

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. *-commutative 100.0%

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

      4. *-un-lft-identity 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 51.5%

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

      1. associate-*r* 51.5%

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

      2. mul-1-neg 51.5%

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

   7. Simplified 51.5%

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

   8. Taylor expanded in y around 0 51.4%

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

      1. mul-1-neg 51.4%

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

      2. unsub-neg 51.4%

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

      3. distribute-lft-out-- 51.5%

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

      4. *-rgt-identity 51.5%

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

   10. Simplified 51.5%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -4 \cdot 10^{-224}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{-79}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.9e-79) (* x (- 1.0 z)) (* y (- 1.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.9e-79) {
		tmp = x * (1.0 - z);
	} else {
		tmp = y * (1.0 - 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 <= (-4.9d-79)) then
        tmp = x * (1.0d0 - z)
    else
        tmp = y * (1.0d0 - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.9e-79) {
		tmp = x * (1.0 - z);
	} else {
		tmp = y * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -4.9e-79:
		tmp = x * (1.0 - z)
	else:
		tmp = y * (1.0 - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.9e-79)
		tmp = Float64(x * Float64(1.0 - z));
	else
		tmp = Float64(y * Float64(1.0 - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -4.9e-79)
		tmp = x * (1.0 - z);
	else
		tmp = y * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.9e-79], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.9000000000000001e-79

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 75.8%

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

      1. *-commutative 75.8%

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

   5. Simplified 75.8%

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

   if -4.9000000000000001e-79 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 58.7%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{-79}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 31.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.9 \cdot 10^{-121}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= y 2.9e-121) x y))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.9e-121) {
		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 (y <= 2.9d-121) 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 (y <= 2.9e-121) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.9e-121:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.9e-121)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.9e-121)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.9e-121], x, y]

Derivation

1. Split input into 2 regimes

2. if y < 2.9e-121

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 53.9%

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

      1. +-commutative 53.9%

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

   5. Simplified 53.9%

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

   6. Taylor expanded in y around 0 35.5%

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

   if 2.9e-121 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 47.2%

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

      1. +-commutative 47.2%

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

   5. Simplified 47.2%

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

   6. Taylor expanded in y around inf 32.9%

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

Alternative 8: 51.2% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in z around 0 51.6%

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

   1. +-commutative 51.6%

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

5. Simplified 51.6%

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

6. Final simplification 51.6%

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

Alternative 9: 26.7% 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%

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

3. Taylor expanded in z around 0 51.6%

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

   1. +-commutative 51.6%

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

5. Simplified 51.6%

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

6. Taylor expanded in y around 0 29.3%

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

Reproduce

herbie shell --seed 2024110 
(FPCore (x y z)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, H"
  :precision binary64
  (* (+ x y) (- 1.0 z)))