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

Percentage Accurate: 100.0% → 100.0%

Time: 6.9s

Alternatives: 7

Speedup: 1.0×

• 21.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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

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

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]

Derivation

1. Initial program 100.0%

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

Alternative 2: 75.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -y \cdot z\\ \mathbf{if}\;1 - z \leq -50000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;1 - z \leq 50000000:\\ \;\;\;\;x + y\\ \mathbf{elif}\;1 - z \leq 5 \cdot 10^{+196}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* y z))))
   (if (<= (- 1.0 z) -50000000000.0)
     t_0
     (if (<= (- 1.0 z) 50000000.0)
       (+ x y)
       (if (<= (- 1.0 z) 5e+196) (* x (- z)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = -(y * z);
	double tmp;
	if ((1.0 - z) <= -50000000000.0) {
		tmp = t_0;
	} else if ((1.0 - z) <= 50000000.0) {
		tmp = x + y;
	} else if ((1.0 - z) <= 5e+196) {
		tmp = x * -z;
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = -(y * z)
    if ((1.0d0 - z) <= (-50000000000.0d0)) then
        tmp = t_0
    else if ((1.0d0 - z) <= 50000000.0d0) then
        tmp = x + y
    else if ((1.0d0 - z) <= 5d+196) then
        tmp = x * -z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -(y * z);
	double tmp;
	if ((1.0 - z) <= -50000000000.0) {
		tmp = t_0;
	} else if ((1.0 - z) <= 50000000.0) {
		tmp = x + y;
	} else if ((1.0 - z) <= 5e+196) {
		tmp = x * -z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -(y * z)
	tmp = 0
	if (1.0 - z) <= -50000000000.0:
		tmp = t_0
	elif (1.0 - z) <= 50000000.0:
		tmp = x + y
	elif (1.0 - z) <= 5e+196:
		tmp = x * -z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-Float64(y * z))
	tmp = 0.0
	if (Float64(1.0 - z) <= -50000000000.0)
		tmp = t_0;
	elseif (Float64(1.0 - z) <= 50000000.0)
		tmp = Float64(x + y);
	elseif (Float64(1.0 - z) <= 5e+196)
		tmp = Float64(x * Float64(-z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -(y * z);
	tmp = 0.0;
	if ((1.0 - z) <= -50000000000.0)
		tmp = t_0;
	elseif ((1.0 - z) <= 50000000.0)
		tmp = x + y;
	elseif ((1.0 - z) <= 5e+196)
		tmp = x * -z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = (-N[(y * z), $MachinePrecision])}, If[LessEqual[N[(1.0 - z), $MachinePrecision], -50000000000.0], t$95$0, If[LessEqual[N[(1.0 - z), $MachinePrecision], 50000000.0], N[(x + y), $MachinePrecision], If[LessEqual[N[(1.0 - z), $MachinePrecision], 5e+196], N[(x * (-z)), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 #s(literal 1 binary64) z) < -5e10 or 4.9999999999999998e196 < (-.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 x around 0

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

      1. distribute-lft-out-- N/A

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

      2. *-rgt-identity N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 47.5

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

   5. Applied rewrites 47.5%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 47.2%

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

   if -5e10 < (-.f64 #s(literal 1 binary64) z) < 5e7

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 94.5

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

   5. Applied rewrites 94.5%

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

   if 5e7 < (-.f64 #s(literal 1 binary64) z) < 4.9999999999999998e196

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(1 - z\right) \]
   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. distribute-lft-out-- N/A

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

      2. *-rgt-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 47.1

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

   5. Applied rewrites 47.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 47.1%

      \[\leadsto -z \cdot x \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - z \leq -50000000000:\\ \;\;\;\;-y \cdot z\\ \mathbf{elif}\;1 - z \leq 50000000:\\ \;\;\;\;x + y\\ \mathbf{elif}\;1 - z \leq 5 \cdot 10^{+196}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;-y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 44.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-255}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{elif}\;x + y \leq 10^{+120}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ x y) -1e-255)
   (- x (* x z))
   (if (<= (+ x y) 1e+120) (+ x y) (- (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x + y) <= -1e-255) {
		tmp = x - (x * z);
	} else if ((x + y) <= 1e+120) {
		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 ((x + y) <= (-1d-255)) then
        tmp = x - (x * z)
    else if ((x + y) <= 1d+120) 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 ((x + y) <= -1e-255) {
		tmp = x - (x * z);
	} else if ((x + y) <= 1e+120) {
		tmp = x + y;
	} else {
		tmp = -(y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x + y) <= -1e-255:
		tmp = x - (x * z)
	elif (x + y) <= 1e+120:
		tmp = x + y
	else:
		tmp = -(y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x + y) <= -1e-255)
		tmp = Float64(x - Float64(x * z));
	elseif (Float64(x + y) <= 1e+120)
		tmp = Float64(x + y);
	else
		tmp = Float64(-Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x + y) <= -1e-255)
		tmp = x - (x * z);
	elseif ((x + y) <= 1e+120)
		tmp = x + y;
	else
		tmp = -(y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x + y), $MachinePrecision], -1e-255], N[(x - N[(x * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x + y), $MachinePrecision], 1e+120], N[(x + y), $MachinePrecision], (-N[(y * z), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x y) < -1e-255

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(1 - z\right) \]
   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. distribute-lft-out-- N/A

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

      2. *-rgt-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 52.3

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

   5. Applied rewrites 52.3%

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

   if -1e-255 < (+.f64 x y) < 9.9999999999999998e119

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 56.8

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

   5. Applied rewrites 56.8%

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

   if 9.9999999999999998e119 < (+.f64 x y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-out-- N/A

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

      2. *-rgt-identity N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 59.1

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

   5. Applied rewrites 59.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 41.2%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-255}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{elif}\;x + y \leq 10^{+120}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(-z\right)\\ \mathbf{if}\;1 - z \leq -50000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;1 - z \leq 50000000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- z))))
   (if (<= (- 1.0 z) -50000000000.0)
     t_0
     (if (<= (- 1.0 z) 50000000.0) (+ x y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * -z;
	double tmp;
	if ((1.0 - z) <= -50000000000.0) {
		tmp = t_0;
	} else if ((1.0 - z) <= 50000000.0) {
		tmp = x + y;
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = x * -z
    if ((1.0d0 - z) <= (-50000000000.0d0)) then
        tmp = t_0
    else if ((1.0d0 - z) <= 50000000.0d0) then
        tmp = x + y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * -z;
	double tmp;
	if ((1.0 - z) <= -50000000000.0) {
		tmp = t_0;
	} else if ((1.0 - z) <= 50000000.0) {
		tmp = x + y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * -z
	tmp = 0
	if (1.0 - z) <= -50000000000.0:
		tmp = t_0
	elif (1.0 - z) <= 50000000.0:
		tmp = x + y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(-z))
	tmp = 0.0
	if (Float64(1.0 - z) <= -50000000000.0)
		tmp = t_0;
	elseif (Float64(1.0 - z) <= 50000000.0)
		tmp = Float64(x + y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * -z;
	tmp = 0.0;
	if ((1.0 - z) <= -50000000000.0)
		tmp = t_0;
	elseif ((1.0 - z) <= 50000000.0)
		tmp = x + y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * (-z)), $MachinePrecision]}, If[LessEqual[N[(1.0 - z), $MachinePrecision], -50000000000.0], t$95$0, If[LessEqual[N[(1.0 - z), $MachinePrecision], 50000000.0], N[(x + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) z) < -5e10 or 5e7 < (-.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 x around inf

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

      1. distribute-lft-out-- N/A

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

      2. *-rgt-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 58.1

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

   5. Applied rewrites 58.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 58.1%

      \[\leadsto -z \cdot x \]

   if -5e10 < (-.f64 #s(literal 1 binary64) z) < 5e7

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 94.5

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

   5. Applied rewrites 94.5%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - z \leq -50000000000:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{elif}\;1 - z \leq 50000000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-273}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ x y) -1e-273) (- x (* x z)) (fma (- z) y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x + y) <= -1e-273) {
		tmp = x - (x * z);
	} else {
		tmp = fma(-z, y, y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x + y) <= -1e-273)
		tmp = Float64(x - Float64(x * z));
	else
		tmp = fma(Float64(-z), y, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x + y), $MachinePrecision], -1e-273], N[(x - N[(x * z), $MachinePrecision]), $MachinePrecision], N[((-z) * y + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -1e-273

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(1 - z\right) \]
   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. distribute-lft-out-- N/A

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

      2. *-rgt-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 52.0

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

   5. Applied rewrites 52.0%

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

   if -1e-273 < (+.f64 x y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-out-- N/A

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

      2. *-rgt-identity N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 50.9

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

   5. Applied rewrites 50.9%

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

   7. Applied rewrites 50.9%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-273}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-273}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ x y) -1e-273) (- x (* x z)) (- y (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x + y) <= -1e-273) {
		tmp = x - (x * 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) <= (-1d-273)) then
        tmp = x - (x * 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) <= -1e-273) {
		tmp = x - (x * z);
	} else {
		tmp = y - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x + y) <= -1e-273:
		tmp = x - (x * z)
	else:
		tmp = y - (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x + y) <= -1e-273)
		tmp = Float64(x - Float64(x * 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) <= -1e-273)
		tmp = x - (x * z);
	else
		tmp = y - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x + y), $MachinePrecision], -1e-273], N[(x - N[(x * z), $MachinePrecision]), $MachinePrecision], N[(y - N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -1e-273

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(1 - z\right) \]
   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. distribute-lft-out-- N/A

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

      2. *-rgt-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 52.0

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

   5. Applied rewrites 52.0%

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

   if -1e-273 < (+.f64 x y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-out-- N/A

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

      2. *-rgt-identity N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 50.9

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

   5. Applied rewrites 50.9%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-273}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.9% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 50.6

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

5. Applied rewrites 50.6%

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

6. Final simplification 50.6%

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

Reproduce

herbie shell --seed 2024220 
(FPCore (x y z)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, H"
  :precision binary64
  (* (+ x y) (- 1.0 z)))