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

Percentage Accurate: 99.9% → 99.9%

Time: 4.7s

Alternatives: 10

Speedup: 1.0×

• 25.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{\left(x + y\right) - z}{t \cdot 2} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ (- (+ x y) z) (* t 2.0)))

C

double code(double x, double y, double z, double t) {
	return ((x + y) - z) / (t * 2.0);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x + y) - z) / (t * 2.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x + y) - z) / (t * 2.0);
}

Python

def code(x, y, z, t):
	return ((x + y) - z) / (t * 2.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x + y) - z) / Float64(t * 2.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x + y) - z) / (t * 2.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x + y), $MachinePrecision] - z), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(x + y\right) - z}{t \cdot 2} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ (- (+ x y) z) (* t 2.0)))

C

double code(double x, double y, double z, double t) {
	return ((x + y) - z) / (t * 2.0);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x + y) - z) / (t * 2.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x + y) - z) / (t * 2.0);
}

Python

def code(x, y, z, t):
	return ((x + y) - z) / (t * 2.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x + y) - z) / Float64(t * 2.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x + y) - z) / (t * 2.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x + y), $MachinePrecision] - z), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(x + y\right) - z}{t \cdot 2} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ (- (+ x y) z) (* t 2.0)))

C

double code(double x, double y, double z, double t) {
	return ((x + y) - z) / (t * 2.0);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x + y) - z) / (t * 2.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x + y) - z) / (t * 2.0);
}

Python

def code(x, y, z, t):
	return ((x + y) - z) / (t * 2.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x + y) - z) / Float64(t * 2.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x + y) - z) / (t * 2.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x + y), $MachinePrecision] - z), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 47.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-32}:\\ \;\;\;\;\frac{x}{t} \cdot 0.5\\ \mathbf{elif}\;x + y \leq 2 \cdot 10^{-35}:\\ \;\;\;\;\frac{-0.5 \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) -2e-32)
   (* (/ x t) 0.5)
   (if (<= (+ x y) 2e-35) (/ (* -0.5 z) t) (/ (* 0.5 y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -2e-32) {
		tmp = (x / t) * 0.5;
	} else if ((x + y) <= 2e-35) {
		tmp = (-0.5 * z) / t;
	} else {
		tmp = (0.5 * y) / t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x + y) <= (-2d-32)) then
        tmp = (x / t) * 0.5d0
    else if ((x + y) <= 2d-35) then
        tmp = ((-0.5d0) * z) / t
    else
        tmp = (0.5d0 * y) / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -2e-32) {
		tmp = (x / t) * 0.5;
	} else if ((x + y) <= 2e-35) {
		tmp = (-0.5 * z) / t;
	} else {
		tmp = (0.5 * y) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + y) <= -2e-32:
		tmp = (x / t) * 0.5
	elif (x + y) <= 2e-35:
		tmp = (-0.5 * z) / t
	else:
		tmp = (0.5 * y) / t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + y) <= -2e-32)
		tmp = Float64(Float64(x / t) * 0.5);
	elseif (Float64(x + y) <= 2e-35)
		tmp = Float64(Float64(-0.5 * z) / t);
	else
		tmp = Float64(Float64(0.5 * y) / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + y) <= -2e-32)
		tmp = (x / t) * 0.5;
	elseif ((x + y) <= 2e-35)
		tmp = (-0.5 * z) / t;
	else
		tmp = (0.5 * y) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + y), $MachinePrecision], -2e-32], N[(N[(x / t), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[N[(x + y), $MachinePrecision], 2e-35], N[(N[(-0.5 * z), $MachinePrecision] / t), $MachinePrecision], N[(N[(0.5 * y), $MachinePrecision] / t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x y) < -2.00000000000000011e-32

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x}{t}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x}{t} \cdot \frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{t} \cdot \frac{1}{2}} \]

      3. lower-/.f64 42.6

         \[\leadsto \color{blue}{\frac{x}{t}} \cdot 0.5 \]

   5. Applied rewrites 42.6%

      \[\leadsto \color{blue}{\frac{x}{t} \cdot 0.5} \]

   if -2.00000000000000011e-32 < (+.f64 x y) < 2.00000000000000002e-35

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{z}{t}} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\color{blue}{1 \cdot z}}{t} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\left(\frac{1}{t} \cdot z\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \frac{1}{t}\right) \cdot z} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{1}{t}\right) \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{t}\right) \cdot z} \]

      6. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{-1}{2}} \cdot \frac{1}{t}\right) \cdot z \]

      7. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot 1}{t}} \cdot z \]

      8. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{2}}}{t} \cdot z \]

      9. lower-/.f64 74.4

         \[\leadsto \color{blue}{\frac{-0.5}{t}} \cdot z \]

   5. Applied rewrites 74.4%

      \[\leadsto \color{blue}{\frac{-0.5}{t} \cdot z} \]
   6. Step-by-step derivation

   7. Applied rewrites 74.7%

      \[\leadsto \frac{-0.5 \cdot z}{\color{blue}{t}} \]

   if 2.00000000000000002e-35 < (+.f64 x y)

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{y}{t}} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\color{blue}{1 \cdot y}}{t} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\frac{1}{t} \cdot y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{t}\right) \cdot y} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{t}\right) \cdot y} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot 1}{t}} \cdot y \]

      6. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2}}}{t} \cdot y \]

      7. lower-/.f64 46.4

         \[\leadsto \color{blue}{\frac{0.5}{t}} \cdot y \]

   5. Applied rewrites 46.4%

      \[\leadsto \color{blue}{\frac{0.5}{t} \cdot y} \]
   6. Step-by-step derivation

   7. Applied rewrites 46.5%

      \[\leadsto \frac{0.5 \cdot y}{\color{blue}{t}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 47.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-32}:\\ \;\;\;\;\frac{x}{t} \cdot 0.5\\ \mathbf{elif}\;x + y \leq 2 \cdot 10^{-35}:\\ \;\;\;\;\frac{-0.5 \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{t} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) -2e-32)
   (* (/ x t) 0.5)
   (if (<= (+ x y) 2e-35) (/ (* -0.5 z) t) (* (/ 0.5 t) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -2e-32) {
		tmp = (x / t) * 0.5;
	} else if ((x + y) <= 2e-35) {
		tmp = (-0.5 * z) / t;
	} else {
		tmp = (0.5 / t) * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x + y) <= (-2d-32)) then
        tmp = (x / t) * 0.5d0
    else if ((x + y) <= 2d-35) then
        tmp = ((-0.5d0) * z) / t
    else
        tmp = (0.5d0 / t) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -2e-32) {
		tmp = (x / t) * 0.5;
	} else if ((x + y) <= 2e-35) {
		tmp = (-0.5 * z) / t;
	} else {
		tmp = (0.5 / t) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + y) <= -2e-32:
		tmp = (x / t) * 0.5
	elif (x + y) <= 2e-35:
		tmp = (-0.5 * z) / t
	else:
		tmp = (0.5 / t) * y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + y) <= -2e-32)
		tmp = Float64(Float64(x / t) * 0.5);
	elseif (Float64(x + y) <= 2e-35)
		tmp = Float64(Float64(-0.5 * z) / t);
	else
		tmp = Float64(Float64(0.5 / t) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + y) <= -2e-32)
		tmp = (x / t) * 0.5;
	elseif ((x + y) <= 2e-35)
		tmp = (-0.5 * z) / t;
	else
		tmp = (0.5 / t) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + y), $MachinePrecision], -2e-32], N[(N[(x / t), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[N[(x + y), $MachinePrecision], 2e-35], N[(N[(-0.5 * z), $MachinePrecision] / t), $MachinePrecision], N[(N[(0.5 / t), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x y) < -2.00000000000000011e-32

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x}{t}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x}{t} \cdot \frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{t} \cdot \frac{1}{2}} \]

      3. lower-/.f64 42.6

         \[\leadsto \color{blue}{\frac{x}{t}} \cdot 0.5 \]

   5. Applied rewrites 42.6%

      \[\leadsto \color{blue}{\frac{x}{t} \cdot 0.5} \]

   if -2.00000000000000011e-32 < (+.f64 x y) < 2.00000000000000002e-35

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{z}{t}} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\color{blue}{1 \cdot z}}{t} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\left(\frac{1}{t} \cdot z\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \frac{1}{t}\right) \cdot z} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{1}{t}\right) \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{t}\right) \cdot z} \]

      6. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{-1}{2}} \cdot \frac{1}{t}\right) \cdot z \]

      7. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot 1}{t}} \cdot z \]

      8. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{2}}}{t} \cdot z \]

      9. lower-/.f64 74.4

         \[\leadsto \color{blue}{\frac{-0.5}{t}} \cdot z \]

   5. Applied rewrites 74.4%

      \[\leadsto \color{blue}{\frac{-0.5}{t} \cdot z} \]
   6. Step-by-step derivation

   7. Applied rewrites 74.7%

      \[\leadsto \frac{-0.5 \cdot z}{\color{blue}{t}} \]

   if 2.00000000000000002e-35 < (+.f64 x y)

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{y}{t}} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\color{blue}{1 \cdot y}}{t} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\frac{1}{t} \cdot y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{t}\right) \cdot y} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{t}\right) \cdot y} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot 1}{t}} \cdot y \]

      6. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2}}}{t} \cdot y \]

      7. lower-/.f64 46.4

         \[\leadsto \color{blue}{\frac{0.5}{t}} \cdot y \]

   5. Applied rewrites 46.4%

      \[\leadsto \color{blue}{\frac{0.5}{t} \cdot y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 47.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-32}:\\ \;\;\;\;\frac{x}{t} \cdot 0.5\\ \mathbf{elif}\;x + y \leq 2 \cdot 10^{-35}:\\ \;\;\;\;\frac{-0.5}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{t} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) -2e-32)
   (* (/ x t) 0.5)
   (if (<= (+ x y) 2e-35) (* (/ -0.5 t) z) (* (/ 0.5 t) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -2e-32) {
		tmp = (x / t) * 0.5;
	} else if ((x + y) <= 2e-35) {
		tmp = (-0.5 / t) * z;
	} else {
		tmp = (0.5 / t) * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x + y) <= (-2d-32)) then
        tmp = (x / t) * 0.5d0
    else if ((x + y) <= 2d-35) then
        tmp = ((-0.5d0) / t) * z
    else
        tmp = (0.5d0 / t) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -2e-32) {
		tmp = (x / t) * 0.5;
	} else if ((x + y) <= 2e-35) {
		tmp = (-0.5 / t) * z;
	} else {
		tmp = (0.5 / t) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + y) <= -2e-32:
		tmp = (x / t) * 0.5
	elif (x + y) <= 2e-35:
		tmp = (-0.5 / t) * z
	else:
		tmp = (0.5 / t) * y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + y) <= -2e-32)
		tmp = Float64(Float64(x / t) * 0.5);
	elseif (Float64(x + y) <= 2e-35)
		tmp = Float64(Float64(-0.5 / t) * z);
	else
		tmp = Float64(Float64(0.5 / t) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + y) <= -2e-32)
		tmp = (x / t) * 0.5;
	elseif ((x + y) <= 2e-35)
		tmp = (-0.5 / t) * z;
	else
		tmp = (0.5 / t) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + y), $MachinePrecision], -2e-32], N[(N[(x / t), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[N[(x + y), $MachinePrecision], 2e-35], N[(N[(-0.5 / t), $MachinePrecision] * z), $MachinePrecision], N[(N[(0.5 / t), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x y) < -2.00000000000000011e-32

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x}{t}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x}{t} \cdot \frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{t} \cdot \frac{1}{2}} \]

      3. lower-/.f64 42.6

         \[\leadsto \color{blue}{\frac{x}{t}} \cdot 0.5 \]

   5. Applied rewrites 42.6%

      \[\leadsto \color{blue}{\frac{x}{t} \cdot 0.5} \]

   if -2.00000000000000011e-32 < (+.f64 x y) < 2.00000000000000002e-35

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{z}{t}} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\color{blue}{1 \cdot z}}{t} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\left(\frac{1}{t} \cdot z\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \frac{1}{t}\right) \cdot z} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{1}{t}\right) \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{t}\right) \cdot z} \]

      6. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{-1}{2}} \cdot \frac{1}{t}\right) \cdot z \]

      7. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot 1}{t}} \cdot z \]

      8. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{2}}}{t} \cdot z \]

      9. lower-/.f64 74.4

         \[\leadsto \color{blue}{\frac{-0.5}{t}} \cdot z \]

   5. Applied rewrites 74.4%

      \[\leadsto \color{blue}{\frac{-0.5}{t} \cdot z} \]

   if 2.00000000000000002e-35 < (+.f64 x y)

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{y}{t}} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\color{blue}{1 \cdot y}}{t} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\frac{1}{t} \cdot y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{t}\right) \cdot y} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{t}\right) \cdot y} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot 1}{t}} \cdot y \]

      6. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2}}}{t} \cdot y \]

      7. lower-/.f64 46.4

         \[\leadsto \color{blue}{\frac{0.5}{t}} \cdot y \]

   5. Applied rewrites 46.4%

      \[\leadsto \color{blue}{\frac{0.5}{t} \cdot y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 47.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-32}:\\ \;\;\;\;x \cdot \frac{0.5}{t}\\ \mathbf{elif}\;x + y \leq 2 \cdot 10^{-35}:\\ \;\;\;\;\frac{-0.5}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{t} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) -2e-32)
   (* x (/ 0.5 t))
   (if (<= (+ x y) 2e-35) (* (/ -0.5 t) z) (* (/ 0.5 t) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -2e-32) {
		tmp = x * (0.5 / t);
	} else if ((x + y) <= 2e-35) {
		tmp = (-0.5 / t) * z;
	} else {
		tmp = (0.5 / t) * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x + y) <= (-2d-32)) then
        tmp = x * (0.5d0 / t)
    else if ((x + y) <= 2d-35) then
        tmp = ((-0.5d0) / t) * z
    else
        tmp = (0.5d0 / t) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -2e-32) {
		tmp = x * (0.5 / t);
	} else if ((x + y) <= 2e-35) {
		tmp = (-0.5 / t) * z;
	} else {
		tmp = (0.5 / t) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + y) <= -2e-32:
		tmp = x * (0.5 / t)
	elif (x + y) <= 2e-35:
		tmp = (-0.5 / t) * z
	else:
		tmp = (0.5 / t) * y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + y) <= -2e-32)
		tmp = Float64(x * Float64(0.5 / t));
	elseif (Float64(x + y) <= 2e-35)
		tmp = Float64(Float64(-0.5 / t) * z);
	else
		tmp = Float64(Float64(0.5 / t) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + y) <= -2e-32)
		tmp = x * (0.5 / t);
	elseif ((x + y) <= 2e-35)
		tmp = (-0.5 / t) * z;
	else
		tmp = (0.5 / t) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + y), $MachinePrecision], -2e-32], N[(x * N[(0.5 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x + y), $MachinePrecision], 2e-35], N[(N[(-0.5 / t), $MachinePrecision] * z), $MachinePrecision], N[(N[(0.5 / t), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x y) < -2.00000000000000011e-32

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x}{t}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x}{t} \cdot \frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{t} \cdot \frac{1}{2}} \]

      3. lower-/.f64 42.6

         \[\leadsto \color{blue}{\frac{x}{t}} \cdot 0.5 \]

   5. Applied rewrites 42.6%

      \[\leadsto \color{blue}{\frac{x}{t} \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 42.6%

      \[\leadsto x \cdot \color{blue}{\frac{0.5}{t}} \]

   if -2.00000000000000011e-32 < (+.f64 x y) < 2.00000000000000002e-35

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{z}{t}} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\color{blue}{1 \cdot z}}{t} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\left(\frac{1}{t} \cdot z\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \frac{1}{t}\right) \cdot z} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{1}{t}\right) \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{t}\right) \cdot z} \]

      6. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{-1}{2}} \cdot \frac{1}{t}\right) \cdot z \]

      7. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot 1}{t}} \cdot z \]

      8. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{2}}}{t} \cdot z \]

      9. lower-/.f64 74.4

         \[\leadsto \color{blue}{\frac{-0.5}{t}} \cdot z \]

   5. Applied rewrites 74.4%

      \[\leadsto \color{blue}{\frac{-0.5}{t} \cdot z} \]

   if 2.00000000000000002e-35 < (+.f64 x y)

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{y}{t}} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\color{blue}{1 \cdot y}}{t} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\frac{1}{t} \cdot y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{t}\right) \cdot y} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{t}\right) \cdot y} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot 1}{t}} \cdot y \]

      6. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2}}}{t} \cdot y \]

      7. lower-/.f64 46.4

         \[\leadsto \color{blue}{\frac{0.5}{t}} \cdot y \]

   5. Applied rewrites 46.4%

      \[\leadsto \color{blue}{\frac{0.5}{t} \cdot y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 47.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-32}:\\ \;\;\;\;x \cdot \frac{0.5}{t}\\ \mathbf{elif}\;x + y \leq 2 \cdot 10^{-35}:\\ \;\;\;\;\frac{-0.5}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) -2e-32)
   (* x (/ 0.5 t))
   (if (<= (+ x y) 2e-35) (* (/ -0.5 t) z) (* (/ y t) 0.5))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -2e-32) {
		tmp = x * (0.5 / t);
	} else if ((x + y) <= 2e-35) {
		tmp = (-0.5 / t) * z;
	} else {
		tmp = (y / t) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x + y) <= (-2d-32)) then
        tmp = x * (0.5d0 / t)
    else if ((x + y) <= 2d-35) then
        tmp = ((-0.5d0) / t) * z
    else
        tmp = (y / t) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -2e-32) {
		tmp = x * (0.5 / t);
	} else if ((x + y) <= 2e-35) {
		tmp = (-0.5 / t) * z;
	} else {
		tmp = (y / t) * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + y) <= -2e-32:
		tmp = x * (0.5 / t)
	elif (x + y) <= 2e-35:
		tmp = (-0.5 / t) * z
	else:
		tmp = (y / t) * 0.5
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + y) <= -2e-32)
		tmp = Float64(x * Float64(0.5 / t));
	elseif (Float64(x + y) <= 2e-35)
		tmp = Float64(Float64(-0.5 / t) * z);
	else
		tmp = Float64(Float64(y / t) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + y) <= -2e-32)
		tmp = x * (0.5 / t);
	elseif ((x + y) <= 2e-35)
		tmp = (-0.5 / t) * z;
	else
		tmp = (y / t) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + y), $MachinePrecision], -2e-32], N[(x * N[(0.5 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x + y), $MachinePrecision], 2e-35], N[(N[(-0.5 / t), $MachinePrecision] * z), $MachinePrecision], N[(N[(y / t), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x y) < -2.00000000000000011e-32

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x}{t}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x}{t} \cdot \frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{t} \cdot \frac{1}{2}} \]

      3. lower-/.f64 42.6

         \[\leadsto \color{blue}{\frac{x}{t}} \cdot 0.5 \]

   5. Applied rewrites 42.6%

      \[\leadsto \color{blue}{\frac{x}{t} \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 42.6%

      \[\leadsto x \cdot \color{blue}{\frac{0.5}{t}} \]

   if -2.00000000000000011e-32 < (+.f64 x y) < 2.00000000000000002e-35

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{z}{t}} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\color{blue}{1 \cdot z}}{t} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\left(\frac{1}{t} \cdot z\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \frac{1}{t}\right) \cdot z} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{1}{t}\right) \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{t}\right) \cdot z} \]

      6. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{-1}{2}} \cdot \frac{1}{t}\right) \cdot z \]

      7. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot 1}{t}} \cdot z \]

      8. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{2}}}{t} \cdot z \]

      9. lower-/.f64 74.4

         \[\leadsto \color{blue}{\frac{-0.5}{t}} \cdot z \]

   5. Applied rewrites 74.4%

      \[\leadsto \color{blue}{\frac{-0.5}{t} \cdot z} \]

   if 2.00000000000000002e-35 < (+.f64 x y)

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x + y}{t}} \]
   4. Step-by-step derivation

      1. div-add-rev N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\frac{x}{t} + \frac{y}{t}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{x}{t} + \frac{y}{t}\right) \cdot \frac{1}{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{x}{t} + \frac{y}{t}\right) \cdot \frac{1}{2}} \]

      4. div-add-rev N/A

         \[\leadsto \color{blue}{\frac{x + y}{t}} \cdot \frac{1}{2} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x + y}{t}} \cdot \frac{1}{2} \]

      6. +-commutative N/A

         \[\leadsto \frac{\color{blue}{y + x}}{t} \cdot \frac{1}{2} \]

      7. lower-+.f64 79.0

         \[\leadsto \frac{\color{blue}{y + x}}{t} \cdot 0.5 \]

   5. Applied rewrites 79.0%

      \[\leadsto \color{blue}{\frac{y + x}{t} \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{y}{t} \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 46.5%

      \[\leadsto \frac{y}{t} \cdot 0.5 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 36.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x + y\right) - z \leq -2 \cdot 10^{-186}:\\ \;\;\;\;x \cdot \frac{0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- (+ x y) z) -2e-186) (* x (/ 0.5 t)) (* (/ y t) 0.5)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x + y) - z) <= -2e-186) {
		tmp = x * (0.5 / t);
	} else {
		tmp = (y / t) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((x + y) - z) <= (-2d-186)) then
        tmp = x * (0.5d0 / t)
    else
        tmp = (y / t) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((x + y) - z) <= -2e-186) {
		tmp = x * (0.5 / t);
	} else {
		tmp = (y / t) * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x + y) - z) <= -2e-186:
		tmp = x * (0.5 / t)
	else:
		tmp = (y / t) * 0.5
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x + y) - z) <= -2e-186)
		tmp = Float64(x * Float64(0.5 / t));
	else
		tmp = Float64(Float64(y / t) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x + y) - z) <= -2e-186)
		tmp = x * (0.5 / t);
	else
		tmp = (y / t) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x + y), $MachinePrecision] - z), $MachinePrecision], -2e-186], N[(x * N[(0.5 / t), $MachinePrecision]), $MachinePrecision], N[(N[(y / t), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x y) z) < -1.9999999999999998e-186

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x}{t}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x}{t} \cdot \frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{t} \cdot \frac{1}{2}} \]

      3. lower-/.f64 38.8

         \[\leadsto \color{blue}{\frac{x}{t}} \cdot 0.5 \]

   5. Applied rewrites 38.8%

      \[\leadsto \color{blue}{\frac{x}{t} \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 38.7%

      \[\leadsto x \cdot \color{blue}{\frac{0.5}{t}} \]

   if -1.9999999999999998e-186 < (-.f64 (+.f64 x y) z)

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x + y}{t}} \]
   4. Step-by-step derivation

      1. div-add-rev N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\frac{x}{t} + \frac{y}{t}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{x}{t} + \frac{y}{t}\right) \cdot \frac{1}{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{x}{t} + \frac{y}{t}\right) \cdot \frac{1}{2}} \]

      4. div-add-rev N/A

         \[\leadsto \color{blue}{\frac{x + y}{t}} \cdot \frac{1}{2} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x + y}{t}} \cdot \frac{1}{2} \]

      6. +-commutative N/A

         \[\leadsto \frac{\color{blue}{y + x}}{t} \cdot \frac{1}{2} \]

      7. lower-+.f64 69.0

         \[\leadsto \frac{\color{blue}{y + x}}{t} \cdot 0.5 \]

   5. Applied rewrites 69.0%

      \[\leadsto \color{blue}{\frac{y + x}{t} \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{y}{t} \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 41.3%

      \[\leadsto \frac{y}{t} \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 69.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-186}:\\ \;\;\;\;\frac{x - z}{t + t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{t + t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) -2e-186) (/ (- x z) (+ t t)) (/ (- y z) (+ t t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -2e-186) {
		tmp = (x - z) / (t + t);
	} else {
		tmp = (y - z) / (t + t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x + y) <= (-2d-186)) then
        tmp = (x - z) / (t + t)
    else
        tmp = (y - z) / (t + t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -2e-186) {
		tmp = (x - z) / (t + t);
	} else {
		tmp = (y - z) / (t + t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + y) <= -2e-186:
		tmp = (x - z) / (t + t)
	else:
		tmp = (y - z) / (t + t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + y) <= -2e-186)
		tmp = Float64(Float64(x - z) / Float64(t + t));
	else
		tmp = Float64(Float64(y - z) / Float64(t + t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + y) <= -2e-186)
		tmp = (x - z) / (t + t);
	else
		tmp = (y - z) / (t + t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + y), $MachinePrecision], -2e-186], N[(N[(x - z), $MachinePrecision] / N[(t + t), $MachinePrecision]), $MachinePrecision], N[(N[(y - z), $MachinePrecision] / N[(t + t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -1.9999999999999998e-186

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{x - z}}{t \cdot 2} \]
   4. Step-by-step derivation

      1. lower--.f64 69.8

         \[\leadsto \frac{\color{blue}{x - z}}{t \cdot 2} \]

   5. Applied rewrites 69.8%

      \[\leadsto \frac{\color{blue}{x - z}}{t \cdot 2} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x - z}{\color{blue}{t \cdot 2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x - z}{\color{blue}{2 \cdot t}} \]

      3. count-2-rev N/A

         \[\leadsto \frac{x - z}{\color{blue}{t + t}} \]

      4. lift-+.f64 69.8

         \[\leadsto \frac{x - z}{\color{blue}{t + t}} \]

   7. Applied rewrites 69.8%

      \[\leadsto \frac{x - z}{\color{blue}{t + t}} \]

   if -1.9999999999999998e-186 < (+.f64 x y)

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{y - z}}{t \cdot 2} \]
   4. Step-by-step derivation

      1. lower--.f64 71.3

         \[\leadsto \frac{\color{blue}{y - z}}{t \cdot 2} \]

   5. Applied rewrites 71.3%

      \[\leadsto \frac{\color{blue}{y - z}}{t \cdot 2} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{y - z}{\color{blue}{t \cdot 2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{y - z}{\color{blue}{2 \cdot t}} \]

      3. count-2-rev N/A

         \[\leadsto \frac{y - z}{\color{blue}{t + t}} \]

      4. lower-+.f64 71.3

         \[\leadsto \frac{y - z}{\color{blue}{t + t}} \]

   7. Applied rewrites 71.3%

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

Alternative 9: 59.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq 2 \cdot 10^{-35}:\\ \;\;\;\;\frac{x - z}{t + t}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) 2e-35) (/ (- x z) (+ t t)) (/ (* 0.5 y) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= 2e-35) {
		tmp = (x - z) / (t + t);
	} else {
		tmp = (0.5 * y) / t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x + y) <= 2d-35) then
        tmp = (x - z) / (t + t)
    else
        tmp = (0.5d0 * y) / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= 2e-35) {
		tmp = (x - z) / (t + t);
	} else {
		tmp = (0.5 * y) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + y) <= 2e-35:
		tmp = (x - z) / (t + t)
	else:
		tmp = (0.5 * y) / t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + y) <= 2e-35)
		tmp = Float64(Float64(x - z) / Float64(t + t));
	else
		tmp = Float64(Float64(0.5 * y) / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + y) <= 2e-35)
		tmp = (x - z) / (t + t);
	else
		tmp = (0.5 * y) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + y), $MachinePrecision], 2e-35], N[(N[(x - z), $MachinePrecision] / N[(t + t), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * y), $MachinePrecision] / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < 2.00000000000000002e-35

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{x - z}}{t \cdot 2} \]
   4. Step-by-step derivation

      1. lower--.f64 74.0

         \[\leadsto \frac{\color{blue}{x - z}}{t \cdot 2} \]

   5. Applied rewrites 74.0%

      \[\leadsto \frac{\color{blue}{x - z}}{t \cdot 2} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x - z}{\color{blue}{t \cdot 2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x - z}{\color{blue}{2 \cdot t}} \]

      3. count-2-rev N/A

         \[\leadsto \frac{x - z}{\color{blue}{t + t}} \]

      4. lift-+.f64 74.0

         \[\leadsto \frac{x - z}{\color{blue}{t + t}} \]

   7. Applied rewrites 74.0%

      \[\leadsto \frac{x - z}{\color{blue}{t + t}} \]

   if 2.00000000000000002e-35 < (+.f64 x y)

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{y}{t}} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\color{blue}{1 \cdot y}}{t} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\frac{1}{t} \cdot y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{t}\right) \cdot y} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{t}\right) \cdot y} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot 1}{t}} \cdot y \]

      6. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2}}}{t} \cdot y \]

      7. lower-/.f64 46.4

         \[\leadsto \color{blue}{\frac{0.5}{t}} \cdot y \]

   5. Applied rewrites 46.4%

      \[\leadsto \color{blue}{\frac{0.5}{t} \cdot y} \]
   6. Step-by-step derivation

   7. Applied rewrites 46.5%

      \[\leadsto \frac{0.5 \cdot y}{\color{blue}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 36.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{0.5}{t} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* x (/ 0.5 t)))

C

double code(double x, double y, double z, double t) {
	return x * (0.5 / t);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x * (0.5d0 / t)
end function

Java

public static double code(double x, double y, double z, double t) {
	return x * (0.5 / t);
}

Python

def code(x, y, z, t):
	return x * (0.5 / t)

Julia

function code(x, y, z, t)
	return Float64(x * Float64(0.5 / t))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x * (0.5 / t);
end

Wolfram

code[x_, y_, z_, t_] := N[(x * N[(0.5 / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x}{t}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\frac{x}{t} \cdot \frac{1}{2}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{t} \cdot \frac{1}{2}} \]

   3. lower-/.f64 38.4

      \[\leadsto \color{blue}{\frac{x}{t}} \cdot 0.5 \]

5. Applied rewrites 38.4%

   \[\leadsto \color{blue}{\frac{x}{t} \cdot 0.5} \]
6. Step-by-step derivation

7. Applied rewrites 38.6%

   \[\leadsto x \cdot \color{blue}{\frac{0.5}{t}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024326 
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, B"
  :precision binary64
  (/ (- (+ x y) z) (* t 2.0)))