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

Percentage Accurate: 99.9% → 99.7%

Time: 7.2s

Alternatives: 9

Speedup: N/A×

• 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: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{t} \cdot \left(x + \left(y - z\right)\right) \end{array} \]

FPCore

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

C

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

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 = (0.5d0 / t) * (x + (y - z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. metadata-eval 99.7

      \[\leadsto \frac{\color{blue}{0.5}}{t} \cdot \left(\left(x + y\right) - z\right) \]

   10. lift--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. associate--l+ N/A

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

   13. lower-+.f64 N/A

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

   14. lower--.f64 99.7

       \[\leadsto \frac{0.5}{t} \cdot \left(x + \color{blue}{\left(y - z\right)}\right) \]

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{0.5}{t} \cdot \left(x + \left(y - z\right)\right)} \]
5. Add Preprocessing

Alternative 2: 37.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x + y\right) - z}{t \cdot 2} \leq 0:\\ \;\;\;\;\frac{0.5 \cdot x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (- (+ x y) z) (* t 2.0)) 0.0) (/ (* 0.5 x) t) (/ (* 0.5 y) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((((x + y) - z) / (t * 2.0)) <= 0.0) {
		tmp = (0.5 * x) / 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) - z) / (t * 2.0d0)) <= 0.0d0) then
        tmp = (0.5d0 * x) / 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) - z) / (t * 2.0)) <= 0.0) {
		tmp = (0.5 * x) / t;
	} else {
		tmp = (0.5 * y) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (((x + y) - z) / (t * 2.0)) <= 0.0:
		tmp = (0.5 * x) / t
	else:
		tmp = (0.5 * y) / t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(Float64(x + y) - z) / Float64(t * 2.0)) <= 0.0)
		tmp = Float64(Float64(0.5 * x) / 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) - z) / (t * 2.0)) <= 0.0)
		tmp = (0.5 * x) / t;
	else
		tmp = (0.5 * y) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(N[(x + y), $MachinePrecision] - z), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(0.5 * x), $MachinePrecision] / t), $MachinePrecision], N[(N[(0.5 * y), $MachinePrecision] / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 x y) z) (*.f64 t #s(literal 2 binary64))) < 0.0

   1. Initial program 99.3%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   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. associate-*l/ N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 45.1

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

   5. Applied rewrites 45.1%

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

   if 0.0 < (/.f64 (-.f64 (+.f64 x y) z) (*.f64 t #s(literal 2 binary64)))

   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. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 32.1

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

   5. Applied rewrites 32.1%

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

Alternative 3: 49.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -5 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.5 \cdot x}{t}\\ \mathbf{elif}\;x + y \leq 10^{-25}:\\ \;\;\;\;\frac{z \cdot -0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) -5e-7)
   (/ (* 0.5 x) t)
   (if (<= (+ x y) 1e-25) (/ (* z -0.5) t) (/ (* 0.5 y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -5e-7) {
		tmp = (0.5 * x) / t;
	} else if ((x + y) <= 1e-25) {
		tmp = (z * -0.5) / 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) <= (-5d-7)) then
        tmp = (0.5d0 * x) / t
    else if ((x + y) <= 1d-25) then
        tmp = (z * (-0.5d0)) / 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) <= -5e-7) {
		tmp = (0.5 * x) / t;
	} else if ((x + y) <= 1e-25) {
		tmp = (z * -0.5) / t;
	} else {
		tmp = (0.5 * y) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + y) <= -5e-7:
		tmp = (0.5 * x) / t
	elif (x + y) <= 1e-25:
		tmp = (z * -0.5) / t
	else:
		tmp = (0.5 * y) / t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + y) <= -5e-7)
		tmp = Float64(Float64(0.5 * x) / t);
	elseif (Float64(x + y) <= 1e-25)
		tmp = Float64(Float64(z * -0.5) / 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) <= -5e-7)
		tmp = (0.5 * x) / t;
	elseif ((x + y) <= 1e-25)
		tmp = (z * -0.5) / t;
	else
		tmp = (0.5 * y) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + y), $MachinePrecision], -5e-7], N[(N[(0.5 * x), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[N[(x + y), $MachinePrecision], 1e-25], N[(N[(z * -0.5), $MachinePrecision] / t), $MachinePrecision], N[(N[(0.5 * y), $MachinePrecision] / t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x y) < -4.99999999999999977e-7

   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. associate-*l/ N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 52.9

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

   5. Applied rewrites 52.9%

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

   if -4.99999999999999977e-7 < (+.f64 x y) < 1.00000000000000004e-25

   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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 66.7

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

   5. Applied rewrites 66.7%

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

   if 1.00000000000000004e-25 < (+.f64 x y)

   1. Initial program 99.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   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. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 36.1

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

   5. Applied rewrites 36.1%

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

Alternative 4: 86.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - z}{t \cdot 2}\\ \mathbf{if}\;z \leq -5.5 \cdot 10^{+103}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+89}:\\ \;\;\;\;\frac{x + y}{t \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x z) (* t 2.0))))
   (if (<= z -5.5e+103) t_1 (if (<= z 4.5e+89) (/ (+ x y) (* t 2.0)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - z) / (t * 2.0);
	double tmp;
	if (z <= -5.5e+103) {
		tmp = t_1;
	} else if (z <= 4.5e+89) {
		tmp = (x + y) / (t * 2.0);
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = (x - z) / (t * 2.0d0)
    if (z <= (-5.5d+103)) then
        tmp = t_1
    else if (z <= 4.5d+89) then
        tmp = (x + y) / (t * 2.0d0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x - z) / (t * 2.0);
	double tmp;
	if (z <= -5.5e+103) {
		tmp = t_1;
	} else if (z <= 4.5e+89) {
		tmp = (x + y) / (t * 2.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - z) / (t * 2.0)
	tmp = 0
	if z <= -5.5e+103:
		tmp = t_1
	elif z <= 4.5e+89:
		tmp = (x + y) / (t * 2.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - z) / Float64(t * 2.0))
	tmp = 0.0
	if (z <= -5.5e+103)
		tmp = t_1;
	elseif (z <= 4.5e+89)
		tmp = Float64(Float64(x + y) / Float64(t * 2.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - z) / (t * 2.0);
	tmp = 0.0;
	if (z <= -5.5e+103)
		tmp = t_1;
	elseif (z <= 4.5e+89)
		tmp = (x + y) / (t * 2.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - z), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.5e+103], t$95$1, If[LessEqual[z, 4.5e+89], N[(N[(x + y), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.50000000000000001e103 or 4.5e89 < z

   1. Initial program 98.9%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   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 81.7

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

   5. Applied rewrites 81.7%

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

   if -5.50000000000000001e103 < z < 4.5e89

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 93.1

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

   5. Applied rewrites 93.1%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+103}:\\ \;\;\;\;\frac{x - z}{t \cdot 2}\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+89}:\\ \;\;\;\;\frac{x + y}{t \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z}{t \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot -0.5}{t}\\ \mathbf{if}\;z \leq -6.7 \cdot 10^{+109}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+146}:\\ \;\;\;\;\frac{x + y}{t \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z -0.5) t)))
   (if (<= z -6.7e+109) t_1 (if (<= z 9e+146) (/ (+ x y) (* t 2.0)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * -0.5) / t;
	double tmp;
	if (z <= -6.7e+109) {
		tmp = t_1;
	} else if (z <= 9e+146) {
		tmp = (x + y) / (t * 2.0);
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = (z * (-0.5d0)) / t
    if (z <= (-6.7d+109)) then
        tmp = t_1
    else if (z <= 9d+146) then
        tmp = (x + y) / (t * 2.0d0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * -0.5) / t;
	double tmp;
	if (z <= -6.7e+109) {
		tmp = t_1;
	} else if (z <= 9e+146) {
		tmp = (x + y) / (t * 2.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * -0.5) / t
	tmp = 0
	if z <= -6.7e+109:
		tmp = t_1
	elif z <= 9e+146:
		tmp = (x + y) / (t * 2.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * -0.5) / t)
	tmp = 0.0
	if (z <= -6.7e+109)
		tmp = t_1;
	elseif (z <= 9e+146)
		tmp = Float64(Float64(x + y) / Float64(t * 2.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * -0.5) / t;
	tmp = 0.0;
	if (z <= -6.7e+109)
		tmp = t_1;
	elseif (z <= 9e+146)
		tmp = (x + y) / (t * 2.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * -0.5), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[z, -6.7e+109], t$95$1, If[LessEqual[z, 9e+146], N[(N[(x + y), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.70000000000000036e109 or 9.00000000000000051e146 < z

   1. Initial program 98.7%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 77.0

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

   5. Applied rewrites 77.0%

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

   if -6.70000000000000036e109 < z < 9.00000000000000051e146

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 90.4

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

   5. Applied rewrites 90.4%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.7 \cdot 10^{+109}:\\ \;\;\;\;\frac{z \cdot -0.5}{t}\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+146}:\\ \;\;\;\;\frac{x + y}{t \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot -0.5}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot -0.5}{t}\\ \mathbf{if}\;z \leq -6.7 \cdot 10^{+109}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+146}:\\ \;\;\;\;\frac{0.5}{t} \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z -0.5) t)))
   (if (<= z -6.7e+109) t_1 (if (<= z 9e+146) (* (/ 0.5 t) (+ x y)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * -0.5) / t;
	double tmp;
	if (z <= -6.7e+109) {
		tmp = t_1;
	} else if (z <= 9e+146) {
		tmp = (0.5 / t) * (x + y);
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = (z * (-0.5d0)) / t
    if (z <= (-6.7d+109)) then
        tmp = t_1
    else if (z <= 9d+146) then
        tmp = (0.5d0 / t) * (x + y)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * -0.5) / t;
	double tmp;
	if (z <= -6.7e+109) {
		tmp = t_1;
	} else if (z <= 9e+146) {
		tmp = (0.5 / t) * (x + y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * -0.5) / t
	tmp = 0
	if z <= -6.7e+109:
		tmp = t_1
	elif z <= 9e+146:
		tmp = (0.5 / t) * (x + y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * -0.5) / t)
	tmp = 0.0
	if (z <= -6.7e+109)
		tmp = t_1;
	elseif (z <= 9e+146)
		tmp = Float64(Float64(0.5 / t) * Float64(x + y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * -0.5) / t;
	tmp = 0.0;
	if (z <= -6.7e+109)
		tmp = t_1;
	elseif (z <= 9e+146)
		tmp = (0.5 / t) * (x + y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * -0.5), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[z, -6.7e+109], t$95$1, If[LessEqual[z, 9e+146], N[(N[(0.5 / t), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.70000000000000036e109 or 9.00000000000000051e146 < z

   1. Initial program 98.7%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 77.0

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

   5. Applied rewrites 77.0%

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

   if -6.70000000000000036e109 < z < 9.00000000000000051e146

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 90.4

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

   5. Applied rewrites 90.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 90.1

         \[\leadsto \color{blue}{\frac{0.5}{t} \cdot \left(y + x\right)} \]

   7. Applied rewrites 90.1%

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

Alternative 7: 69.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -5 \cdot 10^{-236}:\\ \;\;\;\;\frac{x - z}{t \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{t \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) -5e-236) (/ (- x z) (* t 2.0)) (/ (- y z) (* t 2.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -5e-236) {
		tmp = (x - z) / (t * 2.0);
	} else {
		tmp = (y - z) / (t * 2.0);
	}
	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) <= (-5d-236)) then
        tmp = (x - z) / (t * 2.0d0)
    else
        tmp = (y - z) / (t * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -5e-236) {
		tmp = (x - z) / (t * 2.0);
	} else {
		tmp = (y - z) / (t * 2.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + y) <= -5e-236:
		tmp = (x - z) / (t * 2.0)
	else:
		tmp = (y - z) / (t * 2.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + y) <= -5e-236)
		tmp = Float64(Float64(x - z) / Float64(t * 2.0));
	else
		tmp = Float64(Float64(y - z) / Float64(t * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + y) <= -5e-236)
		tmp = (x - z) / (t * 2.0);
	else
		tmp = (y - z) / (t * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + y), $MachinePrecision], -5e-236], N[(N[(x - z), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(y - z), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -4.9999999999999998e-236

   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 70.2

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

   5. Applied rewrites 70.2%

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

   if -4.9999999999999998e-236 < (+.f64 x y)

   1. Initial program 99.2%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   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 61.8

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

   5. Applied rewrites 61.8%

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

Alternative 8: 37.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (0.5 * x) / 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 = (0.5d0 * x) / t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
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. associate-*l/ N/A

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

   3. lower-/.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 46.2

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

5. Applied rewrites 46.2%

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

Alternative 9: 37.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

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 = (0.5d0 / t) * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
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. associate-*l/ N/A

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

   3. lower-/.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 46.2

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

5. Applied rewrites 46.2%

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

7. Applied rewrites 46.0%

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

Reproduce

herbie shell --seed 2024233 
(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)))