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

Percentage Accurate: 99.9% → 99.9%

Time: 4.4s

Alternatives: 7

Speedup: 1.0×

• 26.1% 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. Final simplification 100.0%

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

Alternative 2: 46.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{0.5}{t}\\ t_2 := \frac{z}{t} \cdot -0.5\\ \mathbf{if}\;x \leq -2.2 \cdot 10^{+129}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{+89}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -470:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{-145} \lor \neg \left(x \leq -4.6 \cdot 10^{-191}\right) \land x \leq 3.3 \cdot 10^{-257}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ 0.5 t))) (t_2 (* (/ z t) -0.5)))
   (if (<= x -2.2e+129)
     t_1
     (if (<= x -1.75e+89)
       t_2
       (if (<= x -470.0)
         t_1
         (if (or (<= x -1.25e-145)
                 (and (not (<= x -4.6e-191)) (<= x 3.3e-257)))
           t_2
           (* 0.5 (/ y t))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (0.5 / t);
	double t_2 = (z / t) * -0.5;
	double tmp;
	if (x <= -2.2e+129) {
		tmp = t_1;
	} else if (x <= -1.75e+89) {
		tmp = t_2;
	} else if (x <= -470.0) {
		tmp = t_1;
	} else if ((x <= -1.25e-145) || (!(x <= -4.6e-191) && (x <= 3.3e-257))) {
		tmp = t_2;
	} 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x * (0.5d0 / t)
    t_2 = (z / t) * (-0.5d0)
    if (x <= (-2.2d+129)) then
        tmp = t_1
    else if (x <= (-1.75d+89)) then
        tmp = t_2
    else if (x <= (-470.0d0)) then
        tmp = t_1
    else if ((x <= (-1.25d-145)) .or. (.not. (x <= (-4.6d-191))) .and. (x <= 3.3d-257)) then
        tmp = t_2
    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 t_1 = x * (0.5 / t);
	double t_2 = (z / t) * -0.5;
	double tmp;
	if (x <= -2.2e+129) {
		tmp = t_1;
	} else if (x <= -1.75e+89) {
		tmp = t_2;
	} else if (x <= -470.0) {
		tmp = t_1;
	} else if ((x <= -1.25e-145) || (!(x <= -4.6e-191) && (x <= 3.3e-257))) {
		tmp = t_2;
	} else {
		tmp = 0.5 * (y / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (0.5 / t)
	t_2 = (z / t) * -0.5
	tmp = 0
	if x <= -2.2e+129:
		tmp = t_1
	elif x <= -1.75e+89:
		tmp = t_2
	elif x <= -470.0:
		tmp = t_1
	elif (x <= -1.25e-145) or (not (x <= -4.6e-191) and (x <= 3.3e-257)):
		tmp = t_2
	else:
		tmp = 0.5 * (y / t)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(0.5 / t))
	t_2 = Float64(Float64(z / t) * -0.5)
	tmp = 0.0
	if (x <= -2.2e+129)
		tmp = t_1;
	elseif (x <= -1.75e+89)
		tmp = t_2;
	elseif (x <= -470.0)
		tmp = t_1;
	elseif ((x <= -1.25e-145) || (!(x <= -4.6e-191) && (x <= 3.3e-257)))
		tmp = t_2;
	else
		tmp = Float64(0.5 * Float64(y / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * (0.5 / t);
	t_2 = (z / t) * -0.5;
	tmp = 0.0;
	if (x <= -2.2e+129)
		tmp = t_1;
	elseif (x <= -1.75e+89)
		tmp = t_2;
	elseif (x <= -470.0)
		tmp = t_1;
	elseif ((x <= -1.25e-145) || (~((x <= -4.6e-191)) && (x <= 3.3e-257)))
		tmp = t_2;
	else
		tmp = 0.5 * (y / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(0.5 / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z / t), $MachinePrecision] * -0.5), $MachinePrecision]}, If[LessEqual[x, -2.2e+129], t$95$1, If[LessEqual[x, -1.75e+89], t$95$2, If[LessEqual[x, -470.0], t$95$1, If[Or[LessEqual[x, -1.25e-145], And[N[Not[LessEqual[x, -4.6e-191]], $MachinePrecision], LessEqual[x, 3.3e-257]]], t$95$2, N[(0.5 * N[(y / t), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.1999999999999999e129 or -1.75e89 < x < -470

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0 92.3%

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

      1. associate-*r/ 92.3%

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

      2. +-commutative 92.3%

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

      3. associate-/l* 92.1%

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

      4. +-commutative 92.1%

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

   4. Simplified 92.1%

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

   5. Taylor expanded in y around 0 78.1%

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

      1. associate-*r/ 78.1%

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

      2. associate-*l/ 77.9%

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

      3. *-commutative 77.9%

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

   7. Simplified 77.9%

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

   if -2.1999999999999999e129 < x < -1.75e89 or -470 < x < -1.2499999999999999e-145 or -4.60000000000000021e-191 < x < 3.3e-257

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 56.6%

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

      1. *-commutative 56.6%

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

   4. Simplified 56.6%

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

   if -1.2499999999999999e-145 < x < -4.60000000000000021e-191 or 3.3e-257 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 39.7%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+129}:\\ \;\;\;\;x \cdot \frac{0.5}{t}\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{+89}:\\ \;\;\;\;\frac{z}{t} \cdot -0.5\\ \mathbf{elif}\;x \leq -470:\\ \;\;\;\;x \cdot \frac{0.5}{t}\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{-145} \lor \neg \left(x \leq -4.6 \cdot 10^{-191}\right) \land x \leq 3.3 \cdot 10^{-257}:\\ \;\;\;\;\frac{z}{t} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{t}\\ \end{array} \]

Alternative 3: 46.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot 0.5}{t}\\ t_2 := \frac{z}{t} \cdot -0.5\\ \mathbf{if}\;x \leq -2.65 \cdot 10^{+129}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{+88}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -246:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-144} \lor \neg \left(x \leq -1.1 \cdot 10^{-193}\right) \land x \leq 8 \cdot 10^{-250}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x 0.5) t)) (t_2 (* (/ z t) -0.5)))
   (if (<= x -2.65e+129)
     t_1
     (if (<= x -8.5e+88)
       t_2
       (if (<= x -246.0)
         t_1
         (if (or (<= x -1e-144) (and (not (<= x -1.1e-193)) (<= x 8e-250)))
           t_2
           (* 0.5 (/ y t))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) / t;
	double t_2 = (z / t) * -0.5;
	double tmp;
	if (x <= -2.65e+129) {
		tmp = t_1;
	} else if (x <= -8.5e+88) {
		tmp = t_2;
	} else if (x <= -246.0) {
		tmp = t_1;
	} else if ((x <= -1e-144) || (!(x <= -1.1e-193) && (x <= 8e-250))) {
		tmp = t_2;
	} 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x * 0.5d0) / t
    t_2 = (z / t) * (-0.5d0)
    if (x <= (-2.65d+129)) then
        tmp = t_1
    else if (x <= (-8.5d+88)) then
        tmp = t_2
    else if (x <= (-246.0d0)) then
        tmp = t_1
    else if ((x <= (-1d-144)) .or. (.not. (x <= (-1.1d-193))) .and. (x <= 8d-250)) then
        tmp = t_2
    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 t_1 = (x * 0.5) / t;
	double t_2 = (z / t) * -0.5;
	double tmp;
	if (x <= -2.65e+129) {
		tmp = t_1;
	} else if (x <= -8.5e+88) {
		tmp = t_2;
	} else if (x <= -246.0) {
		tmp = t_1;
	} else if ((x <= -1e-144) || (!(x <= -1.1e-193) && (x <= 8e-250))) {
		tmp = t_2;
	} else {
		tmp = 0.5 * (y / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) / t
	t_2 = (z / t) * -0.5
	tmp = 0
	if x <= -2.65e+129:
		tmp = t_1
	elif x <= -8.5e+88:
		tmp = t_2
	elif x <= -246.0:
		tmp = t_1
	elif (x <= -1e-144) or (not (x <= -1.1e-193) and (x <= 8e-250)):
		tmp = t_2
	else:
		tmp = 0.5 * (y / t)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) / t)
	t_2 = Float64(Float64(z / t) * -0.5)
	tmp = 0.0
	if (x <= -2.65e+129)
		tmp = t_1;
	elseif (x <= -8.5e+88)
		tmp = t_2;
	elseif (x <= -246.0)
		tmp = t_1;
	elseif ((x <= -1e-144) || (!(x <= -1.1e-193) && (x <= 8e-250)))
		tmp = t_2;
	else
		tmp = Float64(0.5 * Float64(y / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) / t;
	t_2 = (z / t) * -0.5;
	tmp = 0.0;
	if (x <= -2.65e+129)
		tmp = t_1;
	elseif (x <= -8.5e+88)
		tmp = t_2;
	elseif (x <= -246.0)
		tmp = t_1;
	elseif ((x <= -1e-144) || (~((x <= -1.1e-193)) && (x <= 8e-250)))
		tmp = t_2;
	else
		tmp = 0.5 * (y / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z / t), $MachinePrecision] * -0.5), $MachinePrecision]}, If[LessEqual[x, -2.65e+129], t$95$1, If[LessEqual[x, -8.5e+88], t$95$2, If[LessEqual[x, -246.0], t$95$1, If[Or[LessEqual[x, -1e-144], And[N[Not[LessEqual[x, -1.1e-193]], $MachinePrecision], LessEqual[x, 8e-250]]], t$95$2, N[(0.5 * N[(y / t), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.6499999999999999e129 or -8.5000000000000005e88 < x < -246

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 78.1%

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

      1. associate-*r/ 78.1%

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

   4. Simplified 78.1%

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

   if -2.6499999999999999e129 < x < -8.5000000000000005e88 or -246 < x < -9.9999999999999995e-145 or -1.09999999999999988e-193 < x < 8.0000000000000004e-250

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 56.6%

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

      1. *-commutative 56.6%

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

   4. Simplified 56.6%

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

   if -9.9999999999999995e-145 < x < -1.09999999999999988e-193 or 8.0000000000000004e-250 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 39.7%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{+129}:\\ \;\;\;\;\frac{x \cdot 0.5}{t}\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{+88}:\\ \;\;\;\;\frac{z}{t} \cdot -0.5\\ \mathbf{elif}\;x \leq -246:\\ \;\;\;\;\frac{x \cdot 0.5}{t}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-144} \lor \neg \left(x \leq -1.1 \cdot 10^{-193}\right) \land x \leq 8 \cdot 10^{-250}:\\ \;\;\;\;\frac{z}{t} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{t}\\ \end{array} \]

Alternative 4: 69.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-237}:\\ \;\;\;\;0.5 \cdot \frac{x - z}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y - z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) -2e-237) (* 0.5 (/ (- x z) t)) (* 0.5 (/ (- y z) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -2e-237) {
		tmp = 0.5 * ((x - z) / t);
	} else {
		tmp = 0.5 * ((y - z) / 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-237)) then
        tmp = 0.5d0 * ((x - z) / t)
    else
        tmp = 0.5d0 * ((y - z) / 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-237) {
		tmp = 0.5 * ((x - z) / t);
	} else {
		tmp = 0.5 * ((y - z) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + y) <= -2e-237:
		tmp = 0.5 * ((x - z) / t)
	else:
		tmp = 0.5 * ((y - z) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + y) <= -2e-237)
		tmp = Float64(0.5 * Float64(Float64(x - z) / t));
	else
		tmp = Float64(0.5 * Float64(Float64(y - z) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + y) <= -2e-237)
		tmp = 0.5 * ((x - z) / t);
	else
		tmp = 0.5 * ((y - z) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + y), $MachinePrecision], -2e-237], N[(0.5 * N[(N[(x - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(y - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -2e-237

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 69.8%

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

   if -2e-237 < (+.f64 x y)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 64.0%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-237}:\\ \;\;\;\;0.5 \cdot \frac{x - z}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y - z}{t}\\ \end{array} \]

Alternative 5: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{+125}:\\ \;\;\;\;0.5 \cdot \frac{x - z}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 4.2e+125) (* 0.5 (/ (- x z) t)) (* 0.5 (/ y t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 4.2e+125) {
		tmp = 0.5 * ((x - 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 (y <= 4.2d+125) then
        tmp = 0.5d0 * ((x - 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 (y <= 4.2e+125) {
		tmp = 0.5 * ((x - z) / t);
	} else {
		tmp = 0.5 * (y / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 4.2e+125:
		tmp = 0.5 * ((x - z) / t)
	else:
		tmp = 0.5 * (y / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 4.2e+125)
		tmp = Float64(0.5 * Float64(Float64(x - z) / t));
	else
		tmp = Float64(0.5 * Float64(y / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 4.2e+125)
		tmp = 0.5 * ((x - z) / t);
	else
		tmp = 0.5 * (y / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 4.2e+125], N[(0.5 * N[(N[(x - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.2000000000000001e125

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 76.2%

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

   if 4.2000000000000001e125 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 68.7%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{+125}:\\ \;\;\;\;0.5 \cdot \frac{x - z}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{t}\\ \end{array} \]

Alternative 6: 46.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.7e-14) (* x (/ 0.5 t)) (* 0.5 (/ y t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.7e-14) {
		tmp = x * (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 <= (-1.7d-14)) then
        tmp = x * (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 <= -1.7e-14) {
		tmp = x * (0.5 / t);
	} else {
		tmp = 0.5 * (y / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1.7e-14:
		tmp = x * (0.5 / t)
	else:
		tmp = 0.5 * (y / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.7e-14)
		tmp = Float64(x * Float64(0.5 / t));
	else
		tmp = Float64(0.5 * Float64(y / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -1.7e-14)
		tmp = x * (0.5 / t);
	else
		tmp = 0.5 * (y / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.7e-14], N[(x * N[(0.5 / t), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.70000000000000001e-14

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 84.6%

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

      1. associate-*r/ 84.6%

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

      2. +-commutative 84.6%

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

      3. associate-/l* 84.4%

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

      4. +-commutative 84.4%

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

   4. Simplified 84.4%

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

   5. Taylor expanded in y around 0 69.4%

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

      1. associate-*r/ 69.4%

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

      2. associate-*l/ 69.2%

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

      3. *-commutative 69.2%

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

   7. Simplified 69.2%

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

   if -1.70000000000000001e-14 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 39.1%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{t}\\ \end{array} \]

Alternative 7: 37.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in y around inf 34.2%

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

3. Final simplification 34.2%

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

Reproduce

herbie shell --seed 2023195 
(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)))