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

Percentage Accurate: 99.9% → 99.9%

Time: 7.4s

Alternatives: 9

Speedup: 1.0×

• 26.7% 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: 81.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot -0.5}{t}\\ t_2 := \frac{x + y}{t \cdot 2}\\ \mathbf{if}\;z \leq -2.45 \cdot 10^{+90}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+14}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 5.3 \cdot 10^{+39}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+132}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z -0.5) t)) (t_2 (/ (+ x y) (* t 2.0))))
   (if (<= z -2.45e+90)
     t_1
     (if (<= z 4e+14)
       t_2
       (if (<= z 5.3e+39) (* z (/ -0.5 t)) (if (<= z 7e+132) t_2 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * -0.5) / t;
	double t_2 = (x + y) / (t * 2.0);
	double tmp;
	if (z <= -2.45e+90) {
		tmp = t_1;
	} else if (z <= 4e+14) {
		tmp = t_2;
	} else if (z <= 5.3e+39) {
		tmp = z * (-0.5 / t);
	} else if (z <= 7e+132) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_1 = (z * (-0.5d0)) / t
    t_2 = (x + y) / (t * 2.0d0)
    if (z <= (-2.45d+90)) then
        tmp = t_1
    else if (z <= 4d+14) then
        tmp = t_2
    else if (z <= 5.3d+39) then
        tmp = z * ((-0.5d0) / t)
    else if (z <= 7d+132) then
        tmp = t_2
    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 t_2 = (x + y) / (t * 2.0);
	double tmp;
	if (z <= -2.45e+90) {
		tmp = t_1;
	} else if (z <= 4e+14) {
		tmp = t_2;
	} else if (z <= 5.3e+39) {
		tmp = z * (-0.5 / t);
	} else if (z <= 7e+132) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * -0.5) / t
	t_2 = (x + y) / (t * 2.0)
	tmp = 0
	if z <= -2.45e+90:
		tmp = t_1
	elif z <= 4e+14:
		tmp = t_2
	elif z <= 5.3e+39:
		tmp = z * (-0.5 / t)
	elif z <= 7e+132:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * -0.5) / t)
	t_2 = Float64(Float64(x + y) / Float64(t * 2.0))
	tmp = 0.0
	if (z <= -2.45e+90)
		tmp = t_1;
	elseif (z <= 4e+14)
		tmp = t_2;
	elseif (z <= 5.3e+39)
		tmp = Float64(z * Float64(-0.5 / t));
	elseif (z <= 7e+132)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * -0.5) / t;
	t_2 = (x + y) / (t * 2.0);
	tmp = 0.0;
	if (z <= -2.45e+90)
		tmp = t_1;
	elseif (z <= 4e+14)
		tmp = t_2;
	elseif (z <= 5.3e+39)
		tmp = z * (-0.5 / t);
	elseif (z <= 7e+132)
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(N[(x + y), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.45e+90], t$95$1, If[LessEqual[z, 4e+14], t$95$2, If[LessEqual[z, 5.3e+39], N[(z * N[(-0.5 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7e+132], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.4500000000000001e90 or 7.00000000000000041e132 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 78.9%

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

      1. associate-*r/ 78.9%

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

   5. Simplified 78.9%

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

   if -2.4500000000000001e90 < z < 4e14 or 5.29999999999999979e39 < z < 7.00000000000000041e132

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 89.1%

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

      1. +-commutative 89.1%

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

   5. Simplified 89.1%

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

   if 4e14 < z < 5.29999999999999979e39

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. associate-*r/ 100.0%

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

      2. associate-*l/ 99.8%

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

      3. associate-*r/ 99.8%

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

      4. associate-*l/ 99.8%

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

      5. distribute-lft-in 99.8%

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

      6. +-commutative 99.8%

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

   5. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{0.5}{t} \cdot \left(\left(y - z\right) + x\right)} \]
   6. Step-by-step derivation

      1. associate-*l/ 100.0%

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

      2. clear-num 99.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{t}{0.5 \cdot \left(\left(y - z\right) + x\right)}}} \]

      3. +-commutative 99.7%

         \[\leadsto \frac{1}{\frac{t}{0.5 \cdot \color{blue}{\left(x + \left(y - z\right)\right)}}} \]

   7. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{t}{0.5 \cdot \left(x + \left(y - z\right)\right)}}} \]

   8. Taylor expanded in z around inf 76.0%

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

      1. associate-*r/ 76.0%

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

      2. *-commutative 76.0%

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

      3. associate-/l* 76.0%

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

   10. Simplified 76.0%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.45 \cdot 10^{+90}:\\ \;\;\;\;\frac{z \cdot -0.5}{t}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+14}:\\ \;\;\;\;\frac{x + y}{t \cdot 2}\\ \mathbf{elif}\;z \leq 5.3 \cdot 10^{+39}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+132}:\\ \;\;\;\;\frac{x + y}{t \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot -0.5}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 47.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+74}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{-222}:\\ \;\;\;\;\frac{z \cdot -0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -6.8e+74)
   (* 0.5 (/ x t))
   (if (<= x 1.06e-222) (/ (* z -0.5) t) (* 0.5 (/ y t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -6.8e+74) {
		tmp = 0.5 * (x / t);
	} else if (x <= 1.06e-222) {
		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 <= (-6.8d+74)) then
        tmp = 0.5d0 * (x / t)
    else if (x <= 1.06d-222) 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 <= -6.8e+74) {
		tmp = 0.5 * (x / t);
	} else if (x <= 1.06e-222) {
		tmp = (z * -0.5) / t;
	} else {
		tmp = 0.5 * (y / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -6.8e+74:
		tmp = 0.5 * (x / t)
	elif x <= 1.06e-222:
		tmp = (z * -0.5) / t
	else:
		tmp = 0.5 * (y / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -6.8e+74)
		tmp = Float64(0.5 * Float64(x / t));
	elseif (x <= 1.06e-222)
		tmp = Float64(Float64(z * -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 <= -6.8e+74)
		tmp = 0.5 * (x / t);
	elseif (x <= 1.06e-222)
		tmp = (z * -0.5) / t;
	else
		tmp = 0.5 * (y / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -6.8e+74], N[(0.5 * N[(x / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.06e-222], N[(N[(z * -0.5), $MachinePrecision] / t), $MachinePrecision], N[(0.5 * N[(y / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.7999999999999998e74

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 72.6%

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

   if -6.7999999999999998e74 < x < 1.06000000000000005e-222

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 56.1%

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

      1. associate-*r/ 56.1%

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

   5. Simplified 56.1%

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

   if 1.06000000000000005e-222 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 33.9%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+74}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{-222}:\\ \;\;\;\;\frac{z \cdot -0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 47.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+74}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-223}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.35e+74)
   (* 0.5 (/ x t))
   (if (<= x 2.05e-223) (* z (/ -0.5 t)) (* 0.5 (/ y t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.35e+74) {
		tmp = 0.5 * (x / t);
	} else if (x <= 2.05e-223) {
		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 <= (-1.35d+74)) then
        tmp = 0.5d0 * (x / t)
    else if (x <= 2.05d-223) 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 <= -1.35e+74) {
		tmp = 0.5 * (x / t);
	} else if (x <= 2.05e-223) {
		tmp = z * (-0.5 / t);
	} else {
		tmp = 0.5 * (y / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1.35e+74:
		tmp = 0.5 * (x / t)
	elif x <= 2.05e-223:
		tmp = z * (-0.5 / t)
	else:
		tmp = 0.5 * (y / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.35e+74)
		tmp = Float64(0.5 * Float64(x / t));
	elseif (x <= 2.05e-223)
		tmp = Float64(z * 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 <= -1.35e+74)
		tmp = 0.5 * (x / t);
	elseif (x <= 2.05e-223)
		tmp = z * (-0.5 / t);
	else
		tmp = 0.5 * (y / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.35e+74], N[(0.5 * N[(x / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.05e-223], N[(z * N[(-0.5 / t), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3499999999999999e74

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 72.6%

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

   if -1.3499999999999999e74 < x < 2.05000000000000007e-223

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. associate-*r/ 100.0%

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

      2. associate-*l/ 99.7%

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

      3. associate-*r/ 99.7%

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

      4. associate-*l/ 99.7%

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

      5. distribute-lft-in 99.8%

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

      6. +-commutative 99.8%

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

   5. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{0.5}{t} \cdot \left(\left(y - z\right) + x\right)} \]
   6. Step-by-step derivation

      1. associate-*l/ 100.0%

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

      2. clear-num 99.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{t}{0.5 \cdot \left(\left(y - z\right) + x\right)}}} \]

      3. +-commutative 99.0%

         \[\leadsto \frac{1}{\frac{t}{0.5 \cdot \color{blue}{\left(x + \left(y - z\right)\right)}}} \]

   7. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{t}{0.5 \cdot \left(x + \left(y - z\right)\right)}}} \]

   8. Taylor expanded in z around inf 56.1%

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

      1. associate-*r/ 56.1%

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

      2. *-commutative 56.1%

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

      3. associate-/l* 56.0%

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

   10. Simplified 56.0%

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

   if 2.05000000000000007e-223 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 33.9%

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

Alternative 5: 76.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.45 \cdot 10^{-59}:\\ \;\;\;\;\frac{x - z}{t \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{t \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -3.45e-59) (/ (- 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 <= -3.45e-59) {
		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 <= (-3.45d-59)) 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 <= -3.45e-59) {
		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 <= -3.45e-59:
		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 (x <= -3.45e-59)
		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 <= -3.45e-59)
		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[x, -3.45e-59], 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 x < -3.44999999999999991e-59

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 84.1%

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

   if -3.44999999999999991e-59 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 79.3%

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

Alternative 6: 78.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{+62}:\\ \;\;\;\;\frac{x - z}{t \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{t \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 3.1e+62) (/ (- x z) (* t 2.0)) (/ (+ x y) (* t 2.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3.1e+62) {
		tmp = (x - z) / (t * 2.0);
	} else {
		tmp = (x + y) / (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 (y <= 3.1d+62) then
        tmp = (x - z) / (t * 2.0d0)
    else
        tmp = (x + y) / (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 (y <= 3.1e+62) {
		tmp = (x - z) / (t * 2.0);
	} else {
		tmp = (x + y) / (t * 2.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 3.1e+62:
		tmp = (x - z) / (t * 2.0)
	else:
		tmp = (x + y) / (t * 2.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 3.1e+62)
		tmp = Float64(Float64(x - z) / Float64(t * 2.0));
	else
		tmp = Float64(Float64(x + y) / Float64(t * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 3.1e+62)
		tmp = (x - z) / (t * 2.0);
	else
		tmp = (x + y) / (t * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 3.1e+62], N[(N[(x - z), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(x + y), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.10000000000000014e62

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 79.9%

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

   if 3.10000000000000014e62 < 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 88.1%

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

      1. +-commutative 88.1%

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

   5. Simplified 88.1%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{+62}:\\ \;\;\;\;\frac{x - z}{t \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{t \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 45.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-63}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -4.8e-63) (* 0.5 (/ x t)) (* 0.5 (/ y t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -4.8e-63) {
		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 <= (-4.8d-63)) 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 <= -4.8e-63) {
		tmp = 0.5 * (x / t);
	} else {
		tmp = 0.5 * (y / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -4.8e-63:
		tmp = 0.5 * (x / t)
	else:
		tmp = 0.5 * (y / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -4.8e-63)
		tmp = Float64(0.5 * Float64(x / 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 <= -4.8e-63)
		tmp = 0.5 * (x / t);
	else
		tmp = 0.5 * (y / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -4.8e-63], N[(0.5 * N[(x / t), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.8000000000000001e-63

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 51.0%

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

   if -4.8000000000000001e-63 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 41.9%

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

Alternative 8: 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 100.0%

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

3. Taylor expanded in x around 0 96.1%

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

   1. associate-*r/ 96.1%

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

   2. associate-*l/ 95.9%

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

   3. associate-*r/ 95.9%

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

   4. associate-*l/ 95.8%

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

   5. distribute-lft-in 99.7%

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

   6. +-commutative 99.7%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 9: 37.1% accurate, 1.8× speedup

Math

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

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(0.5 * Float64(x / t))
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(0.5 * N[(x / 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 34.6%

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

Reproduce

herbie shell --seed 2024111 
(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)))