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

Percentage Accurate: 99.9% → 99.9%

Time: 5.8s

Alternatives: 10

Speedup: 1.0×

• 26.8% 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. Final simplification 100.0%

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

Alternative 2: 45.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.5 \cdot \frac{x}{t}\\ \mathbf{if}\;y \leq -2.5 \cdot 10^{-223}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-202}:\\ \;\;\;\;\frac{z \cdot -0.5}{t}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-38}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot 0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* 0.5 (/ x t))))
   (if (<= y -2.5e-223)
     t_1
     (if (<= y 1.3e-202)
       (/ (* z -0.5) t)
       (if (<= y 8.5e-38) t_1 (/ (* y 0.5) t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 0.5 * (x / t);
	double tmp;
	if (y <= -2.5e-223) {
		tmp = t_1;
	} else if (y <= 1.3e-202) {
		tmp = (z * -0.5) / t;
	} else if (y <= 8.5e-38) {
		tmp = t_1;
	} else {
		tmp = (y * 0.5) / 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) :: tmp
    t_1 = 0.5d0 * (x / t)
    if (y <= (-2.5d-223)) then
        tmp = t_1
    else if (y <= 1.3d-202) then
        tmp = (z * (-0.5d0)) / t
    else if (y <= 8.5d-38) then
        tmp = t_1
    else
        tmp = (y * 0.5d0) / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = 0.5 * (x / t);
	double tmp;
	if (y <= -2.5e-223) {
		tmp = t_1;
	} else if (y <= 1.3e-202) {
		tmp = (z * -0.5) / t;
	} else if (y <= 8.5e-38) {
		tmp = t_1;
	} else {
		tmp = (y * 0.5) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = 0.5 * (x / t)
	tmp = 0
	if y <= -2.5e-223:
		tmp = t_1
	elif y <= 1.3e-202:
		tmp = (z * -0.5) / t
	elif y <= 8.5e-38:
		tmp = t_1
	else:
		tmp = (y * 0.5) / t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(0.5 * Float64(x / t))
	tmp = 0.0
	if (y <= -2.5e-223)
		tmp = t_1;
	elseif (y <= 1.3e-202)
		tmp = Float64(Float64(z * -0.5) / t);
	elseif (y <= 8.5e-38)
		tmp = t_1;
	else
		tmp = Float64(Float64(y * 0.5) / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = 0.5 * (x / t);
	tmp = 0.0;
	if (y <= -2.5e-223)
		tmp = t_1;
	elseif (y <= 1.3e-202)
		tmp = (z * -0.5) / t;
	elseif (y <= 8.5e-38)
		tmp = t_1;
	else
		tmp = (y * 0.5) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(0.5 * N[(x / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.5e-223], t$95$1, If[LessEqual[y, 1.3e-202], N[(N[(z * -0.5), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[y, 8.5e-38], t$95$1, N[(N[(y * 0.5), $MachinePrecision] / t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.50000000000000012e-223 or 1.30000000000000005e-202 < y < 8.50000000000000046e-38

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 38.2%

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

   if -2.50000000000000012e-223 < y < 1.30000000000000005e-202

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 58.1%

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

      1. *-commutative 58.1%

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

      2. associate-*l/ 58.1%

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

   5. Simplified 58.1%

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

   if 8.50000000000000046e-38 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 72.9%

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

      1. associate-*r/ 72.9%

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

   5. Simplified 72.9%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-223}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-202}:\\ \;\;\;\;\frac{z \cdot -0.5}{t}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-38}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot 0.5}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+129} \lor \neg \left(z \leq 4.6 \cdot 10^{+132}\right):\\ \;\;\;\;\frac{z \cdot -0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x + y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -3e+129) (not (<= z 4.6e+132)))
   (/ (* z -0.5) t)
   (* 0.5 (/ (+ x y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -3e+129) || !(z <= 4.6e+132)) {
		tmp = (z * -0.5) / t;
	} else {
		tmp = 0.5 * ((x + 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 ((z <= (-3d+129)) .or. (.not. (z <= 4.6d+132))) then
        tmp = (z * (-0.5d0)) / t
    else
        tmp = 0.5d0 * ((x + y) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -3e+129) || !(z <= 4.6e+132)) {
		tmp = (z * -0.5) / t;
	} else {
		tmp = 0.5 * ((x + y) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -3e+129) or not (z <= 4.6e+132):
		tmp = (z * -0.5) / t
	else:
		tmp = 0.5 * ((x + y) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -3e+129) || !(z <= 4.6e+132))
		tmp = Float64(Float64(z * -0.5) / t);
	else
		tmp = Float64(0.5 * Float64(Float64(x + y) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -3e+129) || ~((z <= 4.6e+132)))
		tmp = (z * -0.5) / t;
	else
		tmp = 0.5 * ((x + y) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -3e+129], N[Not[LessEqual[z, 4.6e+132]], $MachinePrecision]], N[(N[(z * -0.5), $MachinePrecision] / t), $MachinePrecision], N[(0.5 * N[(N[(x + y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.0000000000000003e129 or 4.6000000000000003e132 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 85.2%

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

      1. *-commutative 85.2%

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

      2. associate-*l/ 85.2%

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

   5. Simplified 85.2%

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

   if -3.0000000000000003e129 < z < 4.6000000000000003e132

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 87.6%

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

      1. +-commutative 87.6%

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

   5. Simplified 87.6%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+129} \lor \neg \left(z \leq 4.6 \cdot 10^{+132}\right):\\ \;\;\;\;\frac{z \cdot -0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x + y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-38}:\\ \;\;\;\;\frac{0.5 \cdot \left(x - z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t}{0.5 \cdot \left(y - z\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 9e-38) (/ (* 0.5 (- x z)) t) (/ 1.0 (/ t (* 0.5 (- y z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 9e-38) {
		tmp = (0.5 * (x - z)) / t;
	} else {
		tmp = 1.0 / (t / (0.5 * (y - z)));
	}
	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 <= 9d-38) then
        tmp = (0.5d0 * (x - z)) / t
    else
        tmp = 1.0d0 / (t / (0.5d0 * (y - z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 9e-38) {
		tmp = (0.5 * (x - z)) / t;
	} else {
		tmp = 1.0 / (t / (0.5 * (y - z)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 9e-38:
		tmp = (0.5 * (x - z)) / t
	else:
		tmp = 1.0 / (t / (0.5 * (y - z)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 9e-38)
		tmp = Float64(Float64(0.5 * Float64(x - z)) / t);
	else
		tmp = Float64(1.0 / Float64(t / Float64(0.5 * Float64(y - z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 9e-38)
		tmp = (0.5 * (x - z)) / t;
	else
		tmp = 1.0 / (t / (0.5 * (y - z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 9e-38], N[(N[(0.5 * N[(x - z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(1.0 / N[(t / N[(0.5 * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.00000000000000018e-38

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 97.2%

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

   4. Taylor expanded in y around 0 76.7%

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

      1. associate-*r/ 76.7%

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

   6. Simplified 76.7%

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

   if 9.00000000000000018e-38 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 84.0%

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

      1. *-commutative 84.0%

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

      2. associate-*l/ 84.0%

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

      3. associate-*r/ 83.7%

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

   5. Simplified 83.7%

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

      1. associate-*r/ 84.0%

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

      2. clear-num 83.8%

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

   7. Applied egg-rr 83.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{t}{\left(y - z\right) \cdot 0.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-38}:\\ \;\;\;\;\frac{0.5 \cdot \left(x - z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t}{0.5 \cdot \left(y - z\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{+16}:\\ \;\;\;\;\left(x - z\right) \cdot \frac{0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x + y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 7e+16) (* (- x z) (/ 0.5 t)) (* 0.5 (/ (+ x y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 7e+16) {
		tmp = (x - z) * (0.5 / t);
	} else {
		tmp = 0.5 * ((x + 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 <= 7d+16) then
        tmp = (x - z) * (0.5d0 / t)
    else
        tmp = 0.5d0 * ((x + 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 <= 7e+16) {
		tmp = (x - z) * (0.5 / t);
	} else {
		tmp = 0.5 * ((x + y) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 7e+16:
		tmp = (x - z) * (0.5 / t)
	else:
		tmp = 0.5 * ((x + y) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 7e+16)
		tmp = Float64(Float64(x - z) * Float64(0.5 / t));
	else
		tmp = Float64(0.5 * Float64(Float64(x + y) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 7e+16)
		tmp = (x - z) * (0.5 / t);
	else
		tmp = 0.5 * ((x + y) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 7e+16], N[(N[(x - z), $MachinePrecision] * N[(0.5 / t), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(x + y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7e16

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 76.7%

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

      1. *-commutative 76.7%

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

      2. associate-*l/ 76.7%

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

      3. associate-/l* 76.5%

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

   5. Simplified 76.5%

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

   if 7e16 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0 94.3%

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

      1. +-commutative 94.3%

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

   5. Simplified 94.3%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{+16}:\\ \;\;\;\;\left(x - z\right) \cdot \frac{0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x + y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-38}:\\ \;\;\;\;\left(x - z\right) \cdot \frac{0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 9e-38) (* (- x z) (/ 0.5 t)) (* (- y z) (/ 0.5 t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 9e-38) {
		tmp = (x - z) * (0.5 / t);
	} else {
		tmp = (y - z) * (0.5 / 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 <= 9d-38) then
        tmp = (x - z) * (0.5d0 / t)
    else
        tmp = (y - z) * (0.5d0 / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 9e-38) {
		tmp = (x - z) * (0.5 / t);
	} else {
		tmp = (y - z) * (0.5 / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 9e-38:
		tmp = (x - z) * (0.5 / t)
	else:
		tmp = (y - z) * (0.5 / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 9e-38)
		tmp = Float64(Float64(x - z) * Float64(0.5 / t));
	else
		tmp = Float64(Float64(y - z) * Float64(0.5 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 9e-38)
		tmp = (x - z) * (0.5 / t);
	else
		tmp = (y - z) * (0.5 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 9e-38], N[(N[(x - z), $MachinePrecision] * N[(0.5 / t), $MachinePrecision]), $MachinePrecision], N[(N[(y - z), $MachinePrecision] * N[(0.5 / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.00000000000000018e-38

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 76.7%

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

      1. *-commutative 76.7%

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

      2. associate-*l/ 76.7%

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

      3. associate-/l* 76.5%

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

   5. Simplified 76.5%

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

   if 9.00000000000000018e-38 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 84.0%

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

      1. *-commutative 84.0%

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

      2. associate-*l/ 84.0%

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

      3. associate-*r/ 83.7%

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

   5. Simplified 83.7%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-38}:\\ \;\;\;\;\left(x - z\right) \cdot \frac{0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{0.5}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{-38}:\\ \;\;\;\;\frac{0.5 \cdot \left(x - z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 4.6e-38) (/ (* 0.5 (- x z)) t) (* (- y z) (/ 0.5 t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 4.6e-38) {
		tmp = (0.5 * (x - z)) / t;
	} else {
		tmp = (y - z) * (0.5 / 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 <= 4.6d-38) then
        tmp = (0.5d0 * (x - z)) / t
    else
        tmp = (y - z) * (0.5d0 / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 4.6e-38) {
		tmp = (0.5 * (x - z)) / t;
	} else {
		tmp = (y - z) * (0.5 / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 4.6e-38:
		tmp = (0.5 * (x - z)) / t
	else:
		tmp = (y - z) * (0.5 / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 4.6e-38)
		tmp = Float64(Float64(0.5 * Float64(x - z)) / t);
	else
		tmp = Float64(Float64(y - z) * Float64(0.5 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 4.6e-38)
		tmp = (0.5 * (x - z)) / t;
	else
		tmp = (y - z) * (0.5 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 4.6e-38], N[(N[(0.5 * N[(x - z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(N[(y - z), $MachinePrecision] * N[(0.5 / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.60000000000000003e-38

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 97.2%

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

   4. Taylor expanded in y around 0 76.7%

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

      1. associate-*r/ 76.7%

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

   6. Simplified 76.7%

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

   if 4.60000000000000003e-38 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 84.0%

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

      1. *-commutative 84.0%

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

      2. associate-*l/ 84.0%

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

      3. associate-*r/ 83.7%

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

   5. Simplified 83.7%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{-38}:\\ \;\;\;\;\frac{0.5 \cdot \left(x - z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{0.5}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 45.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-38}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 8e-38) (* 0.5 (/ x t)) (* y (/ 0.5 t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 8e-38) {
		tmp = 0.5 * (x / t);
	} else {
		tmp = y * (0.5 / 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 <= 8d-38) then
        tmp = 0.5d0 * (x / t)
    else
        tmp = y * (0.5d0 / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 8e-38) {
		tmp = 0.5 * (x / t);
	} else {
		tmp = y * (0.5 / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 8e-38:
		tmp = 0.5 * (x / t)
	else:
		tmp = y * (0.5 / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 8e-38)
		tmp = Float64(0.5 * Float64(x / t));
	else
		tmp = Float64(y * Float64(0.5 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 8e-38)
		tmp = 0.5 * (x / t);
	else
		tmp = y * (0.5 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 8e-38], N[(0.5 * N[(x / t), $MachinePrecision]), $MachinePrecision], N[(y * N[(0.5 / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.9999999999999997e-38

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 39.3%

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

   if 7.9999999999999997e-38 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 84.0%

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

      1. *-commutative 84.0%

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

      2. associate-*l/ 84.0%

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

      3. associate-*r/ 83.7%

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

   5. Simplified 83.7%

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

   6. Taylor expanded in y around inf 72.9%

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

      1. associate-*r/ 72.9%

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

      2. associate-*l/ 72.6%

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

      3. *-commutative 72.6%

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

   8. Simplified 72.6%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-38}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{0.5}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 45.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-38}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot 0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 9e-38) (* 0.5 (/ x t)) (/ (* y 0.5) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 9e-38) {
		tmp = 0.5 * (x / t);
	} else {
		tmp = (y * 0.5) / 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 <= 9d-38) then
        tmp = 0.5d0 * (x / t)
    else
        tmp = (y * 0.5d0) / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 9e-38) {
		tmp = 0.5 * (x / t);
	} else {
		tmp = (y * 0.5) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 9e-38:
		tmp = 0.5 * (x / t)
	else:
		tmp = (y * 0.5) / t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 9e-38)
		tmp = Float64(0.5 * Float64(x / t));
	else
		tmp = Float64(Float64(y * 0.5) / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 9e-38)
		tmp = 0.5 * (x / t);
	else
		tmp = (y * 0.5) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 9e-38], N[(0.5 * N[(x / t), $MachinePrecision]), $MachinePrecision], N[(N[(y * 0.5), $MachinePrecision] / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.00000000000000018e-38

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 39.3%

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

   if 9.00000000000000018e-38 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 72.9%

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

      1. associate-*r/ 72.9%

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

   5. Simplified 72.9%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-38}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot 0.5}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 37.0% 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 33.4%

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

4. Final simplification 33.4%

   \[\leadsto 0.5 \cdot \frac{x}{t} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024067 
(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)))