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

Percentage Accurate: 99.9% → 99.9%

Time: 6.8s

Alternatives: 9

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: 80.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+45} \lor \neg \left(z \leq 8.2 \cdot 10^{+130}\right):\\ \;\;\;\;\frac{z \cdot -0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x + y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -7e+45) (not (<= z 8.2e+130)))
   (/ (* z -0.5) t)
   (* 0.5 (/ (+ x y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -7e+45) || !(z <= 8.2e+130)) {
		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 <= (-7d+45)) .or. (.not. (z <= 8.2d+130))) 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 <= -7e+45) || !(z <= 8.2e+130)) {
		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 <= -7e+45) or not (z <= 8.2e+130):
		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 <= -7e+45) || !(z <= 8.2e+130))
		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 <= -7e+45) || ~((z <= 8.2e+130)))
		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, -7e+45], N[Not[LessEqual[z, 8.2e+130]], $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 < -7.00000000000000046e45 or 8.19999999999999955e130 < 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 73.1%

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

      1. *-commutative 73.1%

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

      2. associate-*l/ 73.1%

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

   5. Simplified 73.1%

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

   if -7.00000000000000046e45 < z < 8.19999999999999955e130

   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%

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

      1. +-commutative 89.1%

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

   5. Simplified 89.1%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+45} \lor \neg \left(z \leq 8.2 \cdot 10^{+130}\right):\\ \;\;\;\;\frac{z \cdot -0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x + y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+118}:\\ \;\;\;\;0.5 \cdot \frac{x - z}{t}\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-44}:\\ \;\;\;\;0.5 \cdot \frac{x + y}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1e+118)
   (* 0.5 (/ (- x z) t))
   (if (<= x -7.5e-44) (* 0.5 (/ (+ x y) t)) (* (- y z) (/ 0.5 t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1e+118) {
		tmp = 0.5 * ((x - z) / t);
	} else if (x <= -7.5e-44) {
		tmp = 0.5 * ((x + y) / 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 (x <= (-1d+118)) then
        tmp = 0.5d0 * ((x - z) / t)
    else if (x <= (-7.5d-44)) then
        tmp = 0.5d0 * ((x + y) / 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 (x <= -1e+118) {
		tmp = 0.5 * ((x - z) / t);
	} else if (x <= -7.5e-44) {
		tmp = 0.5 * ((x + y) / t);
	} else {
		tmp = (y - z) * (0.5 / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1e+118:
		tmp = 0.5 * ((x - z) / t)
	elif x <= -7.5e-44:
		tmp = 0.5 * ((x + y) / t)
	else:
		tmp = (y - z) * (0.5 / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1e+118)
		tmp = Float64(0.5 * Float64(Float64(x - z) / t));
	elseif (x <= -7.5e-44)
		tmp = Float64(0.5 * Float64(Float64(x + y) / 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 (x <= -1e+118)
		tmp = 0.5 * ((x - z) / t);
	elseif (x <= -7.5e-44)
		tmp = 0.5 * ((x + y) / t);
	else
		tmp = (y - z) * (0.5 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1e+118], N[(0.5 * N[(N[(x - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -7.5e-44], N[(0.5 * N[(N[(x + y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(y - z), $MachinePrecision] * N[(0.5 / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.99999999999999967e117

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 95.1%

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

   if -9.99999999999999967e117 < x < -7.50000000000000008e-44

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0 93.7%

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

      1. +-commutative 93.7%

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

   5. Simplified 93.7%

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

   if -7.50000000000000008e-44 < 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 75.3%

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

      1. *-commutative 75.3%

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

      2. associate-*l/ 75.3%

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

      3. associate-*r/ 75.1%

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

   5. Simplified 75.1%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+118}:\\ \;\;\;\;0.5 \cdot \frac{x - z}{t}\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-44}:\\ \;\;\;\;0.5 \cdot \frac{x + y}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{0.5}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+118}:\\ \;\;\;\;0.5 \cdot \frac{x - z}{t}\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-44}:\\ \;\;\;\;0.5 \cdot \frac{x + y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \left(y - z\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.05e+118)
   (* 0.5 (/ (- x z) t))
   (if (<= x -4e-44) (* 0.5 (/ (+ x y) t)) (/ (* 0.5 (- y z)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.05e+118) {
		tmp = 0.5 * ((x - z) / t);
	} else if (x <= -4e-44) {
		tmp = 0.5 * ((x + y) / 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 <= (-1.05d+118)) then
        tmp = 0.5d0 * ((x - z) / t)
    else if (x <= (-4d-44)) then
        tmp = 0.5d0 * ((x + y) / 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 <= -1.05e+118) {
		tmp = 0.5 * ((x - z) / t);
	} else if (x <= -4e-44) {
		tmp = 0.5 * ((x + y) / t);
	} else {
		tmp = (0.5 * (y - z)) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1.05e+118:
		tmp = 0.5 * ((x - z) / t)
	elif x <= -4e-44:
		tmp = 0.5 * ((x + y) / t)
	else:
		tmp = (0.5 * (y - z)) / t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.05e+118)
		tmp = Float64(0.5 * Float64(Float64(x - z) / t));
	elseif (x <= -4e-44)
		tmp = Float64(0.5 * Float64(Float64(x + y) / t));
	else
		tmp = Float64(Float64(0.5 * Float64(y - z)) / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -1.05e+118)
		tmp = 0.5 * ((x - z) / t);
	elseif (x <= -4e-44)
		tmp = 0.5 * ((x + y) / t);
	else
		tmp = (0.5 * (y - z)) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.05e+118], N[(0.5 * N[(N[(x - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4e-44], N[(0.5 * N[(N[(x + y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(y - z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05e118

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 95.1%

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

   if -1.05e118 < x < -3.99999999999999981e-44

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0 93.7%

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

      1. +-commutative 93.7%

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

   5. Simplified 93.7%

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

   if -3.99999999999999981e-44 < 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 75.3%

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

      1. associate-*r/ 75.3%

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

   5. Simplified 75.3%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+118}:\\ \;\;\;\;0.5 \cdot \frac{x - z}{t}\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-44}:\\ \;\;\;\;0.5 \cdot \frac{x + y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \left(y - z\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 47.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.25 \cdot 10^{-295}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{elif}\;y \leq 2900000000:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot 0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -3.25e-295)
   (* 0.5 (/ x t))
   (if (<= y 2900000000.0) (* z (/ -0.5 t)) (/ (* y 0.5) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -3.25e-295) {
		tmp = 0.5 * (x / t);
	} else if (y <= 2900000000.0) {
		tmp = z * (-0.5 / 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 <= (-3.25d-295)) then
        tmp = 0.5d0 * (x / t)
    else if (y <= 2900000000.0d0) then
        tmp = z * ((-0.5d0) / 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 <= -3.25e-295) {
		tmp = 0.5 * (x / t);
	} else if (y <= 2900000000.0) {
		tmp = z * (-0.5 / t);
	} else {
		tmp = (y * 0.5) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -3.25e-295:
		tmp = 0.5 * (x / t)
	elif y <= 2900000000.0:
		tmp = z * (-0.5 / t)
	else:
		tmp = (y * 0.5) / t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -3.25e-295)
		tmp = Float64(0.5 * Float64(x / t));
	elseif (y <= 2900000000.0)
		tmp = Float64(z * Float64(-0.5 / 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 <= -3.25e-295)
		tmp = 0.5 * (x / t);
	elseif (y <= 2900000000.0)
		tmp = z * (-0.5 / t);
	else
		tmp = (y * 0.5) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -3.25e-295], N[(0.5 * N[(x / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2900000000.0], N[(z * N[(-0.5 / t), $MachinePrecision]), $MachinePrecision], N[(N[(y * 0.5), $MachinePrecision] / t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.2499999999999999e-295

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 42.0%

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

   if -3.2499999999999999e-295 < y < 2.9e9

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 47.3%

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

      1. *-commutative 47.3%

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

      2. associate-*l/ 47.3%

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

      3. associate-*r/ 47.2%

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

   5. Simplified 47.2%

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

   if 2.9e9 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 69.9%

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

      1. associate-*r/ 69.9%

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

   5. Simplified 69.9%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.25 \cdot 10^{-295}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{elif}\;y \leq 2900000000:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot 0.5}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 47.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.1 \cdot 10^{-294}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{elif}\;y \leq 180000000:\\ \;\;\;\;\frac{z \cdot -0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot 0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -5.1e-294)
   (* 0.5 (/ x t))
   (if (<= y 180000000.0) (/ (* z -0.5) t) (/ (* y 0.5) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -5.1e-294) {
		tmp = 0.5 * (x / t);
	} else if (y <= 180000000.0) {
		tmp = (z * -0.5) / 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 <= (-5.1d-294)) then
        tmp = 0.5d0 * (x / t)
    else if (y <= 180000000.0d0) then
        tmp = (z * (-0.5d0)) / 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 <= -5.1e-294) {
		tmp = 0.5 * (x / t);
	} else if (y <= 180000000.0) {
		tmp = (z * -0.5) / t;
	} else {
		tmp = (y * 0.5) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -5.1e-294:
		tmp = 0.5 * (x / t)
	elif y <= 180000000.0:
		tmp = (z * -0.5) / t
	else:
		tmp = (y * 0.5) / t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -5.1e-294)
		tmp = Float64(0.5 * Float64(x / t));
	elseif (y <= 180000000.0)
		tmp = Float64(Float64(z * -0.5) / 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 <= -5.1e-294)
		tmp = 0.5 * (x / t);
	elseif (y <= 180000000.0)
		tmp = (z * -0.5) / t;
	else
		tmp = (y * 0.5) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -5.1e-294], N[(0.5 * N[(x / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 180000000.0], N[(N[(z * -0.5), $MachinePrecision] / t), $MachinePrecision], N[(N[(y * 0.5), $MachinePrecision] / t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.10000000000000007e-294

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 42.0%

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

   if -5.10000000000000007e-294 < y < 1.8e8

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 47.3%

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

      1. *-commutative 47.3%

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

      2. associate-*l/ 47.3%

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

   5. Simplified 47.3%

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

   if 1.8e8 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 69.9%

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

      1. associate-*r/ 69.9%

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

   5. Simplified 69.9%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.1 \cdot 10^{-294}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{elif}\;y \leq 180000000:\\ \;\;\;\;\frac{z \cdot -0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot 0.5}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2200:\\ \;\;\;\;0.5 \cdot \frac{x - z}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x + y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 2200.0) (* 0.5 (/ (- x z) t)) (* 0.5 (/ (+ x y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2200.0) {
		tmp = 0.5 * ((x - z) / 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 <= 2200.0d0) then
        tmp = 0.5d0 * ((x - z) / 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 <= 2200.0) {
		tmp = 0.5 * ((x - z) / t);
	} else {
		tmp = 0.5 * ((x + y) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 2200.0:
		tmp = 0.5 * ((x - z) / t)
	else:
		tmp = 0.5 * ((x + y) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2200.0)
		tmp = Float64(0.5 * Float64(Float64(x - z) / 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 <= 2200.0)
		tmp = 0.5 * ((x - z) / t);
	else
		tmp = 0.5 * ((x + y) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 2200.0], N[(0.5 * N[(N[(x - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(x + y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2200

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 81.1%

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

   if 2200 < 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 85.8%

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

      1. +-commutative 85.8%

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

   5. Simplified 85.8%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2200:\\ \;\;\;\;0.5 \cdot \frac{x - z}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x + y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 45.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-44}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -4e-44) (* 0.5 (/ x t)) (* z (/ -0.5 t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -4e-44) {
		tmp = 0.5 * (x / t);
	} else {
		tmp = 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 (x <= (-4d-44)) then
        tmp = 0.5d0 * (x / t)
    else
        tmp = 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 (x <= -4e-44) {
		tmp = 0.5 * (x / t);
	} else {
		tmp = z * (-0.5 / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -4e-44:
		tmp = 0.5 * (x / t)
	else:
		tmp = z * (-0.5 / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -4e-44)
		tmp = Float64(0.5 * Float64(x / t));
	else
		tmp = Float64(z * Float64(-0.5 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -4e-44)
		tmp = 0.5 * (x / t);
	else
		tmp = z * (-0.5 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -4e-44], N[(0.5 * N[(x / t), $MachinePrecision]), $MachinePrecision], N[(z * N[(-0.5 / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.99999999999999981e-44

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 62.6%

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

   if -3.99999999999999981e-44 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 43.3%

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

      1. *-commutative 43.3%

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

      2. associate-*l/ 43.3%

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

      3. associate-*r/ 43.1%

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

   5. Simplified 43.1%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-44}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 36.9% 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 40.7%

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

4. Final simplification 40.7%

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

Reproduce

herbie shell --seed 2024071 
(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)))