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

Percentage Accurate: 99.9% → 99.9%

Time: 5.1s

Alternatives: 8

Speedup: 1.0×

• 25.9% 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: 44.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{t \cdot 2}\\ \mathbf{if}\;x \leq -1.05 \cdot 10^{+59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{+26}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-75}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -6.3 \cdot 10^{-173}:\\ \;\;\;\;\frac{z}{t} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* t 2.0))))
   (if (<= x -1.05e+59)
     t_1
     (if (<= x -2.6e+26)
       (* z (/ -0.5 t))
       (if (<= x -6e-75)
         t_1
         (if (<= x -6.3e-173) (* (/ z t) -0.5) (/ y (* t 2.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x / (t * 2.0);
	double tmp;
	if (x <= -1.05e+59) {
		tmp = t_1;
	} else if (x <= -2.6e+26) {
		tmp = z * (-0.5 / t);
	} else if (x <= -6e-75) {
		tmp = t_1;
	} else if (x <= -6.3e-173) {
		tmp = (z / t) * -0.5;
	} else {
		tmp = 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) :: t_1
    real(8) :: tmp
    t_1 = x / (t * 2.0d0)
    if (x <= (-1.05d+59)) then
        tmp = t_1
    else if (x <= (-2.6d+26)) then
        tmp = z * ((-0.5d0) / t)
    else if (x <= (-6d-75)) then
        tmp = t_1
    else if (x <= (-6.3d-173)) then
        tmp = (z / t) * (-0.5d0)
    else
        tmp = y / (t * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x / (t * 2.0);
	double tmp;
	if (x <= -1.05e+59) {
		tmp = t_1;
	} else if (x <= -2.6e+26) {
		tmp = z * (-0.5 / t);
	} else if (x <= -6e-75) {
		tmp = t_1;
	} else if (x <= -6.3e-173) {
		tmp = (z / t) * -0.5;
	} else {
		tmp = y / (t * 2.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x / (t * 2.0)
	tmp = 0
	if x <= -1.05e+59:
		tmp = t_1
	elif x <= -2.6e+26:
		tmp = z * (-0.5 / t)
	elif x <= -6e-75:
		tmp = t_1
	elif x <= -6.3e-173:
		tmp = (z / t) * -0.5
	else:
		tmp = y / (t * 2.0)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x / Float64(t * 2.0))
	tmp = 0.0
	if (x <= -1.05e+59)
		tmp = t_1;
	elseif (x <= -2.6e+26)
		tmp = Float64(z * Float64(-0.5 / t));
	elseif (x <= -6e-75)
		tmp = t_1;
	elseif (x <= -6.3e-173)
		tmp = Float64(Float64(z / t) * -0.5);
	else
		tmp = Float64(y / Float64(t * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x / (t * 2.0);
	tmp = 0.0;
	if (x <= -1.05e+59)
		tmp = t_1;
	elseif (x <= -2.6e+26)
		tmp = z * (-0.5 / t);
	elseif (x <= -6e-75)
		tmp = t_1;
	elseif (x <= -6.3e-173)
		tmp = (z / t) * -0.5;
	else
		tmp = y / (t * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(t * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.05e+59], t$95$1, If[LessEqual[x, -2.6e+26], N[(z * N[(-0.5 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -6e-75], t$95$1, If[LessEqual[x, -6.3e-173], N[(N[(z / t), $MachinePrecision] * -0.5), $MachinePrecision], N[(y / N[(t * 2.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.04999999999999992e59 or -2.60000000000000002e26 < x < -5.9999999999999997e-75

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 65.5%

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

   if -1.04999999999999992e59 < x < -2.60000000000000002e26

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 61.4%

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

      1. associate-*r/ 61.4%

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

      2. metadata-eval 61.4%

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

      3. distribute-lft-neg-in 61.4%

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

      4. *-commutative 61.4%

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

      5. distribute-neg-frac 61.4%

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

      6. associate-*r/ 61.4%

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

      7. distribute-rgt-neg-in 61.4%

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

      8. distribute-neg-frac 61.4%

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

      9. metadata-eval 61.4%

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

   5. Simplified 61.4%

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

   if -5.9999999999999997e-75 < x < -6.29999999999999968e-173

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 50.2%

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

      1. *-commutative 50.2%

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

   5. Simplified 50.2%

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

   if -6.29999999999999968e-173 < 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 46.3%

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

Alternative 3: 81.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+109}:\\ \;\;\;\;\frac{z}{t} \cdot -0.5\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+184}:\\ \;\;\;\;\frac{x + y}{t \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot -0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -3.8e+109)
   (* (/ z t) -0.5)
   (if (<= z 7e+184) (/ (+ x y) (* t 2.0)) (/ (* z -0.5) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.8e+109) {
		tmp = (z / t) * -0.5;
	} else if (z <= 7e+184) {
		tmp = (x + y) / (t * 2.0);
	} 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 (z <= (-3.8d+109)) then
        tmp = (z / t) * (-0.5d0)
    else if (z <= 7d+184) then
        tmp = (x + y) / (t * 2.0d0)
    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 (z <= -3.8e+109) {
		tmp = (z / t) * -0.5;
	} else if (z <= 7e+184) {
		tmp = (x + y) / (t * 2.0);
	} else {
		tmp = (z * -0.5) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -3.8e+109:
		tmp = (z / t) * -0.5
	elif z <= 7e+184:
		tmp = (x + y) / (t * 2.0)
	else:
		tmp = (z * -0.5) / t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -3.8e+109)
		tmp = Float64(Float64(z / t) * -0.5);
	elseif (z <= 7e+184)
		tmp = Float64(Float64(x + y) / Float64(t * 2.0));
	else
		tmp = Float64(Float64(z * -0.5) / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -3.8e+109)
		tmp = (z / t) * -0.5;
	elseif (z <= 7e+184)
		tmp = (x + y) / (t * 2.0);
	else
		tmp = (z * -0.5) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -3.8e+109], N[(N[(z / t), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[z, 7e+184], N[(N[(x + y), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(z * -0.5), $MachinePrecision] / t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.80000000000000039e109

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 76.4%

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

      1. *-commutative 76.4%

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

   5. Simplified 76.4%

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

   if -3.80000000000000039e109 < z < 6.99999999999999956e184

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 84.9%

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

      1. +-commutative 84.9%

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

   5. Simplified 84.9%

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

   if 6.99999999999999956e184 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 89.1%

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

      1. distribute-lft-out 89.1%

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

   5. Simplified 89.1%

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

   6. Taylor expanded in z around inf 77.1%

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

      1. associate-*r/ 81.8%

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

      2. *-commutative 81.8%

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

   8. Simplified 81.8%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+109}:\\ \;\;\;\;\frac{z}{t} \cdot -0.5\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+184}:\\ \;\;\;\;\frac{x + y}{t \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot -0.5}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-11} \lor \neg \left(z \leq 1.6 \cdot 10^{+31}\right):\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -4.8e-11) (not (<= z 1.6e+31)))
   (* z (/ -0.5 t))
   (* x (/ 0.5 t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -4.8e-11) || !(z <= 1.6e+31)) {
		tmp = z * (-0.5 / t);
	} else {
		tmp = x * (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 ((z <= (-4.8d-11)) .or. (.not. (z <= 1.6d+31))) then
        tmp = z * ((-0.5d0) / t)
    else
        tmp = x * (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 ((z <= -4.8e-11) || !(z <= 1.6e+31)) {
		tmp = z * (-0.5 / t);
	} else {
		tmp = x * (0.5 / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -4.8e-11) or not (z <= 1.6e+31):
		tmp = z * (-0.5 / t)
	else:
		tmp = x * (0.5 / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -4.8e-11) || !(z <= 1.6e+31))
		tmp = Float64(z * Float64(-0.5 / t));
	else
		tmp = Float64(x * Float64(0.5 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -4.8e-11) || ~((z <= 1.6e+31)))
		tmp = z * (-0.5 / t);
	else
		tmp = x * (0.5 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -4.8e-11], N[Not[LessEqual[z, 1.6e+31]], $MachinePrecision]], N[(z * N[(-0.5 / t), $MachinePrecision]), $MachinePrecision], N[(x * N[(0.5 / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.8000000000000002e-11 or 1.6e31 < 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 63.3%

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

      1. associate-*r/ 64.1%

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

      2. metadata-eval 64.1%

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

      3. distribute-lft-neg-in 64.1%

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

      4. *-commutative 64.1%

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

      5. distribute-neg-frac 64.1%

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

      6. associate-*r/ 63.9%

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

      7. distribute-rgt-neg-in 63.9%

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

      8. distribute-neg-frac 63.9%

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

      9. metadata-eval 63.9%

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

   5. Simplified 63.9%

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

   if -4.8000000000000002e-11 < z < 1.6e31

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 97.4%

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

      1. distribute-lft-out 97.4%

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

   5. Simplified 97.4%

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

   6. Taylor expanded in x around inf 52.5%

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

      1. associate-*r/ 52.5%

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

      2. *-commutative 52.5%

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

      3. associate-*r/ 52.4%

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

   8. Simplified 52.4%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-11} \lor \neg \left(z \leq 1.6 \cdot 10^{+31}\right):\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{0.5}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{-11}:\\ \;\;\;\;\frac{z}{t} \cdot -0.5\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+35}:\\ \;\;\;\;\frac{x}{t \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -3.3e-11)
   (* (/ z t) -0.5)
   (if (<= z 1.2e+35) (/ x (* t 2.0)) (* z (/ -0.5 t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.3e-11) {
		tmp = (z / t) * -0.5;
	} else if (z <= 1.2e+35) {
		tmp = x / (t * 2.0);
	} 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 (z <= (-3.3d-11)) then
        tmp = (z / t) * (-0.5d0)
    else if (z <= 1.2d+35) then
        tmp = x / (t * 2.0d0)
    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 (z <= -3.3e-11) {
		tmp = (z / t) * -0.5;
	} else if (z <= 1.2e+35) {
		tmp = x / (t * 2.0);
	} else {
		tmp = z * (-0.5 / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -3.3e-11:
		tmp = (z / t) * -0.5
	elif z <= 1.2e+35:
		tmp = x / (t * 2.0)
	else:
		tmp = z * (-0.5 / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -3.3e-11)
		tmp = Float64(Float64(z / t) * -0.5);
	elseif (z <= 1.2e+35)
		tmp = Float64(x / Float64(t * 2.0));
	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 (z <= -3.3e-11)
		tmp = (z / t) * -0.5;
	elseif (z <= 1.2e+35)
		tmp = x / (t * 2.0);
	else
		tmp = z * (-0.5 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -3.3e-11], N[(N[(z / t), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[z, 1.2e+35], N[(x / N[(t * 2.0), $MachinePrecision]), $MachinePrecision], N[(z * N[(-0.5 / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.3000000000000002e-11

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 67.3%

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

      1. *-commutative 67.3%

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

   5. Simplified 67.3%

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

   if -3.3000000000000002e-11 < z < 1.20000000000000007e35

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 52.5%

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

   if 1.20000000000000007e35 < 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 59.1%

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

      1. associate-*r/ 60.6%

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

      2. metadata-eval 60.6%

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

      3. distribute-lft-neg-in 60.6%

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

      4. *-commutative 60.6%

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

      5. distribute-neg-frac 60.6%

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

      6. associate-*r/ 60.5%

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

      7. distribute-rgt-neg-in 60.5%

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

      8. distribute-neg-frac 60.5%

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

      9. metadata-eval 60.5%

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

   5. Simplified 60.5%

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

Alternative 6: 55.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{z}{t} \cdot -0.5\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+33}:\\ \;\;\;\;x \cdot \frac{0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -8.2e-11)
   (* (/ z t) -0.5)
   (if (<= z 7.5e+33) (* x (/ 0.5 t)) (* z (/ -0.5 t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -8.2e-11) {
		tmp = (z / t) * -0.5;
	} else if (z <= 7.5e+33) {
		tmp = x * (0.5 / 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 (z <= (-8.2d-11)) then
        tmp = (z / t) * (-0.5d0)
    else if (z <= 7.5d+33) then
        tmp = x * (0.5d0 / 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 (z <= -8.2e-11) {
		tmp = (z / t) * -0.5;
	} else if (z <= 7.5e+33) {
		tmp = x * (0.5 / t);
	} else {
		tmp = z * (-0.5 / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -8.2e-11:
		tmp = (z / t) * -0.5
	elif z <= 7.5e+33:
		tmp = x * (0.5 / t)
	else:
		tmp = z * (-0.5 / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -8.2e-11)
		tmp = Float64(Float64(z / t) * -0.5);
	elseif (z <= 7.5e+33)
		tmp = Float64(x * Float64(0.5 / 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 (z <= -8.2e-11)
		tmp = (z / t) * -0.5;
	elseif (z <= 7.5e+33)
		tmp = x * (0.5 / t);
	else
		tmp = z * (-0.5 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -8.2e-11], N[(N[(z / t), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[z, 7.5e+33], N[(x * N[(0.5 / t), $MachinePrecision]), $MachinePrecision], N[(z * N[(-0.5 / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.2000000000000001e-11

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 67.3%

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

      1. *-commutative 67.3%

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

   5. Simplified 67.3%

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

   if -8.2000000000000001e-11 < z < 7.50000000000000046e33

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 97.4%

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

      1. distribute-lft-out 97.4%

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

   5. Simplified 97.4%

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

   6. Taylor expanded in x around inf 52.5%

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

      1. associate-*r/ 52.5%

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

      2. *-commutative 52.5%

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

      3. associate-*r/ 52.4%

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

   8. Simplified 52.4%

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

   if 7.50000000000000046e33 < 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 59.1%

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

      1. associate-*r/ 60.6%

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

      2. metadata-eval 60.6%

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

      3. distribute-lft-neg-in 60.6%

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

      4. *-commutative 60.6%

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

      5. distribute-neg-frac 60.6%

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

      6. associate-*r/ 60.5%

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

      7. distribute-rgt-neg-in 60.5%

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

      8. distribute-neg-frac 60.5%

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

      9. metadata-eval 60.5%

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

   5. Simplified 60.5%

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

Alternative 7: 75.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-75}:\\ \;\;\;\;\frac{x + y}{t \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{t \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.9e-75) (/ (+ x y) (* t 2.0)) (/ (- y z) (* t 2.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.9e-75) {
		tmp = (x + y) / (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 <= (-1.9d-75)) then
        tmp = (x + y) / (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 <= -1.9e-75) {
		tmp = (x + y) / (t * 2.0);
	} else {
		tmp = (y - z) / (t * 2.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1.9e-75:
		tmp = (x + y) / (t * 2.0)
	else:
		tmp = (y - z) / (t * 2.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.9e-75)
		tmp = Float64(Float64(x + y) / 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 <= -1.9e-75)
		tmp = (x + y) / (t * 2.0);
	else
		tmp = (y - z) / (t * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.9e-75], N[(N[(x + y), $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 < -1.89999999999999997e-75

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 85.7%

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

      1. +-commutative 85.7%

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

   5. Simplified 85.7%

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

   if -1.89999999999999997e-75 < 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 78.1%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-75}:\\ \;\;\;\;\frac{x + y}{t \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{t \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 38.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(x * N[(0.5 / 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 0 97.0%

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

   1. distribute-lft-out 97.0%

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

5. Simplified 97.0%

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

6. Taylor expanded in x around inf 43.0%

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

   1. associate-*r/ 43.0%

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

   2. *-commutative 43.0%

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

   3. associate-*r/ 42.9%

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

8. Simplified 42.9%

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

Reproduce

herbie shell --seed 2024084 
(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)))