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

Percentage Accurate: 99.9% → 99.9%

Time: 6.0s

Alternatives: 8

Speedup: 1.0×

• 25.6% 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

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

FPCore

(FPCore (x y z t) :precision binary64 (/ (- (+ y x) z) (* 2.0 t)))

C

double code(double x, double y, double z, double t) {
	return ((y + x) - z) / (2.0 * 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 = ((y + x) - z) / (2.0d0 * t)
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((y + x) - z) / (2.0 * t);
}

Python

def code(x, y, z, t):
	return ((y + x) - z) / (2.0 * t)

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(y + x) - z) / Float64(2.0 * t))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((y + x) - z) / (2.0 * t);
end

Wolfram

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

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

Alternative 2: 48.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + x \leq -5 \cdot 10^{+70}:\\ \;\;\;\;\frac{x}{t} \cdot 0.5\\ \mathbf{elif}\;y + x \leq 5 \cdot 10^{-45}:\\ \;\;\;\;\frac{-z}{2 \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ y x) -5e+70)
   (* (/ x t) 0.5)
   (if (<= (+ y x) 5e-45) (/ (- z) (* 2.0 t)) (* (/ y t) 0.5))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y + x) <= -5e+70) {
		tmp = (x / t) * 0.5;
	} else if ((y + x) <= 5e-45) {
		tmp = -z / (2.0 * t);
	} else {
		tmp = (y / t) * 0.5;
	}
	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 + x) <= (-5d+70)) then
        tmp = (x / t) * 0.5d0
    else if ((y + x) <= 5d-45) then
        tmp = -z / (2.0d0 * t)
    else
        tmp = (y / t) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y + x) <= -5e+70) {
		tmp = (x / t) * 0.5;
	} else if ((y + x) <= 5e-45) {
		tmp = -z / (2.0 * t);
	} else {
		tmp = (y / t) * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y + x) <= -5e+70:
		tmp = (x / t) * 0.5
	elif (y + x) <= 5e-45:
		tmp = -z / (2.0 * t)
	else:
		tmp = (y / t) * 0.5
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(y + x) <= -5e+70)
		tmp = Float64(Float64(x / t) * 0.5);
	elseif (Float64(y + x) <= 5e-45)
		tmp = Float64(Float64(-z) / Float64(2.0 * t));
	else
		tmp = Float64(Float64(y / t) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y + x) <= -5e+70)
		tmp = (x / t) * 0.5;
	elseif ((y + x) <= 5e-45)
		tmp = -z / (2.0 * t);
	else
		tmp = (y / t) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(y + x), $MachinePrecision], -5e+70], N[(N[(x / t), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[N[(y + x), $MachinePrecision], 5e-45], N[((-z) / N[(2.0 * t), $MachinePrecision]), $MachinePrecision], N[(N[(y / t), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x y) < -5.0000000000000002e70

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x}{t}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x}{t} \cdot \frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{t} \cdot \frac{1}{2}} \]

      3. lower-/.f64 45.4

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

   5. Applied rewrites 45.4%

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

   if -5.0000000000000002e70 < (+.f64 x y) < 4.99999999999999976e-45

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

      \[\leadsto \frac{\color{blue}{-1 \cdot z}}{t \cdot 2} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(z\right)}}{t \cdot 2} \]

      2. lower-neg.f64 72.3

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

   5. Applied rewrites 72.3%

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

   if 4.99999999999999976e-45 < (+.f64 x 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

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x + y}{t}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(x + y\right)}{t}} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2}}{t} \cdot \left(x + y\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot 1}}{t} \cdot \left(x + y\right) \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{t}\right)} \cdot \left(x + y\right) \]

      5. distribute-lft-out N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{t}\right) \cdot x + \left(\frac{1}{2} \cdot \frac{1}{t}\right) \cdot y} \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\frac{1}{t} \cdot x\right)} + \left(\frac{1}{2} \cdot \frac{1}{t}\right) \cdot y \]

      7. associate-*l* N/A

         \[\leadsto \frac{1}{2} \cdot \left(\frac{1}{t} \cdot x\right) + \color{blue}{\frac{1}{2} \cdot \left(\frac{1}{t} \cdot y\right)} \]

      8. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\frac{1}{t} \cdot x + \frac{1}{t} \cdot y\right)} \]

      9. associate-*l/ N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\frac{1 \cdot x}{t}} + \frac{1}{t} \cdot y\right) \]

      10. *-lft-identity N/A

          \[\leadsto \frac{1}{2} \cdot \left(\frac{\color{blue}{x}}{t} + \frac{1}{t} \cdot y\right) \]

      11. associate-*l/ N/A

          \[\leadsto \frac{1}{2} \cdot \left(\frac{x}{t} + \color{blue}{\frac{1 \cdot y}{t}}\right) \]

      12. *-lft-identity N/A

          \[\leadsto \frac{1}{2} \cdot \left(\frac{x}{t} + \frac{\color{blue}{y}}{t}\right) \]

      13. *-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{x}{t} + \frac{y}{t}\right) \cdot \frac{1}{2}} \]

      14. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{x}{t} + \frac{y}{t}\right) \cdot \frac{1}{2}} \]

   5. Applied rewrites 79.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{y}{t} \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.3%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + x \leq -5 \cdot 10^{+70}:\\ \;\;\;\;\frac{x}{t} \cdot 0.5\\ \mathbf{elif}\;y + x \leq 5 \cdot 10^{-45}:\\ \;\;\;\;\frac{-z}{2 \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 48.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + x \leq -5 \cdot 10^{+70}:\\ \;\;\;\;\frac{x}{t} \cdot 0.5\\ \mathbf{elif}\;y + x \leq 5 \cdot 10^{-45}:\\ \;\;\;\;\frac{-0.5}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ y x) -5e+70)
   (* (/ x t) 0.5)
   (if (<= (+ y x) 5e-45) (* (/ -0.5 t) z) (* (/ y t) 0.5))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y + x) <= -5e+70) {
		tmp = (x / t) * 0.5;
	} else if ((y + x) <= 5e-45) {
		tmp = (-0.5 / t) * z;
	} else {
		tmp = (y / t) * 0.5;
	}
	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 + x) <= (-5d+70)) then
        tmp = (x / t) * 0.5d0
    else if ((y + x) <= 5d-45) then
        tmp = ((-0.5d0) / t) * z
    else
        tmp = (y / t) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y + x) <= -5e+70) {
		tmp = (x / t) * 0.5;
	} else if ((y + x) <= 5e-45) {
		tmp = (-0.5 / t) * z;
	} else {
		tmp = (y / t) * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y + x) <= -5e+70:
		tmp = (x / t) * 0.5
	elif (y + x) <= 5e-45:
		tmp = (-0.5 / t) * z
	else:
		tmp = (y / t) * 0.5
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(y + x) <= -5e+70)
		tmp = Float64(Float64(x / t) * 0.5);
	elseif (Float64(y + x) <= 5e-45)
		tmp = Float64(Float64(-0.5 / t) * z);
	else
		tmp = Float64(Float64(y / t) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y + x) <= -5e+70)
		tmp = (x / t) * 0.5;
	elseif ((y + x) <= 5e-45)
		tmp = (-0.5 / t) * z;
	else
		tmp = (y / t) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(y + x), $MachinePrecision], -5e+70], N[(N[(x / t), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[N[(y + x), $MachinePrecision], 5e-45], N[(N[(-0.5 / t), $MachinePrecision] * z), $MachinePrecision], N[(N[(y / t), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x y) < -5.0000000000000002e70

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x}{t}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x}{t} \cdot \frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{t} \cdot \frac{1}{2}} \]

      3. lower-/.f64 45.4

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

   5. Applied rewrites 45.4%

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

   if -5.0000000000000002e70 < (+.f64 x y) < 4.99999999999999976e-45

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{z}{t}} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\color{blue}{1 \cdot z}}{t} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\left(\frac{1}{t} \cdot z\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \frac{1}{t}\right) \cdot z} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{1}{t}\right) \cdot z \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{t}\right)\right)} \cdot z \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{t}\right)\right) \cdot z} \]

      7. associate-*r/ N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{2} \cdot 1}{t}}\right)\right) \cdot z \]

      8. metadata-eval N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{2}}}{t}\right)\right) \cdot z \]

      9. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{t}} \cdot z \]

      10. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{2}}}{t} \cdot z \]

      11. lower-/.f64 72.0

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

   5. Applied rewrites 72.0%

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

   if 4.99999999999999976e-45 < (+.f64 x 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

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x + y}{t}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(x + y\right)}{t}} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2}}{t} \cdot \left(x + y\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot 1}}{t} \cdot \left(x + y\right) \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{t}\right)} \cdot \left(x + y\right) \]

      5. distribute-lft-out N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{t}\right) \cdot x + \left(\frac{1}{2} \cdot \frac{1}{t}\right) \cdot y} \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\frac{1}{t} \cdot x\right)} + \left(\frac{1}{2} \cdot \frac{1}{t}\right) \cdot y \]

      7. associate-*l* N/A

         \[\leadsto \frac{1}{2} \cdot \left(\frac{1}{t} \cdot x\right) + \color{blue}{\frac{1}{2} \cdot \left(\frac{1}{t} \cdot y\right)} \]

      8. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\frac{1}{t} \cdot x + \frac{1}{t} \cdot y\right)} \]

      9. associate-*l/ N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\frac{1 \cdot x}{t}} + \frac{1}{t} \cdot y\right) \]

      10. *-lft-identity N/A

          \[\leadsto \frac{1}{2} \cdot \left(\frac{\color{blue}{x}}{t} + \frac{1}{t} \cdot y\right) \]

      11. associate-*l/ N/A

          \[\leadsto \frac{1}{2} \cdot \left(\frac{x}{t} + \color{blue}{\frac{1 \cdot y}{t}}\right) \]

      12. *-lft-identity N/A

          \[\leadsto \frac{1}{2} \cdot \left(\frac{x}{t} + \frac{\color{blue}{y}}{t}\right) \]

      13. *-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{x}{t} + \frac{y}{t}\right) \cdot \frac{1}{2}} \]

      14. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{x}{t} + \frac{y}{t}\right) \cdot \frac{1}{2}} \]

   5. Applied rewrites 79.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{y}{t} \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.3%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + x \leq -5 \cdot 10^{+70}:\\ \;\;\;\;\frac{x}{t} \cdot 0.5\\ \mathbf{elif}\;y + x \leq 5 \cdot 10^{-45}:\\ \;\;\;\;\frac{-0.5}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-z}{2 \cdot t}\\ \mathbf{if}\;z \leq -2.1 \cdot 10^{+55}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+113}:\\ \;\;\;\;0.5 \cdot \frac{y + x}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- z) (* 2.0 t))))
   (if (<= z -2.1e+55) t_1 (if (<= z 4.1e+113) (* 0.5 (/ (+ y x) t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -z / (2.0 * t);
	double tmp;
	if (z <= -2.1e+55) {
		tmp = t_1;
	} else if (z <= 4.1e+113) {
		tmp = 0.5 * ((y + x) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -z / (2.0d0 * t)
    if (z <= (-2.1d+55)) then
        tmp = t_1
    else if (z <= 4.1d+113) then
        tmp = 0.5d0 * ((y + x) / t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = -z / (2.0 * t);
	double tmp;
	if (z <= -2.1e+55) {
		tmp = t_1;
	} else if (z <= 4.1e+113) {
		tmp = 0.5 * ((y + x) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = -z / (2.0 * t)
	tmp = 0
	if z <= -2.1e+55:
		tmp = t_1
	elif z <= 4.1e+113:
		tmp = 0.5 * ((y + x) / t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(-z) / Float64(2.0 * t))
	tmp = 0.0
	if (z <= -2.1e+55)
		tmp = t_1;
	elseif (z <= 4.1e+113)
		tmp = Float64(0.5 * Float64(Float64(y + x) / t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = -z / (2.0 * t);
	tmp = 0.0;
	if (z <= -2.1e+55)
		tmp = t_1;
	elseif (z <= 4.1e+113)
		tmp = 0.5 * ((y + x) / t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[((-z) / N[(2.0 * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.1e+55], t$95$1, If[LessEqual[z, 4.1e+113], N[(0.5 * N[(N[(y + x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.1000000000000001e55 or 4.09999999999999993e113 < 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

      \[\leadsto \frac{\color{blue}{-1 \cdot z}}{t \cdot 2} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(z\right)}}{t \cdot 2} \]

      2. lower-neg.f64 77.2

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

   5. Applied rewrites 77.2%

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

   if -2.1000000000000001e55 < z < 4.09999999999999993e113

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x + y}{t}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(x + y\right)}{t}} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2}}{t} \cdot \left(x + y\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot 1}}{t} \cdot \left(x + y\right) \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{t}\right)} \cdot \left(x + y\right) \]

      5. distribute-lft-out N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{t}\right) \cdot x + \left(\frac{1}{2} \cdot \frac{1}{t}\right) \cdot y} \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\frac{1}{t} \cdot x\right)} + \left(\frac{1}{2} \cdot \frac{1}{t}\right) \cdot y \]

      7. associate-*l* N/A

         \[\leadsto \frac{1}{2} \cdot \left(\frac{1}{t} \cdot x\right) + \color{blue}{\frac{1}{2} \cdot \left(\frac{1}{t} \cdot y\right)} \]

      8. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\frac{1}{t} \cdot x + \frac{1}{t} \cdot y\right)} \]

      9. associate-*l/ N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\frac{1 \cdot x}{t}} + \frac{1}{t} \cdot y\right) \]

      10. *-lft-identity N/A

          \[\leadsto \frac{1}{2} \cdot \left(\frac{\color{blue}{x}}{t} + \frac{1}{t} \cdot y\right) \]

      11. associate-*l/ N/A

          \[\leadsto \frac{1}{2} \cdot \left(\frac{x}{t} + \color{blue}{\frac{1 \cdot y}{t}}\right) \]

      12. *-lft-identity N/A

          \[\leadsto \frac{1}{2} \cdot \left(\frac{x}{t} + \frac{\color{blue}{y}}{t}\right) \]

      13. *-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{x}{t} + \frac{y}{t}\right) \cdot \frac{1}{2}} \]

      14. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{x}{t} + \frac{y}{t}\right) \cdot \frac{1}{2}} \]

   5. Applied rewrites 91.1%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+55}:\\ \;\;\;\;\frac{-z}{2 \cdot t}\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+113}:\\ \;\;\;\;0.5 \cdot \frac{y + x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{2 \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + x \leq -2 \cdot 10^{-128}:\\ \;\;\;\;\frac{x - z}{2 \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{2 \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ y x) -2e-128) (/ (- x z) (* 2.0 t)) (/ (- y z) (* 2.0 t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y + x) <= -2e-128) {
		tmp = (x - z) / (2.0 * t);
	} else {
		tmp = (y - z) / (2.0 * 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 + x) <= (-2d-128)) then
        tmp = (x - z) / (2.0d0 * t)
    else
        tmp = (y - z) / (2.0d0 * t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y + x) <= -2e-128) {
		tmp = (x - z) / (2.0 * t);
	} else {
		tmp = (y - z) / (2.0 * t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y + x) <= -2e-128:
		tmp = (x - z) / (2.0 * t)
	else:
		tmp = (y - z) / (2.0 * t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(y + x) <= -2e-128)
		tmp = Float64(Float64(x - z) / Float64(2.0 * t));
	else
		tmp = Float64(Float64(y - z) / Float64(2.0 * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y + x) <= -2e-128)
		tmp = (x - z) / (2.0 * t);
	else
		tmp = (y - z) / (2.0 * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(y + x), $MachinePrecision], -2e-128], N[(N[(x - z), $MachinePrecision] / N[(2.0 * t), $MachinePrecision]), $MachinePrecision], N[(N[(y - z), $MachinePrecision] / N[(2.0 * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -2.00000000000000011e-128

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 65.7

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

   5. Applied rewrites 65.7%

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

   if -2.00000000000000011e-128 < (+.f64 x y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 73.1

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

   5. Applied rewrites 73.1%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + x \leq -2 \cdot 10^{-128}:\\ \;\;\;\;\frac{x - z}{2 \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{2 \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + x \leq 5 \cdot 10^{-45}:\\ \;\;\;\;\frac{x - z}{2 \cdot t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y + x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ y x) 5e-45) (/ (- x z) (* 2.0 t)) (* 0.5 (/ (+ y x) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y + x) <= 5e-45) {
		tmp = (x - z) / (2.0 * t);
	} else {
		tmp = 0.5 * ((y + x) / 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 + x) <= 5d-45) then
        tmp = (x - z) / (2.0d0 * t)
    else
        tmp = 0.5d0 * ((y + x) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y + x) <= 5e-45) {
		tmp = (x - z) / (2.0 * t);
	} else {
		tmp = 0.5 * ((y + x) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y + x) <= 5e-45:
		tmp = (x - z) / (2.0 * t)
	else:
		tmp = 0.5 * ((y + x) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(y + x) <= 5e-45)
		tmp = Float64(Float64(x - z) / Float64(2.0 * t));
	else
		tmp = Float64(0.5 * Float64(Float64(y + x) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y + x) <= 5e-45)
		tmp = (x - z) / (2.0 * t);
	else
		tmp = 0.5 * ((y + x) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(y + x), $MachinePrecision], 5e-45], N[(N[(x - z), $MachinePrecision] / N[(2.0 * t), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(y + x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < 4.99999999999999976e-45

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 71.8

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

   5. Applied rewrites 71.8%

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

   if 4.99999999999999976e-45 < (+.f64 x 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

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x + y}{t}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(x + y\right)}{t}} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2}}{t} \cdot \left(x + y\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot 1}}{t} \cdot \left(x + y\right) \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{t}\right)} \cdot \left(x + y\right) \]

      5. distribute-lft-out N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{t}\right) \cdot x + \left(\frac{1}{2} \cdot \frac{1}{t}\right) \cdot y} \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\frac{1}{t} \cdot x\right)} + \left(\frac{1}{2} \cdot \frac{1}{t}\right) \cdot y \]

      7. associate-*l* N/A

         \[\leadsto \frac{1}{2} \cdot \left(\frac{1}{t} \cdot x\right) + \color{blue}{\frac{1}{2} \cdot \left(\frac{1}{t} \cdot y\right)} \]

      8. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\frac{1}{t} \cdot x + \frac{1}{t} \cdot y\right)} \]

      9. associate-*l/ N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\frac{1 \cdot x}{t}} + \frac{1}{t} \cdot y\right) \]

      10. *-lft-identity N/A

          \[\leadsto \frac{1}{2} \cdot \left(\frac{\color{blue}{x}}{t} + \frac{1}{t} \cdot y\right) \]

      11. associate-*l/ N/A

          \[\leadsto \frac{1}{2} \cdot \left(\frac{x}{t} + \color{blue}{\frac{1 \cdot y}{t}}\right) \]

      12. *-lft-identity N/A

          \[\leadsto \frac{1}{2} \cdot \left(\frac{x}{t} + \frac{\color{blue}{y}}{t}\right) \]

      13. *-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{x}{t} + \frac{y}{t}\right) \cdot \frac{1}{2}} \]

      14. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{x}{t} + \frac{y}{t}\right) \cdot \frac{1}{2}} \]

   5. Applied rewrites 79.5%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + x \leq 5 \cdot 10^{-45}:\\ \;\;\;\;\frac{x - z}{2 \cdot t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y + x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 42.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + x \leq 5 \cdot 10^{-45}:\\ \;\;\;\;\frac{-0.5}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ y x) 5e-45) (* (/ -0.5 t) z) (* (/ y t) 0.5)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y + x) <= 5e-45) {
		tmp = (-0.5 / t) * z;
	} else {
		tmp = (y / t) * 0.5;
	}
	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 + x) <= 5d-45) then
        tmp = ((-0.5d0) / t) * z
    else
        tmp = (y / t) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y + x) <= 5e-45) {
		tmp = (-0.5 / t) * z;
	} else {
		tmp = (y / t) * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y + x) <= 5e-45:
		tmp = (-0.5 / t) * z
	else:
		tmp = (y / t) * 0.5
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(y + x) <= 5e-45)
		tmp = Float64(Float64(-0.5 / t) * z);
	else
		tmp = Float64(Float64(y / t) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y + x) <= 5e-45)
		tmp = (-0.5 / t) * z;
	else
		tmp = (y / t) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(y + x), $MachinePrecision], 5e-45], N[(N[(-0.5 / t), $MachinePrecision] * z), $MachinePrecision], N[(N[(y / t), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < 4.99999999999999976e-45

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{z}{t}} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\color{blue}{1 \cdot z}}{t} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\left(\frac{1}{t} \cdot z\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \frac{1}{t}\right) \cdot z} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{1}{t}\right) \cdot z \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{t}\right)\right)} \cdot z \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{t}\right)\right) \cdot z} \]

      7. associate-*r/ N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{2} \cdot 1}{t}}\right)\right) \cdot z \]

      8. metadata-eval N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{2}}}{t}\right)\right) \cdot z \]

      9. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{t}} \cdot z \]

      10. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{2}}}{t} \cdot z \]

      11. lower-/.f64 43.7

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

   5. Applied rewrites 43.7%

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

   if 4.99999999999999976e-45 < (+.f64 x 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

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x + y}{t}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(x + y\right)}{t}} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2}}{t} \cdot \left(x + y\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot 1}}{t} \cdot \left(x + y\right) \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{t}\right)} \cdot \left(x + y\right) \]

      5. distribute-lft-out N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{t}\right) \cdot x + \left(\frac{1}{2} \cdot \frac{1}{t}\right) \cdot y} \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\frac{1}{t} \cdot x\right)} + \left(\frac{1}{2} \cdot \frac{1}{t}\right) \cdot y \]

      7. associate-*l* N/A

         \[\leadsto \frac{1}{2} \cdot \left(\frac{1}{t} \cdot x\right) + \color{blue}{\frac{1}{2} \cdot \left(\frac{1}{t} \cdot y\right)} \]

      8. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\frac{1}{t} \cdot x + \frac{1}{t} \cdot y\right)} \]

      9. associate-*l/ N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\frac{1 \cdot x}{t}} + \frac{1}{t} \cdot y\right) \]

      10. *-lft-identity N/A

          \[\leadsto \frac{1}{2} \cdot \left(\frac{\color{blue}{x}}{t} + \frac{1}{t} \cdot y\right) \]

      11. associate-*l/ N/A

          \[\leadsto \frac{1}{2} \cdot \left(\frac{x}{t} + \color{blue}{\frac{1 \cdot y}{t}}\right) \]

      12. *-lft-identity N/A

          \[\leadsto \frac{1}{2} \cdot \left(\frac{x}{t} + \frac{\color{blue}{y}}{t}\right) \]

      13. *-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{x}{t} + \frac{y}{t}\right) \cdot \frac{1}{2}} \]

      14. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{x}{t} + \frac{y}{t}\right) \cdot \frac{1}{2}} \]

   5. Applied rewrites 79.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{y}{t} \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.3%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + x \leq 5 \cdot 10^{-45}:\\ \;\;\;\;\frac{-0.5}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 36.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

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 = (y / t) * 0.5d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x + y}{t}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(x + y\right)}{t}} \]

   2. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{t} \cdot \left(x + y\right)} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot 1}}{t} \cdot \left(x + y\right) \]

   4. associate-*r/ N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{t}\right)} \cdot \left(x + y\right) \]

   5. distribute-lft-out N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{t}\right) \cdot x + \left(\frac{1}{2} \cdot \frac{1}{t}\right) \cdot y} \]

   6. associate-*l* N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\frac{1}{t} \cdot x\right)} + \left(\frac{1}{2} \cdot \frac{1}{t}\right) \cdot y \]

   7. associate-*l* N/A

      \[\leadsto \frac{1}{2} \cdot \left(\frac{1}{t} \cdot x\right) + \color{blue}{\frac{1}{2} \cdot \left(\frac{1}{t} \cdot y\right)} \]

   8. distribute-lft-out N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\frac{1}{t} \cdot x + \frac{1}{t} \cdot y\right)} \]

   9. associate-*l/ N/A

      \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\frac{1 \cdot x}{t}} + \frac{1}{t} \cdot y\right) \]

   10. *-lft-identity N/A

       \[\leadsto \frac{1}{2} \cdot \left(\frac{\color{blue}{x}}{t} + \frac{1}{t} \cdot y\right) \]

   11. associate-*l/ N/A

       \[\leadsto \frac{1}{2} \cdot \left(\frac{x}{t} + \color{blue}{\frac{1 \cdot y}{t}}\right) \]

   12. *-lft-identity N/A

       \[\leadsto \frac{1}{2} \cdot \left(\frac{x}{t} + \frac{\color{blue}{y}}{t}\right) \]

   13. *-commutative N/A

       \[\leadsto \color{blue}{\left(\frac{x}{t} + \frac{y}{t}\right) \cdot \frac{1}{2}} \]

   14. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(\frac{x}{t} + \frac{y}{t}\right) \cdot \frac{1}{2}} \]

5. Applied rewrites 71.3%

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

6. Taylor expanded in x around 0

   \[\leadsto \frac{y}{t} \cdot \frac{1}{2} \]
7. Step-by-step derivation

8. Applied rewrites 41.7%

   \[\leadsto \frac{y}{t} \cdot 0.5 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024294 
(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)))