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

Percentage Accurate: 99.9% → 99.9%

Time: 6.9s

Alternatives: 10

Speedup: 1.0×

• 26.5% 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: 50.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -50000000000:\\ \;\;\;\;\frac{x}{t} \cdot 0.5\\ \mathbf{elif}\;x + y \leq 2000000000000:\\ \;\;\;\;\frac{-0.5 \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) -50000000000.0)
   (* (/ x t) 0.5)
   (if (<= (+ x y) 2000000000000.0) (/ (* -0.5 z) t) (* (/ y t) 0.5))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -50000000000.0) {
		tmp = (x / t) * 0.5;
	} else if ((x + y) <= 2000000000000.0) {
		tmp = (-0.5 * z) / 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 ((x + y) <= (-50000000000.0d0)) then
        tmp = (x / t) * 0.5d0
    else if ((x + y) <= 2000000000000.0d0) then
        tmp = ((-0.5d0) * z) / 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 ((x + y) <= -50000000000.0) {
		tmp = (x / t) * 0.5;
	} else if ((x + y) <= 2000000000000.0) {
		tmp = (-0.5 * z) / t;
	} else {
		tmp = (y / t) * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + y) <= -50000000000.0:
		tmp = (x / t) * 0.5
	elif (x + y) <= 2000000000000.0:
		tmp = (-0.5 * z) / t
	else:
		tmp = (y / t) * 0.5
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + y) <= -50000000000.0)
		tmp = Float64(Float64(x / t) * 0.5);
	elseif (Float64(x + y) <= 2000000000000.0)
		tmp = Float64(Float64(-0.5 * z) / 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 ((x + y) <= -50000000000.0)
		tmp = (x / t) * 0.5;
	elseif ((x + y) <= 2000000000000.0)
		tmp = (-0.5 * z) / t;
	else
		tmp = (y / t) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + y), $MachinePrecision], -50000000000.0], N[(N[(x / t), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[N[(x + y), $MachinePrecision], 2000000000000.0], N[(N[(-0.5 * z), $MachinePrecision] / t), $MachinePrecision], N[(N[(y / t), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x y) < -5e10

   1. Initial program 99.9%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   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 40.0

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

   5. Applied rewrites 40.0%

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

   if -5e10 < (+.f64 x y) < 2e12

   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 67.9

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

   5. Applied rewrites 67.9%

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

   7. Applied rewrites 68.2%

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

   if 2e12 < (+.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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 85.1

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 44.1%

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

Alternative 3: 50.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -50000000000:\\ \;\;\;\;\frac{x}{t} \cdot 0.5\\ \mathbf{elif}\;x + y \leq 2000000000000:\\ \;\;\;\;\frac{-0.5}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) -50000000000.0)
   (* (/ x t) 0.5)
   (if (<= (+ x y) 2000000000000.0) (* (/ -0.5 t) z) (* (/ y t) 0.5))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -50000000000.0) {
		tmp = (x / t) * 0.5;
	} else if ((x + y) <= 2000000000000.0) {
		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 ((x + y) <= (-50000000000.0d0)) then
        tmp = (x / t) * 0.5d0
    else if ((x + y) <= 2000000000000.0d0) 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 ((x + y) <= -50000000000.0) {
		tmp = (x / t) * 0.5;
	} else if ((x + y) <= 2000000000000.0) {
		tmp = (-0.5 / t) * z;
	} else {
		tmp = (y / t) * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + y) <= -50000000000.0:
		tmp = (x / t) * 0.5
	elif (x + y) <= 2000000000000.0:
		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(x + y) <= -50000000000.0)
		tmp = Float64(Float64(x / t) * 0.5);
	elseif (Float64(x + y) <= 2000000000000.0)
		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 ((x + y) <= -50000000000.0)
		tmp = (x / t) * 0.5;
	elseif ((x + y) <= 2000000000000.0)
		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[(x + y), $MachinePrecision], -50000000000.0], N[(N[(x / t), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[N[(x + y), $MachinePrecision], 2000000000000.0], 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) < -5e10

   1. Initial program 99.9%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   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 40.0

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

   5. Applied rewrites 40.0%

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

   if -5e10 < (+.f64 x y) < 2e12

   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 67.9

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

   5. Applied rewrites 67.9%

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

   if 2e12 < (+.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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 85.1

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 44.1%

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

Alternative 4: 81.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+69} \lor \neg \left(z \leq 3.6 \cdot 10^{+137}\right):\\ \;\;\;\;\frac{-0.5 \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{t} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.15e+69) (not (<= z 3.6e+137)))
   (/ (* -0.5 z) t)
   (* (/ (+ y x) t) 0.5)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.15e+69) || !(z <= 3.6e+137)) {
		tmp = (-0.5 * z) / t;
	} else {
		tmp = ((y + x) / 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 ((z <= (-1.15d+69)) .or. (.not. (z <= 3.6d+137))) then
        tmp = ((-0.5d0) * z) / t
    else
        tmp = ((y + x) / 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 ((z <= -1.15e+69) || !(z <= 3.6e+137)) {
		tmp = (-0.5 * z) / t;
	} else {
		tmp = ((y + x) / t) * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.15e+69) or not (z <= 3.6e+137):
		tmp = (-0.5 * z) / t
	else:
		tmp = ((y + x) / t) * 0.5
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.15e+69) || !(z <= 3.6e+137))
		tmp = Float64(Float64(-0.5 * z) / t);
	else
		tmp = Float64(Float64(Float64(y + x) / t) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.15e+69) || ~((z <= 3.6e+137)))
		tmp = (-0.5 * z) / t;
	else
		tmp = ((y + x) / t) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.15e+69], N[Not[LessEqual[z, 3.6e+137]], $MachinePrecision]], N[(N[(-0.5 * z), $MachinePrecision] / t), $MachinePrecision], N[(N[(N[(y + x), $MachinePrecision] / t), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.15000000000000008e69 or 3.6e137 < 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

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{z}{t}} \]
   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 77.9

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

   5. Applied rewrites 77.9%

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

   7. Applied rewrites 78.1%

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

   if -1.15000000000000008e69 < z < 3.6e137

   1. Initial program 99.9%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 84.7

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

   5. Applied rewrites 84.7%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+69} \lor \neg \left(z \leq 3.6 \cdot 10^{+137}\right):\\ \;\;\;\;\frac{-0.5 \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{t} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-104}:\\ \;\;\;\;\frac{x - z}{t \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{t \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) -1e-104) (/ (- x z) (* t 2.0)) (/ (- y z) (* t 2.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -1e-104) {
		tmp = (x - z) / (t * 2.0);
	} else {
		tmp = (y - z) / (t * 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x + y) <= (-1d-104)) then
        tmp = (x - z) / (t * 2.0d0)
    else
        tmp = (y - z) / (t * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -1e-104) {
		tmp = (x - z) / (t * 2.0);
	} else {
		tmp = (y - z) / (t * 2.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + y) <= -1e-104:
		tmp = (x - z) / (t * 2.0)
	else:
		tmp = (y - z) / (t * 2.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + y) <= -1e-104)
		tmp = Float64(Float64(x - z) / Float64(t * 2.0));
	else
		tmp = Float64(Float64(y - z) / Float64(t * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + y) <= -1e-104)
		tmp = (x - z) / (t * 2.0);
	else
		tmp = (y - z) / (t * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + y), $MachinePrecision], -1e-104], N[(N[(x - z), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(y - z), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -9.99999999999999927e-105

   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 65.1

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

   5. Applied rewrites 65.1%

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

   if -9.99999999999999927e-105 < (+.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 68.1

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

   5. Applied rewrites 68.1%

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

Alternative 6: 69.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -5 \cdot 10^{-253}:\\ \;\;\;\;\frac{x - z}{t \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{t} \cdot \left(y - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) -5e-253) (/ (- x z) (* t 2.0)) (* (/ 0.5 t) (- y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -5e-253) {
		tmp = (x - z) / (t * 2.0);
	} else {
		tmp = (0.5 / t) * (y - z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x + y) <= (-5d-253)) then
        tmp = (x - z) / (t * 2.0d0)
    else
        tmp = (0.5d0 / t) * (y - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -5e-253) {
		tmp = (x - z) / (t * 2.0);
	} else {
		tmp = (0.5 / t) * (y - z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + y) <= -5e-253:
		tmp = (x - z) / (t * 2.0)
	else:
		tmp = (0.5 / t) * (y - z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + y) <= -5e-253)
		tmp = Float64(Float64(x - z) / Float64(t * 2.0));
	else
		tmp = Float64(Float64(0.5 / t) * Float64(y - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + y) <= -5e-253)
		tmp = (x - z) / (t * 2.0);
	else
		tmp = (0.5 / t) * (y - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + y), $MachinePrecision], -5e-253], N[(N[(x - z), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / t), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -4.99999999999999971e-253

   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 68.8

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

   5. Applied rewrites 68.8%

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

   if -4.99999999999999971e-253 < (+.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 63.9

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

   5. Applied rewrites 63.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 63.8

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

   7. Applied rewrites 63.8%

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

Alternative 7: 78.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+35}:\\ \;\;\;\;\frac{y + x}{t} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{t} \cdot \left(y - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -2.6e+35) (* (/ (+ y x) t) 0.5) (* (/ 0.5 t) (- y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.6e+35) {
		tmp = ((y + x) / t) * 0.5;
	} else {
		tmp = (0.5 / t) * (y - z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-2.6d+35)) then
        tmp = ((y + x) / t) * 0.5d0
    else
        tmp = (0.5d0 / t) * (y - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.6e+35) {
		tmp = ((y + x) / t) * 0.5;
	} else {
		tmp = (0.5 / t) * (y - z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -2.6e+35:
		tmp = ((y + x) / t) * 0.5
	else:
		tmp = (0.5 / t) * (y - z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -2.6e+35)
		tmp = Float64(Float64(Float64(y + x) / t) * 0.5);
	else
		tmp = Float64(Float64(0.5 / t) * Float64(y - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -2.6e+35)
		tmp = ((y + x) / t) * 0.5;
	else
		tmp = (0.5 / t) * (y - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -2.6e+35], N[(N[(N[(y + x), $MachinePrecision] / t), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(0.5 / t), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.60000000000000007e35

   1. Initial program 99.9%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 88.9

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

   5. Applied rewrites 88.9%

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

   if -2.60000000000000007e35 < 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

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

      1. lower--.f64 75.9

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

   5. Applied rewrites 75.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 75.7

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

   7. Applied rewrites 75.7%

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

Alternative 8: 43.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq 2000000000000:\\ \;\;\;\;\frac{-0.5}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) 2000000000000.0) (* (/ -0.5 t) z) (* (/ y t) 0.5)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= 2000000000000.0) {
		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 ((x + y) <= 2000000000000.0d0) 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 ((x + y) <= 2000000000000.0) {
		tmp = (-0.5 / t) * z;
	} else {
		tmp = (y / t) * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + y) <= 2000000000000.0:
		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(x + y) <= 2000000000000.0)
		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 ((x + y) <= 2000000000000.0)
		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[(x + y), $MachinePrecision], 2000000000000.0], 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) < 2e12

   1. Initial program 99.9%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   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 44.7

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

   5. Applied rewrites 44.7%

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

   if 2e12 < (+.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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 85.1

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 44.1%

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

Alternative 9: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{t} \cdot \left(\left(y + x\right) - z\right) \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (0.5d0 / t) * ((y + x) - z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. metadata-eval 99.7

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 99.7

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

4. Applied rewrites 99.7%

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

Alternative 10: 38.0% 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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 70.6

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

5. Applied rewrites 70.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 41.1%

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

Reproduce

herbie shell --seed 2024318 
(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)))