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

Percentage Accurate: 100.0% → 99.7%

Time: 4.3s

Alternatives: 9

Speedup: N/A×

• 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: 100.0% 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: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

   \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
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. Final simplification 99.7%

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

Alternative 2: 49.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -200000000000:\\ \;\;\;\;\frac{x}{t} \cdot 0.5\\ \mathbf{elif}\;x + y \leq 10^{-11}:\\ \;\;\;\;\frac{-0.5 \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) -200000000000.0)
   (* (/ x t) 0.5)
   (if (<= (+ x y) 1e-11) (/ (* -0.5 z) t) (* (/ y t) 0.5))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -200000000000.0) {
		tmp = (x / t) * 0.5;
	} else if ((x + y) <= 1e-11) {
		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) <= (-200000000000.0d0)) then
        tmp = (x / t) * 0.5d0
    else if ((x + y) <= 1d-11) 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) <= -200000000000.0) {
		tmp = (x / t) * 0.5;
	} else if ((x + y) <= 1e-11) {
		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) <= -200000000000.0:
		tmp = (x / t) * 0.5
	elif (x + y) <= 1e-11:
		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) <= -200000000000.0)
		tmp = Float64(Float64(x / t) * 0.5);
	elseif (Float64(x + y) <= 1e-11)
		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) <= -200000000000.0)
		tmp = (x / t) * 0.5;
	elseif ((x + y) <= 1e-11)
		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], -200000000000.0], N[(N[(x / t), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[N[(x + y), $MachinePrecision], 1e-11], 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) < -2e11

   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 49.8

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

   5. Applied rewrites 49.8%

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

   if -2e11 < (+.f64 x y) < 9.99999999999999939e-12

   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 74.2

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

   5. Applied rewrites 74.2%

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

   7. Applied rewrites 74.4%

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

   if 9.99999999999999939e-12 < (+.f64 x y)

   1. Initial program 98.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   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 76.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 47.5%

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

Alternative 3: 49.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -200000000000:\\ \;\;\;\;\frac{x}{t} \cdot 0.5\\ \mathbf{elif}\;x + y \leq 10^{-11}:\\ \;\;\;\;\frac{-0.5}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) -200000000000.0)
   (* (/ x t) 0.5)
   (if (<= (+ x y) 1e-11) (* (/ -0.5 t) z) (* (/ y t) 0.5))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -200000000000.0) {
		tmp = (x / t) * 0.5;
	} else if ((x + y) <= 1e-11) {
		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) <= (-200000000000.0d0)) then
        tmp = (x / t) * 0.5d0
    else if ((x + y) <= 1d-11) 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) <= -200000000000.0) {
		tmp = (x / t) * 0.5;
	} else if ((x + y) <= 1e-11) {
		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) <= -200000000000.0:
		tmp = (x / t) * 0.5
	elif (x + y) <= 1e-11:
		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) <= -200000000000.0)
		tmp = Float64(Float64(x / t) * 0.5);
	elseif (Float64(x + y) <= 1e-11)
		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) <= -200000000000.0)
		tmp = (x / t) * 0.5;
	elseif ((x + y) <= 1e-11)
		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], -200000000000.0], N[(N[(x / t), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[N[(x + y), $MachinePrecision], 1e-11], 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) < -2e11

   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 49.8

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

   5. Applied rewrites 49.8%

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

   if -2e11 < (+.f64 x y) < 9.99999999999999939e-12

   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 74.2

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

   5. Applied rewrites 74.2%

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

   if 9.99999999999999939e-12 < (+.f64 x y)

   1. Initial program 98.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   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 76.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 47.5%

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

Alternative 4: 81.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-0.5 \cdot z}{t}\\ \mathbf{if}\;z \leq -5.45 \cdot 10^{+160}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{+73}:\\ \;\;\;\;\frac{x + y}{t} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* -0.5 z) t)))
   (if (<= z -5.45e+160) t_1 (if (<= z 8.6e+73) (* (/ (+ x y) t) 0.5) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (-0.5 * z) / t;
	double tmp;
	if (z <= -5.45e+160) {
		tmp = t_1;
	} else if (z <= 8.6e+73) {
		tmp = ((x + y) / t) * 0.5;
	} 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 = ((-0.5d0) * z) / t
    if (z <= (-5.45d+160)) then
        tmp = t_1
    else if (z <= 8.6d+73) then
        tmp = ((x + y) / t) * 0.5d0
    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 = (-0.5 * z) / t;
	double tmp;
	if (z <= -5.45e+160) {
		tmp = t_1;
	} else if (z <= 8.6e+73) {
		tmp = ((x + y) / t) * 0.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (-0.5 * z) / t
	tmp = 0
	if z <= -5.45e+160:
		tmp = t_1
	elif z <= 8.6e+73:
		tmp = ((x + y) / t) * 0.5
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(-0.5 * z) / t)
	tmp = 0.0
	if (z <= -5.45e+160)
		tmp = t_1;
	elseif (z <= 8.6e+73)
		tmp = Float64(Float64(Float64(x + y) / t) * 0.5);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (-0.5 * z) / t;
	tmp = 0.0;
	if (z <= -5.45e+160)
		tmp = t_1;
	elseif (z <= 8.6e+73)
		tmp = ((x + y) / t) * 0.5;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(-0.5 * z), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[z, -5.45e+160], t$95$1, If[LessEqual[z, 8.6e+73], N[(N[(N[(x + y), $MachinePrecision] / t), $MachinePrecision] * 0.5), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.45e160 or 8.60000000000000026e73 < z

   1. Initial program 98.6%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   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 85.8

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

   5. Applied rewrites 85.8%

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

   7. Applied rewrites 86.1%

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

   if -5.45e160 < z < 8.60000000000000026e73

   1. Initial program 99.5%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   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 85.0%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.45 \cdot 10^{+160}:\\ \;\;\;\;\frac{-0.5 \cdot z}{t}\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{+73}:\\ \;\;\;\;\frac{x + y}{t} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-0.5}{t} \cdot z\\ \mathbf{if}\;z \leq -3 \cdot 10^{+64}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 460:\\ \;\;\;\;\frac{y}{t} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ -0.5 t) z)))
   (if (<= z -3e+64) t_1 (if (<= z 460.0) (* (/ y t) 0.5) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (-0.5 / t) * z;
	double tmp;
	if (z <= -3e+64) {
		tmp = t_1;
	} else if (z <= 460.0) {
		tmp = (y / t) * 0.5;
	} 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 = ((-0.5d0) / t) * z
    if (z <= (-3d+64)) then
        tmp = t_1
    else if (z <= 460.0d0) then
        tmp = (y / t) * 0.5d0
    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 = (-0.5 / t) * z;
	double tmp;
	if (z <= -3e+64) {
		tmp = t_1;
	} else if (z <= 460.0) {
		tmp = (y / t) * 0.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (-0.5 / t) * z
	tmp = 0
	if z <= -3e+64:
		tmp = t_1
	elif z <= 460.0:
		tmp = (y / t) * 0.5
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(-0.5 / t) * z)
	tmp = 0.0
	if (z <= -3e+64)
		tmp = t_1;
	elseif (z <= 460.0)
		tmp = Float64(Float64(y / t) * 0.5);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (-0.5 / t) * z;
	tmp = 0.0;
	if (z <= -3e+64)
		tmp = t_1;
	elseif (z <= 460.0)
		tmp = (y / t) * 0.5;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(-0.5 / t), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -3e+64], t$95$1, If[LessEqual[z, 460.0], N[(N[(y / t), $MachinePrecision] * 0.5), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.0000000000000002e64 or 460 < z

   1. Initial program 99.1%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   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 68.6

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

   5. Applied rewrites 68.6%

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

   if -3.0000000000000002e64 < z < 460

   1. Initial program 99.3%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   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 91.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 50.1%

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

Alternative 6: 69.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-199}:\\ \;\;\;\;\frac{x - z}{2 \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{t} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) -1e-199) (/ (- x z) (* 2.0 t)) (* (/ (- y z) t) 0.5)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -1e-199) {
		tmp = (x - z) / (2.0 * t);
	} else {
		tmp = ((y - z) / 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) <= (-1d-199)) then
        tmp = (x - z) / (2.0d0 * t)
    else
        tmp = ((y - z) / 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) <= -1e-199) {
		tmp = (x - z) / (2.0 * t);
	} else {
		tmp = ((y - z) / t) * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + y) <= -1e-199:
		tmp = (x - z) / (2.0 * t)
	else:
		tmp = ((y - z) / t) * 0.5
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + y) <= -1e-199)
		tmp = Float64(Float64(x - z) / Float64(2.0 * t));
	else
		tmp = Float64(Float64(Float64(y - z) / t) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + y) <= -1e-199)
		tmp = (x - z) / (2.0 * t);
	else
		tmp = ((y - z) / t) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + y), $MachinePrecision], -1e-199], N[(N[(x - z), $MachinePrecision] / N[(2.0 * t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y - z), $MachinePrecision] / t), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -9.99999999999999982e-200

   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 68.7

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

   5. Applied rewrites 68.7%

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

   if -9.99999999999999982e-200 < (+.f64 x y)

   1. Initial program 98.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 78.1

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

   5. Applied rewrites 78.1%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-199}:\\ \;\;\;\;\frac{x - z}{2 \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{t} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-199}:\\ \;\;\;\;\left(x - z\right) \cdot \frac{0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{t} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) -1e-199) (* (- x z) (/ 0.5 t)) (* (/ (- y z) t) 0.5)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -1e-199) {
		tmp = (x - z) * (0.5 / t);
	} else {
		tmp = ((y - z) / 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) <= (-1d-199)) then
        tmp = (x - z) * (0.5d0 / t)
    else
        tmp = ((y - z) / 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) <= -1e-199) {
		tmp = (x - z) * (0.5 / t);
	} else {
		tmp = ((y - z) / t) * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + y) <= -1e-199:
		tmp = (x - z) * (0.5 / t)
	else:
		tmp = ((y - z) / t) * 0.5
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + y) <= -1e-199)
		tmp = Float64(Float64(x - z) * Float64(0.5 / t));
	else
		tmp = Float64(Float64(Float64(y - z) / t) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + y) <= -1e-199)
		tmp = (x - z) * (0.5 / t);
	else
		tmp = ((y - z) / t) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + y), $MachinePrecision], -1e-199], N[(N[(x - z), $MachinePrecision] * N[(0.5 / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y - z), $MachinePrecision] / t), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -9.99999999999999982e-200

   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 68.7

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

   5. Applied rewrites 68.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 68.5

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

   7. Applied rewrites 68.5%

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

   if -9.99999999999999982e-200 < (+.f64 x y)

   1. Initial program 98.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 78.1

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

   5. Applied rewrites 78.1%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-199}:\\ \;\;\;\;\left(x - z\right) \cdot \frac{0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{t} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -200000000000:\\ \;\;\;\;\frac{x + y}{t} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{t} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) -200000000000.0) (* (/ (+ x y) t) 0.5) (* (/ (- y z) t) 0.5)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -200000000000.0) {
		tmp = ((x + y) / t) * 0.5;
	} else {
		tmp = ((y - z) / 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) <= (-200000000000.0d0)) then
        tmp = ((x + y) / t) * 0.5d0
    else
        tmp = ((y - z) / 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) <= -200000000000.0) {
		tmp = ((x + y) / t) * 0.5;
	} else {
		tmp = ((y - z) / t) * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + y) <= -200000000000.0:
		tmp = ((x + y) / t) * 0.5
	else:
		tmp = ((y - z) / t) * 0.5
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + y) <= -200000000000.0)
		tmp = Float64(Float64(Float64(x + y) / t) * 0.5);
	else
		tmp = Float64(Float64(Float64(y - z) / t) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + y) <= -200000000000.0)
		tmp = ((x + y) / t) * 0.5;
	else
		tmp = ((y - z) / t) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + y), $MachinePrecision], -200000000000.0], N[(N[(N[(x + y), $MachinePrecision] / t), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(y - z), $MachinePrecision] / t), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -2e11

   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 87.3%

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

   if -2e11 < (+.f64 x y)

   1. Initial program 98.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 78.7

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

   5. Applied rewrites 78.7%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -200000000000:\\ \;\;\;\;\frac{x + y}{t} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{t} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 36.7% 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 99.2%

   \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
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 68.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 37.4%

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

Reproduce

herbie shell --seed 2024308 
(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)))