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

Percentage Accurate: 99.9% → 99.9%

Time: 9.0s

Alternatives: 10

Speedup: 1.0×

• 25.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x + y) - z) / (t * 2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x + y) - z) / (t * 2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x + y) - z) / (t * 2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 48.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -5 \cdot 10^{-69}:\\ \;\;\;\;\frac{x \cdot 0.5}{t}\\ \mathbf{elif}\;x + y \leq 8 \cdot 10^{+48}:\\ \;\;\;\;\frac{z \cdot -0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot 0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) -5e-69)
   (/ (* x 0.5) t)
   (if (<= (+ x y) 8e+48) (/ (* z -0.5) t) (/ (* y 0.5) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -5e-69) {
		tmp = (x * 0.5) / t;
	} else if ((x + y) <= 8e+48) {
		tmp = (z * -0.5) / t;
	} else {
		tmp = (y * 0.5) / t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x + y) <= (-5d-69)) then
        tmp = (x * 0.5d0) / t
    else if ((x + y) <= 8d+48) then
        tmp = (z * (-0.5d0)) / t
    else
        tmp = (y * 0.5d0) / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -5e-69) {
		tmp = (x * 0.5) / t;
	} else if ((x + y) <= 8e+48) {
		tmp = (z * -0.5) / t;
	} else {
		tmp = (y * 0.5) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + y) <= -5e-69:
		tmp = (x * 0.5) / t
	elif (x + y) <= 8e+48:
		tmp = (z * -0.5) / t
	else:
		tmp = (y * 0.5) / t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + y) <= -5e-69)
		tmp = Float64(Float64(x * 0.5) / t);
	elseif (Float64(x + y) <= 8e+48)
		tmp = Float64(Float64(z * -0.5) / t);
	else
		tmp = Float64(Float64(y * 0.5) / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + y) <= -5e-69)
		tmp = (x * 0.5) / t;
	elseif ((x + y) <= 8e+48)
		tmp = (z * -0.5) / t;
	else
		tmp = (y * 0.5) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + y), $MachinePrecision], -5e-69], N[(N[(x * 0.5), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[N[(x + y), $MachinePrecision], 8e+48], N[(N[(z * -0.5), $MachinePrecision] / t), $MachinePrecision], N[(N[(y * 0.5), $MachinePrecision] / t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x y) < -5.00000000000000033e-69

   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. associate-*l/ N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 36.1

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

   5. Applied rewrites 36.1%

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

   if -5.00000000000000033e-69 < (+.f64 x y) < 8.00000000000000035e48

   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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 78.4

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

   5. Applied rewrites 78.4%

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

   if 8.00000000000000035e48 < (+.f64 x y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 45.5

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

   5. Applied rewrites 45.5%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -5 \cdot 10^{-69}:\\ \;\;\;\;\frac{x \cdot 0.5}{t}\\ \mathbf{elif}\;x + y \leq 8 \cdot 10^{+48}:\\ \;\;\;\;\frac{z \cdot -0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot 0.5}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 48.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -5 \cdot 10^{-69}:\\ \;\;\;\;\frac{x \cdot 0.5}{t}\\ \mathbf{elif}\;x + y \leq 8 \cdot 10^{+48}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot 0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) -5e-69)
   (/ (* x 0.5) t)
   (if (<= (+ x y) 8e+48) (* z (/ -0.5 t)) (/ (* y 0.5) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -5e-69) {
		tmp = (x * 0.5) / t;
	} else if ((x + y) <= 8e+48) {
		tmp = z * (-0.5 / t);
	} else {
		tmp = (y * 0.5) / t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x + y) <= (-5d-69)) then
        tmp = (x * 0.5d0) / t
    else if ((x + y) <= 8d+48) then
        tmp = z * ((-0.5d0) / t)
    else
        tmp = (y * 0.5d0) / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -5e-69) {
		tmp = (x * 0.5) / t;
	} else if ((x + y) <= 8e+48) {
		tmp = z * (-0.5 / t);
	} else {
		tmp = (y * 0.5) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + y) <= -5e-69:
		tmp = (x * 0.5) / t
	elif (x + y) <= 8e+48:
		tmp = z * (-0.5 / t)
	else:
		tmp = (y * 0.5) / t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + y) <= -5e-69)
		tmp = Float64(Float64(x * 0.5) / t);
	elseif (Float64(x + y) <= 8e+48)
		tmp = Float64(z * Float64(-0.5 / t));
	else
		tmp = Float64(Float64(y * 0.5) / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + y) <= -5e-69)
		tmp = (x * 0.5) / t;
	elseif ((x + y) <= 8e+48)
		tmp = z * (-0.5 / t);
	else
		tmp = (y * 0.5) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + y), $MachinePrecision], -5e-69], N[(N[(x * 0.5), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[N[(x + y), $MachinePrecision], 8e+48], N[(z * N[(-0.5 / t), $MachinePrecision]), $MachinePrecision], N[(N[(y * 0.5), $MachinePrecision] / t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x y) < -5.00000000000000033e-69

   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. associate-*l/ N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 36.1

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

   5. Applied rewrites 36.1%

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

   if -5.00000000000000033e-69 < (+.f64 x y) < 8.00000000000000035e48

   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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 78.4

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

   5. Applied rewrites 78.4%

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

   7. Applied rewrites 78.3%

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

   if 8.00000000000000035e48 < (+.f64 x y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 45.5

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

   5. Applied rewrites 45.5%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -5 \cdot 10^{-69}:\\ \;\;\;\;\frac{x \cdot 0.5}{t}\\ \mathbf{elif}\;x + y \leq 8 \cdot 10^{+48}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot 0.5}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-115}:\\ \;\;\;\;\frac{x - z}{t \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{t \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) -2e-115) (/ (- 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) <= -2e-115) {
		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) <= (-2d-115)) 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) <= -2e-115) {
		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) <= -2e-115:
		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) <= -2e-115)
		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) <= -2e-115)
		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], -2e-115], 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) < -2.0000000000000001e-115

   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 62.9

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

   5. Applied rewrites 62.9%

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

   if -2.0000000000000001e-115 < (+.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 73.2

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

   5. Applied rewrites 73.2%

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

Alternative 5: 69.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-115}:\\ \;\;\;\;\frac{x - z}{t \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{t} \cdot \left(z - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) -2e-115) (/ (- x z) (* t 2.0)) (* (/ -0.5 t) (- z y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -2e-115) {
		tmp = (x - z) / (t * 2.0);
	} else {
		tmp = (-0.5 / t) * (z - y);
	}
	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) <= (-2d-115)) then
        tmp = (x - z) / (t * 2.0d0)
    else
        tmp = ((-0.5d0) / t) * (z - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -2e-115) {
		tmp = (x - z) / (t * 2.0);
	} else {
		tmp = (-0.5 / t) * (z - y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + y) <= -2e-115:
		tmp = (x - z) / (t * 2.0)
	else:
		tmp = (-0.5 / t) * (z - y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + y) <= -2e-115)
		tmp = Float64(Float64(x - z) / Float64(t * 2.0));
	else
		tmp = Float64(Float64(-0.5 / t) * Float64(z - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + y) <= -2e-115)
		tmp = (x - z) / (t * 2.0);
	else
		tmp = (-0.5 / t) * (z - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + y), $MachinePrecision], -2e-115], N[(N[(x - z), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 / t), $MachinePrecision] * N[(z - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -2.0000000000000001e-115

   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 62.9

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

   5. Applied rewrites 62.9%

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

   if -2.0000000000000001e-115 < (+.f64 x y)

   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. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-/r* N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. neg-sub0 N/A

          \[\leadsto \frac{\frac{-1}{2}}{t} \cdot \color{blue}{\left(0 - \left(\left(x + y\right) - z\right)\right)} \]

      18. lift--.f64 N/A

          \[\leadsto \frac{\frac{-1}{2}}{t} \cdot \left(0 - \color{blue}{\left(\left(x + y\right) - z\right)}\right) \]

      19. sub-neg N/A

          \[\leadsto \frac{\frac{-1}{2}}{t} \cdot \left(0 - \color{blue}{\left(\left(x + y\right) + \left(\mathsf{neg}\left(z\right)\right)\right)}\right) \]

      20. +-commutative N/A

          \[\leadsto \frac{\frac{-1}{2}}{t} \cdot \left(0 - \color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) + \left(x + y\right)\right)}\right) \]

      21. associate--r+ N/A

          \[\leadsto \frac{\frac{-1}{2}}{t} \cdot \color{blue}{\left(\left(0 - \left(\mathsf{neg}\left(z\right)\right)\right) - \left(x + y\right)\right)} \]

      22. neg-sub0 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{-1}{2}}{t} \cdot \color{blue}{\left(z - y\right)} \]
   6. Step-by-step derivation

      1. lower--.f64 73.0

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

   7. Applied rewrites 73.0%

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

Alternative 6: 69.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-115}:\\ \;\;\;\;\left(x - z\right) \cdot \frac{0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{t} \cdot \left(z - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) -2e-115) (* (- x z) (/ 0.5 t)) (* (/ -0.5 t) (- z y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -2e-115) {
		tmp = (x - z) * (0.5 / t);
	} else {
		tmp = (-0.5 / t) * (z - y);
	}
	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) <= (-2d-115)) then
        tmp = (x - z) * (0.5d0 / t)
    else
        tmp = ((-0.5d0) / t) * (z - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -2e-115) {
		tmp = (x - z) * (0.5 / t);
	} else {
		tmp = (-0.5 / t) * (z - y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + y) <= -2e-115:
		tmp = (x - z) * (0.5 / t)
	else:
		tmp = (-0.5 / t) * (z - y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + y) <= -2e-115)
		tmp = Float64(Float64(x - z) * Float64(0.5 / t));
	else
		tmp = Float64(Float64(-0.5 / t) * Float64(z - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + y) <= -2e-115)
		tmp = (x - z) * (0.5 / t);
	else
		tmp = (-0.5 / t) * (z - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + y), $MachinePrecision], -2e-115], N[(N[(x - z), $MachinePrecision] * N[(0.5 / t), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 / t), $MachinePrecision] * N[(z - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -2.0000000000000001e-115

   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 62.9

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

   5. Applied rewrites 62.9%

      \[\leadsto \frac{\color{blue}{x - z}}{t \cdot 2} \]
   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. lower-*.f64 N/A

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

      9. lower-/.f64 62.7

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

   7. Applied rewrites 62.7%

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

   if -2.0000000000000001e-115 < (+.f64 x y)

   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. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-/r* N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. neg-sub0 N/A

          \[\leadsto \frac{\frac{-1}{2}}{t} \cdot \color{blue}{\left(0 - \left(\left(x + y\right) - z\right)\right)} \]

      18. lift--.f64 N/A

          \[\leadsto \frac{\frac{-1}{2}}{t} \cdot \left(0 - \color{blue}{\left(\left(x + y\right) - z\right)}\right) \]

      19. sub-neg N/A

          \[\leadsto \frac{\frac{-1}{2}}{t} \cdot \left(0 - \color{blue}{\left(\left(x + y\right) + \left(\mathsf{neg}\left(z\right)\right)\right)}\right) \]

      20. +-commutative N/A

          \[\leadsto \frac{\frac{-1}{2}}{t} \cdot \left(0 - \color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) + \left(x + y\right)\right)}\right) \]

      21. associate--r+ N/A

          \[\leadsto \frac{\frac{-1}{2}}{t} \cdot \color{blue}{\left(\left(0 - \left(\mathsf{neg}\left(z\right)\right)\right) - \left(x + y\right)\right)} \]

      22. neg-sub0 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{-1}{2}}{t} \cdot \color{blue}{\left(z - y\right)} \]
   6. Step-by-step derivation

      1. lower--.f64 73.0

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

   7. Applied rewrites 73.0%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-115}:\\ \;\;\;\;\left(x - z\right) \cdot \frac{0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{t} \cdot \left(z - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -5 \cdot 10^{-69}:\\ \;\;\;\;\frac{x \cdot 0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{t} \cdot \left(z - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) -5e-69) (/ (* x 0.5) t) (* (/ -0.5 t) (- z y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -5e-69) {
		tmp = (x * 0.5) / t;
	} else {
		tmp = (-0.5 / t) * (z - y);
	}
	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-69)) then
        tmp = (x * 0.5d0) / t
    else
        tmp = ((-0.5d0) / t) * (z - y)
    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-69) {
		tmp = (x * 0.5) / t;
	} else {
		tmp = (-0.5 / t) * (z - y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + y) <= -5e-69:
		tmp = (x * 0.5) / t
	else:
		tmp = (-0.5 / t) * (z - y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + y) <= -5e-69)
		tmp = Float64(Float64(x * 0.5) / t);
	else
		tmp = Float64(Float64(-0.5 / t) * Float64(z - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + y) <= -5e-69)
		tmp = (x * 0.5) / t;
	else
		tmp = (-0.5 / t) * (z - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + y), $MachinePrecision], -5e-69], N[(N[(x * 0.5), $MachinePrecision] / t), $MachinePrecision], N[(N[(-0.5 / t), $MachinePrecision] * N[(z - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -5.00000000000000033e-69

   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. associate-*l/ N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 36.1

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

   5. Applied rewrites 36.1%

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

   if -5.00000000000000033e-69 < (+.f64 x y)

   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. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-/r* N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. neg-sub0 N/A

          \[\leadsto \frac{\frac{-1}{2}}{t} \cdot \color{blue}{\left(0 - \left(\left(x + y\right) - z\right)\right)} \]

      18. lift--.f64 N/A

          \[\leadsto \frac{\frac{-1}{2}}{t} \cdot \left(0 - \color{blue}{\left(\left(x + y\right) - z\right)}\right) \]

      19. sub-neg N/A

          \[\leadsto \frac{\frac{-1}{2}}{t} \cdot \left(0 - \color{blue}{\left(\left(x + y\right) + \left(\mathsf{neg}\left(z\right)\right)\right)}\right) \]

      20. +-commutative N/A

          \[\leadsto \frac{\frac{-1}{2}}{t} \cdot \left(0 - \color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) + \left(x + y\right)\right)}\right) \]

      21. associate--r+ N/A

          \[\leadsto \frac{\frac{-1}{2}}{t} \cdot \color{blue}{\left(\left(0 - \left(\mathsf{neg}\left(z\right)\right)\right) - \left(x + y\right)\right)} \]

      22. neg-sub0 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{-1}{2}}{t} \cdot \color{blue}{\left(z - y\right)} \]
   6. Step-by-step derivation

      1. lower--.f64 73.8

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

   7. Applied rewrites 73.8%

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

4. Final simplification 57.9%

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

Alternative 8: 42.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -5 \cdot 10^{-69}:\\ \;\;\;\;\frac{x \cdot 0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) -5e-69) (/ (* x 0.5) t) (* z (/ -0.5 t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -5e-69) {
		tmp = (x * 0.5) / t;
	} else {
		tmp = z * (-0.5 / t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x + y) <= (-5d-69)) then
        tmp = (x * 0.5d0) / t
    else
        tmp = z * ((-0.5d0) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -5e-69) {
		tmp = (x * 0.5) / t;
	} else {
		tmp = z * (-0.5 / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + y) <= -5e-69:
		tmp = (x * 0.5) / t
	else:
		tmp = z * (-0.5 / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + y) <= -5e-69)
		tmp = Float64(Float64(x * 0.5) / t);
	else
		tmp = Float64(z * Float64(-0.5 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + y) <= -5e-69)
		tmp = (x * 0.5) / t;
	else
		tmp = z * (-0.5 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + y), $MachinePrecision], -5e-69], N[(N[(x * 0.5), $MachinePrecision] / t), $MachinePrecision], N[(z * N[(-0.5 / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -5.00000000000000033e-69

   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. associate-*l/ N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 36.1

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

   5. Applied rewrites 36.1%

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

   if -5.00000000000000033e-69 < (+.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 inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 45.4

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

   5. Applied rewrites 45.4%

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

   7. Applied rewrites 45.4%

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

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -5 \cdot 10^{-69}:\\ \;\;\;\;\frac{x \cdot 0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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) * (z - (x + y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(-0.5 / t), $MachinePrecision] * N[(z - N[(x + y), $MachinePrecision]), $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. frac-2neg N/A

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

   4. associate-/r/ N/A

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

   5. lower-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. associate-/r* N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f64 N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. neg-sub0 N/A

       \[\leadsto \frac{\frac{-1}{2}}{t} \cdot \color{blue}{\left(0 - \left(\left(x + y\right) - z\right)\right)} \]

   18. lift--.f64 N/A

       \[\leadsto \frac{\frac{-1}{2}}{t} \cdot \left(0 - \color{blue}{\left(\left(x + y\right) - z\right)}\right) \]

   19. sub-neg N/A

       \[\leadsto \frac{\frac{-1}{2}}{t} \cdot \left(0 - \color{blue}{\left(\left(x + y\right) + \left(\mathsf{neg}\left(z\right)\right)\right)}\right) \]

   20. +-commutative N/A

       \[\leadsto \frac{\frac{-1}{2}}{t} \cdot \left(0 - \color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) + \left(x + y\right)\right)}\right) \]

   21. associate--r+ N/A

       \[\leadsto \frac{\frac{-1}{2}}{t} \cdot \color{blue}{\left(\left(0 - \left(\mathsf{neg}\left(z\right)\right)\right) - \left(x + y\right)\right)} \]

   22. neg-sub0 N/A

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

   23. remove-double-neg N/A

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

   24. lower--.f64 99.7

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

4. Applied rewrites 99.7%

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

Alternative 10: 38.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in z around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 38.4

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

5. Applied rewrites 38.4%

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

7. Applied rewrites 38.3%

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

8. Final simplification 38.3%

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

Reproduce

herbie shell --seed 2024228 
(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)))