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

Percentage Accurate: 99.9% → 99.9%

Time: 9.2s

Alternatives: 12

Speedup: 1.0×

• 26.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

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 99.9%

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

Alternative 2: 49.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-19}:\\ \;\;\;\;\frac{x \cdot 0.5}{t}\\ \mathbf{elif}\;x + y \leq 0.005:\\ \;\;\;\;\frac{z \cdot -0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot 0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) -2e-19)
   (/ (* x 0.5) t)
   (if (<= (+ x y) 0.005) (/ (* z -0.5) t) (/ (* y 0.5) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -2e-19) {
		tmp = (x * 0.5) / t;
	} else if ((x + y) <= 0.005) {
		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) <= (-2d-19)) then
        tmp = (x * 0.5d0) / t
    else if ((x + y) <= 0.005d0) 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) <= -2e-19) {
		tmp = (x * 0.5) / t;
	} else if ((x + y) <= 0.005) {
		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) <= -2e-19:
		tmp = (x * 0.5) / t
	elif (x + y) <= 0.005:
		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) <= -2e-19)
		tmp = Float64(Float64(x * 0.5) / t);
	elseif (Float64(x + y) <= 0.005)
		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) <= -2e-19)
		tmp = (x * 0.5) / t;
	elseif ((x + y) <= 0.005)
		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], -2e-19], N[(N[(x * 0.5), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[N[(x + y), $MachinePrecision], 0.005], 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) < -2e-19

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. 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 50.1

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

   5. Simplified 50.1%

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

   if -2e-19 < (+.f64 x y) < 0.0050000000000000001

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. 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 72.9

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

   5. Simplified 72.9%

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

   if 0.0050000000000000001 < (+.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 44.7

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

   5. Simplified 44.7%

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

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-19}:\\ \;\;\;\;\frac{x \cdot 0.5}{t}\\ \mathbf{elif}\;x + y \leq 0.005:\\ \;\;\;\;\frac{z \cdot -0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot 0.5}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 49.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-19}:\\ \;\;\;\;\frac{x \cdot 0.5}{t}\\ \mathbf{elif}\;x + y \leq 0.005:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot 0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) -2e-19)
   (/ (* x 0.5) t)
   (if (<= (+ x y) 0.005) (* z (/ -0.5 t)) (/ (* y 0.5) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -2e-19) {
		tmp = (x * 0.5) / t;
	} else if ((x + y) <= 0.005) {
		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) <= (-2d-19)) then
        tmp = (x * 0.5d0) / t
    else if ((x + y) <= 0.005d0) 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) <= -2e-19) {
		tmp = (x * 0.5) / t;
	} else if ((x + y) <= 0.005) {
		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) <= -2e-19:
		tmp = (x * 0.5) / t
	elif (x + y) <= 0.005:
		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) <= -2e-19)
		tmp = Float64(Float64(x * 0.5) / t);
	elseif (Float64(x + y) <= 0.005)
		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) <= -2e-19)
		tmp = (x * 0.5) / t;
	elseif ((x + y) <= 0.005)
		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], -2e-19], N[(N[(x * 0.5), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[N[(x + y), $MachinePrecision], 0.005], 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) < -2e-19

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. 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 50.1

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

   5. Simplified 50.1%

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

   if -2e-19 < (+.f64 x y) < 0.0050000000000000001

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. 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 72.9

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

   5. Simplified 72.9%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 72.8

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

   7. Applied egg-rr 72.8%

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

   if 0.0050000000000000001 < (+.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 44.7

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

   5. Simplified 44.7%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-19}:\\ \;\;\;\;\frac{x \cdot 0.5}{t}\\ \mathbf{elif}\;x + y \leq 0.005:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot 0.5}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 49.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-19}:\\ \;\;\;\;\frac{x \cdot 0.5}{t}\\ \mathbf{elif}\;x + y \leq 0.005:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) -2e-19)
   (/ (* x 0.5) t)
   (if (<= (+ x y) 0.005) (* z (/ -0.5 t)) (* y (/ 0.5 t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -2e-19) {
		tmp = (x * 0.5) / t;
	} else if ((x + y) <= 0.005) {
		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) <= (-2d-19)) then
        tmp = (x * 0.5d0) / t
    else if ((x + y) <= 0.005d0) 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) <= -2e-19) {
		tmp = (x * 0.5) / t;
	} else if ((x + y) <= 0.005) {
		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) <= -2e-19:
		tmp = (x * 0.5) / t
	elif (x + y) <= 0.005:
		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) <= -2e-19)
		tmp = Float64(Float64(x * 0.5) / t);
	elseif (Float64(x + y) <= 0.005)
		tmp = Float64(z * Float64(-0.5 / t));
	else
		tmp = Float64(y * Float64(0.5 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + y) <= -2e-19)
		tmp = (x * 0.5) / t;
	elseif ((x + y) <= 0.005)
		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], -2e-19], N[(N[(x * 0.5), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[N[(x + y), $MachinePrecision], 0.005], N[(z * N[(-0.5 / t), $MachinePrecision]), $MachinePrecision], N[(y * N[(0.5 / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x y) < -2e-19

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. 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 50.1

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

   5. Simplified 50.1%

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

   if -2e-19 < (+.f64 x y) < 0.0050000000000000001

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. 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 72.9

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

   5. Simplified 72.9%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 72.8

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

   7. Applied egg-rr 72.8%

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

   if 0.0050000000000000001 < (+.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 44.7

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

   5. Simplified 44.7%

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

      1. lift-*.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. div-inv N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

      10. div-inv N/A

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

      11. clear-num N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 44.5

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

   7. Applied egg-rr 44.5%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-19}:\\ \;\;\;\;\frac{x \cdot 0.5}{t}\\ \mathbf{elif}\;x + y \leq 0.005:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{0.5}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 48.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-19}:\\ \;\;\;\;x \cdot \frac{0.5}{t}\\ \mathbf{elif}\;x + y \leq 0.005:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) -2e-19)
   (* x (/ 0.5 t))
   (if (<= (+ x y) 0.005) (* z (/ -0.5 t)) (* y (/ 0.5 t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -2e-19) {
		tmp = x * (0.5 / t);
	} else if ((x + y) <= 0.005) {
		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) <= (-2d-19)) then
        tmp = x * (0.5d0 / t)
    else if ((x + y) <= 0.005d0) 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) <= -2e-19) {
		tmp = x * (0.5 / t);
	} else if ((x + y) <= 0.005) {
		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) <= -2e-19:
		tmp = x * (0.5 / t)
	elif (x + y) <= 0.005:
		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) <= -2e-19)
		tmp = Float64(x * Float64(0.5 / t));
	elseif (Float64(x + y) <= 0.005)
		tmp = Float64(z * Float64(-0.5 / t));
	else
		tmp = Float64(y * Float64(0.5 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + y) <= -2e-19)
		tmp = x * (0.5 / t);
	elseif ((x + y) <= 0.005)
		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], -2e-19], N[(x * N[(0.5 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x + y), $MachinePrecision], 0.005], N[(z * N[(-0.5 / t), $MachinePrecision]), $MachinePrecision], N[(y * N[(0.5 / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x y) < -2e-19

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. 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 50.1

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

   5. Simplified 50.1%

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

      1. associate-*l/ N/A

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

      2. lift-/.f64 N/A

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

      3. lower-*.f64 50.0

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

   7. Applied egg-rr 50.0%

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

   if -2e-19 < (+.f64 x y) < 0.0050000000000000001

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. 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 72.9

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

   5. Simplified 72.9%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 72.8

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

   7. Applied egg-rr 72.8%

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

   if 0.0050000000000000001 < (+.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 44.7

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

   5. Simplified 44.7%

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

      1. lift-*.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. div-inv N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

      10. div-inv N/A

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

      11. clear-num N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 44.5

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

   7. Applied egg-rr 44.5%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-19}:\\ \;\;\;\;x \cdot \frac{0.5}{t}\\ \mathbf{elif}\;x + y \leq 0.005:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{0.5}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-95}:\\ \;\;\;\;\frac{x - z}{t \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{t \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) -1e-95) (/ (- x z) (* t 2.0)) (/ (- y z) (* t 2.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -1e-95) {
		tmp = (x - z) / (t * 2.0);
	} else {
		tmp = (y - z) / (t * 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x + y) <= (-1d-95)) then
        tmp = (x - z) / (t * 2.0d0)
    else
        tmp = (y - z) / (t * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -1e-95) {
		tmp = (x - z) / (t * 2.0);
	} else {
		tmp = (y - z) / (t * 2.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + y) <= -1e-95:
		tmp = (x - z) / (t * 2.0)
	else:
		tmp = (y - z) / (t * 2.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + y) <= -1e-95)
		tmp = Float64(Float64(x - z) / Float64(t * 2.0));
	else
		tmp = Float64(Float64(y - z) / Float64(t * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + y) <= -1e-95)
		tmp = (x - z) / (t * 2.0);
	else
		tmp = (y - z) / (t * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + y), $MachinePrecision], -1e-95], N[(N[(x - z), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(y - z), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -9.99999999999999989e-96

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 71.3

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

   5. Simplified 71.3%

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

   if -9.99999999999999989e-96 < (+.f64 x y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 68.4

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

   5. Simplified 68.4%

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

Alternative 7: 75.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq 0.005:\\ \;\;\;\;\frac{x - z}{t \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{t \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) 0.005) (/ (- x z) (* t 2.0)) (/ (+ x y) (* t 2.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= 0.005) {
		tmp = (x - z) / (t * 2.0);
	} else {
		tmp = (x + y) / (t * 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x + y) <= 0.005d0) then
        tmp = (x - z) / (t * 2.0d0)
    else
        tmp = (x + y) / (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) <= 0.005) {
		tmp = (x - z) / (t * 2.0);
	} else {
		tmp = (x + y) / (t * 2.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + y) <= 0.005:
		tmp = (x - z) / (t * 2.0)
	else:
		tmp = (x + y) / (t * 2.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + y) <= 0.005)
		tmp = Float64(Float64(x - z) / Float64(t * 2.0));
	else
		tmp = Float64(Float64(x + y) / Float64(t * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + y) <= 0.005)
		tmp = (x - z) / (t * 2.0);
	else
		tmp = (x + y) / (t * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + y), $MachinePrecision], 0.005], N[(N[(x - z), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(x + y), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < 0.0050000000000000001

   1. Initial program 99.9%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   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 75.8

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

   5. Simplified 75.8%

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

   if 0.0050000000000000001 < (+.f64 x y)

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 85.9

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

   5. Simplified 85.9%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq 0.005:\\ \;\;\;\;\frac{x - z}{t \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{t \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq 0.005:\\ \;\;\;\;\frac{0.5}{t} \cdot \left(x - z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{t \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) 0.005) (* (/ 0.5 t) (- x z)) (/ (+ x y) (* t 2.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= 0.005) {
		tmp = (0.5 / t) * (x - z);
	} else {
		tmp = (x + y) / (t * 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x + y) <= 0.005d0) then
        tmp = (0.5d0 / t) * (x - z)
    else
        tmp = (x + y) / (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) <= 0.005) {
		tmp = (0.5 / t) * (x - z);
	} else {
		tmp = (x + y) / (t * 2.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + y) <= 0.005:
		tmp = (0.5 / t) * (x - z)
	else:
		tmp = (x + y) / (t * 2.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + y) <= 0.005)
		tmp = Float64(Float64(0.5 / t) * Float64(x - z));
	else
		tmp = Float64(Float64(x + y) / Float64(t * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + y) <= 0.005)
		tmp = (0.5 / t) * (x - z);
	else
		tmp = (x + y) / (t * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + y), $MachinePrecision], 0.005], N[(N[(0.5 / t), $MachinePrecision] * N[(x - z), $MachinePrecision]), $MachinePrecision], N[(N[(x + y), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < 0.0050000000000000001

   1. Initial program 99.9%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   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 75.8

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

   5. Simplified 75.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. lower-*.f64 75.7

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

   7. Applied egg-rr 75.7%

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

   if 0.0050000000000000001 < (+.f64 x y)

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 85.9

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

   5. Simplified 85.9%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq 0.005:\\ \;\;\;\;\frac{0.5}{t} \cdot \left(x - z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{t \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq 0.005:\\ \;\;\;\;\frac{0.5}{t} \cdot \left(x - z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot 0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) 0.005) (* (/ 0.5 t) (- x z)) (/ (* y 0.5) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= 0.005) {
		tmp = (0.5 / t) * (x - z);
	} 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) <= 0.005d0) then
        tmp = (0.5d0 / t) * (x - z)
    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) <= 0.005) {
		tmp = (0.5 / t) * (x - z);
	} else {
		tmp = (y * 0.5) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + y) <= 0.005:
		tmp = (0.5 / t) * (x - z)
	else:
		tmp = (y * 0.5) / t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + y) <= 0.005)
		tmp = Float64(Float64(0.5 / t) * Float64(x - z));
	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) <= 0.005)
		tmp = (0.5 / t) * (x - z);
	else
		tmp = (y * 0.5) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + y), $MachinePrecision], 0.005], N[(N[(0.5 / t), $MachinePrecision] * N[(x - z), $MachinePrecision]), $MachinePrecision], N[(N[(y * 0.5), $MachinePrecision] / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < 0.0050000000000000001

   1. Initial program 99.9%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   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 75.8

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

   5. Simplified 75.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. lower-*.f64 75.7

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

   7. Applied egg-rr 75.7%

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

   if 0.0050000000000000001 < (+.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 44.7

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

   5. Simplified 44.7%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq 0.005:\\ \;\;\;\;\frac{0.5}{t} \cdot \left(x - z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot 0.5}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 44.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-19}:\\ \;\;\;\;x \cdot \frac{0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) -2e-19) (* x (/ 0.5 t)) (* z (/ -0.5 t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -2e-19) {
		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) <= (-2d-19)) 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) <= -2e-19) {
		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) <= -2e-19:
		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) <= -2e-19)
		tmp = Float64(x * Float64(0.5 / t));
	else
		tmp = Float64(z * Float64(-0.5 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + y) <= -2e-19)
		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], -2e-19], N[(x * N[(0.5 / t), $MachinePrecision]), $MachinePrecision], N[(z * N[(-0.5 / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -2e-19

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. 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 50.1

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

   5. Simplified 50.1%

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

      1. associate-*l/ N/A

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

      2. lift-/.f64 N/A

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

      3. lower-*.f64 50.0

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

   7. Applied egg-rr 50.0%

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

   if -2e-19 < (+.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 39.0

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

   5. Simplified 39.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 38.9

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

   7. Applied egg-rr 38.9%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-19}:\\ \;\;\;\;x \cdot \frac{0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. div-inv N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

   11. metadata-eval 99.7

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

   12. lift--.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. associate--l+ N/A

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

   15. lower-+.f64 N/A

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

   16. lower--.f64 99.7

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

4. Applied egg-rr 99.7%

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

Alternative 12: 38.5% 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 99.9%

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

3. Taylor expanded in z around inf

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

   1. 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 32.2

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

5. Simplified 32.2%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 32.1

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

7. Applied egg-rr 32.1%

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

8. Final simplification 32.1%

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

Reproduce

herbie shell --seed 2024207 
(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)))