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

Percentage Accurate: 99.9% → 99.9%

Time: 7.9s

Alternatives: 11

Speedup: 1.0×

• 26.4% 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: 87.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.75 \cdot 10^{+14} \lor \neg \left(z \leq 5.3 \cdot 10^{+76}\right):\\ \;\;\;\;\left(y - z\right) \cdot \frac{0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{t \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -2.75e+14) (not (<= z 5.3e+76)))
   (* (- y z) (/ 0.5 t))
   (/ (+ x y) (* t 2.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.75e+14) || !(z <= 5.3e+76)) {
		tmp = (y - z) * (0.5 / t);
	} 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 ((z <= (-2.75d+14)) .or. (.not. (z <= 5.3d+76))) then
        tmp = (y - z) * (0.5d0 / t)
    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 ((z <= -2.75e+14) || !(z <= 5.3e+76)) {
		tmp = (y - z) * (0.5 / t);
	} else {
		tmp = (x + y) / (t * 2.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -2.75e+14) or not (z <= 5.3e+76):
		tmp = (y - z) * (0.5 / t)
	else:
		tmp = (x + y) / (t * 2.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -2.75e+14) || !(z <= 5.3e+76))
		tmp = Float64(Float64(y - z) * Float64(0.5 / t));
	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 ((z <= -2.75e+14) || ~((z <= 5.3e+76)))
		tmp = (y - z) * (0.5 / t);
	else
		tmp = (x + y) / (t * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -2.75e+14], N[Not[LessEqual[z, 5.3e+76]], $MachinePrecision]], N[(N[(y - z), $MachinePrecision] * N[(0.5 / t), $MachinePrecision]), $MachinePrecision], N[(N[(x + y), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.75e14 or 5.30000000000000015e76 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 81.7%

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

      1. *-commutative 81.7%

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

      2. associate-*l/ 81.7%

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

      3. associate-*r/ 81.6%

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

   5. Simplified 81.6%

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

   if -2.75e14 < z < 5.30000000000000015e76

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 90.4%

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

      1. +-commutative 90.4%

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

   5. Simplified 90.4%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.75 \cdot 10^{+14} \lor \neg \left(z \leq 5.3 \cdot 10^{+76}\right):\\ \;\;\;\;\left(y - z\right) \cdot \frac{0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{t \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 46.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-275}:\\ \;\;\;\;\frac{x}{t \cdot 2}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+41}:\\ \;\;\;\;\frac{z \cdot -0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -8.5e-275)
   (/ x (* t 2.0))
   (if (<= y 1.6e+41) (/ (* z -0.5) t) (/ y (* t 2.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -8.5e-275) {
		tmp = x / (t * 2.0);
	} else if (y <= 1.6e+41) {
		tmp = (z * -0.5) / t;
	} else {
		tmp = 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 (y <= (-8.5d-275)) then
        tmp = x / (t * 2.0d0)
    else if (y <= 1.6d+41) then
        tmp = (z * (-0.5d0)) / t
    else
        tmp = 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 (y <= -8.5e-275) {
		tmp = x / (t * 2.0);
	} else if (y <= 1.6e+41) {
		tmp = (z * -0.5) / t;
	} else {
		tmp = y / (t * 2.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -8.5e-275:
		tmp = x / (t * 2.0)
	elif y <= 1.6e+41:
		tmp = (z * -0.5) / t
	else:
		tmp = y / (t * 2.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -8.5e-275)
		tmp = Float64(x / Float64(t * 2.0));
	elseif (y <= 1.6e+41)
		tmp = Float64(Float64(z * -0.5) / t);
	else
		tmp = Float64(y / Float64(t * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -8.5e-275)
		tmp = x / (t * 2.0);
	elseif (y <= 1.6e+41)
		tmp = (z * -0.5) / t;
	else
		tmp = y / (t * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -8.5e-275], N[(x / N[(t * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.6e+41], N[(N[(z * -0.5), $MachinePrecision] / t), $MachinePrecision], N[(y / N[(t * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.49999999999999952e-275

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 35.1%

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

   if -8.49999999999999952e-275 < y < 1.60000000000000005e41

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 50.0%

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

      1. *-commutative 50.0%

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

      2. associate-*l/ 50.0%

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

   5. Simplified 50.0%

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

   if 1.60000000000000005e41 < 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 63.3%

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

Alternative 4: 46.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-275}:\\ \;\;\;\;\frac{x}{t \cdot 2}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+38}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -8.5e-275)
   (/ x (* t 2.0))
   (if (<= y 1.35e+38) (* z (/ -0.5 t)) (/ y (* t 2.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -8.5e-275) {
		tmp = x / (t * 2.0);
	} else if (y <= 1.35e+38) {
		tmp = z * (-0.5 / t);
	} else {
		tmp = 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 (y <= (-8.5d-275)) then
        tmp = x / (t * 2.0d0)
    else if (y <= 1.35d+38) then
        tmp = z * ((-0.5d0) / t)
    else
        tmp = 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 (y <= -8.5e-275) {
		tmp = x / (t * 2.0);
	} else if (y <= 1.35e+38) {
		tmp = z * (-0.5 / t);
	} else {
		tmp = y / (t * 2.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -8.5e-275:
		tmp = x / (t * 2.0)
	elif y <= 1.35e+38:
		tmp = z * (-0.5 / t)
	else:
		tmp = y / (t * 2.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -8.5e-275)
		tmp = Float64(x / Float64(t * 2.0));
	elseif (y <= 1.35e+38)
		tmp = Float64(z * Float64(-0.5 / t));
	else
		tmp = Float64(y / Float64(t * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -8.5e-275)
		tmp = x / (t * 2.0);
	elseif (y <= 1.35e+38)
		tmp = z * (-0.5 / t);
	else
		tmp = y / (t * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -8.5e-275], N[(x / N[(t * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.35e+38], N[(z * N[(-0.5 / t), $MachinePrecision]), $MachinePrecision], N[(y / N[(t * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.49999999999999952e-275

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 35.1%

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

   if -8.49999999999999952e-275 < y < 1.34999999999999998e38

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 50.6%

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

      1. *-commutative 50.6%

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

      2. associate-*l/ 50.6%

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

   5. Simplified 50.6%

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

      1. *-commutative 50.6%

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

      2. associate-*l/ 50.4%

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

   7. Applied egg-rr 50.4%

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

   if 1.34999999999999998e38 < 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 62.5%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-275}:\\ \;\;\;\;\frac{x}{t \cdot 2}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+38}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 46.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-275}:\\ \;\;\;\;\frac{x}{t \cdot 2}\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+38}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -8.5e-275)
   (/ x (* t 2.0))
   (if (<= y 9.2e+38) (* z (/ -0.5 t)) (* y (/ 0.5 t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -8.5e-275) {
		tmp = x / (t * 2.0);
	} else if (y <= 9.2e+38) {
		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 (y <= (-8.5d-275)) then
        tmp = x / (t * 2.0d0)
    else if (y <= 9.2d+38) 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 (y <= -8.5e-275) {
		tmp = x / (t * 2.0);
	} else if (y <= 9.2e+38) {
		tmp = z * (-0.5 / t);
	} else {
		tmp = y * (0.5 / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -8.5e-275:
		tmp = x / (t * 2.0)
	elif y <= 9.2e+38:
		tmp = z * (-0.5 / t)
	else:
		tmp = y * (0.5 / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -8.5e-275)
		tmp = Float64(x / Float64(t * 2.0));
	elseif (y <= 9.2e+38)
		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 (y <= -8.5e-275)
		tmp = x / (t * 2.0);
	elseif (y <= 9.2e+38)
		tmp = z * (-0.5 / t);
	else
		tmp = y * (0.5 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -8.5e-275], N[(x / N[(t * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.2e+38], 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 y < -8.49999999999999952e-275

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 35.1%

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

   if -8.49999999999999952e-275 < y < 9.2000000000000005e38

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 50.0%

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

      1. *-commutative 50.0%

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

      2. associate-*l/ 50.0%

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

   5. Simplified 50.0%

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

      1. *-commutative 50.0%

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

      2. associate-*l/ 49.9%

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

   7. Applied egg-rr 49.9%

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

   if 9.2000000000000005e38 < 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 89.0%

      \[\leadsto \color{blue}{z \cdot \left(0.5 \cdot \frac{\frac{x}{t} + \frac{y}{t}}{z} - 0.5 \cdot \frac{1}{t}\right)} \]
   4. Step-by-step derivation

      1. sub-neg 89.0%

         \[\leadsto z \cdot \color{blue}{\left(0.5 \cdot \frac{\frac{x}{t} + \frac{y}{t}}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right)} \]

      2. associate-*r/ 89.0%

         \[\leadsto z \cdot \left(\color{blue}{\frac{0.5 \cdot \left(\frac{x}{t} + \frac{y}{t}\right)}{z}} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      3. distribute-rgt-in 89.0%

         \[\leadsto z \cdot \left(\frac{\color{blue}{\frac{x}{t} \cdot 0.5 + \frac{y}{t} \cdot 0.5}}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      4. associate-*l/ 89.0%

         \[\leadsto z \cdot \left(\frac{\color{blue}{\frac{x \cdot 0.5}{t}} + \frac{y}{t} \cdot 0.5}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      5. associate-*r/ 89.0%

         \[\leadsto z \cdot \left(\frac{\color{blue}{x \cdot \frac{0.5}{t}} + \frac{y}{t} \cdot 0.5}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      6. metadata-eval 89.0%

         \[\leadsto z \cdot \left(\frac{x \cdot \frac{\color{blue}{0.5 \cdot 1}}{t} + \frac{y}{t} \cdot 0.5}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      7. associate-*r/ 89.0%

         \[\leadsto z \cdot \left(\frac{x \cdot \color{blue}{\left(0.5 \cdot \frac{1}{t}\right)} + \frac{y}{t} \cdot 0.5}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      8. associate-*l/ 89.0%

         \[\leadsto z \cdot \left(\frac{x \cdot \left(0.5 \cdot \frac{1}{t}\right) + \color{blue}{\frac{y \cdot 0.5}{t}}}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      9. associate-*r/ 88.8%

         \[\leadsto z \cdot \left(\frac{x \cdot \left(0.5 \cdot \frac{1}{t}\right) + \color{blue}{y \cdot \frac{0.5}{t}}}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      10. metadata-eval 88.8%

          \[\leadsto z \cdot \left(\frac{x \cdot \left(0.5 \cdot \frac{1}{t}\right) + y \cdot \frac{\color{blue}{0.5 \cdot 1}}{t}}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      11. associate-*r/ 88.8%

          \[\leadsto z \cdot \left(\frac{x \cdot \left(0.5 \cdot \frac{1}{t}\right) + y \cdot \color{blue}{\left(0.5 \cdot \frac{1}{t}\right)}}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      12. distribute-rgt-in 90.3%

          \[\leadsto z \cdot \left(\frac{\color{blue}{\left(0.5 \cdot \frac{1}{t}\right) \cdot \left(x + y\right)}}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      13. associate-*r/ 90.3%

          \[\leadsto z \cdot \left(\frac{\color{blue}{\frac{0.5 \cdot 1}{t}} \cdot \left(x + y\right)}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      14. metadata-eval 90.3%

          \[\leadsto z \cdot \left(\frac{\frac{\color{blue}{0.5}}{t} \cdot \left(x + y\right)}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      15. associate-*l/ 90.5%

          \[\leadsto z \cdot \left(\frac{\color{blue}{\frac{0.5 \cdot \left(x + y\right)}{t}}}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      16. associate-*r/ 90.5%

          \[\leadsto z \cdot \left(\frac{\color{blue}{0.5 \cdot \frac{x + y}{t}}}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      17. +-commutative 90.5%

          \[\leadsto z \cdot \left(\frac{0.5 \cdot \frac{\color{blue}{y + x}}{t}}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      18. associate-*r/ 90.5%

          \[\leadsto z \cdot \left(\frac{0.5 \cdot \frac{y + x}{t}}{z} + \left(-\color{blue}{\frac{0.5 \cdot 1}{t}}\right)\right) \]

      19. metadata-eval 90.5%

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

   5. Simplified 90.5%

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

   6. Taylor expanded in t around 0 80.9%

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

   7. Taylor expanded in y around inf 63.3%

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

      1. *-commutative 63.3%

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

      2. associate-*l/ 63.3%

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

      3. associate-*r/ 63.1%

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

   9. Simplified 63.1%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-275}:\\ \;\;\;\;\frac{x}{t \cdot 2}\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+38}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{0.5}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 46.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-275}:\\ \;\;\;\;x \cdot \frac{0.5}{t}\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+40}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -8.5e-275)
   (* x (/ 0.5 t))
   (if (<= y 3.1e+40) (* z (/ -0.5 t)) (* y (/ 0.5 t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -8.5e-275) {
		tmp = x * (0.5 / t);
	} else if (y <= 3.1e+40) {
		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 (y <= (-8.5d-275)) then
        tmp = x * (0.5d0 / t)
    else if (y <= 3.1d+40) 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 (y <= -8.5e-275) {
		tmp = x * (0.5 / t);
	} else if (y <= 3.1e+40) {
		tmp = z * (-0.5 / t);
	} else {
		tmp = y * (0.5 / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -8.5e-275:
		tmp = x * (0.5 / t)
	elif y <= 3.1e+40:
		tmp = z * (-0.5 / t)
	else:
		tmp = y * (0.5 / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -8.5e-275)
		tmp = Float64(x * Float64(0.5 / t));
	elseif (y <= 3.1e+40)
		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 (y <= -8.5e-275)
		tmp = x * (0.5 / t);
	elseif (y <= 3.1e+40)
		tmp = z * (-0.5 / t);
	else
		tmp = y * (0.5 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -8.5e-275], N[(x * N[(0.5 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.1e+40], 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 y < -8.49999999999999952e-275

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 35.1%

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

      1. clear-num 35.0%

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

      2. associate-/r/ 35.0%

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

      3. *-commutative 35.0%

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

      4. associate-/r* 35.0%

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

      5. metadata-eval 35.0%

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

   5. Applied egg-rr 35.0%

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

   if -8.49999999999999952e-275 < y < 3.0999999999999998e40

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 50.0%

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

      1. *-commutative 50.0%

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

      2. associate-*l/ 50.0%

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

   5. Simplified 50.0%

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

      1. *-commutative 50.0%

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

      2. associate-*l/ 49.9%

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

   7. Applied egg-rr 49.9%

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

   if 3.0999999999999998e40 < 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 89.0%

      \[\leadsto \color{blue}{z \cdot \left(0.5 \cdot \frac{\frac{x}{t} + \frac{y}{t}}{z} - 0.5 \cdot \frac{1}{t}\right)} \]
   4. Step-by-step derivation

      1. sub-neg 89.0%

         \[\leadsto z \cdot \color{blue}{\left(0.5 \cdot \frac{\frac{x}{t} + \frac{y}{t}}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right)} \]

      2. associate-*r/ 89.0%

         \[\leadsto z \cdot \left(\color{blue}{\frac{0.5 \cdot \left(\frac{x}{t} + \frac{y}{t}\right)}{z}} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      3. distribute-rgt-in 89.0%

         \[\leadsto z \cdot \left(\frac{\color{blue}{\frac{x}{t} \cdot 0.5 + \frac{y}{t} \cdot 0.5}}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      4. associate-*l/ 89.0%

         \[\leadsto z \cdot \left(\frac{\color{blue}{\frac{x \cdot 0.5}{t}} + \frac{y}{t} \cdot 0.5}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      5. associate-*r/ 89.0%

         \[\leadsto z \cdot \left(\frac{\color{blue}{x \cdot \frac{0.5}{t}} + \frac{y}{t} \cdot 0.5}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      6. metadata-eval 89.0%

         \[\leadsto z \cdot \left(\frac{x \cdot \frac{\color{blue}{0.5 \cdot 1}}{t} + \frac{y}{t} \cdot 0.5}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      7. associate-*r/ 89.0%

         \[\leadsto z \cdot \left(\frac{x \cdot \color{blue}{\left(0.5 \cdot \frac{1}{t}\right)} + \frac{y}{t} \cdot 0.5}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      8. associate-*l/ 89.0%

         \[\leadsto z \cdot \left(\frac{x \cdot \left(0.5 \cdot \frac{1}{t}\right) + \color{blue}{\frac{y \cdot 0.5}{t}}}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      9. associate-*r/ 88.8%

         \[\leadsto z \cdot \left(\frac{x \cdot \left(0.5 \cdot \frac{1}{t}\right) + \color{blue}{y \cdot \frac{0.5}{t}}}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      10. metadata-eval 88.8%

          \[\leadsto z \cdot \left(\frac{x \cdot \left(0.5 \cdot \frac{1}{t}\right) + y \cdot \frac{\color{blue}{0.5 \cdot 1}}{t}}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      11. associate-*r/ 88.8%

          \[\leadsto z \cdot \left(\frac{x \cdot \left(0.5 \cdot \frac{1}{t}\right) + y \cdot \color{blue}{\left(0.5 \cdot \frac{1}{t}\right)}}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      12. distribute-rgt-in 90.3%

          \[\leadsto z \cdot \left(\frac{\color{blue}{\left(0.5 \cdot \frac{1}{t}\right) \cdot \left(x + y\right)}}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      13. associate-*r/ 90.3%

          \[\leadsto z \cdot \left(\frac{\color{blue}{\frac{0.5 \cdot 1}{t}} \cdot \left(x + y\right)}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      14. metadata-eval 90.3%

          \[\leadsto z \cdot \left(\frac{\frac{\color{blue}{0.5}}{t} \cdot \left(x + y\right)}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      15. associate-*l/ 90.5%

          \[\leadsto z \cdot \left(\frac{\color{blue}{\frac{0.5 \cdot \left(x + y\right)}{t}}}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      16. associate-*r/ 90.5%

          \[\leadsto z \cdot \left(\frac{\color{blue}{0.5 \cdot \frac{x + y}{t}}}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      17. +-commutative 90.5%

          \[\leadsto z \cdot \left(\frac{0.5 \cdot \frac{\color{blue}{y + x}}{t}}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      18. associate-*r/ 90.5%

          \[\leadsto z \cdot \left(\frac{0.5 \cdot \frac{y + x}{t}}{z} + \left(-\color{blue}{\frac{0.5 \cdot 1}{t}}\right)\right) \]

      19. metadata-eval 90.5%

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

   5. Simplified 90.5%

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

   6. Taylor expanded in t around 0 80.9%

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

   7. Taylor expanded in y around inf 63.3%

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

      1. *-commutative 63.3%

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

      2. associate-*l/ 63.3%

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

      3. associate-*r/ 63.1%

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

   9. Simplified 63.1%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-275}:\\ \;\;\;\;x \cdot \frac{0.5}{t}\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+40}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{0.5}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.8 \cdot 10^{-81}:\\ \;\;\;\;\frac{x - z}{t \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{t \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 7.8e-81) (/ (- x z) (* t 2.0)) (/ (- y z) (* t 2.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 7.8e-81) {
		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 (y <= 7.8d-81) 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 (y <= 7.8e-81) {
		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 y <= 7.8e-81:
		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 (y <= 7.8e-81)
		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 (y <= 7.8e-81)
		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[y, 7.8e-81], 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 y < 7.7999999999999997e-81

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 78.7%

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

   if 7.7999999999999997e-81 < 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 85.3%

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

Alternative 8: 76.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-84}:\\ \;\;\;\;\frac{x - z}{t \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 8e-84) (/ (- x z) (* t 2.0)) (* (- y z) (/ 0.5 t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 8e-84) {
		tmp = (x - z) / (t * 2.0);
	} else {
		tmp = (y - 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 (y <= 8d-84) then
        tmp = (x - z) / (t * 2.0d0)
    else
        tmp = (y - 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 (y <= 8e-84) {
		tmp = (x - z) / (t * 2.0);
	} else {
		tmp = (y - z) * (0.5 / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 8e-84:
		tmp = (x - z) / (t * 2.0)
	else:
		tmp = (y - z) * (0.5 / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 8e-84)
		tmp = Float64(Float64(x - z) / Float64(t * 2.0));
	else
		tmp = Float64(Float64(y - z) * Float64(0.5 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 8e-84)
		tmp = (x - z) / (t * 2.0);
	else
		tmp = (y - z) * (0.5 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 8e-84], N[(N[(x - z), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(y - z), $MachinePrecision] * N[(0.5 / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.0000000000000003e-84

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 78.7%

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

   if 8.0000000000000003e-84 < 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 85.3%

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

      1. *-commutative 85.3%

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

      2. associate-*l/ 85.3%

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

      3. associate-*r/ 85.1%

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

   5. Simplified 85.1%

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

Alternative 9: 75.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{+89}:\\ \;\;\;\;\frac{x}{t \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.85e+89) (/ x (* t 2.0)) (* (- y z) (/ 0.5 t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.85e+89) {
		tmp = x / (t * 2.0);
	} else {
		tmp = (y - 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 <= (-1.85d+89)) then
        tmp = x / (t * 2.0d0)
    else
        tmp = (y - 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 <= -1.85e+89) {
		tmp = x / (t * 2.0);
	} else {
		tmp = (y - z) * (0.5 / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1.85e+89:
		tmp = x / (t * 2.0)
	else:
		tmp = (y - z) * (0.5 / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.85e+89)
		tmp = Float64(x / Float64(t * 2.0));
	else
		tmp = Float64(Float64(y - z) * Float64(0.5 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -1.85e+89)
		tmp = x / (t * 2.0);
	else
		tmp = (y - z) * (0.5 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.85e+89], N[(x / N[(t * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(y - z), $MachinePrecision] * N[(0.5 / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.8499999999999999e89

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 75.3%

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

   if -1.8499999999999999e89 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 78.4%

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

      1. *-commutative 78.4%

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

      2. associate-*l/ 78.4%

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

      3. associate-*r/ 78.1%

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

   5. Simplified 78.1%

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

Alternative 10: 46.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{+39}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 2.2e+39) (* z (/ -0.5 t)) (* y (/ 0.5 t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.2e+39) {
		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 (y <= 2.2d+39) 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 (y <= 2.2e+39) {
		tmp = z * (-0.5 / t);
	} else {
		tmp = y * (0.5 / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 2.2e+39:
		tmp = z * (-0.5 / t)
	else:
		tmp = y * (0.5 / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2.2e+39)
		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 (y <= 2.2e+39)
		tmp = z * (-0.5 / t);
	else
		tmp = y * (0.5 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 2.2e+39], N[(z * N[(-0.5 / t), $MachinePrecision]), $MachinePrecision], N[(y * N[(0.5 / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.2000000000000001e39

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 44.6%

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

      1. *-commutative 44.6%

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

      2. associate-*l/ 44.6%

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

   5. Simplified 44.6%

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

      1. *-commutative 44.6%

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

      2. associate-*l/ 44.5%

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

   7. Applied egg-rr 44.5%

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

   if 2.2000000000000001e39 < 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 89.0%

      \[\leadsto \color{blue}{z \cdot \left(0.5 \cdot \frac{\frac{x}{t} + \frac{y}{t}}{z} - 0.5 \cdot \frac{1}{t}\right)} \]
   4. Step-by-step derivation

      1. sub-neg 89.0%

         \[\leadsto z \cdot \color{blue}{\left(0.5 \cdot \frac{\frac{x}{t} + \frac{y}{t}}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right)} \]

      2. associate-*r/ 89.0%

         \[\leadsto z \cdot \left(\color{blue}{\frac{0.5 \cdot \left(\frac{x}{t} + \frac{y}{t}\right)}{z}} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      3. distribute-rgt-in 89.0%

         \[\leadsto z \cdot \left(\frac{\color{blue}{\frac{x}{t} \cdot 0.5 + \frac{y}{t} \cdot 0.5}}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      4. associate-*l/ 89.0%

         \[\leadsto z \cdot \left(\frac{\color{blue}{\frac{x \cdot 0.5}{t}} + \frac{y}{t} \cdot 0.5}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      5. associate-*r/ 89.0%

         \[\leadsto z \cdot \left(\frac{\color{blue}{x \cdot \frac{0.5}{t}} + \frac{y}{t} \cdot 0.5}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      6. metadata-eval 89.0%

         \[\leadsto z \cdot \left(\frac{x \cdot \frac{\color{blue}{0.5 \cdot 1}}{t} + \frac{y}{t} \cdot 0.5}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      7. associate-*r/ 89.0%

         \[\leadsto z \cdot \left(\frac{x \cdot \color{blue}{\left(0.5 \cdot \frac{1}{t}\right)} + \frac{y}{t} \cdot 0.5}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      8. associate-*l/ 89.0%

         \[\leadsto z \cdot \left(\frac{x \cdot \left(0.5 \cdot \frac{1}{t}\right) + \color{blue}{\frac{y \cdot 0.5}{t}}}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      9. associate-*r/ 88.8%

         \[\leadsto z \cdot \left(\frac{x \cdot \left(0.5 \cdot \frac{1}{t}\right) + \color{blue}{y \cdot \frac{0.5}{t}}}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      10. metadata-eval 88.8%

          \[\leadsto z \cdot \left(\frac{x \cdot \left(0.5 \cdot \frac{1}{t}\right) + y \cdot \frac{\color{blue}{0.5 \cdot 1}}{t}}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      11. associate-*r/ 88.8%

          \[\leadsto z \cdot \left(\frac{x \cdot \left(0.5 \cdot \frac{1}{t}\right) + y \cdot \color{blue}{\left(0.5 \cdot \frac{1}{t}\right)}}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      12. distribute-rgt-in 90.3%

          \[\leadsto z \cdot \left(\frac{\color{blue}{\left(0.5 \cdot \frac{1}{t}\right) \cdot \left(x + y\right)}}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      13. associate-*r/ 90.3%

          \[\leadsto z \cdot \left(\frac{\color{blue}{\frac{0.5 \cdot 1}{t}} \cdot \left(x + y\right)}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      14. metadata-eval 90.3%

          \[\leadsto z \cdot \left(\frac{\frac{\color{blue}{0.5}}{t} \cdot \left(x + y\right)}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      15. associate-*l/ 90.5%

          \[\leadsto z \cdot \left(\frac{\color{blue}{\frac{0.5 \cdot \left(x + y\right)}{t}}}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      16. associate-*r/ 90.5%

          \[\leadsto z \cdot \left(\frac{\color{blue}{0.5 \cdot \frac{x + y}{t}}}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      17. +-commutative 90.5%

          \[\leadsto z \cdot \left(\frac{0.5 \cdot \frac{\color{blue}{y + x}}{t}}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

      18. associate-*r/ 90.5%

          \[\leadsto z \cdot \left(\frac{0.5 \cdot \frac{y + x}{t}}{z} + \left(-\color{blue}{\frac{0.5 \cdot 1}{t}}\right)\right) \]

      19. metadata-eval 90.5%

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

   5. Simplified 90.5%

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

   6. Taylor expanded in t around 0 80.9%

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

   7. Taylor expanded in y around inf 63.3%

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

      1. *-commutative 63.3%

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

      2. associate-*l/ 63.3%

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

      3. associate-*r/ 63.1%

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

   9. Simplified 63.1%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{+39}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{0.5}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 37.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(y * 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 91.0%

   \[\leadsto \color{blue}{z \cdot \left(0.5 \cdot \frac{\frac{x}{t} + \frac{y}{t}}{z} - 0.5 \cdot \frac{1}{t}\right)} \]
4. Step-by-step derivation

   1. sub-neg 91.0%

      \[\leadsto z \cdot \color{blue}{\left(0.5 \cdot \frac{\frac{x}{t} + \frac{y}{t}}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right)} \]

   2. associate-*r/ 91.0%

      \[\leadsto z \cdot \left(\color{blue}{\frac{0.5 \cdot \left(\frac{x}{t} + \frac{y}{t}\right)}{z}} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

   3. distribute-rgt-in 91.0%

      \[\leadsto z \cdot \left(\frac{\color{blue}{\frac{x}{t} \cdot 0.5 + \frac{y}{t} \cdot 0.5}}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

   4. associate-*l/ 91.0%

      \[\leadsto z \cdot \left(\frac{\color{blue}{\frac{x \cdot 0.5}{t}} + \frac{y}{t} \cdot 0.5}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

   5. associate-*r/ 90.9%

      \[\leadsto z \cdot \left(\frac{\color{blue}{x \cdot \frac{0.5}{t}} + \frac{y}{t} \cdot 0.5}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

   6. metadata-eval 90.9%

      \[\leadsto z \cdot \left(\frac{x \cdot \frac{\color{blue}{0.5 \cdot 1}}{t} + \frac{y}{t} \cdot 0.5}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

   7. associate-*r/ 90.9%

      \[\leadsto z \cdot \left(\frac{x \cdot \color{blue}{\left(0.5 \cdot \frac{1}{t}\right)} + \frac{y}{t} \cdot 0.5}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

   8. associate-*l/ 90.9%

      \[\leadsto z \cdot \left(\frac{x \cdot \left(0.5 \cdot \frac{1}{t}\right) + \color{blue}{\frac{y \cdot 0.5}{t}}}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

   9. associate-*r/ 90.8%

      \[\leadsto z \cdot \left(\frac{x \cdot \left(0.5 \cdot \frac{1}{t}\right) + \color{blue}{y \cdot \frac{0.5}{t}}}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

   10. metadata-eval 90.8%

       \[\leadsto z \cdot \left(\frac{x \cdot \left(0.5 \cdot \frac{1}{t}\right) + y \cdot \frac{\color{blue}{0.5 \cdot 1}}{t}}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

   11. associate-*r/ 90.8%

       \[\leadsto z \cdot \left(\frac{x \cdot \left(0.5 \cdot \frac{1}{t}\right) + y \cdot \color{blue}{\left(0.5 \cdot \frac{1}{t}\right)}}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

   12. distribute-rgt-in 91.2%

       \[\leadsto z \cdot \left(\frac{\color{blue}{\left(0.5 \cdot \frac{1}{t}\right) \cdot \left(x + y\right)}}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

   13. associate-*r/ 91.2%

       \[\leadsto z \cdot \left(\frac{\color{blue}{\frac{0.5 \cdot 1}{t}} \cdot \left(x + y\right)}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

   14. metadata-eval 91.2%

       \[\leadsto z \cdot \left(\frac{\frac{\color{blue}{0.5}}{t} \cdot \left(x + y\right)}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

   15. associate-*l/ 91.4%

       \[\leadsto z \cdot \left(\frac{\color{blue}{\frac{0.5 \cdot \left(x + y\right)}{t}}}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

   16. associate-*r/ 91.4%

       \[\leadsto z \cdot \left(\frac{\color{blue}{0.5 \cdot \frac{x + y}{t}}}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

   17. +-commutative 91.4%

       \[\leadsto z \cdot \left(\frac{0.5 \cdot \frac{\color{blue}{y + x}}{t}}{z} + \left(-0.5 \cdot \frac{1}{t}\right)\right) \]

   18. associate-*r/ 91.4%

       \[\leadsto z \cdot \left(\frac{0.5 \cdot \frac{y + x}{t}}{z} + \left(-\color{blue}{\frac{0.5 \cdot 1}{t}}\right)\right) \]

   19. metadata-eval 91.4%

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

5. Simplified 91.4%

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

6. Taylor expanded in t around 0 84.1%

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

7. Taylor expanded in y around inf 37.2%

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

   1. *-commutative 37.2%

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

   2. associate-*l/ 37.2%

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

   3. associate-*r/ 37.0%

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

9. Simplified 37.0%

   \[\leadsto \color{blue}{y \cdot \frac{0.5}{t}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024181 
(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)))