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

Percentage Accurate: 99.9% → 99.9%

Time: 11.5s

Alternatives: 12

Speedup: 1.0×

• 26.2% 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

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

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 + Float64(y - z)) / 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 + N[(y - z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. associate-/r* N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x + y\right) - z}{t}\right), \color{blue}{2}\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x + y\right) - z\right), t\right), 2\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(y + x\right) - z\right), t\right), 2\right) \]

   5. associate--l+ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y + \left(x - z\right)\right), t\right), 2\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x - z\right) + y\right), t\right), 2\right) \]

   7. associate-+l- N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x - \left(z - y\right)\right), t\right), 2\right) \]

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), t\right), 2\right) \]

   9. --lowering--.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), t\right), 2\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{\frac{x - \left(z - y\right)}{t}}{2}} \]
4. Add Preprocessing

5. Final simplification 100.0%

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

Alternative 2: 80.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot -0.5}{t}\\ \mathbf{if}\;z \leq -3.7 \cdot 10^{+85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.44 \cdot 10^{+169}:\\ \;\;\;\;\frac{\frac{x + y}{t}}{2}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z -0.5) t)))
   (if (<= z -3.7e+85) t_1 (if (<= z 1.44e+169) (/ (/ (+ x y) t) 2.0) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * -0.5) / t;
	double tmp;
	if (z <= -3.7e+85) {
		tmp = t_1;
	} else if (z <= 1.44e+169) {
		tmp = ((x + y) / t) / 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * (-0.5d0)) / t
    if (z <= (-3.7d+85)) then
        tmp = t_1
    else if (z <= 1.44d+169) then
        tmp = ((x + y) / t) / 2.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * -0.5) / t;
	double tmp;
	if (z <= -3.7e+85) {
		tmp = t_1;
	} else if (z <= 1.44e+169) {
		tmp = ((x + y) / t) / 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * -0.5) / t
	tmp = 0
	if z <= -3.7e+85:
		tmp = t_1
	elif z <= 1.44e+169:
		tmp = ((x + y) / t) / 2.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * -0.5) / t)
	tmp = 0.0
	if (z <= -3.7e+85)
		tmp = t_1;
	elseif (z <= 1.44e+169)
		tmp = Float64(Float64(Float64(x + y) / t) / 2.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * -0.5) / t;
	tmp = 0.0;
	if (z <= -3.7e+85)
		tmp = t_1;
	elseif (z <= 1.44e+169)
		tmp = ((x + y) / t) / 2.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * -0.5), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[z, -3.7e+85], t$95$1, If[LessEqual[z, 1.44e+169], N[(N[(N[(x + y), $MachinePrecision] / t), $MachinePrecision] / 2.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.7000000000000002e85 or 1.44e169 < z

   1. Initial program 98.6%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x + y\right) - z}{t}\right), \color{blue}{2}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x + y\right) - z\right), t\right), 2\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(y + x\right) - z\right), t\right), 2\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y + \left(x - z\right)\right), t\right), 2\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x - z\right) + y\right), t\right), 2\right) \]

      7. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x - \left(z - y\right)\right), t\right), 2\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), t\right), 2\right) \]

      9. --lowering--.f64 99.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), t\right), 2\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{\frac{x - \left(z - y\right)}{t}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(\mathsf{neg}\left(z\right)\right)}{t} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(z\right)\right)\right), \color{blue}{t}\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(-1 \cdot z\right)\right), t\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot -1\right) \cdot z\right), t\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot z\right), t\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(z \cdot \frac{-1}{2}\right), t\right) \]

      10. *-lowering-*.f64 72.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \frac{-1}{2}\right), t\right) \]

   7. Simplified 72.4%

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

   if -3.7000000000000002e85 < z < 1.44e169

   1. Initial program 100.0%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x + y\right) - z}{t}\right), \color{blue}{2}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x + y\right) - z\right), t\right), 2\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(y + x\right) - z\right), t\right), 2\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y + \left(x - z\right)\right), t\right), 2\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x - z\right) + y\right), t\right), 2\right) \]

      7. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x - \left(z - y\right)\right), t\right), 2\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), t\right), 2\right) \]

      9. --lowering--.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), t\right), 2\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\frac{x - \left(z - y\right)}{t}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{x + y}{t}\right)}, 2\right) \]
   6. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 \cdot x + y}{t}\right), 2\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(-1 \cdot -1\right) \cdot x + y}{t}\right), 2\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(-1 \cdot x\right) + y}{t}\right), 2\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(-1 \cdot x\right) + 1 \cdot y}{t}\right), 2\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(-1 \cdot x\right) + \left(-1 \cdot -1\right) \cdot y}{t}\right), 2\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(-1 \cdot x\right) + -1 \cdot \left(-1 \cdot y\right)}{t}\right), 2\right) \]

      7. distribute-lft-out N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(-1 \cdot x + -1 \cdot y\right)}{t}\right), 2\right) \]

      8. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(-1 \cdot \left(x + y\right)\right)}{t}\right), 2\right) \]

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(\left(-1 \cdot \left(x + y\right)\right) \cdot 1\right)}{t}\right), 2\right) \]

      10. *-inverses N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(\left(-1 \cdot \left(x + y\right)\right) \cdot \frac{z}{z}\right)}{t}\right), 2\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \frac{\left(-1 \cdot \left(x + y\right)\right) \cdot z}{z}}{t}\right), 2\right) \]

      12. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(\frac{-1 \cdot \left(x + y\right)}{z} \cdot z\right)}{t}\right), 2\right) \]

      13. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(\left(-1 \cdot \frac{x + y}{z}\right) \cdot z\right)}{t}\right), 2\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(z \cdot \left(-1 \cdot \frac{x + y}{z}\right)\right)}{t}\right), 2\right) \]

      15. neg-mul-1 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(z \cdot \left(-1 \cdot \frac{x + y}{z}\right)\right)}{t}\right), 2\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(z \cdot \left(-1 \cdot \frac{x + y}{z}\right)\right)\right), t\right), 2\right) \]

   7. Simplified 88.5%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{+85}:\\ \;\;\;\;\frac{z \cdot -0.5}{t}\\ \mathbf{elif}\;z \leq 1.44 \cdot 10^{+169}:\\ \;\;\;\;\frac{\frac{x + y}{t}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot -0.5}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot -0.5}{t}\\ \mathbf{if}\;z \leq -2.8 \cdot 10^{+86}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.9 \cdot 10^{+168}:\\ \;\;\;\;\frac{0.5}{t} \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z -0.5) t)))
   (if (<= z -2.8e+86) t_1 (if (<= z 5.9e+168) (* (/ 0.5 t) (+ x y)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * -0.5) / t;
	double tmp;
	if (z <= -2.8e+86) {
		tmp = t_1;
	} else if (z <= 5.9e+168) {
		tmp = (0.5 / t) * (x + y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * (-0.5d0)) / t
    if (z <= (-2.8d+86)) then
        tmp = t_1
    else if (z <= 5.9d+168) then
        tmp = (0.5d0 / t) * (x + y)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * -0.5) / t;
	double tmp;
	if (z <= -2.8e+86) {
		tmp = t_1;
	} else if (z <= 5.9e+168) {
		tmp = (0.5 / t) * (x + y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * -0.5) / t
	tmp = 0
	if z <= -2.8e+86:
		tmp = t_1
	elif z <= 5.9e+168:
		tmp = (0.5 / t) * (x + y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * -0.5) / t)
	tmp = 0.0
	if (z <= -2.8e+86)
		tmp = t_1;
	elseif (z <= 5.9e+168)
		tmp = Float64(Float64(0.5 / t) * Float64(x + y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * -0.5) / t;
	tmp = 0.0;
	if (z <= -2.8e+86)
		tmp = t_1;
	elseif (z <= 5.9e+168)
		tmp = (0.5 / t) * (x + y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * -0.5), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[z, -2.8e+86], t$95$1, If[LessEqual[z, 5.9e+168], N[(N[(0.5 / t), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.80000000000000004e86 or 5.89999999999999986e168 < z

   1. Initial program 98.6%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x + y\right) - z}{t}\right), \color{blue}{2}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x + y\right) - z\right), t\right), 2\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(y + x\right) - z\right), t\right), 2\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y + \left(x - z\right)\right), t\right), 2\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x - z\right) + y\right), t\right), 2\right) \]

      7. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x - \left(z - y\right)\right), t\right), 2\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), t\right), 2\right) \]

      9. --lowering--.f64 99.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), t\right), 2\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{\frac{x - \left(z - y\right)}{t}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(\mathsf{neg}\left(z\right)\right)}{t} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(z\right)\right)\right), \color{blue}{t}\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(-1 \cdot z\right)\right), t\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot -1\right) \cdot z\right), t\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot z\right), t\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(z \cdot \frac{-1}{2}\right), t\right) \]

      10. *-lowering-*.f64 72.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \frac{-1}{2}\right), t\right) \]

   7. Simplified 72.4%

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

   if -2.80000000000000004e86 < z < 5.89999999999999986e168

   1. Initial program 100.0%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x + y\right) - z}{t}\right), \color{blue}{2}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x + y\right) - z\right), t\right), 2\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(y + x\right) - z\right), t\right), 2\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y + \left(x - z\right)\right), t\right), 2\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x - z\right) + y\right), t\right), 2\right) \]

      7. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x - \left(z - y\right)\right), t\right), 2\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), t\right), 2\right) \]

      9. --lowering--.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), t\right), 2\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\frac{x - \left(z - y\right)}{t}}{2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l/ N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2 \cdot t}\right), \color{blue}{\left(x - \left(z - y\right)\right)}\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{t}\right), \left(\color{blue}{x} - \left(z - y\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2}\right), t\right), \left(\color{blue}{x} - \left(z - y\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \left(x - \left(z - y\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \left(x + \color{blue}{\left(\mathsf{neg}\left(\left(z - y\right)\right)\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\left(z - y\right)\right)\right)}\right)\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \left(0 - \color{blue}{\left(z - y\right)}\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \left(0 - \left(z + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \left(0 - \left(\left(\mathsf{neg}\left(y\right)\right) + \color{blue}{z}\right)\right)\right)\right) \]

      13. associate--r+ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \left(\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - \color{blue}{z}\right)\right)\right) \]

      14. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) - z\right)\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \left(y - z\right)\right)\right) \]

      16. --lowering--.f64 99.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(y, \color{blue}{z}\right)\right)\right) \]

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in z around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \color{blue}{\left(x + y\right)}\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \left(y + \color{blue}{x}\right)\right) \]

      2. +-lowering-+.f64 88.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(y, \color{blue}{x}\right)\right) \]

   9. Simplified 88.4%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+86}:\\ \;\;\;\;\frac{z \cdot -0.5}{t}\\ \mathbf{elif}\;z \leq 5.9 \cdot 10^{+168}:\\ \;\;\;\;\frac{0.5}{t} \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot -0.5}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 47.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+64}:\\ \;\;\;\;\frac{\frac{x}{t}}{2}\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-110}:\\ \;\;\;\;\frac{z \cdot -0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -2.5e+64)
   (/ (/ x t) 2.0)
   (if (<= x -4.8e-110) (/ (* z -0.5) t) (/ (/ y t) 2.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.5e+64) {
		tmp = (x / t) / 2.0;
	} else if (x <= -4.8e-110) {
		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 (x <= (-2.5d+64)) then
        tmp = (x / t) / 2.0d0
    else if (x <= (-4.8d-110)) 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 (x <= -2.5e+64) {
		tmp = (x / t) / 2.0;
	} else if (x <= -4.8e-110) {
		tmp = (z * -0.5) / t;
	} else {
		tmp = (y / t) / 2.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -2.5e+64:
		tmp = (x / t) / 2.0
	elif x <= -4.8e-110:
		tmp = (z * -0.5) / t
	else:
		tmp = (y / t) / 2.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -2.5e+64)
		tmp = Float64(Float64(x / t) / 2.0);
	elseif (x <= -4.8e-110)
		tmp = Float64(Float64(z * -0.5) / t);
	else
		tmp = Float64(Float64(y / t) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -2.5e+64)
		tmp = (x / t) / 2.0;
	elseif (x <= -4.8e-110)
		tmp = (z * -0.5) / t;
	else
		tmp = (y / t) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -2.5e+64], N[(N[(x / t), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -4.8e-110], N[(N[(z * -0.5), $MachinePrecision] / t), $MachinePrecision], N[(N[(y / t), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.5e64

   1. Initial program 100.0%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x + y\right) - z}{t}\right), \color{blue}{2}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x + y\right) - z\right), t\right), 2\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(y + x\right) - z\right), t\right), 2\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y + \left(x - z\right)\right), t\right), 2\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x - z\right) + y\right), t\right), 2\right) \]

      7. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x - \left(z - y\right)\right), t\right), 2\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), t\right), 2\right) \]

      9. --lowering--.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), t\right), 2\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\frac{x - \left(z - y\right)}{t}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{x}{t}\right)}, 2\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 87.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, t\right), 2\right) \]

   7. Simplified 87.2%

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

   if -2.5e64 < x < -4.80000000000000013e-110

   1. Initial program 99.9%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x + y\right) - z}{t}\right), \color{blue}{2}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x + y\right) - z\right), t\right), 2\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(y + x\right) - z\right), t\right), 2\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y + \left(x - z\right)\right), t\right), 2\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x - z\right) + y\right), t\right), 2\right) \]

      7. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x - \left(z - y\right)\right), t\right), 2\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), t\right), 2\right) \]

      9. --lowering--.f64 99.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), t\right), 2\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{\frac{x - \left(z - y\right)}{t}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(\mathsf{neg}\left(z\right)\right)}{t} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(z\right)\right)\right), \color{blue}{t}\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(-1 \cdot z\right)\right), t\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot -1\right) \cdot z\right), t\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot z\right), t\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(z \cdot \frac{-1}{2}\right), t\right) \]

      10. *-lowering-*.f64 43.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \frac{-1}{2}\right), t\right) \]

   7. Simplified 43.8%

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

   if -4.80000000000000013e-110 < x

   1. Initial program 99.4%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x + y\right) - z}{t}\right), \color{blue}{2}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x + y\right) - z\right), t\right), 2\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(y + x\right) - z\right), t\right), 2\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y + \left(x - z\right)\right), t\right), 2\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x - z\right) + y\right), t\right), 2\right) \]

      7. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x - \left(z - y\right)\right), t\right), 2\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), t\right), 2\right) \]

      9. --lowering--.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), t\right), 2\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\frac{x - \left(z - y\right)}{t}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{y}{t}\right)}, 2\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 46.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, t\right), 2\right) \]

   7. Simplified 46.9%

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

Alternative 5: 47.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+64}:\\ \;\;\;\;\frac{\frac{x}{t}}{2}\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-110}:\\ \;\;\;\;\frac{z \cdot -0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -2.5e+64)
   (/ (/ x t) 2.0)
   (if (<= x -3.5e-110) (/ (* z -0.5) t) (* y (/ 0.5 t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.5e+64) {
		tmp = (x / t) / 2.0;
	} else if (x <= -3.5e-110) {
		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 <= (-2.5d+64)) then
        tmp = (x / t) / 2.0d0
    else if (x <= (-3.5d-110)) 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 <= -2.5e+64) {
		tmp = (x / t) / 2.0;
	} else if (x <= -3.5e-110) {
		tmp = (z * -0.5) / t;
	} else {
		tmp = y * (0.5 / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -2.5e+64:
		tmp = (x / t) / 2.0
	elif x <= -3.5e-110:
		tmp = (z * -0.5) / t
	else:
		tmp = y * (0.5 / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -2.5e+64)
		tmp = Float64(Float64(x / t) / 2.0);
	elseif (x <= -3.5e-110)
		tmp = Float64(Float64(z * -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 <= -2.5e+64)
		tmp = (x / t) / 2.0;
	elseif (x <= -3.5e-110)
		tmp = (z * -0.5) / t;
	else
		tmp = y * (0.5 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -2.5e+64], N[(N[(x / t), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -3.5e-110], N[(N[(z * -0.5), $MachinePrecision] / t), $MachinePrecision], N[(y * N[(0.5 / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.5e64

   1. Initial program 100.0%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x + y\right) - z}{t}\right), \color{blue}{2}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x + y\right) - z\right), t\right), 2\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(y + x\right) - z\right), t\right), 2\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y + \left(x - z\right)\right), t\right), 2\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x - z\right) + y\right), t\right), 2\right) \]

      7. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x - \left(z - y\right)\right), t\right), 2\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), t\right), 2\right) \]

      9. --lowering--.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), t\right), 2\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\frac{x - \left(z - y\right)}{t}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{x}{t}\right)}, 2\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 87.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, t\right), 2\right) \]

   7. Simplified 87.2%

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

   if -2.5e64 < x < -3.49999999999999974e-110

   1. Initial program 99.9%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x + y\right) - z}{t}\right), \color{blue}{2}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x + y\right) - z\right), t\right), 2\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(y + x\right) - z\right), t\right), 2\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y + \left(x - z\right)\right), t\right), 2\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x - z\right) + y\right), t\right), 2\right) \]

      7. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x - \left(z - y\right)\right), t\right), 2\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), t\right), 2\right) \]

      9. --lowering--.f64 99.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), t\right), 2\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{\frac{x - \left(z - y\right)}{t}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(\mathsf{neg}\left(z\right)\right)}{t} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(z\right)\right)\right), \color{blue}{t}\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(-1 \cdot z\right)\right), t\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot -1\right) \cdot z\right), t\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot z\right), t\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(z \cdot \frac{-1}{2}\right), t\right) \]

      10. *-lowering-*.f64 43.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \frac{-1}{2}\right), t\right) \]

   7. Simplified 43.8%

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

   if -3.49999999999999974e-110 < x

   1. Initial program 99.4%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x + y\right) - z}{t}\right), \color{blue}{2}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x + y\right) - z\right), t\right), 2\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(y + x\right) - z\right), t\right), 2\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y + \left(x - z\right)\right), t\right), 2\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x - z\right) + y\right), t\right), 2\right) \]

      7. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x - \left(z - y\right)\right), t\right), 2\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), t\right), 2\right) \]

      9. --lowering--.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), t\right), 2\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\frac{x - \left(z - y\right)}{t}}{2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l/ N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2 \cdot t}\right), \color{blue}{\left(x - \left(z - y\right)\right)}\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{t}\right), \left(\color{blue}{x} - \left(z - y\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2}\right), t\right), \left(\color{blue}{x} - \left(z - y\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \left(x - \left(z - y\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \left(x + \color{blue}{\left(\mathsf{neg}\left(\left(z - y\right)\right)\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\left(z - y\right)\right)\right)}\right)\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \left(0 - \color{blue}{\left(z - y\right)}\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \left(0 - \left(z + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \left(0 - \left(\left(\mathsf{neg}\left(y\right)\right) + \color{blue}{z}\right)\right)\right)\right) \]

      13. associate--r+ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \left(\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - \color{blue}{z}\right)\right)\right) \]

      14. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) - z\right)\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \left(y - z\right)\right)\right) \]

      16. --lowering--.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(y, \color{blue}{z}\right)\right)\right) \]

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \color{blue}{y}\right) \]
   8. Step-by-step derivation

   9. Simplified 46.8%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+64}:\\ \;\;\;\;\frac{\frac{x}{t}}{2}\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-110}:\\ \;\;\;\;\frac{z \cdot -0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{0.5}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 47.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+64}:\\ \;\;\;\;\frac{0.5}{\frac{t}{x}}\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-110}:\\ \;\;\;\;\frac{z \cdot -0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -2.6e+64)
   (/ 0.5 (/ t x))
   (if (<= x -4e-110) (/ (* z -0.5) t) (* y (/ 0.5 t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.6e+64) {
		tmp = 0.5 / (t / x);
	} else if (x <= -4e-110) {
		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 <= (-2.6d+64)) then
        tmp = 0.5d0 / (t / x)
    else if (x <= (-4d-110)) 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 <= -2.6e+64) {
		tmp = 0.5 / (t / x);
	} else if (x <= -4e-110) {
		tmp = (z * -0.5) / t;
	} else {
		tmp = y * (0.5 / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -2.6e+64:
		tmp = 0.5 / (t / x)
	elif x <= -4e-110:
		tmp = (z * -0.5) / t
	else:
		tmp = y * (0.5 / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -2.6e+64)
		tmp = Float64(0.5 / Float64(t / x));
	elseif (x <= -4e-110)
		tmp = Float64(Float64(z * -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 <= -2.6e+64)
		tmp = 0.5 / (t / x);
	elseif (x <= -4e-110)
		tmp = (z * -0.5) / t;
	else
		tmp = y * (0.5 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -2.6e+64], N[(0.5 / N[(t / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4e-110], N[(N[(z * -0.5), $MachinePrecision] / t), $MachinePrecision], N[(y * N[(0.5 / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.59999999999999997e64

   1. Initial program 100.0%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x + y\right) - z}{t}\right), \color{blue}{2}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x + y\right) - z\right), t\right), 2\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(y + x\right) - z\right), t\right), 2\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y + \left(x - z\right)\right), t\right), 2\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x - z\right) + y\right), t\right), 2\right) \]

      7. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x - \left(z - y\right)\right), t\right), 2\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), t\right), 2\right) \]

      9. --lowering--.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), t\right), 2\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\frac{x - \left(z - y\right)}{t}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{x}{t}\right)}, 2\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 87.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, t\right), 2\right) \]

   7. Simplified 87.2%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. metadata-eval N/A

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

      4. associate-*l/ N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{t}{x}\right)}\right) \]

      7. /-lowering-/.f64 87.0%

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(t, \color{blue}{x}\right)\right) \]

   9. Applied egg-rr 87.0%

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

   if -2.59999999999999997e64 < x < -4.0000000000000002e-110

   1. Initial program 99.9%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x + y\right) - z}{t}\right), \color{blue}{2}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x + y\right) - z\right), t\right), 2\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(y + x\right) - z\right), t\right), 2\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y + \left(x - z\right)\right), t\right), 2\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x - z\right) + y\right), t\right), 2\right) \]

      7. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x - \left(z - y\right)\right), t\right), 2\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), t\right), 2\right) \]

      9. --lowering--.f64 99.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), t\right), 2\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{\frac{x - \left(z - y\right)}{t}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(\mathsf{neg}\left(z\right)\right)}{t} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(z\right)\right)\right), \color{blue}{t}\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(-1 \cdot z\right)\right), t\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot -1\right) \cdot z\right), t\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot z\right), t\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(z \cdot \frac{-1}{2}\right), t\right) \]

      10. *-lowering-*.f64 43.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \frac{-1}{2}\right), t\right) \]

   7. Simplified 43.8%

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

   if -4.0000000000000002e-110 < x

   1. Initial program 99.4%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x + y\right) - z}{t}\right), \color{blue}{2}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x + y\right) - z\right), t\right), 2\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(y + x\right) - z\right), t\right), 2\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y + \left(x - z\right)\right), t\right), 2\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x - z\right) + y\right), t\right), 2\right) \]

      7. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x - \left(z - y\right)\right), t\right), 2\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), t\right), 2\right) \]

      9. --lowering--.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), t\right), 2\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\frac{x - \left(z - y\right)}{t}}{2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l/ N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2 \cdot t}\right), \color{blue}{\left(x - \left(z - y\right)\right)}\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{t}\right), \left(\color{blue}{x} - \left(z - y\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2}\right), t\right), \left(\color{blue}{x} - \left(z - y\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \left(x - \left(z - y\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \left(x + \color{blue}{\left(\mathsf{neg}\left(\left(z - y\right)\right)\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\left(z - y\right)\right)\right)}\right)\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \left(0 - \color{blue}{\left(z - y\right)}\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \left(0 - \left(z + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \left(0 - \left(\left(\mathsf{neg}\left(y\right)\right) + \color{blue}{z}\right)\right)\right)\right) \]

      13. associate--r+ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \left(\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - \color{blue}{z}\right)\right)\right) \]

      14. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) - z\right)\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \left(y - z\right)\right)\right) \]

      16. --lowering--.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(y, \color{blue}{z}\right)\right)\right) \]

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \color{blue}{y}\right) \]
   8. Step-by-step derivation

   9. Simplified 46.8%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+64}:\\ \;\;\;\;\frac{0.5}{\frac{t}{x}}\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-110}:\\ \;\;\;\;\frac{z \cdot -0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{0.5}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 47.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+64}:\\ \;\;\;\;\frac{0.5}{\frac{t}{x}}\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{-110}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -3.6e+64)
   (/ 0.5 (/ t x))
   (if (<= x -3.3e-110) (* z (/ -0.5 t)) (* y (/ 0.5 t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -3.6e+64) {
		tmp = 0.5 / (t / x);
	} else if (x <= -3.3e-110) {
		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 <= (-3.6d+64)) then
        tmp = 0.5d0 / (t / x)
    else if (x <= (-3.3d-110)) 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 <= -3.6e+64) {
		tmp = 0.5 / (t / x);
	} else if (x <= -3.3e-110) {
		tmp = z * (-0.5 / t);
	} else {
		tmp = y * (0.5 / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -3.6e+64:
		tmp = 0.5 / (t / x)
	elif x <= -3.3e-110:
		tmp = z * (-0.5 / t)
	else:
		tmp = y * (0.5 / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -3.6e+64)
		tmp = Float64(0.5 / Float64(t / x));
	elseif (x <= -3.3e-110)
		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 <= -3.6e+64)
		tmp = 0.5 / (t / x);
	elseif (x <= -3.3e-110)
		tmp = z * (-0.5 / t);
	else
		tmp = y * (0.5 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -3.6e+64], N[(0.5 / N[(t / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.3e-110], 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 x < -3.60000000000000014e64

   1. Initial program 100.0%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x + y\right) - z}{t}\right), \color{blue}{2}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x + y\right) - z\right), t\right), 2\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(y + x\right) - z\right), t\right), 2\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y + \left(x - z\right)\right), t\right), 2\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x - z\right) + y\right), t\right), 2\right) \]

      7. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x - \left(z - y\right)\right), t\right), 2\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), t\right), 2\right) \]

      9. --lowering--.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), t\right), 2\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\frac{x - \left(z - y\right)}{t}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{x}{t}\right)}, 2\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 87.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, t\right), 2\right) \]

   7. Simplified 87.2%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. metadata-eval N/A

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

      4. associate-*l/ N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{t}{x}\right)}\right) \]

      7. /-lowering-/.f64 87.0%

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(t, \color{blue}{x}\right)\right) \]

   9. Applied egg-rr 87.0%

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

   if -3.60000000000000014e64 < x < -3.2999999999999999e-110

   1. Initial program 99.9%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x + y\right) - z}{t}\right), \color{blue}{2}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x + y\right) - z\right), t\right), 2\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(y + x\right) - z\right), t\right), 2\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y + \left(x - z\right)\right), t\right), 2\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x - z\right) + y\right), t\right), 2\right) \]

      7. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x - \left(z - y\right)\right), t\right), 2\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), t\right), 2\right) \]

      9. --lowering--.f64 99.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), t\right), 2\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{\frac{x - \left(z - y\right)}{t}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(\mathsf{neg}\left(z\right)\right)}{t} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(z\right)\right)\right), \color{blue}{t}\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(-1 \cdot z\right)\right), t\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot -1\right) \cdot z\right), t\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot z\right), t\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(z \cdot \frac{-1}{2}\right), t\right) \]

      10. *-lowering-*.f64 43.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \frac{-1}{2}\right), t\right) \]

   7. Simplified 43.8%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{-1}{2}}{t}\right), \color{blue}{z}\right) \]

      4. /-lowering-/.f64 43.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, t\right), z\right) \]

   9. Applied egg-rr 43.7%

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

   if -3.2999999999999999e-110 < x

   1. Initial program 99.4%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x + y\right) - z}{t}\right), \color{blue}{2}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x + y\right) - z\right), t\right), 2\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(y + x\right) - z\right), t\right), 2\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y + \left(x - z\right)\right), t\right), 2\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x - z\right) + y\right), t\right), 2\right) \]

      7. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x - \left(z - y\right)\right), t\right), 2\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), t\right), 2\right) \]

      9. --lowering--.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), t\right), 2\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\frac{x - \left(z - y\right)}{t}}{2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l/ N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2 \cdot t}\right), \color{blue}{\left(x - \left(z - y\right)\right)}\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{t}\right), \left(\color{blue}{x} - \left(z - y\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2}\right), t\right), \left(\color{blue}{x} - \left(z - y\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \left(x - \left(z - y\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \left(x + \color{blue}{\left(\mathsf{neg}\left(\left(z - y\right)\right)\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\left(z - y\right)\right)\right)}\right)\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \left(0 - \color{blue}{\left(z - y\right)}\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \left(0 - \left(z + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \left(0 - \left(\left(\mathsf{neg}\left(y\right)\right) + \color{blue}{z}\right)\right)\right)\right) \]

      13. associate--r+ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \left(\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - \color{blue}{z}\right)\right)\right) \]

      14. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) - z\right)\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \left(y - z\right)\right)\right) \]

      16. --lowering--.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(y, \color{blue}{z}\right)\right)\right) \]

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \color{blue}{y}\right) \]
   8. Step-by-step derivation

   9. Simplified 46.8%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+64}:\\ \;\;\;\;\frac{0.5}{\frac{t}{x}}\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{-110}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{0.5}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \frac{-0.5}{t}\\ \mathbf{if}\;z \leq -128000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7.3 \cdot 10^{+50}:\\ \;\;\;\;y \cdot \frac{0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (/ -0.5 t))))
   (if (<= z -128000.0) t_1 (if (<= z 7.3e+50) (* y (/ 0.5 t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = z * (-0.5 / t);
	double tmp;
	if (z <= -128000.0) {
		tmp = t_1;
	} else if (z <= 7.3e+50) {
		tmp = y * (0.5 / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = z * ((-0.5d0) / t)
    if (z <= (-128000.0d0)) then
        tmp = t_1
    else if (z <= 7.3d+50) then
        tmp = y * (0.5d0 / t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = z * (-0.5 / t);
	double tmp;
	if (z <= -128000.0) {
		tmp = t_1;
	} else if (z <= 7.3e+50) {
		tmp = y * (0.5 / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = z * (-0.5 / t)
	tmp = 0
	if z <= -128000.0:
		tmp = t_1
	elif z <= 7.3e+50:
		tmp = y * (0.5 / t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(z * Float64(-0.5 / t))
	tmp = 0.0
	if (z <= -128000.0)
		tmp = t_1;
	elseif (z <= 7.3e+50)
		tmp = Float64(y * Float64(0.5 / t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = z * (-0.5 / t);
	tmp = 0.0;
	if (z <= -128000.0)
		tmp = t_1;
	elseif (z <= 7.3e+50)
		tmp = y * (0.5 / t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[(-0.5 / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -128000.0], t$95$1, If[LessEqual[z, 7.3e+50], N[(y * N[(0.5 / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -128000 or 7.3000000000000003e50 < z

   1. Initial program 99.0%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x + y\right) - z}{t}\right), \color{blue}{2}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x + y\right) - z\right), t\right), 2\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(y + x\right) - z\right), t\right), 2\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y + \left(x - z\right)\right), t\right), 2\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x - z\right) + y\right), t\right), 2\right) \]

      7. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x - \left(z - y\right)\right), t\right), 2\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), t\right), 2\right) \]

      9. --lowering--.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), t\right), 2\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\frac{x - \left(z - y\right)}{t}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(\mathsf{neg}\left(z\right)\right)}{t} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(z\right)\right)\right), \color{blue}{t}\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(-1 \cdot z\right)\right), t\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot -1\right) \cdot z\right), t\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot z\right), t\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(z \cdot \frac{-1}{2}\right), t\right) \]

      10. *-lowering-*.f64 61.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \frac{-1}{2}\right), t\right) \]

   7. Simplified 61.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{-1}{2}}{t}\right), \color{blue}{z}\right) \]

      4. /-lowering-/.f64 61.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, t\right), z\right) \]

   9. Applied egg-rr 61.3%

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

   if -128000 < z < 7.3000000000000003e50

   1. Initial program 99.9%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x + y\right) - z}{t}\right), \color{blue}{2}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x + y\right) - z\right), t\right), 2\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(y + x\right) - z\right), t\right), 2\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y + \left(x - z\right)\right), t\right), 2\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x - z\right) + y\right), t\right), 2\right) \]

      7. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x - \left(z - y\right)\right), t\right), 2\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), t\right), 2\right) \]

      9. --lowering--.f64 99.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), t\right), 2\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{\frac{x - \left(z - y\right)}{t}}{2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l/ N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2 \cdot t}\right), \color{blue}{\left(x - \left(z - y\right)\right)}\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{t}\right), \left(\color{blue}{x} - \left(z - y\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2}\right), t\right), \left(\color{blue}{x} - \left(z - y\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \left(x - \left(z - y\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \left(x + \color{blue}{\left(\mathsf{neg}\left(\left(z - y\right)\right)\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\left(z - y\right)\right)\right)}\right)\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \left(0 - \color{blue}{\left(z - y\right)}\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \left(0 - \left(z + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \left(0 - \left(\left(\mathsf{neg}\left(y\right)\right) + \color{blue}{z}\right)\right)\right)\right) \]

      13. associate--r+ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \left(\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - \color{blue}{z}\right)\right)\right) \]

      14. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) - z\right)\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \left(y - z\right)\right)\right) \]

      16. --lowering--.f64 99.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(y, \color{blue}{z}\right)\right)\right) \]

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \color{blue}{y}\right) \]
   8. Step-by-step derivation

   9. Simplified 51.5%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -128000:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \mathbf{elif}\;z \leq 7.3 \cdot 10^{+50}:\\ \;\;\;\;y \cdot \frac{0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-113}:\\ \;\;\;\;\frac{\frac{x - z}{t}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y - z}{t}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) -1e-113) (/ (/ (- 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-113) {
		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-113)) 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-113) {
		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-113:
		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-113)
		tmp = Float64(Float64(Float64(x - z) / t) / 2.0);
	else
		tmp = Float64(Float64(Float64(y - z) / t) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + y) <= -1e-113)
		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-113], N[(N[(N[(x - z), $MachinePrecision] / t), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(y - z), $MachinePrecision] / t), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -9.99999999999999979e-114

   1. Initial program 99.9%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x + y\right) - z}{t}\right), \color{blue}{2}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x + y\right) - z\right), t\right), 2\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(y + x\right) - z\right), t\right), 2\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y + \left(x - z\right)\right), t\right), 2\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x - z\right) + y\right), t\right), 2\right) \]

      7. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x - \left(z - y\right)\right), t\right), 2\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), t\right), 2\right) \]

      9. --lowering--.f64 99.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), t\right), 2\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{\frac{x - \left(z - y\right)}{t}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{x - z}{t}\right)}, 2\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x - z\right), t\right), 2\right) \]

      2. --lowering--.f64 62.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, z\right), t\right), 2\right) \]

   7. Simplified 62.7%

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

   if -9.99999999999999979e-114 < (+.f64 x y)

   1. Initial program 99.4%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x + y\right) - z}{t}\right), \color{blue}{2}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x + y\right) - z\right), t\right), 2\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(y + x\right) - z\right), t\right), 2\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y + \left(x - z\right)\right), t\right), 2\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x - z\right) + y\right), t\right), 2\right) \]

      7. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x - \left(z - y\right)\right), t\right), 2\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), t\right), 2\right) \]

      9. --lowering--.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), t\right), 2\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\frac{x - \left(z - y\right)}{t}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{y - z}{t}\right)}, 2\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y - z\right), t\right), 2\right) \]

      2. --lowering--.f64 72.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, z\right), t\right), 2\right) \]

   7. Simplified 72.2%

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

Alternative 10: 73.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq 1.3 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{x - z}{t}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + y}{t}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) 1.3e-52) (/ (/ (- 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) <= 1.3e-52) {
		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) <= 1.3d-52) 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) <= 1.3e-52) {
		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) <= 1.3e-52:
		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) <= 1.3e-52)
		tmp = Float64(Float64(Float64(x - z) / t) / 2.0);
	else
		tmp = Float64(Float64(Float64(x + y) / t) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + y) <= 1.3e-52)
		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], 1.3e-52], N[(N[(N[(x - z), $MachinePrecision] / t), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(x + y), $MachinePrecision] / t), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < 1.2999999999999999e-52

   1. Initial program 99.9%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x + y\right) - z}{t}\right), \color{blue}{2}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x + y\right) - z\right), t\right), 2\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(y + x\right) - z\right), t\right), 2\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y + \left(x - z\right)\right), t\right), 2\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x - z\right) + y\right), t\right), 2\right) \]

      7. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x - \left(z - y\right)\right), t\right), 2\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), t\right), 2\right) \]

      9. --lowering--.f64 99.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), t\right), 2\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{\frac{x - \left(z - y\right)}{t}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{x - z}{t}\right)}, 2\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x - z\right), t\right), 2\right) \]

      2. --lowering--.f64 70.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, z\right), t\right), 2\right) \]

   7. Simplified 70.3%

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

   if 1.2999999999999999e-52 < (+.f64 x y)

   1. Initial program 99.2%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x + y\right) - z}{t}\right), \color{blue}{2}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x + y\right) - z\right), t\right), 2\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(y + x\right) - z\right), t\right), 2\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y + \left(x - z\right)\right), t\right), 2\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x - z\right) + y\right), t\right), 2\right) \]

      7. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x - \left(z - y\right)\right), t\right), 2\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), t\right), 2\right) \]

      9. --lowering--.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), t\right), 2\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\frac{x - \left(z - y\right)}{t}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{x + y}{t}\right)}, 2\right) \]
   6. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 \cdot x + y}{t}\right), 2\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(-1 \cdot -1\right) \cdot x + y}{t}\right), 2\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(-1 \cdot x\right) + y}{t}\right), 2\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(-1 \cdot x\right) + 1 \cdot y}{t}\right), 2\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(-1 \cdot x\right) + \left(-1 \cdot -1\right) \cdot y}{t}\right), 2\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(-1 \cdot x\right) + -1 \cdot \left(-1 \cdot y\right)}{t}\right), 2\right) \]

      7. distribute-lft-out N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(-1 \cdot x + -1 \cdot y\right)}{t}\right), 2\right) \]

      8. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(-1 \cdot \left(x + y\right)\right)}{t}\right), 2\right) \]

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(\left(-1 \cdot \left(x + y\right)\right) \cdot 1\right)}{t}\right), 2\right) \]

      10. *-inverses N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(\left(-1 \cdot \left(x + y\right)\right) \cdot \frac{z}{z}\right)}{t}\right), 2\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \frac{\left(-1 \cdot \left(x + y\right)\right) \cdot z}{z}}{t}\right), 2\right) \]

      12. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(\frac{-1 \cdot \left(x + y\right)}{z} \cdot z\right)}{t}\right), 2\right) \]

      13. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(\left(-1 \cdot \frac{x + y}{z}\right) \cdot z\right)}{t}\right), 2\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(z \cdot \left(-1 \cdot \frac{x + y}{z}\right)\right)}{t}\right), 2\right) \]

      15. neg-mul-1 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(z \cdot \left(-1 \cdot \frac{x + y}{z}\right)\right)}{t}\right), 2\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(z \cdot \left(-1 \cdot \frac{x + y}{z}\right)\right)\right), t\right), 2\right) \]

   7. Simplified 77.6%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq 1.3 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{x - z}{t}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + y}{t}}{2}\\ \end{array} \]
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 99.6%

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

   1. associate-/r* N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x + y\right) - z}{t}\right), \color{blue}{2}\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x + y\right) - z\right), t\right), 2\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(y + x\right) - z\right), t\right), 2\right) \]

   5. associate--l+ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y + \left(x - z\right)\right), t\right), 2\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x - z\right) + y\right), t\right), 2\right) \]

   7. associate-+l- N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x - \left(z - y\right)\right), t\right), 2\right) \]

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), t\right), 2\right) \]

   9. --lowering--.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), t\right), 2\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{\frac{x - \left(z - y\right)}{t}}{2}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/l/ N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2 \cdot t}\right), \color{blue}{\left(x - \left(z - y\right)\right)}\right) \]

   5. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{t}\right), \left(\color{blue}{x} - \left(z - y\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2}\right), t\right), \left(\color{blue}{x} - \left(z - y\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \left(x - \left(z - y\right)\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \left(x + \color{blue}{\left(\mathsf{neg}\left(\left(z - y\right)\right)\right)}\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\left(z - y\right)\right)\right)}\right)\right) \]

   10. neg-sub0 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \left(0 - \color{blue}{\left(z - y\right)}\right)\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \left(0 - \left(z + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right)\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \left(0 - \left(\left(\mathsf{neg}\left(y\right)\right) + \color{blue}{z}\right)\right)\right)\right) \]

   13. associate--r+ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \left(\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - \color{blue}{z}\right)\right)\right) \]

   14. neg-sub0 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) - z\right)\right)\right) \]

   15. remove-double-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \left(y - z\right)\right)\right) \]

   16. --lowering--.f64 99.7%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, t\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(y, \color{blue}{z}\right)\right)\right) \]

6. Applied egg-rr 99.7%

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

Alternative 12: 37.5% accurate, 1.8× 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 99.6%

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

   1. associate-/r* N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x + y\right) - z}{t}\right), \color{blue}{2}\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x + y\right) - z\right), t\right), 2\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(y + x\right) - z\right), t\right), 2\right) \]

   5. associate--l+ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y + \left(x - z\right)\right), t\right), 2\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x - z\right) + y\right), t\right), 2\right) \]

   7. associate-+l- N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x - \left(z - y\right)\right), t\right), 2\right) \]

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), t\right), 2\right) \]

   9. --lowering--.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), t\right), 2\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{\frac{x - \left(z - y\right)}{t}}{2}} \]
4. Add Preprocessing

5. Taylor expanded in z around inf

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

   1. associate-*r/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r* N/A

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

   4. mul-1-neg N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \left(\mathsf{neg}\left(z\right)\right)}{t} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(z\right)\right)\right), \color{blue}{t}\right) \]

   6. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(-1 \cdot z\right)\right), t\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot -1\right) \cdot z\right), t\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot z\right), t\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(z \cdot \frac{-1}{2}\right), t\right) \]

   10. *-lowering-*.f64 35.0%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \frac{-1}{2}\right), t\right) \]

7. Simplified 35.0%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{-1}{2}}{t}\right), \color{blue}{z}\right) \]

   4. /-lowering-/.f64 34.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, t\right), z\right) \]

9. Applied egg-rr 34.9%

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

10. Final simplification 34.9%

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

Reproduce

herbie shell --seed 2024138 
(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)))