Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2

Percentage Accurate: 61.1% → 88.6%

Time: 19.1s

Alternatives: 15

Speedup: 18.6×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = ((x * y) * z) / sqrt(((z * z) - (t * a)))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((x * y) * z) / Math.sqrt(((z * z) - (t * a)));
}

Python

def code(x, y, z, t, a):
	return ((x * y) * z) / math.sqrt(((z * z) - (t * a)))

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = ((x * y) * z) / sqrt(((z * z) - (t * a)))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((x * y) * z) / Math.sqrt(((z * z) - (t * a)));
}

Python

def code(x, y, z, t, a):
	return ((x * y) * z) / math.sqrt(((z * z) - (t * a)))

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := \sqrt{z \cdot z - a \cdot t}\\ t_2 := x \cdot \left(-y\right)\\ \mathbf{if}\;z \leq -8.8 \cdot 10^{+114}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -3.3 \cdot 10^{-147}:\\ \;\;\;\;\frac{x}{t_1} \cdot \left(z \cdot y\right)\\ \mathbf{elif}\;z \leq -3.35 \cdot 10^{-180}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+26}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{t_1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (sqrt (- (* z z) (* a t)))) (t_2 (* x (- y))))
   (if (<= z -8.8e+114)
     t_2
     (if (<= z -3.3e-147)
       (* (/ x t_1) (* z y))
       (if (<= z -3.35e-180)
         t_2
         (if (<= z 3.2e+26) (* y (/ (* z x) t_1)) (* x y)))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = sqrt(((z * z) - (a * t)));
	double t_2 = x * -y;
	double tmp;
	if (z <= -8.8e+114) {
		tmp = t_2;
	} else if (z <= -3.3e-147) {
		tmp = (x / t_1) * (z * y);
	} else if (z <= -3.35e-180) {
		tmp = t_2;
	} else if (z <= 3.2e+26) {
		tmp = y * ((z * x) / t_1);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt(((z * z) - (a * t)))
    t_2 = x * -y
    if (z <= (-8.8d+114)) then
        tmp = t_2
    else if (z <= (-3.3d-147)) then
        tmp = (x / t_1) * (z * y)
    else if (z <= (-3.35d-180)) then
        tmp = t_2
    else if (z <= 3.2d+26) then
        tmp = y * ((z * x) / t_1)
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.sqrt(((z * z) - (a * t)));
	double t_2 = x * -y;
	double tmp;
	if (z <= -8.8e+114) {
		tmp = t_2;
	} else if (z <= -3.3e-147) {
		tmp = (x / t_1) * (z * y);
	} else if (z <= -3.35e-180) {
		tmp = t_2;
	} else if (z <= 3.2e+26) {
		tmp = y * ((z * x) / t_1);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	t_1 = math.sqrt(((z * z) - (a * t)))
	t_2 = x * -y
	tmp = 0
	if z <= -8.8e+114:
		tmp = t_2
	elif z <= -3.3e-147:
		tmp = (x / t_1) * (z * y)
	elif z <= -3.35e-180:
		tmp = t_2
	elif z <= 3.2e+26:
		tmp = y * ((z * x) / t_1)
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = sqrt(Float64(Float64(z * z) - Float64(a * t)))
	t_2 = Float64(x * Float64(-y))
	tmp = 0.0
	if (z <= -8.8e+114)
		tmp = t_2;
	elseif (z <= -3.3e-147)
		tmp = Float64(Float64(x / t_1) * Float64(z * y));
	elseif (z <= -3.35e-180)
		tmp = t_2;
	elseif (z <= 3.2e+26)
		tmp = Float64(y * Float64(Float64(z * x) / t_1));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = sqrt(((z * z) - (a * t)));
	t_2 = x * -y;
	tmp = 0.0;
	if (z <= -8.8e+114)
		tmp = t_2;
	elseif (z <= -3.3e-147)
		tmp = (x / t_1) * (z * y);
	elseif (z <= -3.35e-180)
		tmp = t_2;
	elseif (z <= 3.2e+26)
		tmp = y * ((z * x) / t_1);
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(x * (-y)), $MachinePrecision]}, If[LessEqual[z, -8.8e+114], t$95$2, If[LessEqual[z, -3.3e-147], N[(N[(x / t$95$1), $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -3.35e-180], t$95$2, If[LessEqual[z, 3.2e+26], N[(y * N[(N[(z * x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -8.8000000000000001e114 or -3.29999999999999987e-147 < z < -3.3499999999999999e-180

   1. Initial program 27.2%

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

      1. *-commutative 27.2%

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

      2. associate-*l* 26.4%

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

      3. associate-*r/ 28.4%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 28.4%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around -inf 94.9%

      \[\leadsto y \cdot \color{blue}{\left(-1 \cdot x\right)} \]
   5. Step-by-step derivation

      1. neg-mul-1 94.9%

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

   6. Simplified 94.9%

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

   if -8.8000000000000001e114 < z < -3.29999999999999987e-147

   1. Initial program 90.8%

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

      1. *-commutative 90.8%

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

      2. associate-*l* 89.3%

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

      3. associate-*r/ 91.9%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 91.9%

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

      1. *-commutative 91.9%

         \[\leadsto \color{blue}{\frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}} \cdot y} \]

      2. associate-/l* 94.5%

         \[\leadsto \color{blue}{\frac{x}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \cdot y \]

      3. associate-/r/ 95.7%

         \[\leadsto \color{blue}{\frac{x}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}{y}}} \]

      4. associate-/l/ 89.6%

         \[\leadsto \frac{x}{\color{blue}{\frac{\sqrt{z \cdot z - t \cdot a}}{y \cdot z}}} \]

      5. *-commutative 89.6%

         \[\leadsto \frac{x}{\frac{\sqrt{z \cdot z - t \cdot a}}{\color{blue}{z \cdot y}}} \]

   5. Applied egg-rr 89.6%

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

      1. associate-/r/ 89.4%

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

   7. Applied egg-rr 89.4%

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

   if -3.3499999999999999e-180 < z < 3.20000000000000029e26

   1. Initial program 77.8%

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

      1. *-commutative 77.8%

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

      2. associate-*l* 74.1%

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

      3. associate-*r/ 75.6%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 75.6%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   if 3.20000000000000029e26 < z

   1. Initial program 36.8%

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

      1. *-commutative 36.8%

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

      2. associate-*l* 34.8%

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

      3. associate-*r/ 35.1%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 35.1%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around inf 96.9%

      \[\leadsto y \cdot \color{blue}{x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.8 \cdot 10^{+114}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq -3.3 \cdot 10^{-147}:\\ \;\;\;\;\frac{x}{\sqrt{z \cdot z - a \cdot t}} \cdot \left(z \cdot y\right)\\ \mathbf{elif}\;z \leq -3.35 \cdot 10^{-180}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+26}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 2: 88.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := \sqrt{z \cdot z - a \cdot t}\\ t_2 := x \cdot \left(-y\right)\\ \mathbf{if}\;z \leq -5.3 \cdot 10^{+106}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -3.3 \cdot 10^{-147}:\\ \;\;\;\;z \cdot \left(y \cdot \frac{x}{t_1}\right)\\ \mathbf{elif}\;z \leq -3.35 \cdot 10^{-180}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+26}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{t_1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (sqrt (- (* z z) (* a t)))) (t_2 (* x (- y))))
   (if (<= z -5.3e+106)
     t_2
     (if (<= z -3.3e-147)
       (* z (* y (/ x t_1)))
       (if (<= z -3.35e-180)
         t_2
         (if (<= z 3.8e+26) (* y (/ (* z x) t_1)) (* x y)))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = sqrt(((z * z) - (a * t)));
	double t_2 = x * -y;
	double tmp;
	if (z <= -5.3e+106) {
		tmp = t_2;
	} else if (z <= -3.3e-147) {
		tmp = z * (y * (x / t_1));
	} else if (z <= -3.35e-180) {
		tmp = t_2;
	} else if (z <= 3.8e+26) {
		tmp = y * ((z * x) / t_1);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt(((z * z) - (a * t)))
    t_2 = x * -y
    if (z <= (-5.3d+106)) then
        tmp = t_2
    else if (z <= (-3.3d-147)) then
        tmp = z * (y * (x / t_1))
    else if (z <= (-3.35d-180)) then
        tmp = t_2
    else if (z <= 3.8d+26) then
        tmp = y * ((z * x) / t_1)
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.sqrt(((z * z) - (a * t)));
	double t_2 = x * -y;
	double tmp;
	if (z <= -5.3e+106) {
		tmp = t_2;
	} else if (z <= -3.3e-147) {
		tmp = z * (y * (x / t_1));
	} else if (z <= -3.35e-180) {
		tmp = t_2;
	} else if (z <= 3.8e+26) {
		tmp = y * ((z * x) / t_1);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	t_1 = math.sqrt(((z * z) - (a * t)))
	t_2 = x * -y
	tmp = 0
	if z <= -5.3e+106:
		tmp = t_2
	elif z <= -3.3e-147:
		tmp = z * (y * (x / t_1))
	elif z <= -3.35e-180:
		tmp = t_2
	elif z <= 3.8e+26:
		tmp = y * ((z * x) / t_1)
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = sqrt(Float64(Float64(z * z) - Float64(a * t)))
	t_2 = Float64(x * Float64(-y))
	tmp = 0.0
	if (z <= -5.3e+106)
		tmp = t_2;
	elseif (z <= -3.3e-147)
		tmp = Float64(z * Float64(y * Float64(x / t_1)));
	elseif (z <= -3.35e-180)
		tmp = t_2;
	elseif (z <= 3.8e+26)
		tmp = Float64(y * Float64(Float64(z * x) / t_1));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = sqrt(((z * z) - (a * t)));
	t_2 = x * -y;
	tmp = 0.0;
	if (z <= -5.3e+106)
		tmp = t_2;
	elseif (z <= -3.3e-147)
		tmp = z * (y * (x / t_1));
	elseif (z <= -3.35e-180)
		tmp = t_2;
	elseif (z <= 3.8e+26)
		tmp = y * ((z * x) / t_1);
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(x * (-y)), $MachinePrecision]}, If[LessEqual[z, -5.3e+106], t$95$2, If[LessEqual[z, -3.3e-147], N[(z * N[(y * N[(x / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -3.35e-180], t$95$2, If[LessEqual[z, 3.8e+26], N[(y * N[(N[(z * x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -5.3e106 or -3.29999999999999987e-147 < z < -3.3499999999999999e-180

   1. Initial program 30.5%

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

      1. *-commutative 30.5%

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

      2. associate-*l* 29.8%

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

      3. associate-*r/ 33.1%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 33.1%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around -inf 95.2%

      \[\leadsto y \cdot \color{blue}{\left(-1 \cdot x\right)} \]
   5. Step-by-step derivation

      1. neg-mul-1 95.2%

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

   6. Simplified 95.2%

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

   if -5.3e106 < z < -3.29999999999999987e-147

   1. Initial program 91.6%

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

      1. associate-*l/ 93.5%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\sqrt{z \cdot z - t \cdot a}} \cdot z} \]

   3. Simplified 93.5%

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

      1. associate-/l* 93.5%

         \[\leadsto \color{blue}{\frac{x}{\frac{\sqrt{z \cdot z - t \cdot a}}{y}}} \cdot z \]

      2. associate-/r/ 90.6%

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

   5. Applied egg-rr 90.6%

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

   if -3.3499999999999999e-180 < z < 3.8000000000000002e26

   1. Initial program 77.8%

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

      1. *-commutative 77.8%

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

      2. associate-*l* 74.1%

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

      3. associate-*r/ 75.6%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 75.6%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   if 3.8000000000000002e26 < z

   1. Initial program 36.8%

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

      1. *-commutative 36.8%

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

      2. associate-*l* 34.8%

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

      3. associate-*r/ 35.1%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 35.1%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around inf 96.9%

      \[\leadsto y \cdot \color{blue}{x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.3 \cdot 10^{+106}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq -3.3 \cdot 10^{-147}:\\ \;\;\;\;z \cdot \left(y \cdot \frac{x}{\sqrt{z \cdot z - a \cdot t}}\right)\\ \mathbf{elif}\;z \leq -3.35 \cdot 10^{-180}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+26}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 3: 89.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+19}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{+26}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.4e+19)
   (* x (- y))
   (if (<= z 2.15e+26) (* y (/ (* z x) (sqrt (- (* z z) (* a t))))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.4e+19) {
		tmp = x * -y;
	} else if (z <= 2.15e+26) {
		tmp = y * ((z * x) / sqrt(((z * z) - (a * t))));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.4d+19)) then
        tmp = x * -y
    else if (z <= 2.15d+26) then
        tmp = y * ((z * x) / sqrt(((z * z) - (a * t))))
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.4e+19) {
		tmp = x * -y;
	} else if (z <= 2.15e+26) {
		tmp = y * ((z * x) / Math.sqrt(((z * z) - (a * t))));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.4e+19:
		tmp = x * -y
	elif z <= 2.15e+26:
		tmp = y * ((z * x) / math.sqrt(((z * z) - (a * t))))
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.4e+19)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 2.15e+26)
		tmp = Float64(y * Float64(Float64(z * x) / sqrt(Float64(Float64(z * z) - Float64(a * t)))));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.4e+19)
		tmp = x * -y;
	elseif (z <= 2.15e+26)
		tmp = y * ((z * x) / sqrt(((z * z) - (a * t))));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.4e+19], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 2.15e+26], N[(y * N[(N[(z * x), $MachinePrecision] / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.4e19

   1. Initial program 50.7%

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

      1. *-commutative 50.7%

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

      2. associate-*l* 49.4%

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

      3. associate-*r/ 52.9%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 52.9%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around -inf 92.0%

      \[\leadsto y \cdot \color{blue}{\left(-1 \cdot x\right)} \]
   5. Step-by-step derivation

      1. neg-mul-1 92.0%

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

   6. Simplified 92.0%

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

   if -1.4e19 < z < 2.1499999999999999e26

   1. Initial program 81.5%

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

      1. *-commutative 81.5%

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

      2. associate-*l* 79.0%

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

      3. associate-*r/ 79.9%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 79.9%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   if 2.1499999999999999e26 < z

   1. Initial program 36.8%

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

      1. *-commutative 36.8%

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

      2. associate-*l* 34.8%

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

      3. associate-*r/ 35.1%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 35.1%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around inf 96.9%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+19}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{+26}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 4: 90.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{a}{z} \cdot \frac{t}{z}, -1\right) \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+26}:\\ \;\;\;\;\frac{z}{\sqrt{z \cdot z - a \cdot t}} \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.4e+154)
   (* (fma -0.5 (* (/ a z) (/ t z)) -1.0) (* x y))
   (if (<= z 3.8e+26) (* (/ z (sqrt (- (* z z) (* a t)))) (* x y)) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.4e+154) {
		tmp = fma(-0.5, ((a / z) * (t / z)), -1.0) * (x * y);
	} else if (z <= 3.8e+26) {
		tmp = (z / sqrt(((z * z) - (a * t)))) * (x * y);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.4e+154)
		tmp = Float64(fma(-0.5, Float64(Float64(a / z) * Float64(t / z)), -1.0) * Float64(x * y));
	elseif (z <= 3.8e+26)
		tmp = Float64(Float64(z / sqrt(Float64(Float64(z * z) - Float64(a * t)))) * Float64(x * y));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.4e+154], N[(N[(-0.5 * N[(N[(a / z), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.8e+26], N[(N[(z / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.4e154

   1. Initial program 10.0%

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

      1. associate-/l* 10.6%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

   3. Simplified 10.6%

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

      1. clear-num 10.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}{x \cdot y}}} \]

      2. associate-/r/ 10.6%

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

      3. clear-num 10.6%

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

   5. Applied egg-rr 10.6%

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

   6. Taylor expanded in z around -inf 83.8%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \frac{a \cdot t}{{z}^{2}} - 1\right)} \cdot \left(x \cdot y\right) \]
   7. Step-by-step derivation

      1. fma-neg 83.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{a \cdot t}{{z}^{2}}, -1\right)} \cdot \left(x \cdot y\right) \]

      2. unpow2 83.8%

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

      3. times-frac 100.0%

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

      4. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   if -1.4e154 < z < 3.8000000000000002e26

   1. Initial program 81.7%

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

      1. associate-/l* 84.9%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

   3. Simplified 84.9%

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

      1. clear-num 84.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}{x \cdot y}}} \]

      2. associate-/r/ 84.9%

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

      3. clear-num 84.9%

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

   5. Applied egg-rr 84.9%

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

   if 3.8000000000000002e26 < z

   1. Initial program 36.8%

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

      1. *-commutative 36.8%

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

      2. associate-*l* 34.8%

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

      3. associate-*r/ 35.1%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 35.1%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around inf 96.9%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{a}{z} \cdot \frac{t}{z}, -1\right) \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+26}:\\ \;\;\;\;\frac{z}{\sqrt{z \cdot z - a \cdot t}} \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 5: 90.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+121}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+26}:\\ \;\;\;\;\frac{z}{\sqrt{z \cdot z - a \cdot t}} \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1e+121)
   (* x (- y))
   (if (<= z 3.4e+26) (* (/ z (sqrt (- (* z z) (* a t)))) (* x y)) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1e+121) {
		tmp = x * -y;
	} else if (z <= 3.4e+26) {
		tmp = (z / sqrt(((z * z) - (a * t)))) * (x * y);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1d+121)) then
        tmp = x * -y
    else if (z <= 3.4d+26) then
        tmp = (z / sqrt(((z * z) - (a * t)))) * (x * y)
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1e+121) {
		tmp = x * -y;
	} else if (z <= 3.4e+26) {
		tmp = (z / Math.sqrt(((z * z) - (a * t)))) * (x * y);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1e+121:
		tmp = x * -y
	elif z <= 3.4e+26:
		tmp = (z / math.sqrt(((z * z) - (a * t)))) * (x * y)
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1e+121)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 3.4e+26)
		tmp = Float64(Float64(z / sqrt(Float64(Float64(z * z) - Float64(a * t)))) * Float64(x * y));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1e+121)
		tmp = x * -y;
	elseif (z <= 3.4e+26)
		tmp = (z / sqrt(((z * z) - (a * t)))) * (x * y);
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1e+121], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 3.4e+26], N[(N[(z / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.00000000000000004e121

   1. Initial program 27.6%

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

      1. *-commutative 27.6%

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

      2. associate-*l* 27.4%

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

      3. associate-*r/ 29.5%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 29.5%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around -inf 95.7%

      \[\leadsto y \cdot \color{blue}{\left(-1 \cdot x\right)} \]
   5. Step-by-step derivation

      1. neg-mul-1 95.7%

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

   6. Simplified 95.7%

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

   if -1.00000000000000004e121 < z < 3.4000000000000003e26

   1. Initial program 82.0%

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

      1. associate-/l* 84.9%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

   3. Simplified 84.9%

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

      1. clear-num 84.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}{x \cdot y}}} \]

      2. associate-/r/ 84.9%

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

      3. clear-num 84.9%

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

   5. Applied egg-rr 84.9%

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

   if 3.4000000000000003e26 < z

   1. Initial program 36.8%

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

      1. *-commutative 36.8%

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

      2. associate-*l* 34.8%

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

      3. associate-*r/ 35.1%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 35.1%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around inf 96.9%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+121}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+26}:\\ \;\;\;\;\frac{z}{\sqrt{z \cdot z - a \cdot t}} \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 6: 83.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{+56}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq -3.35 \cdot 10^{-180}:\\ \;\;\;\;\frac{x \cdot y}{\frac{0.5 \cdot \frac{a \cdot t}{z} - z}{z}}\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-106}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.7e+56)
   (* x (- y))
   (if (<= z -3.35e-180)
     (/ (* x y) (/ (- (* 0.5 (/ (* a t) z)) z) z))
     (if (<= z 7.2e-106)
       (* y (/ (* z x) (sqrt (* a (- t)))))
       (/ (* x y) (+ 1.0 (* -0.5 (/ a (/ (* z z) t)))))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.7e+56) {
		tmp = x * -y;
	} else if (z <= -3.35e-180) {
		tmp = (x * y) / (((0.5 * ((a * t) / z)) - z) / z);
	} else if (z <= 7.2e-106) {
		tmp = y * ((z * x) / sqrt((a * -t)));
	} else {
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-3.7d+56)) then
        tmp = x * -y
    else if (z <= (-3.35d-180)) then
        tmp = (x * y) / (((0.5d0 * ((a * t) / z)) - z) / z)
    else if (z <= 7.2d-106) then
        tmp = y * ((z * x) / sqrt((a * -t)))
    else
        tmp = (x * y) / (1.0d0 + ((-0.5d0) * (a / ((z * z) / t))))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.7e+56) {
		tmp = x * -y;
	} else if (z <= -3.35e-180) {
		tmp = (x * y) / (((0.5 * ((a * t) / z)) - z) / z);
	} else if (z <= 7.2e-106) {
		tmp = y * ((z * x) / Math.sqrt((a * -t)));
	} else {
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.7e+56:
		tmp = x * -y
	elif z <= -3.35e-180:
		tmp = (x * y) / (((0.5 * ((a * t) / z)) - z) / z)
	elif z <= 7.2e-106:
		tmp = y * ((z * x) / math.sqrt((a * -t)))
	else:
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.7e+56)
		tmp = Float64(x * Float64(-y));
	elseif (z <= -3.35e-180)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(0.5 * Float64(Float64(a * t) / z)) - z) / z));
	elseif (z <= 7.2e-106)
		tmp = Float64(y * Float64(Float64(z * x) / sqrt(Float64(a * Float64(-t)))));
	else
		tmp = Float64(Float64(x * y) / Float64(1.0 + Float64(-0.5 * Float64(a / Float64(Float64(z * z) / t)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -3.7e+56)
		tmp = x * -y;
	elseif (z <= -3.35e-180)
		tmp = (x * y) / (((0.5 * ((a * t) / z)) - z) / z);
	elseif (z <= 7.2e-106)
		tmp = y * ((z * x) / sqrt((a * -t)));
	else
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.7e+56], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, -3.35e-180], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(0.5 * N[(N[(a * t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.2e-106], N[(y * N[(N[(z * x), $MachinePrecision] / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(a / N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -3.69999999999999997e56

   1. Initial program 44.2%

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

      1. *-commutative 44.2%

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

      2. associate-*l* 44.0%

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

      3. associate-*r/ 47.9%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 47.9%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around -inf 92.2%

      \[\leadsto y \cdot \color{blue}{\left(-1 \cdot x\right)} \]
   5. Step-by-step derivation

      1. neg-mul-1 92.2%

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

   6. Simplified 92.2%

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

   if -3.69999999999999997e56 < z < -3.3499999999999999e-180

   1. Initial program 89.5%

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

      1. associate-/l* 89.7%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

   3. Simplified 89.7%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

   4. Taylor expanded in z around -inf 83.9%

      \[\leadsto \frac{x \cdot y}{\frac{\color{blue}{0.5 \cdot \frac{a \cdot t}{z} + -1 \cdot z}}{z}} \]

   if -3.3499999999999999e-180 < z < 7.20000000000000025e-106

   1. Initial program 64.6%

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

      1. *-commutative 64.6%

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

      2. associate-*l* 58.3%

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

      3. associate-*r/ 60.8%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 60.8%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around 0 60.8%

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

      1. associate-*r* 60.8%

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

      2. neg-mul-1 60.8%

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

   6. Simplified 60.8%

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

   if 7.20000000000000025e-106 < z

   1. Initial program 55.6%

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

      1. associate-/l* 57.1%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

   3. Simplified 57.1%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

   4. Taylor expanded in z around inf 84.7%

      \[\leadsto \frac{x \cdot y}{\color{blue}{1 + -0.5 \cdot \frac{a \cdot t}{{z}^{2}}}} \]
   5. Step-by-step derivation

      1. unpow2 84.7%

         \[\leadsto \frac{x \cdot y}{1 + -0.5 \cdot \frac{a \cdot t}{\color{blue}{z \cdot z}}} \]

      2. associate-/l* 89.4%

         \[\leadsto \frac{x \cdot y}{1 + -0.5 \cdot \color{blue}{\frac{a}{\frac{z \cdot z}{t}}}} \]

   6. Simplified 89.4%

      \[\leadsto \frac{x \cdot y}{\color{blue}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{+56}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq -3.35 \cdot 10^{-180}:\\ \;\;\;\;\frac{x \cdot y}{\frac{0.5 \cdot \frac{a \cdot t}{z} - z}{z}}\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-106}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}\\ \end{array} \]

Alternative 7: 77.4% accurate, 4.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := x \cdot \left(-y\right)\\ \mathbf{if}\;z \leq -210000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -9 \cdot 10^{-132}:\\ \;\;\;\;y \cdot \frac{z}{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right) - z}{x}}\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{-195}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+23}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z + -0.5 \cdot \frac{a \cdot t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* x (- y))))
   (if (<= z -210000.0)
     t_1
     (if (<= z -9e-132)
       (* y (/ z (/ (- (* 0.5 (* a (/ t z))) z) x)))
       (if (<= z -2.2e-195)
         t_1
         (if (<= z 2.1e+23)
           (* y (/ (* z x) (+ z (* -0.5 (/ (* a t) z)))))
           (* x y)))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = x * -y;
	double tmp;
	if (z <= -210000.0) {
		tmp = t_1;
	} else if (z <= -9e-132) {
		tmp = y * (z / (((0.5 * (a * (t / z))) - z) / x));
	} else if (z <= -2.2e-195) {
		tmp = t_1;
	} else if (z <= 2.1e+23) {
		tmp = y * ((z * x) / (z + (-0.5 * ((a * t) / z))));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * -y
    if (z <= (-210000.0d0)) then
        tmp = t_1
    else if (z <= (-9d-132)) then
        tmp = y * (z / (((0.5d0 * (a * (t / z))) - z) / x))
    else if (z <= (-2.2d-195)) then
        tmp = t_1
    else if (z <= 2.1d+23) then
        tmp = y * ((z * x) / (z + ((-0.5d0) * ((a * t) / z))))
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x * -y;
	double tmp;
	if (z <= -210000.0) {
		tmp = t_1;
	} else if (z <= -9e-132) {
		tmp = y * (z / (((0.5 * (a * (t / z))) - z) / x));
	} else if (z <= -2.2e-195) {
		tmp = t_1;
	} else if (z <= 2.1e+23) {
		tmp = y * ((z * x) / (z + (-0.5 * ((a * t) / z))));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	t_1 = x * -y
	tmp = 0
	if z <= -210000.0:
		tmp = t_1
	elif z <= -9e-132:
		tmp = y * (z / (((0.5 * (a * (t / z))) - z) / x))
	elif z <= -2.2e-195:
		tmp = t_1
	elif z <= 2.1e+23:
		tmp = y * ((z * x) / (z + (-0.5 * ((a * t) / z))))
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = Float64(x * Float64(-y))
	tmp = 0.0
	if (z <= -210000.0)
		tmp = t_1;
	elseif (z <= -9e-132)
		tmp = Float64(y * Float64(z / Float64(Float64(Float64(0.5 * Float64(a * Float64(t / z))) - z) / x)));
	elseif (z <= -2.2e-195)
		tmp = t_1;
	elseif (z <= 2.1e+23)
		tmp = Float64(y * Float64(Float64(z * x) / Float64(z + Float64(-0.5 * Float64(Float64(a * t) / z)))));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = x * -y;
	tmp = 0.0;
	if (z <= -210000.0)
		tmp = t_1;
	elseif (z <= -9e-132)
		tmp = y * (z / (((0.5 * (a * (t / z))) - z) / x));
	elseif (z <= -2.2e-195)
		tmp = t_1;
	elseif (z <= 2.1e+23)
		tmp = y * ((z * x) / (z + (-0.5 * ((a * t) / z))));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x * (-y)), $MachinePrecision]}, If[LessEqual[z, -210000.0], t$95$1, If[LessEqual[z, -9e-132], N[(y * N[(z / N[(N[(N[(0.5 * N[(a * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -2.2e-195], t$95$1, If[LessEqual[z, 2.1e+23], N[(y * N[(N[(z * x), $MachinePrecision] / N[(z + N[(-0.5 * N[(N[(a * t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -2.1e5 or -8.9999999999999999e-132 < z < -2.20000000000000005e-195

   1. Initial program 52.6%

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

      1. *-commutative 52.6%

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

      2. associate-*l* 50.1%

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

      3. associate-*r/ 53.1%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 53.1%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around -inf 89.5%

      \[\leadsto y \cdot \color{blue}{\left(-1 \cdot x\right)} \]
   5. Step-by-step derivation

      1. neg-mul-1 89.5%

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

   6. Simplified 89.5%

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

   if -2.1e5 < z < -8.9999999999999999e-132

   1. Initial program 96.8%

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

      1. *-commutative 96.8%

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

      2. associate-*l* 96.7%

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

      3. associate-*r/ 96.8%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 96.8%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around -inf 82.5%

      \[\leadsto y \cdot \frac{x \cdot z}{\color{blue}{0.5 \cdot \frac{a \cdot t}{z} + -1 \cdot z}} \]

   5. Taylor expanded in x around 0 82.5%

      \[\leadsto y \cdot \color{blue}{\frac{z \cdot x}{0.5 \cdot \frac{a \cdot t}{z} + -1 \cdot z}} \]
   6. Step-by-step derivation

      1. associate-/l* 82.2%

         \[\leadsto y \cdot \color{blue}{\frac{z}{\frac{0.5 \cdot \frac{a \cdot t}{z} + -1 \cdot z}{x}}} \]

      2. neg-mul-1 82.2%

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

      3. +-commutative 82.2%

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

      4. neg-mul-1 82.2%

         \[\leadsto y \cdot \frac{z}{\frac{\color{blue}{-1 \cdot z} + 0.5 \cdot \frac{a \cdot t}{z}}{x}} \]

      5. neg-mul-1 82.2%

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

      6. +-commutative 82.2%

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

      7. associate-*r/ 82.2%

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

      8. unsub-neg 82.2%

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

   7. Simplified 82.2%

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

   if -2.20000000000000005e-195 < z < 2.1000000000000001e23

   1. Initial program 77.5%

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

      1. *-commutative 77.5%

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

      2. associate-*l* 75.2%

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

      3. associate-*r/ 76.7%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 76.7%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around inf 49.9%

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

   if 2.1000000000000001e23 < z

   1. Initial program 36.8%

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

      1. *-commutative 36.8%

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

      2. associate-*l* 34.8%

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

      3. associate-*r/ 35.1%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 35.1%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around inf 96.9%

      \[\leadsto y \cdot \color{blue}{x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -210000:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq -9 \cdot 10^{-132}:\\ \;\;\;\;y \cdot \frac{z}{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right) - z}{x}}\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{-195}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+23}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z + -0.5 \cdot \frac{a \cdot t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 8: 77.3% accurate, 5.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := x \cdot \left(-y\right)\\ \mathbf{if}\;z \leq -180000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -3 \cdot 10^{-135}:\\ \;\;\;\;y \cdot \frac{z}{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right) - z}{x}}\\ \mathbf{elif}\;z \leq -2.05 \cdot 10^{-194}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* x (- y))))
   (if (<= z -180000.0)
     t_1
     (if (<= z -3e-135)
       (* y (/ z (/ (- (* 0.5 (* a (/ t z))) z) x)))
       (if (<= z -2.05e-194)
         t_1
         (/ (* x y) (+ 1.0 (* -0.5 (/ a (/ (* z z) t))))))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = x * -y;
	double tmp;
	if (z <= -180000.0) {
		tmp = t_1;
	} else if (z <= -3e-135) {
		tmp = y * (z / (((0.5 * (a * (t / z))) - z) / x));
	} else if (z <= -2.05e-194) {
		tmp = t_1;
	} else {
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * -y
    if (z <= (-180000.0d0)) then
        tmp = t_1
    else if (z <= (-3d-135)) then
        tmp = y * (z / (((0.5d0 * (a * (t / z))) - z) / x))
    else if (z <= (-2.05d-194)) then
        tmp = t_1
    else
        tmp = (x * y) / (1.0d0 + ((-0.5d0) * (a / ((z * z) / t))))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x * -y;
	double tmp;
	if (z <= -180000.0) {
		tmp = t_1;
	} else if (z <= -3e-135) {
		tmp = y * (z / (((0.5 * (a * (t / z))) - z) / x));
	} else if (z <= -2.05e-194) {
		tmp = t_1;
	} else {
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	t_1 = x * -y
	tmp = 0
	if z <= -180000.0:
		tmp = t_1
	elif z <= -3e-135:
		tmp = y * (z / (((0.5 * (a * (t / z))) - z) / x))
	elif z <= -2.05e-194:
		tmp = t_1
	else:
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = Float64(x * Float64(-y))
	tmp = 0.0
	if (z <= -180000.0)
		tmp = t_1;
	elseif (z <= -3e-135)
		tmp = Float64(y * Float64(z / Float64(Float64(Float64(0.5 * Float64(a * Float64(t / z))) - z) / x)));
	elseif (z <= -2.05e-194)
		tmp = t_1;
	else
		tmp = Float64(Float64(x * y) / Float64(1.0 + Float64(-0.5 * Float64(a / Float64(Float64(z * z) / t)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = x * -y;
	tmp = 0.0;
	if (z <= -180000.0)
		tmp = t_1;
	elseif (z <= -3e-135)
		tmp = y * (z / (((0.5 * (a * (t / z))) - z) / x));
	elseif (z <= -2.05e-194)
		tmp = t_1;
	else
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x * (-y)), $MachinePrecision]}, If[LessEqual[z, -180000.0], t$95$1, If[LessEqual[z, -3e-135], N[(y * N[(z / N[(N[(N[(0.5 * N[(a * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -2.05e-194], t$95$1, N[(N[(x * y), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(a / N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.8e5 or -3.00000000000000012e-135 < z < -2.0500000000000001e-194

   1. Initial program 52.6%

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

      1. *-commutative 52.6%

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

      2. associate-*l* 50.1%

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

      3. associate-*r/ 53.1%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 53.1%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around -inf 89.5%

      \[\leadsto y \cdot \color{blue}{\left(-1 \cdot x\right)} \]
   5. Step-by-step derivation

      1. neg-mul-1 89.5%

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

   6. Simplified 89.5%

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

   if -1.8e5 < z < -3.00000000000000012e-135

   1. Initial program 96.8%

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

      1. *-commutative 96.8%

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

      2. associate-*l* 96.7%

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

      3. associate-*r/ 96.8%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 96.8%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around -inf 82.5%

      \[\leadsto y \cdot \frac{x \cdot z}{\color{blue}{0.5 \cdot \frac{a \cdot t}{z} + -1 \cdot z}} \]

   5. Taylor expanded in x around 0 82.5%

      \[\leadsto y \cdot \color{blue}{\frac{z \cdot x}{0.5 \cdot \frac{a \cdot t}{z} + -1 \cdot z}} \]
   6. Step-by-step derivation

      1. associate-/l* 82.2%

         \[\leadsto y \cdot \color{blue}{\frac{z}{\frac{0.5 \cdot \frac{a \cdot t}{z} + -1 \cdot z}{x}}} \]

      2. neg-mul-1 82.2%

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

      3. +-commutative 82.2%

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

      4. neg-mul-1 82.2%

         \[\leadsto y \cdot \frac{z}{\frac{\color{blue}{-1 \cdot z} + 0.5 \cdot \frac{a \cdot t}{z}}{x}} \]

      5. neg-mul-1 82.2%

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

      6. +-commutative 82.2%

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

      7. associate-*r/ 82.2%

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

      8. unsub-neg 82.2%

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

   7. Simplified 82.2%

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

   if -2.0500000000000001e-194 < z

   1. Initial program 58.1%

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

      1. associate-/l* 59.3%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

   3. Simplified 59.3%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

   4. Taylor expanded in z around inf 67.2%

      \[\leadsto \frac{x \cdot y}{\color{blue}{1 + -0.5 \cdot \frac{a \cdot t}{{z}^{2}}}} \]
   5. Step-by-step derivation

      1. unpow2 67.2%

         \[\leadsto \frac{x \cdot y}{1 + -0.5 \cdot \frac{a \cdot t}{\color{blue}{z \cdot z}}} \]

      2. associate-/l* 71.3%

         \[\leadsto \frac{x \cdot y}{1 + -0.5 \cdot \color{blue}{\frac{a}{\frac{z \cdot z}{t}}}} \]

   6. Simplified 71.3%

      \[\leadsto \frac{x \cdot y}{\color{blue}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -180000:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq -3 \cdot 10^{-135}:\\ \;\;\;\;y \cdot \frac{z}{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right) - z}{x}}\\ \mathbf{elif}\;z \leq -2.05 \cdot 10^{-194}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}\\ \end{array} \]

Alternative 9: 77.4% accurate, 5.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := x \cdot \left(-y\right)\\ \mathbf{if}\;z \leq -2.05 \cdot 10^{+49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -9.2 \cdot 10^{-137}:\\ \;\;\;\;\frac{z \cdot \left(x \cdot y\right)}{0.5 \cdot \left(a \cdot \frac{t}{z}\right) - z}\\ \mathbf{elif}\;z \leq -9 \cdot 10^{-195}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* x (- y))))
   (if (<= z -2.05e+49)
     t_1
     (if (<= z -9.2e-137)
       (/ (* z (* x y)) (- (* 0.5 (* a (/ t z))) z))
       (if (<= z -9e-195)
         t_1
         (/ (* x y) (+ 1.0 (* -0.5 (/ a (/ (* z z) t))))))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = x * -y;
	double tmp;
	if (z <= -2.05e+49) {
		tmp = t_1;
	} else if (z <= -9.2e-137) {
		tmp = (z * (x * y)) / ((0.5 * (a * (t / z))) - z);
	} else if (z <= -9e-195) {
		tmp = t_1;
	} else {
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * -y
    if (z <= (-2.05d+49)) then
        tmp = t_1
    else if (z <= (-9.2d-137)) then
        tmp = (z * (x * y)) / ((0.5d0 * (a * (t / z))) - z)
    else if (z <= (-9d-195)) then
        tmp = t_1
    else
        tmp = (x * y) / (1.0d0 + ((-0.5d0) * (a / ((z * z) / t))))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x * -y;
	double tmp;
	if (z <= -2.05e+49) {
		tmp = t_1;
	} else if (z <= -9.2e-137) {
		tmp = (z * (x * y)) / ((0.5 * (a * (t / z))) - z);
	} else if (z <= -9e-195) {
		tmp = t_1;
	} else {
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	t_1 = x * -y
	tmp = 0
	if z <= -2.05e+49:
		tmp = t_1
	elif z <= -9.2e-137:
		tmp = (z * (x * y)) / ((0.5 * (a * (t / z))) - z)
	elif z <= -9e-195:
		tmp = t_1
	else:
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = Float64(x * Float64(-y))
	tmp = 0.0
	if (z <= -2.05e+49)
		tmp = t_1;
	elseif (z <= -9.2e-137)
		tmp = Float64(Float64(z * Float64(x * y)) / Float64(Float64(0.5 * Float64(a * Float64(t / z))) - z));
	elseif (z <= -9e-195)
		tmp = t_1;
	else
		tmp = Float64(Float64(x * y) / Float64(1.0 + Float64(-0.5 * Float64(a / Float64(Float64(z * z) / t)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = x * -y;
	tmp = 0.0;
	if (z <= -2.05e+49)
		tmp = t_1;
	elseif (z <= -9.2e-137)
		tmp = (z * (x * y)) / ((0.5 * (a * (t / z))) - z);
	elseif (z <= -9e-195)
		tmp = t_1;
	else
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x * (-y)), $MachinePrecision]}, If[LessEqual[z, -2.05e+49], t$95$1, If[LessEqual[z, -9.2e-137], N[(N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * N[(a * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -9e-195], t$95$1, N[(N[(x * y), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(a / N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.05e49 or -9.20000000000000032e-137 < z < -9e-195

   1. Initial program 44.7%

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

      1. *-commutative 44.7%

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

      2. associate-*l* 43.0%

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

      3. associate-*r/ 46.5%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 46.5%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around -inf 89.9%

      \[\leadsto y \cdot \color{blue}{\left(-1 \cdot x\right)} \]
   5. Step-by-step derivation

      1. neg-mul-1 89.9%

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

   6. Simplified 89.9%

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

   if -2.05e49 < z < -9.20000000000000032e-137

   1. Initial program 97.7%

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

      1. *-commutative 97.7%

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

      2. associate-*l* 95.5%

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

      3. associate-*r/ 95.6%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 95.6%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around -inf 82.6%

      \[\leadsto y \cdot \frac{x \cdot z}{\color{blue}{0.5 \cdot \frac{a \cdot t}{z} + -1 \cdot z}} \]

   5. Taylor expanded in y around 0 82.4%

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

      1. fma-def 82.4%

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

      2. associate-*r/ 82.4%

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

      3. neg-mul-1 82.4%

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

      4. associate-*r* 83.0%

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

      5. *-commutative 83.0%

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

      6. associate-*r* 84.7%

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

      7. fma-udef 84.7%

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

      8. unsub-neg 84.7%

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

   7. Simplified 84.7%

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

   if -9e-195 < z

   1. Initial program 58.1%

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

      1. associate-/l* 59.3%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

   3. Simplified 59.3%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

   4. Taylor expanded in z around inf 67.2%

      \[\leadsto \frac{x \cdot y}{\color{blue}{1 + -0.5 \cdot \frac{a \cdot t}{{z}^{2}}}} \]
   5. Step-by-step derivation

      1. unpow2 67.2%

         \[\leadsto \frac{x \cdot y}{1 + -0.5 \cdot \frac{a \cdot t}{\color{blue}{z \cdot z}}} \]

      2. associate-/l* 71.3%

         \[\leadsto \frac{x \cdot y}{1 + -0.5 \cdot \color{blue}{\frac{a}{\frac{z \cdot z}{t}}}} \]

   6. Simplified 71.3%

      \[\leadsto \frac{x \cdot y}{\color{blue}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.05 \cdot 10^{+49}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq -9.2 \cdot 10^{-137}:\\ \;\;\;\;\frac{z \cdot \left(x \cdot y\right)}{0.5 \cdot \left(a \cdot \frac{t}{z}\right) - z}\\ \mathbf{elif}\;z \leq -9 \cdot 10^{-195}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}\\ \end{array} \]

Alternative 10: 76.9% accurate, 5.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -200000:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-176}:\\ \;\;\;\;y \cdot \frac{z}{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right) - z}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -200000.0)
   (* x (- y))
   (if (<= z 2.2e-176) (* y (/ z (/ (- (* 0.5 (* a (/ t z))) z) x))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -200000.0) {
		tmp = x * -y;
	} else if (z <= 2.2e-176) {
		tmp = y * (z / (((0.5 * (a * (t / z))) - z) / x));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-200000.0d0)) then
        tmp = x * -y
    else if (z <= 2.2d-176) then
        tmp = y * (z / (((0.5d0 * (a * (t / z))) - z) / x))
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -200000.0) {
		tmp = x * -y;
	} else if (z <= 2.2e-176) {
		tmp = y * (z / (((0.5 * (a * (t / z))) - z) / x));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -200000.0:
		tmp = x * -y
	elif z <= 2.2e-176:
		tmp = y * (z / (((0.5 * (a * (t / z))) - z) / x))
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -200000.0)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 2.2e-176)
		tmp = Float64(y * Float64(z / Float64(Float64(Float64(0.5 * Float64(a * Float64(t / z))) - z) / x)));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -200000.0)
		tmp = x * -y;
	elseif (z <= 2.2e-176)
		tmp = y * (z / (((0.5 * (a * (t / z))) - z) / x));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -200000.0], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 2.2e-176], N[(y * N[(z / N[(N[(N[(0.5 * N[(a * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2e5

   1. Initial program 53.5%

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

      1. *-commutative 53.5%

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

      2. associate-*l* 52.2%

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

      3. associate-*r/ 55.5%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 55.5%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around -inf 91.4%

      \[\leadsto y \cdot \color{blue}{\left(-1 \cdot x\right)} \]
   5. Step-by-step derivation

      1. neg-mul-1 91.4%

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

   6. Simplified 91.4%

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

   if -2e5 < z < 2.1999999999999999e-176

   1. Initial program 77.0%

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

      1. *-commutative 77.0%

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

      2. associate-*l* 71.8%

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

      3. associate-*r/ 73.1%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 73.1%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around -inf 56.5%

      \[\leadsto y \cdot \frac{x \cdot z}{\color{blue}{0.5 \cdot \frac{a \cdot t}{z} + -1 \cdot z}} \]

   5. Taylor expanded in x around 0 56.5%

      \[\leadsto y \cdot \color{blue}{\frac{z \cdot x}{0.5 \cdot \frac{a \cdot t}{z} + -1 \cdot z}} \]
   6. Step-by-step derivation

      1. associate-/l* 59.3%

         \[\leadsto y \cdot \color{blue}{\frac{z}{\frac{0.5 \cdot \frac{a \cdot t}{z} + -1 \cdot z}{x}}} \]

      2. neg-mul-1 59.3%

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

      3. +-commutative 59.3%

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

      4. neg-mul-1 59.3%

         \[\leadsto y \cdot \frac{z}{\frac{\color{blue}{-1 \cdot z} + 0.5 \cdot \frac{a \cdot t}{z}}{x}} \]

      5. neg-mul-1 59.3%

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

      6. +-commutative 59.3%

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

      7. associate-*r/ 59.5%

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

      8. unsub-neg 59.5%

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

   7. Simplified 59.5%

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

   if 2.1999999999999999e-176 < z

   1. Initial program 55.5%

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

      1. *-commutative 55.5%

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

      2. associate-*l* 55.2%

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

      3. associate-*r/ 55.5%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 55.5%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around inf 82.1%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -200000:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-176}:\\ \;\;\;\;y \cdot \frac{z}{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right) - z}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 11: 78.3% accurate, 5.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+51}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-282}:\\ \;\;\;\;z \cdot \frac{x \cdot y}{0.5 \cdot \frac{a \cdot t}{z} - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.2e+51)
   (* x (- y))
   (if (<= z 1.45e-282)
     (* z (/ (* x y) (- (* 0.5 (/ (* a t) z)) z)))
     (/ (* x y) (+ 1.0 (* -0.5 (/ a (/ (* z z) t))))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.2e+51) {
		tmp = x * -y;
	} else if (z <= 1.45e-282) {
		tmp = z * ((x * y) / ((0.5 * ((a * t) / z)) - z));
	} else {
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-5.2d+51)) then
        tmp = x * -y
    else if (z <= 1.45d-282) then
        tmp = z * ((x * y) / ((0.5d0 * ((a * t) / z)) - z))
    else
        tmp = (x * y) / (1.0d0 + ((-0.5d0) * (a / ((z * z) / t))))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.2e+51) {
		tmp = x * -y;
	} else if (z <= 1.45e-282) {
		tmp = z * ((x * y) / ((0.5 * ((a * t) / z)) - z));
	} else {
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -5.2e+51:
		tmp = x * -y
	elif z <= 1.45e-282:
		tmp = z * ((x * y) / ((0.5 * ((a * t) / z)) - z))
	else:
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.2e+51)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 1.45e-282)
		tmp = Float64(z * Float64(Float64(x * y) / Float64(Float64(0.5 * Float64(Float64(a * t) / z)) - z)));
	else
		tmp = Float64(Float64(x * y) / Float64(1.0 + Float64(-0.5 * Float64(a / Float64(Float64(z * z) / t)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -5.2e+51)
		tmp = x * -y;
	elseif (z <= 1.45e-282)
		tmp = z * ((x * y) / ((0.5 * ((a * t) / z)) - z));
	else
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.2e+51], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 1.45e-282], N[(z * N[(N[(x * y), $MachinePrecision] / N[(N[(0.5 * N[(N[(a * t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(a / N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.2000000000000002e51

   1. Initial program 44.9%

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

      1. *-commutative 44.9%

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

      2. associate-*l* 44.7%

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

      3. associate-*r/ 48.5%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 48.5%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around -inf 92.3%

      \[\leadsto y \cdot \color{blue}{\left(-1 \cdot x\right)} \]
   5. Step-by-step derivation

      1. neg-mul-1 92.3%

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

   6. Simplified 92.3%

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

   if -5.2000000000000002e51 < z < 1.44999999999999999e-282

   1. Initial program 83.1%

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

      1. associate-*l/ 81.1%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\sqrt{z \cdot z - t \cdot a}} \cdot z} \]

   3. Simplified 81.1%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\sqrt{z \cdot z - t \cdot a}} \cdot z} \]

   4. Taylor expanded in z around -inf 68.5%

      \[\leadsto \frac{x \cdot y}{\color{blue}{0.5 \cdot \frac{a \cdot t}{z} + -1 \cdot z}} \cdot z \]

   if 1.44999999999999999e-282 < z

   1. Initial program 56.9%

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

      1. associate-/l* 58.4%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

   3. Simplified 58.4%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

   4. Taylor expanded in z around inf 74.0%

      \[\leadsto \frac{x \cdot y}{\color{blue}{1 + -0.5 \cdot \frac{a \cdot t}{{z}^{2}}}} \]
   5. Step-by-step derivation

      1. unpow2 74.0%

         \[\leadsto \frac{x \cdot y}{1 + -0.5 \cdot \frac{a \cdot t}{\color{blue}{z \cdot z}}} \]

      2. associate-/l* 77.9%

         \[\leadsto \frac{x \cdot y}{1 + -0.5 \cdot \color{blue}{\frac{a}{\frac{z \cdot z}{t}}}} \]

   6. Simplified 77.9%

      \[\leadsto \frac{x \cdot y}{\color{blue}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+51}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-282}:\\ \;\;\;\;z \cdot \frac{x \cdot y}{0.5 \cdot \frac{a \cdot t}{z} - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}\\ \end{array} \]

Alternative 12: 78.5% accurate, 5.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+56}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-297}:\\ \;\;\;\;\frac{x \cdot y}{\frac{0.5 \cdot \frac{a \cdot t}{z} - z}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -6.5e+56)
   (* x (- y))
   (if (<= z 3.2e-297)
     (/ (* x y) (/ (- (* 0.5 (/ (* a t) z)) z) z))
     (/ (* x y) (+ 1.0 (* -0.5 (/ a (/ (* z z) t))))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6.5e+56) {
		tmp = x * -y;
	} else if (z <= 3.2e-297) {
		tmp = (x * y) / (((0.5 * ((a * t) / z)) - z) / z);
	} else {
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-6.5d+56)) then
        tmp = x * -y
    else if (z <= 3.2d-297) then
        tmp = (x * y) / (((0.5d0 * ((a * t) / z)) - z) / z)
    else
        tmp = (x * y) / (1.0d0 + ((-0.5d0) * (a / ((z * z) / t))))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6.5e+56) {
		tmp = x * -y;
	} else if (z <= 3.2e-297) {
		tmp = (x * y) / (((0.5 * ((a * t) / z)) - z) / z);
	} else {
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -6.5e+56:
		tmp = x * -y
	elif z <= 3.2e-297:
		tmp = (x * y) / (((0.5 * ((a * t) / z)) - z) / z)
	else:
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -6.5e+56)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 3.2e-297)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(0.5 * Float64(Float64(a * t) / z)) - z) / z));
	else
		tmp = Float64(Float64(x * y) / Float64(1.0 + Float64(-0.5 * Float64(a / Float64(Float64(z * z) / t)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -6.5e+56)
		tmp = x * -y;
	elseif (z <= 3.2e-297)
		tmp = (x * y) / (((0.5 * ((a * t) / z)) - z) / z);
	else
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -6.5e+56], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 3.2e-297], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(0.5 * N[(N[(a * t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(a / N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.5000000000000001e56

   1. Initial program 44.2%

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

      1. *-commutative 44.2%

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

      2. associate-*l* 44.0%

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

      3. associate-*r/ 47.9%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 47.9%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around -inf 92.2%

      \[\leadsto y \cdot \color{blue}{\left(-1 \cdot x\right)} \]
   5. Step-by-step derivation

      1. neg-mul-1 92.2%

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

   6. Simplified 92.2%

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

   if -6.5000000000000001e56 < z < 3.19999999999999972e-297

   1. Initial program 85.5%

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

      1. associate-/l* 85.6%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

   3. Simplified 85.6%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

   4. Taylor expanded in z around -inf 72.2%

      \[\leadsto \frac{x \cdot y}{\frac{\color{blue}{0.5 \cdot \frac{a \cdot t}{z} + -1 \cdot z}}{z}} \]

   if 3.19999999999999972e-297 < z

   1. Initial program 55.9%

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

      1. associate-/l* 57.4%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

   3. Simplified 57.4%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

   4. Taylor expanded in z around inf 72.6%

      \[\leadsto \frac{x \cdot y}{\color{blue}{1 + -0.5 \cdot \frac{a \cdot t}{{z}^{2}}}} \]
   5. Step-by-step derivation

      1. unpow2 72.6%

         \[\leadsto \frac{x \cdot y}{1 + -0.5 \cdot \frac{a \cdot t}{\color{blue}{z \cdot z}}} \]

      2. associate-/l* 76.4%

         \[\leadsto \frac{x \cdot y}{1 + -0.5 \cdot \color{blue}{\frac{a}{\frac{z \cdot z}{t}}}} \]

   6. Simplified 76.4%

      \[\leadsto \frac{x \cdot y}{\color{blue}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+56}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-297}:\\ \;\;\;\;\frac{x \cdot y}{\frac{0.5 \cdot \frac{a \cdot t}{z} - z}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}\\ \end{array} \]

Alternative 13: 75.5% accurate, 9.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-44}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-173}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot x\right)}{-z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4e-44)
   (* x (- y))
   (if (<= z 6.6e-173) (/ (* y (* z x)) (- z)) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4e-44) {
		tmp = x * -y;
	} else if (z <= 6.6e-173) {
		tmp = (y * (z * x)) / -z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-4d-44)) then
        tmp = x * -y
    else if (z <= 6.6d-173) then
        tmp = (y * (z * x)) / -z
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4e-44) {
		tmp = x * -y;
	} else if (z <= 6.6e-173) {
		tmp = (y * (z * x)) / -z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -4e-44:
		tmp = x * -y
	elif z <= 6.6e-173:
		tmp = (y * (z * x)) / -z
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4e-44)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 6.6e-173)
		tmp = Float64(Float64(y * Float64(z * x)) / Float64(-z));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -4e-44)
		tmp = x * -y;
	elseif (z <= 6.6e-173)
		tmp = (y * (z * x)) / -z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4e-44], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 6.6e-173], N[(N[(y * N[(z * x), $MachinePrecision]), $MachinePrecision] / (-z)), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.99999999999999981e-44

   1. Initial program 56.8%

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

      1. *-commutative 56.8%

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

      2. associate-*l* 55.6%

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

      3. associate-*r/ 58.7%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 58.7%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around -inf 91.1%

      \[\leadsto y \cdot \color{blue}{\left(-1 \cdot x\right)} \]
   5. Step-by-step derivation

      1. neg-mul-1 91.1%

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

   6. Simplified 91.1%

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

   if -3.99999999999999981e-44 < z < 6.6000000000000006e-173

   1. Initial program 74.5%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]

   2. Taylor expanded in z around -inf 45.2%

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

      1. neg-mul-1 45.2%

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

   4. Simplified 45.2%

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

   5. Taylor expanded in x around 0 45.2%

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

   if 6.6000000000000006e-173 < z

   1. Initial program 55.5%

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

      1. *-commutative 55.5%

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

      2. associate-*l* 55.2%

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

      3. associate-*r/ 55.5%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 55.5%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around inf 82.1%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-44}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-173}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot x\right)}{-z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 14: 73.1% accurate, 18.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-290}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5e-290) (* x (- y)) (* x y)))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5e-290) {
		tmp = x * -y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-5d-290)) then
        tmp = x * -y
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5e-290) {
		tmp = x * -y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -5e-290:
		tmp = x * -y
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5e-290)
		tmp = Float64(x * Float64(-y));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -5e-290)
		tmp = x * -y;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5e-290], N[(x * (-y)), $MachinePrecision], N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.0000000000000001e-290

   1. Initial program 63.7%

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

      1. *-commutative 63.7%

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

      2. associate-*l* 60.0%

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

      3. associate-*r/ 62.7%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 62.7%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around -inf 77.7%

      \[\leadsto y \cdot \color{blue}{\left(-1 \cdot x\right)} \]
   5. Step-by-step derivation

      1. neg-mul-1 77.7%

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

   6. Simplified 77.7%

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

   if -5.0000000000000001e-290 < z

   1. Initial program 57.1%

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

      1. *-commutative 57.1%

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

      2. associate-*l* 57.3%

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

      3. associate-*r/ 57.5%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 57.5%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around inf 71.3%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-290}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 15: 43.3% accurate, 37.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ x \cdot y \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a) :precision binary64 (* x y))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	return x * y;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x * y
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	return x * y;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	return x * y

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	return Float64(x * y)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y, z, t, a)
	tmp = x * y;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := N[(x * y), $MachinePrecision]

Derivation

1. Initial program 60.8%

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

   1. *-commutative 60.8%

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

   2. associate-*l* 58.8%

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

   3. associate-*r/ 60.4%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

3. Simplified 60.4%

   \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

4. Taylor expanded in z around inf 40.7%

   \[\leadsto y \cdot \color{blue}{x} \]

5. Final simplification 40.7%

   \[\leadsto x \cdot y \]

Developer target: 89.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z < -3.1921305903852764 \cdot 10^{+46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z < 5.976268120920894 \cdot 10^{+90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (< z -3.1921305903852764e+46)
   (- (* y x))
   (if (< z 5.976268120920894e+90)
     (/ (* x z) (/ (sqrt (- (* z z) (* a t))) y))
     (* y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z < -3.1921305903852764e+46) {
		tmp = -(y * x);
	} else if (z < 5.976268120920894e+90) {
		tmp = (x * z) / (sqrt(((z * z) - (a * t))) / y);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z < (-3.1921305903852764d+46)) then
        tmp = -(y * x)
    else if (z < 5.976268120920894d+90) then
        tmp = (x * z) / (sqrt(((z * z) - (a * t))) / y)
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z < -3.1921305903852764e+46) {
		tmp = -(y * x);
	} else if (z < 5.976268120920894e+90) {
		tmp = (x * z) / (Math.sqrt(((z * z) - (a * t))) / y);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z < -3.1921305903852764e+46:
		tmp = -(y * x)
	elif z < 5.976268120920894e+90:
		tmp = (x * z) / (math.sqrt(((z * z) - (a * t))) / y)
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z < -3.1921305903852764e+46)
		tmp = Float64(-Float64(y * x));
	elseif (z < 5.976268120920894e+90)
		tmp = Float64(Float64(x * z) / Float64(sqrt(Float64(Float64(z * z) - Float64(a * t))) / y));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z < -3.1921305903852764e+46)
		tmp = -(y * x);
	elseif (z < 5.976268120920894e+90)
		tmp = (x * z) / (sqrt(((z * z) - (a * t))) / y);
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Less[z, -3.1921305903852764e+46], (-N[(y * x), $MachinePrecision]), If[Less[z, 5.976268120920894e+90], N[(N[(x * z), $MachinePrecision] / N[(N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Reproduce

herbie shell --seed 2023196 
(FPCore (x y z t a)
  :name "Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< z -3.1921305903852764e+46) (- (* y x)) (if (< z 5.976268120920894e+90) (/ (* x z) (/ (sqrt (- (* z z) (* a t))) y)) (* y x)))

  (/ (* (* x y) z) (sqrt (- (* z z) (* t a)))))