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

Percentage Accurate: 61.0% → 90.7%

Time: 18.6s

Alternatives: 19

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: 61.0% 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: 90.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.16 \cdot 10^{+154}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+54}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

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 -1.16e+154)
   (* x (- y))
   (if (<= z 4.2e+54) (* y (* x (/ z (sqrt (- (* z z) (* t a)))))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.16e+154) {
		tmp = x * -y;
	} else if (z <= 4.2e+54) {
		tmp = y * (x * (z / sqrt(((z * z) - (t * a)))));
	} 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 <= (-1.16d+154)) then
        tmp = x * -y
    else if (z <= 4.2d+54) then
        tmp = y * (x * (z / sqrt(((z * z) - (t * a)))))
    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 <= -1.16e+154) {
		tmp = x * -y;
	} else if (z <= 4.2e+54) {
		tmp = y * (x * (z / Math.sqrt(((z * z) - (t * a)))));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.16e+154:
		tmp = x * -y
	elif z <= 4.2e+54:
		tmp = y * (x * (z / math.sqrt(((z * z) - (t * a)))))
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.16e+154)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 4.2e+54)
		tmp = Float64(y * Float64(x * Float64(z / sqrt(Float64(Float64(z * z) - Float64(t * a))))));
	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 <= -1.16e+154)
		tmp = x * -y;
	elseif (z <= 4.2e+54)
		tmp = y * (x * (z / sqrt(((z * z) - (t * a)))));
	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, -1.16e+154], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 4.2e+54], N[(y * N[(x * N[(z / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.16000000000000001e154

   1. Initial program 19.0%

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

      1. associate-*l* 18.5%

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

      2. associate-*r/ 18.7%

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

      3. *-commutative 18.7%

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

      4. associate-/l* 19.6%

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

   3. Simplified 19.6%

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

   4. Taylor expanded in z around -inf 100.0%

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

      1. neg-mul-1 100.0%

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

   6. Simplified 100.0%

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

   if -1.16000000000000001e154 < z < 4.19999999999999972e54

   1. Initial program 85.8%

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

      1. associate-*l* 84.3%

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

      2. associate-*r/ 87.6%

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

      3. *-commutative 87.6%

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

      4. associate-*l/ 88.5%

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

      5. associate-*r* 87.8%

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

   3. Applied egg-rr 87.8%

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

   if 4.19999999999999972e54 < z

   1. Initial program 53.2%

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

      1. associate-*l* 51.4%

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

      2. associate-*r/ 55.8%

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

      3. *-commutative 55.8%

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

      4. associate-/l* 50.3%

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

   3. Simplified 50.3%

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

   4. Taylor expanded in z around inf 97.1%

      \[\leadsto \color{blue}{x \cdot y} \]
   5. Step-by-step derivation

      1. *-commutative 97.1%

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

   6. Simplified 97.1%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.16 \cdot 10^{+154}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+54}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 2: 80.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := y \cdot \left(x \cdot \frac{z}{0.5 \cdot \frac{a}{\frac{z}{t}} - z}\right)\\ t_2 := \sqrt{a \cdot \left(-t\right)}\\ t_3 := \frac{z \cdot x}{\frac{t_2}{y}}\\ \mathbf{if}\;z \leq -1.15 \cdot 10^{-24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.52 \cdot 10^{-75}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -4.5 \cdot 10^{-138}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.15 \cdot 10^{-279}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{t_2}\right)\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-57}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{z + -0.5 \cdot \frac{t \cdot a}{z}}\right)\\ \end{array} \end{array} \]

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 (* y (* x (/ z (- (* 0.5 (/ a (/ z t))) z)))))
        (t_2 (sqrt (* a (- t))))
        (t_3 (/ (* z x) (/ t_2 y))))
   (if (<= z -1.15e-24)
     t_1
     (if (<= z -1.52e-75)
       t_3
       (if (<= z -4.5e-138)
         t_1
         (if (<= z 3.15e-279)
           (* y (* x (/ z t_2)))
           (if (<= z 1.1e-57)
             t_3
             (* y (* x (/ z (+ z (* -0.5 (/ (* t a) z)))))))))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (x * (z / ((0.5 * (a / (z / t))) - z)));
	double t_2 = sqrt((a * -t));
	double t_3 = (z * x) / (t_2 / y);
	double tmp;
	if (z <= -1.15e-24) {
		tmp = t_1;
	} else if (z <= -1.52e-75) {
		tmp = t_3;
	} else if (z <= -4.5e-138) {
		tmp = t_1;
	} else if (z <= 3.15e-279) {
		tmp = y * (x * (z / t_2));
	} else if (z <= 1.1e-57) {
		tmp = t_3;
	} else {
		tmp = y * (x * (z / (z + (-0.5 * ((t * a) / z)))));
	}
	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) :: t_3
    real(8) :: tmp
    t_1 = y * (x * (z / ((0.5d0 * (a / (z / t))) - z)))
    t_2 = sqrt((a * -t))
    t_3 = (z * x) / (t_2 / y)
    if (z <= (-1.15d-24)) then
        tmp = t_1
    else if (z <= (-1.52d-75)) then
        tmp = t_3
    else if (z <= (-4.5d-138)) then
        tmp = t_1
    else if (z <= 3.15d-279) then
        tmp = y * (x * (z / t_2))
    else if (z <= 1.1d-57) then
        tmp = t_3
    else
        tmp = y * (x * (z / (z + ((-0.5d0) * ((t * a) / z)))))
    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 = y * (x * (z / ((0.5 * (a / (z / t))) - z)));
	double t_2 = Math.sqrt((a * -t));
	double t_3 = (z * x) / (t_2 / y);
	double tmp;
	if (z <= -1.15e-24) {
		tmp = t_1;
	} else if (z <= -1.52e-75) {
		tmp = t_3;
	} else if (z <= -4.5e-138) {
		tmp = t_1;
	} else if (z <= 3.15e-279) {
		tmp = y * (x * (z / t_2));
	} else if (z <= 1.1e-57) {
		tmp = t_3;
	} else {
		tmp = y * (x * (z / (z + (-0.5 * ((t * a) / z)))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	t_1 = y * (x * (z / ((0.5 * (a / (z / t))) - z)))
	t_2 = math.sqrt((a * -t))
	t_3 = (z * x) / (t_2 / y)
	tmp = 0
	if z <= -1.15e-24:
		tmp = t_1
	elif z <= -1.52e-75:
		tmp = t_3
	elif z <= -4.5e-138:
		tmp = t_1
	elif z <= 3.15e-279:
		tmp = y * (x * (z / t_2))
	elif z <= 1.1e-57:
		tmp = t_3
	else:
		tmp = y * (x * (z / (z + (-0.5 * ((t * a) / z)))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(x * Float64(z / Float64(Float64(0.5 * Float64(a / Float64(z / t))) - z))))
	t_2 = sqrt(Float64(a * Float64(-t)))
	t_3 = Float64(Float64(z * x) / Float64(t_2 / y))
	tmp = 0.0
	if (z <= -1.15e-24)
		tmp = t_1;
	elseif (z <= -1.52e-75)
		tmp = t_3;
	elseif (z <= -4.5e-138)
		tmp = t_1;
	elseif (z <= 3.15e-279)
		tmp = Float64(y * Float64(x * Float64(z / t_2)));
	elseif (z <= 1.1e-57)
		tmp = t_3;
	else
		tmp = Float64(y * Float64(x * Float64(z / Float64(z + Float64(-0.5 * Float64(Float64(t * a) / z))))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = y * (x * (z / ((0.5 * (a / (z / t))) - z)));
	t_2 = sqrt((a * -t));
	t_3 = (z * x) / (t_2 / y);
	tmp = 0.0;
	if (z <= -1.15e-24)
		tmp = t_1;
	elseif (z <= -1.52e-75)
		tmp = t_3;
	elseif (z <= -4.5e-138)
		tmp = t_1;
	elseif (z <= 3.15e-279)
		tmp = y * (x * (z / t_2));
	elseif (z <= 1.1e-57)
		tmp = t_3;
	else
		tmp = y * (x * (z / (z + (-0.5 * ((t * a) / z)))));
	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[(y * N[(x * N[(z / N[(N[(0.5 * N[(a / N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(z * x), $MachinePrecision] / N[(t$95$2 / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.15e-24], t$95$1, If[LessEqual[z, -1.52e-75], t$95$3, If[LessEqual[z, -4.5e-138], t$95$1, If[LessEqual[z, 3.15e-279], N[(y * N[(x * N[(z / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.1e-57], t$95$3, N[(y * N[(x * N[(z / N[(z + N[(-0.5 * N[(N[(t * a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.1500000000000001e-24 or -1.5200000000000001e-75 < z < -4.50000000000000008e-138

   1. Initial program 52.2%

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

      1. associate-*l* 49.7%

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

      2. associate-*r/ 54.9%

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

      3. *-commutative 54.9%

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

      4. associate-*l/ 57.4%

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

      5. associate-*r* 57.4%

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

   3. Applied egg-rr 57.4%

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

   4. Taylor expanded in z around -inf 90.2%

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

      1. neg-mul-1 90.2%

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

      2. +-commutative 90.2%

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

      3. unsub-neg 90.2%

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

      4. associate-/l* 92.2%

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

   6. Simplified 92.2%

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

   if -1.1500000000000001e-24 < z < -1.5200000000000001e-75 or 3.1499999999999999e-279 < z < 1.09999999999999999e-57

   1. Initial program 82.1%

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

      1. associate-*l* 84.6%

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

      2. associate-*r/ 81.9%

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

      3. *-commutative 81.9%

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

      4. associate-/l* 81.9%

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

   3. Simplified 81.9%

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

   4. Taylor expanded in z around 0 76.3%

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

      1. mul-1-neg 76.3%

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

      2. distribute-rgt-neg-out 76.3%

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

   6. Simplified 76.3%

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

      1. associate-*r/ 76.9%

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

      2. *-commutative 76.9%

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

   8. Applied egg-rr 76.9%

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

   if -4.50000000000000008e-138 < z < 3.1499999999999999e-279

   1. Initial program 85.7%

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

      1. associate-*l* 83.1%

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

      2. associate-*r/ 85.1%

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

      3. *-commutative 85.1%

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

      4. associate-/l* 85.5%

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

   3. Simplified 85.5%

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

   4. Taylor expanded in z around 0 83.3%

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

      1. mul-1-neg 83.3%

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

      2. distribute-rgt-neg-out 83.3%

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

   6. Simplified 83.3%

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

      1. expm1-log1p-u 71.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \frac{z}{\frac{\sqrt{a \cdot \left(-t\right)}}{y}}\right)\right)} \]

      2. expm1-udef 50.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{z}{\frac{\sqrt{a \cdot \left(-t\right)}}{y}}\right)} - 1} \]

      3. associate-/r/ 50.9%

         \[\leadsto e^{\mathsf{log1p}\left(x \cdot \color{blue}{\left(\frac{z}{\sqrt{a \cdot \left(-t\right)}} \cdot y\right)}\right)} - 1 \]

   8. Applied egg-rr 50.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \left(\frac{z}{\sqrt{a \cdot \left(-t\right)}} \cdot y\right)\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 71.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \left(\frac{z}{\sqrt{a \cdot \left(-t\right)}} \cdot y\right)\right)\right)} \]

      2. expm1-log1p 83.3%

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

      3. associate-*r* 83.2%

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

      4. *-commutative 83.2%

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

   10. Simplified 83.2%

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

   if 1.09999999999999999e-57 < z

   1. Initial program 63.5%

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

      1. associate-*l* 62.2%

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

      2. associate-*r/ 65.5%

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

      3. *-commutative 65.5%

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

      4. associate-*l/ 66.9%

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

      5. associate-*r* 66.9%

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

   3. Applied egg-rr 66.9%

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

   4. Taylor expanded in z around inf 89.1%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-24}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{0.5 \cdot \frac{a}{\frac{z}{t}} - z}\right)\\ \mathbf{elif}\;z \leq -1.52 \cdot 10^{-75}:\\ \;\;\;\;\frac{z \cdot x}{\frac{\sqrt{a \cdot \left(-t\right)}}{y}}\\ \mathbf{elif}\;z \leq -4.5 \cdot 10^{-138}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{0.5 \cdot \frac{a}{\frac{z}{t}} - z}\right)\\ \mathbf{elif}\;z \leq 3.15 \cdot 10^{-279}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{\sqrt{a \cdot \left(-t\right)}}\right)\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-57}:\\ \;\;\;\;\frac{z \cdot x}{\frac{\sqrt{a \cdot \left(-t\right)}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{z + -0.5 \cdot \frac{t \cdot a}{z}}\right)\\ \end{array} \]

Alternative 3: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := x \cdot \left(y \cdot \frac{z}{\sqrt{a \cdot \left(-t\right)}}\right)\\ t_2 := y \cdot \left(x \cdot \frac{z}{0.5 \cdot \frac{a}{\frac{z}{t}} - z}\right)\\ \mathbf{if}\;z \leq -1.15 \cdot 10^{-24}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.15 \cdot 10^{-75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.8 \cdot 10^{-138}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.08 \cdot 10^{-57}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{z + -0.5 \cdot \frac{t \cdot a}{z}}\right)\\ \end{array} \end{array} \]

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 (/ z (sqrt (* a (- t)))))))
        (t_2 (* y (* x (/ z (- (* 0.5 (/ a (/ z t))) z))))))
   (if (<= z -1.15e-24)
     t_2
     (if (<= z -1.15e-75)
       t_1
       (if (<= z -2.8e-138)
         t_2
         (if (<= z 1.08e-57)
           t_1
           (* y (* x (/ z (+ z (* -0.5 (/ (* t a) z))))))))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = x * (y * (z / sqrt((a * -t))));
	double t_2 = y * (x * (z / ((0.5 * (a / (z / t))) - z)));
	double tmp;
	if (z <= -1.15e-24) {
		tmp = t_2;
	} else if (z <= -1.15e-75) {
		tmp = t_1;
	} else if (z <= -2.8e-138) {
		tmp = t_2;
	} else if (z <= 1.08e-57) {
		tmp = t_1;
	} else {
		tmp = y * (x * (z / (z + (-0.5 * ((t * a) / z)))));
	}
	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 = x * (y * (z / sqrt((a * -t))))
    t_2 = y * (x * (z / ((0.5d0 * (a / (z / t))) - z)))
    if (z <= (-1.15d-24)) then
        tmp = t_2
    else if (z <= (-1.15d-75)) then
        tmp = t_1
    else if (z <= (-2.8d-138)) then
        tmp = t_2
    else if (z <= 1.08d-57) then
        tmp = t_1
    else
        tmp = y * (x * (z / (z + ((-0.5d0) * ((t * a) / z)))))
    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 * (z / Math.sqrt((a * -t))));
	double t_2 = y * (x * (z / ((0.5 * (a / (z / t))) - z)));
	double tmp;
	if (z <= -1.15e-24) {
		tmp = t_2;
	} else if (z <= -1.15e-75) {
		tmp = t_1;
	} else if (z <= -2.8e-138) {
		tmp = t_2;
	} else if (z <= 1.08e-57) {
		tmp = t_1;
	} else {
		tmp = y * (x * (z / (z + (-0.5 * ((t * a) / z)))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	t_1 = x * (y * (z / math.sqrt((a * -t))))
	t_2 = y * (x * (z / ((0.5 * (a / (z / t))) - z)))
	tmp = 0
	if z <= -1.15e-24:
		tmp = t_2
	elif z <= -1.15e-75:
		tmp = t_1
	elif z <= -2.8e-138:
		tmp = t_2
	elif z <= 1.08e-57:
		tmp = t_1
	else:
		tmp = y * (x * (z / (z + (-0.5 * ((t * a) / z)))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = Float64(x * Float64(y * Float64(z / sqrt(Float64(a * Float64(-t))))))
	t_2 = Float64(y * Float64(x * Float64(z / Float64(Float64(0.5 * Float64(a / Float64(z / t))) - z))))
	tmp = 0.0
	if (z <= -1.15e-24)
		tmp = t_2;
	elseif (z <= -1.15e-75)
		tmp = t_1;
	elseif (z <= -2.8e-138)
		tmp = t_2;
	elseif (z <= 1.08e-57)
		tmp = t_1;
	else
		tmp = Float64(y * Float64(x * Float64(z / Float64(z + Float64(-0.5 * Float64(Float64(t * a) / z))))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = x * (y * (z / sqrt((a * -t))));
	t_2 = y * (x * (z / ((0.5 * (a / (z / t))) - z)));
	tmp = 0.0;
	if (z <= -1.15e-24)
		tmp = t_2;
	elseif (z <= -1.15e-75)
		tmp = t_1;
	elseif (z <= -2.8e-138)
		tmp = t_2;
	elseif (z <= 1.08e-57)
		tmp = t_1;
	else
		tmp = y * (x * (z / (z + (-0.5 * ((t * a) / z)))));
	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 * N[(y * N[(z / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(x * N[(z / N[(N[(0.5 * N[(a / N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.15e-24], t$95$2, If[LessEqual[z, -1.15e-75], t$95$1, If[LessEqual[z, -2.8e-138], t$95$2, If[LessEqual[z, 1.08e-57], t$95$1, N[(y * N[(x * N[(z / N[(z + N[(-0.5 * N[(N[(t * a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.1500000000000001e-24 or -1.15e-75 < z < -2.80000000000000001e-138

   1. Initial program 52.2%

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

      1. associate-*l* 49.7%

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

      2. associate-*r/ 54.9%

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

      3. *-commutative 54.9%

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

      4. associate-*l/ 57.4%

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

      5. associate-*r* 57.4%

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

   3. Applied egg-rr 57.4%

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

   4. Taylor expanded in z around -inf 90.2%

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

      1. neg-mul-1 90.2%

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

      2. +-commutative 90.2%

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

      3. unsub-neg 90.2%

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

      4. associate-/l* 92.2%

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

   6. Simplified 92.2%

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

   if -1.1500000000000001e-24 < z < -1.15e-75 or -2.80000000000000001e-138 < z < 1.08e-57

   1. Initial program 83.8%

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

      1. associate-*l* 83.9%

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

      2. associate-*r/ 83.5%

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

      3. *-commutative 83.5%

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

      4. associate-/l* 83.7%

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

   3. Simplified 83.7%

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

   4. Taylor expanded in z around 0 79.8%

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

      1. mul-1-neg 79.8%

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

      2. distribute-rgt-neg-out 79.8%

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

   6. Simplified 79.8%

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

      1. associate-/r/ 78.4%

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

   8. Applied egg-rr 78.4%

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

   if 1.08e-57 < z

   1. Initial program 63.5%

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

      1. associate-*l* 62.2%

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

      2. associate-*r/ 65.5%

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

      3. *-commutative 65.5%

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

      4. associate-*l/ 66.9%

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

      5. associate-*r* 66.9%

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

   3. Applied egg-rr 66.9%

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

   4. Taylor expanded in z around inf 89.1%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-24}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{0.5 \cdot \frac{a}{\frac{z}{t}} - z}\right)\\ \mathbf{elif}\;z \leq -1.15 \cdot 10^{-75}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{a \cdot \left(-t\right)}}\right)\\ \mathbf{elif}\;z \leq -2.8 \cdot 10^{-138}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{0.5 \cdot \frac{a}{\frac{z}{t}} - z}\right)\\ \mathbf{elif}\;z \leq 1.08 \cdot 10^{-57}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{a \cdot \left(-t\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{z + -0.5 \cdot \frac{t \cdot a}{z}}\right)\\ \end{array} \]

Alternative 4: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := y \cdot \left(x \cdot \frac{z}{0.5 \cdot \frac{a}{\frac{z}{t}} - z}\right)\\ t_2 := \sqrt{a \cdot \left(-t\right)}\\ \mathbf{if}\;z \leq -1.75 \cdot 10^{-24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -5.8 \cdot 10^{-76}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{t_2}\right)\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{-138}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-57}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{t_2}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{z + -0.5 \cdot \frac{t \cdot a}{z}}\right)\\ \end{array} \end{array} \]

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 (* y (* x (/ z (- (* 0.5 (/ a (/ z t))) z)))))
        (t_2 (sqrt (* a (- t)))))
   (if (<= z -1.75e-24)
     t_1
     (if (<= z -5.8e-76)
       (* x (* y (/ z t_2)))
       (if (<= z -1.5e-138)
         t_1
         (if (<= z 7.5e-57)
           (* x (/ (* z y) t_2))
           (* y (* x (/ z (+ z (* -0.5 (/ (* t a) z))))))))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (x * (z / ((0.5 * (a / (z / t))) - z)));
	double t_2 = sqrt((a * -t));
	double tmp;
	if (z <= -1.75e-24) {
		tmp = t_1;
	} else if (z <= -5.8e-76) {
		tmp = x * (y * (z / t_2));
	} else if (z <= -1.5e-138) {
		tmp = t_1;
	} else if (z <= 7.5e-57) {
		tmp = x * ((z * y) / t_2);
	} else {
		tmp = y * (x * (z / (z + (-0.5 * ((t * a) / z)))));
	}
	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 = y * (x * (z / ((0.5d0 * (a / (z / t))) - z)))
    t_2 = sqrt((a * -t))
    if (z <= (-1.75d-24)) then
        tmp = t_1
    else if (z <= (-5.8d-76)) then
        tmp = x * (y * (z / t_2))
    else if (z <= (-1.5d-138)) then
        tmp = t_1
    else if (z <= 7.5d-57) then
        tmp = x * ((z * y) / t_2)
    else
        tmp = y * (x * (z / (z + ((-0.5d0) * ((t * a) / z)))))
    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 = y * (x * (z / ((0.5 * (a / (z / t))) - z)));
	double t_2 = Math.sqrt((a * -t));
	double tmp;
	if (z <= -1.75e-24) {
		tmp = t_1;
	} else if (z <= -5.8e-76) {
		tmp = x * (y * (z / t_2));
	} else if (z <= -1.5e-138) {
		tmp = t_1;
	} else if (z <= 7.5e-57) {
		tmp = x * ((z * y) / t_2);
	} else {
		tmp = y * (x * (z / (z + (-0.5 * ((t * a) / z)))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	t_1 = y * (x * (z / ((0.5 * (a / (z / t))) - z)))
	t_2 = math.sqrt((a * -t))
	tmp = 0
	if z <= -1.75e-24:
		tmp = t_1
	elif z <= -5.8e-76:
		tmp = x * (y * (z / t_2))
	elif z <= -1.5e-138:
		tmp = t_1
	elif z <= 7.5e-57:
		tmp = x * ((z * y) / t_2)
	else:
		tmp = y * (x * (z / (z + (-0.5 * ((t * a) / z)))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(x * Float64(z / Float64(Float64(0.5 * Float64(a / Float64(z / t))) - z))))
	t_2 = sqrt(Float64(a * Float64(-t)))
	tmp = 0.0
	if (z <= -1.75e-24)
		tmp = t_1;
	elseif (z <= -5.8e-76)
		tmp = Float64(x * Float64(y * Float64(z / t_2)));
	elseif (z <= -1.5e-138)
		tmp = t_1;
	elseif (z <= 7.5e-57)
		tmp = Float64(x * Float64(Float64(z * y) / t_2));
	else
		tmp = Float64(y * Float64(x * Float64(z / Float64(z + Float64(-0.5 * Float64(Float64(t * a) / z))))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = y * (x * (z / ((0.5 * (a / (z / t))) - z)));
	t_2 = sqrt((a * -t));
	tmp = 0.0;
	if (z <= -1.75e-24)
		tmp = t_1;
	elseif (z <= -5.8e-76)
		tmp = x * (y * (z / t_2));
	elseif (z <= -1.5e-138)
		tmp = t_1;
	elseif (z <= 7.5e-57)
		tmp = x * ((z * y) / t_2);
	else
		tmp = y * (x * (z / (z + (-0.5 * ((t * a) / z)))));
	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[(y * N[(x * N[(z / N[(N[(0.5 * N[(a / N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -1.75e-24], t$95$1, If[LessEqual[z, -5.8e-76], N[(x * N[(y * N[(z / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -1.5e-138], t$95$1, If[LessEqual[z, 7.5e-57], N[(x * N[(N[(z * y), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], N[(y * N[(x * N[(z / N[(z + N[(-0.5 * N[(N[(t * a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.7499999999999998e-24 or -5.8000000000000003e-76 < z < -1.5e-138

   1. Initial program 52.2%

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

      1. associate-*l* 49.7%

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

      2. associate-*r/ 54.9%

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

      3. *-commutative 54.9%

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

      4. associate-*l/ 57.4%

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

      5. associate-*r* 57.4%

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

   3. Applied egg-rr 57.4%

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

   4. Taylor expanded in z around -inf 90.2%

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

      1. neg-mul-1 90.2%

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

      2. +-commutative 90.2%

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

      3. unsub-neg 90.2%

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

      4. associate-/l* 92.2%

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

   6. Simplified 92.2%

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

   if -1.7499999999999998e-24 < z < -5.8000000000000003e-76

   1. Initial program 89.3%

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

      1. associate-*l* 89.5%

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

      2. associate-*r/ 89.2%

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

      3. *-commutative 89.2%

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

      4. associate-/l* 89.0%

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

   3. Simplified 89.0%

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

   4. Taylor expanded in z around 0 82.4%

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

      1. mul-1-neg 82.4%

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

      2. distribute-rgt-neg-out 82.4%

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

   6. Simplified 82.4%

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

      1. associate-/r/ 82.4%

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

   8. Applied egg-rr 82.4%

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

   if -1.5e-138 < z < 7.49999999999999973e-57

   1. Initial program 83.0%

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

      1. associate-*l* 83.1%

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

      2. associate-*r/ 82.6%

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

   3. Simplified 82.6%

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

   4. Taylor expanded in z around 0 79.1%

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

      1. mul-1-neg 79.4%

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

      2. distribute-rgt-neg-out 79.4%

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

   6. Simplified 79.1%

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

   if 7.49999999999999973e-57 < z

   1. Initial program 63.5%

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

      1. associate-*l* 62.2%

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

      2. associate-*r/ 65.5%

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

      3. *-commutative 65.5%

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

      4. associate-*l/ 66.9%

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

      5. associate-*r* 66.9%

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

   3. Applied egg-rr 66.9%

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

   4. Taylor expanded in z around inf 89.1%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{-24}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{0.5 \cdot \frac{a}{\frac{z}{t}} - z}\right)\\ \mathbf{elif}\;z \leq -5.8 \cdot 10^{-76}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{a \cdot \left(-t\right)}}\right)\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{-138}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{0.5 \cdot \frac{a}{\frac{z}{t}} - z}\right)\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-57}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{z + -0.5 \cdot \frac{t \cdot a}{z}}\right)\\ \end{array} \]

Alternative 5: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := y \cdot \left(x \cdot \frac{z}{0.5 \cdot \frac{a}{\frac{z}{t}} - z}\right)\\ t_2 := \frac{z}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{if}\;z \leq -1.58 \cdot 10^{-24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -3.8 \cdot 10^{-76}:\\ \;\;\;\;x \cdot \left(y \cdot t_2\right)\\ \mathbf{elif}\;z \leq -3.7 \cdot 10^{-138}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-58}:\\ \;\;\;\;y \cdot \left(x \cdot t_2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{z + -0.5 \cdot \frac{t \cdot a}{z}}\right)\\ \end{array} \end{array} \]

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 (* y (* x (/ z (- (* 0.5 (/ a (/ z t))) z)))))
        (t_2 (/ z (sqrt (* a (- t))))))
   (if (<= z -1.58e-24)
     t_1
     (if (<= z -3.8e-76)
       (* x (* y t_2))
       (if (<= z -3.7e-138)
         t_1
         (if (<= z 2.4e-58)
           (* y (* x t_2))
           (* y (* x (/ z (+ z (* -0.5 (/ (* t a) z))))))))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (x * (z / ((0.5 * (a / (z / t))) - z)));
	double t_2 = z / sqrt((a * -t));
	double tmp;
	if (z <= -1.58e-24) {
		tmp = t_1;
	} else if (z <= -3.8e-76) {
		tmp = x * (y * t_2);
	} else if (z <= -3.7e-138) {
		tmp = t_1;
	} else if (z <= 2.4e-58) {
		tmp = y * (x * t_2);
	} else {
		tmp = y * (x * (z / (z + (-0.5 * ((t * a) / z)))));
	}
	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 = y * (x * (z / ((0.5d0 * (a / (z / t))) - z)))
    t_2 = z / sqrt((a * -t))
    if (z <= (-1.58d-24)) then
        tmp = t_1
    else if (z <= (-3.8d-76)) then
        tmp = x * (y * t_2)
    else if (z <= (-3.7d-138)) then
        tmp = t_1
    else if (z <= 2.4d-58) then
        tmp = y * (x * t_2)
    else
        tmp = y * (x * (z / (z + ((-0.5d0) * ((t * a) / z)))))
    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 = y * (x * (z / ((0.5 * (a / (z / t))) - z)));
	double t_2 = z / Math.sqrt((a * -t));
	double tmp;
	if (z <= -1.58e-24) {
		tmp = t_1;
	} else if (z <= -3.8e-76) {
		tmp = x * (y * t_2);
	} else if (z <= -3.7e-138) {
		tmp = t_1;
	} else if (z <= 2.4e-58) {
		tmp = y * (x * t_2);
	} else {
		tmp = y * (x * (z / (z + (-0.5 * ((t * a) / z)))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	t_1 = y * (x * (z / ((0.5 * (a / (z / t))) - z)))
	t_2 = z / math.sqrt((a * -t))
	tmp = 0
	if z <= -1.58e-24:
		tmp = t_1
	elif z <= -3.8e-76:
		tmp = x * (y * t_2)
	elif z <= -3.7e-138:
		tmp = t_1
	elif z <= 2.4e-58:
		tmp = y * (x * t_2)
	else:
		tmp = y * (x * (z / (z + (-0.5 * ((t * a) / z)))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(x * Float64(z / Float64(Float64(0.5 * Float64(a / Float64(z / t))) - z))))
	t_2 = Float64(z / sqrt(Float64(a * Float64(-t))))
	tmp = 0.0
	if (z <= -1.58e-24)
		tmp = t_1;
	elseif (z <= -3.8e-76)
		tmp = Float64(x * Float64(y * t_2));
	elseif (z <= -3.7e-138)
		tmp = t_1;
	elseif (z <= 2.4e-58)
		tmp = Float64(y * Float64(x * t_2));
	else
		tmp = Float64(y * Float64(x * Float64(z / Float64(z + Float64(-0.5 * Float64(Float64(t * a) / z))))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = y * (x * (z / ((0.5 * (a / (z / t))) - z)));
	t_2 = z / sqrt((a * -t));
	tmp = 0.0;
	if (z <= -1.58e-24)
		tmp = t_1;
	elseif (z <= -3.8e-76)
		tmp = x * (y * t_2);
	elseif (z <= -3.7e-138)
		tmp = t_1;
	elseif (z <= 2.4e-58)
		tmp = y * (x * t_2);
	else
		tmp = y * (x * (z / (z + (-0.5 * ((t * a) / z)))));
	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[(y * N[(x * N[(z / N[(N[(0.5 * N[(a / N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.58e-24], t$95$1, If[LessEqual[z, -3.8e-76], N[(x * N[(y * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -3.7e-138], t$95$1, If[LessEqual[z, 2.4e-58], N[(y * N[(x * t$95$2), $MachinePrecision]), $MachinePrecision], N[(y * N[(x * N[(z / N[(z + N[(-0.5 * N[(N[(t * a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.5799999999999999e-24 or -3.8000000000000002e-76 < z < -3.69999999999999991e-138

   1. Initial program 52.2%

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

      1. associate-*l* 49.7%

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

      2. associate-*r/ 54.9%

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

      3. *-commutative 54.9%

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

      4. associate-*l/ 57.4%

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

      5. associate-*r* 57.4%

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

   3. Applied egg-rr 57.4%

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

   4. Taylor expanded in z around -inf 90.2%

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

      1. neg-mul-1 90.2%

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

      2. +-commutative 90.2%

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

      3. unsub-neg 90.2%

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

      4. associate-/l* 92.2%

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

   6. Simplified 92.2%

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

   if -1.5799999999999999e-24 < z < -3.8000000000000002e-76

   1. Initial program 89.3%

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

      1. associate-*l* 89.5%

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

      2. associate-*r/ 89.2%

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

      3. *-commutative 89.2%

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

      4. associate-/l* 89.0%

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

   3. Simplified 89.0%

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

   4. Taylor expanded in z around 0 82.4%

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

      1. mul-1-neg 82.4%

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

      2. distribute-rgt-neg-out 82.4%

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

   6. Simplified 82.4%

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

      1. associate-/r/ 82.4%

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

   8. Applied egg-rr 82.4%

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

   if -3.69999999999999991e-138 < z < 2.4000000000000001e-58

   1. Initial program 83.0%

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

      1. associate-*l* 83.1%

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

      2. associate-*r/ 82.6%

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

      3. *-commutative 82.6%

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

      4. associate-/l* 82.9%

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

   3. Simplified 82.9%

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

   4. Taylor expanded in z around 0 79.4%

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

      1. mul-1-neg 79.4%

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

      2. distribute-rgt-neg-out 79.4%

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

   6. Simplified 79.4%

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

      1. expm1-log1p-u 66.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \frac{z}{\frac{\sqrt{a \cdot \left(-t\right)}}{y}}\right)\right)} \]

      2. expm1-udef 41.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{z}{\frac{\sqrt{a \cdot \left(-t\right)}}{y}}\right)} - 1} \]

      3. associate-/r/ 41.1%

         \[\leadsto e^{\mathsf{log1p}\left(x \cdot \color{blue}{\left(\frac{z}{\sqrt{a \cdot \left(-t\right)}} \cdot y\right)}\right)} - 1 \]

   8. Applied egg-rr 41.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \left(\frac{z}{\sqrt{a \cdot \left(-t\right)}} \cdot y\right)\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 64.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \left(\frac{z}{\sqrt{a \cdot \left(-t\right)}} \cdot y\right)\right)\right)} \]

      2. expm1-log1p 77.9%

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

      3. associate-*r* 76.3%

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

      4. *-commutative 76.3%

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

   10. Simplified 76.3%

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

   if 2.4000000000000001e-58 < z

   1. Initial program 63.5%

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

      1. associate-*l* 62.2%

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

      2. associate-*r/ 65.5%

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

      3. *-commutative 65.5%

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

      4. associate-*l/ 66.9%

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

      5. associate-*r* 66.9%

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

   3. Applied egg-rr 66.9%

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

   4. Taylor expanded in z around inf 89.1%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.58 \cdot 10^{-24}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{0.5 \cdot \frac{a}{\frac{z}{t}} - z}\right)\\ \mathbf{elif}\;z \leq -3.8 \cdot 10^{-76}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{a \cdot \left(-t\right)}}\right)\\ \mathbf{elif}\;z \leq -3.7 \cdot 10^{-138}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{0.5 \cdot \frac{a}{\frac{z}{t}} - z}\right)\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-58}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{\sqrt{a \cdot \left(-t\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{z + -0.5 \cdot \frac{t \cdot a}{z}}\right)\\ \end{array} \]

Alternative 6: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := y \cdot \left(x \cdot \frac{z}{0.5 \cdot \frac{a}{\frac{z}{t}} - z}\right)\\ t_2 := \sqrt{a \cdot \left(-t\right)}\\ \mathbf{if}\;z \leq -1.15 \cdot 10^{-24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.35 \cdot 10^{-76}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{t_2}\right)\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{-140}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-58}:\\ \;\;\;\;z \cdot \frac{x \cdot y}{t_2}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{z + -0.5 \cdot \frac{t \cdot a}{z}}\right)\\ \end{array} \end{array} \]

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 (* y (* x (/ z (- (* 0.5 (/ a (/ z t))) z)))))
        (t_2 (sqrt (* a (- t)))))
   (if (<= z -1.15e-24)
     t_1
     (if (<= z -1.35e-76)
       (* x (* y (/ z t_2)))
       (if (<= z -1.65e-140)
         t_1
         (if (<= z 3.7e-58)
           (* z (/ (* x y) t_2))
           (* y (* x (/ z (+ z (* -0.5 (/ (* t a) z))))))))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (x * (z / ((0.5 * (a / (z / t))) - z)));
	double t_2 = sqrt((a * -t));
	double tmp;
	if (z <= -1.15e-24) {
		tmp = t_1;
	} else if (z <= -1.35e-76) {
		tmp = x * (y * (z / t_2));
	} else if (z <= -1.65e-140) {
		tmp = t_1;
	} else if (z <= 3.7e-58) {
		tmp = z * ((x * y) / t_2);
	} else {
		tmp = y * (x * (z / (z + (-0.5 * ((t * a) / z)))));
	}
	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 = y * (x * (z / ((0.5d0 * (a / (z / t))) - z)))
    t_2 = sqrt((a * -t))
    if (z <= (-1.15d-24)) then
        tmp = t_1
    else if (z <= (-1.35d-76)) then
        tmp = x * (y * (z / t_2))
    else if (z <= (-1.65d-140)) then
        tmp = t_1
    else if (z <= 3.7d-58) then
        tmp = z * ((x * y) / t_2)
    else
        tmp = y * (x * (z / (z + ((-0.5d0) * ((t * a) / z)))))
    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 = y * (x * (z / ((0.5 * (a / (z / t))) - z)));
	double t_2 = Math.sqrt((a * -t));
	double tmp;
	if (z <= -1.15e-24) {
		tmp = t_1;
	} else if (z <= -1.35e-76) {
		tmp = x * (y * (z / t_2));
	} else if (z <= -1.65e-140) {
		tmp = t_1;
	} else if (z <= 3.7e-58) {
		tmp = z * ((x * y) / t_2);
	} else {
		tmp = y * (x * (z / (z + (-0.5 * ((t * a) / z)))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	t_1 = y * (x * (z / ((0.5 * (a / (z / t))) - z)))
	t_2 = math.sqrt((a * -t))
	tmp = 0
	if z <= -1.15e-24:
		tmp = t_1
	elif z <= -1.35e-76:
		tmp = x * (y * (z / t_2))
	elif z <= -1.65e-140:
		tmp = t_1
	elif z <= 3.7e-58:
		tmp = z * ((x * y) / t_2)
	else:
		tmp = y * (x * (z / (z + (-0.5 * ((t * a) / z)))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(x * Float64(z / Float64(Float64(0.5 * Float64(a / Float64(z / t))) - z))))
	t_2 = sqrt(Float64(a * Float64(-t)))
	tmp = 0.0
	if (z <= -1.15e-24)
		tmp = t_1;
	elseif (z <= -1.35e-76)
		tmp = Float64(x * Float64(y * Float64(z / t_2)));
	elseif (z <= -1.65e-140)
		tmp = t_1;
	elseif (z <= 3.7e-58)
		tmp = Float64(z * Float64(Float64(x * y) / t_2));
	else
		tmp = Float64(y * Float64(x * Float64(z / Float64(z + Float64(-0.5 * Float64(Float64(t * a) / z))))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = y * (x * (z / ((0.5 * (a / (z / t))) - z)));
	t_2 = sqrt((a * -t));
	tmp = 0.0;
	if (z <= -1.15e-24)
		tmp = t_1;
	elseif (z <= -1.35e-76)
		tmp = x * (y * (z / t_2));
	elseif (z <= -1.65e-140)
		tmp = t_1;
	elseif (z <= 3.7e-58)
		tmp = z * ((x * y) / t_2);
	else
		tmp = y * (x * (z / (z + (-0.5 * ((t * a) / z)))));
	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[(y * N[(x * N[(z / N[(N[(0.5 * N[(a / N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -1.15e-24], t$95$1, If[LessEqual[z, -1.35e-76], N[(x * N[(y * N[(z / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -1.65e-140], t$95$1, If[LessEqual[z, 3.7e-58], N[(z * N[(N[(x * y), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], N[(y * N[(x * N[(z / N[(z + N[(-0.5 * N[(N[(t * a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.1500000000000001e-24 or -1.35e-76 < z < -1.64999999999999994e-140

   1. Initial program 52.7%

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

      1. associate-*l* 50.2%

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

      2. associate-*r/ 55.4%

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

      3. *-commutative 55.4%

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

      4. associate-*l/ 57.8%

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

      5. associate-*r* 57.9%

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

   3. Applied egg-rr 57.9%

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

   4. Taylor expanded in z around -inf 89.3%

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

      1. neg-mul-1 89.3%

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

      2. +-commutative 89.3%

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

      3. unsub-neg 89.3%

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

      4. associate-/l* 91.3%

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

   6. Simplified 91.3%

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

   if -1.1500000000000001e-24 < z < -1.35e-76

   1. Initial program 89.3%

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

      1. associate-*l* 89.5%

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

      2. associate-*r/ 89.2%

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

      3. *-commutative 89.2%

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

      4. associate-/l* 89.0%

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

   3. Simplified 89.0%

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

   4. Taylor expanded in z around 0 82.4%

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

      1. mul-1-neg 82.4%

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

      2. distribute-rgt-neg-out 82.4%

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

   6. Simplified 82.4%

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

      1. associate-/r/ 82.4%

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

   8. Applied egg-rr 82.4%

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

   if -1.64999999999999994e-140 < z < 3.7000000000000003e-58

   1. Initial program 82.8%

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

      1. associate-*l/ 85.3%

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

   3. Simplified 85.3%

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

   4. Taylor expanded in z around 0 82.1%

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

      1. mul-1-neg 79.0%

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

      2. distribute-rgt-neg-out 79.0%

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

   6. Simplified 82.1%

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

   if 3.7000000000000003e-58 < z

   1. Initial program 63.5%

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

      1. associate-*l* 62.2%

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

      2. associate-*r/ 65.5%

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

      3. *-commutative 65.5%

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

      4. associate-*l/ 66.9%

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

      5. associate-*r* 66.9%

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

   3. Applied egg-rr 66.9%

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

   4. Taylor expanded in z around inf 89.1%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-24}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{0.5 \cdot \frac{a}{\frac{z}{t}} - z}\right)\\ \mathbf{elif}\;z \leq -1.35 \cdot 10^{-76}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{a \cdot \left(-t\right)}}\right)\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{-140}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{0.5 \cdot \frac{a}{\frac{z}{t}} - z}\right)\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-58}:\\ \;\;\;\;z \cdot \frac{x \cdot y}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{z + -0.5 \cdot \frac{t \cdot a}{z}}\right)\\ \end{array} \]

Alternative 7: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+115}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+54}:\\ \;\;\;\;x \cdot \frac{z}{\frac{\sqrt{z \cdot z - t \cdot a}}{y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

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 -2e+115)
   (* x (- y))
   (if (<= z 4e+54) (* x (/ z (/ (sqrt (- (* z z) (* t a))) y))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2e+115) {
		tmp = x * -y;
	} else if (z <= 4e+54) {
		tmp = x * (z / (sqrt(((z * z) - (t * a))) / 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 <= (-2d+115)) then
        tmp = x * -y
    else if (z <= 4d+54) then
        tmp = x * (z / (sqrt(((z * z) - (t * a))) / 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 <= -2e+115) {
		tmp = x * -y;
	} else if (z <= 4e+54) {
		tmp = x * (z / (Math.sqrt(((z * z) - (t * a))) / y));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -2e+115:
		tmp = x * -y
	elif z <= 4e+54:
		tmp = x * (z / (math.sqrt(((z * z) - (t * a))) / 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 <= -2e+115)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 4e+54)
		tmp = Float64(x * Float64(z / Float64(sqrt(Float64(Float64(z * z) - Float64(t * a))) / 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 <= -2e+115)
		tmp = x * -y;
	elseif (z <= 4e+54)
		tmp = x * (z / (sqrt(((z * z) - (t * a))) / 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, -2e+115], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 4e+54], N[(x * N[(z / N[(N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2e115

   1. Initial program 32.3%

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

      1. associate-*l* 30.3%

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

      2. associate-*r/ 31.9%

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

      3. *-commutative 31.9%

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

      4. associate-/l* 32.7%

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

   3. Simplified 32.7%

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

   4. Taylor expanded in z around -inf 100.0%

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

      1. neg-mul-1 100.0%

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

   6. Simplified 100.0%

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

   if -2e115 < z < 4.0000000000000003e54

   1. Initial program 85.4%

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

      1. associate-*l* 84.4%

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

      2. associate-*r/ 87.4%

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

      3. *-commutative 87.4%

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

      4. associate-/l* 87.5%

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

   3. Simplified 87.5%

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

   if 4.0000000000000003e54 < z

   1. Initial program 53.2%

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

      1. associate-*l* 51.4%

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

      2. associate-*r/ 55.8%

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

      3. *-commutative 55.8%

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

      4. associate-/l* 50.3%

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

   3. Simplified 50.3%

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

   4. Taylor expanded in z around inf 97.1%

      \[\leadsto \color{blue}{x \cdot y} \]
   5. Step-by-step derivation

      1. *-commutative 97.1%

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

   6. Simplified 97.1%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+115}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+54}:\\ \;\;\;\;x \cdot \frac{z}{\frac{\sqrt{z \cdot z - t \cdot a}}{y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 8: 89.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+67}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+54}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

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 -1.9e+67)
   (* x (- y))
   (if (<= z 3.1e+54) (* x (/ (* z y) (sqrt (- (* z z) (* t a))))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.9e+67) {
		tmp = x * -y;
	} else if (z <= 3.1e+54) {
		tmp = x * ((z * y) / sqrt(((z * z) - (t * a))));
	} 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 <= (-1.9d+67)) then
        tmp = x * -y
    else if (z <= 3.1d+54) then
        tmp = x * ((z * y) / sqrt(((z * z) - (t * a))))
    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 <= -1.9e+67) {
		tmp = x * -y;
	} else if (z <= 3.1e+54) {
		tmp = x * ((z * y) / Math.sqrt(((z * z) - (t * a))));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.9e+67:
		tmp = x * -y
	elif z <= 3.1e+54:
		tmp = x * ((z * y) / math.sqrt(((z * z) - (t * a))))
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.9e+67)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 3.1e+54)
		tmp = Float64(x * Float64(Float64(z * y) / sqrt(Float64(Float64(z * z) - Float64(t * a)))));
	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 <= -1.9e+67)
		tmp = x * -y;
	elseif (z <= 3.1e+54)
		tmp = x * ((z * y) / sqrt(((z * z) - (t * a))));
	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, -1.9e+67], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 3.1e+54], N[(x * N[(N[(z * y), $MachinePrecision] / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.9000000000000001e67

   1. Initial program 37.5%

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

      1. associate-*l* 34.1%

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

      2. associate-*r/ 35.7%

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

      3. *-commutative 35.7%

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

      4. associate-/l* 37.9%

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

   3. Simplified 37.9%

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

   4. Taylor expanded in z around -inf 100.0%

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

      1. neg-mul-1 100.0%

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

   6. Simplified 100.0%

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

   if -1.9000000000000001e67 < z < 3.0999999999999999e54

   1. Initial program 84.8%

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

      1. associate-*l* 84.6%

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

      2. associate-*r/ 87.6%

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

   3. Simplified 87.6%

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

   if 3.0999999999999999e54 < z

   1. Initial program 53.2%

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

      1. associate-*l* 51.4%

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

      2. associate-*r/ 55.8%

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

      3. *-commutative 55.8%

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

      4. associate-/l* 50.3%

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

   3. Simplified 50.3%

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

   4. Taylor expanded in z around inf 97.1%

      \[\leadsto \color{blue}{x \cdot y} \]
   5. Step-by-step derivation

      1. *-commutative 97.1%

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

   6. Simplified 97.1%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+67}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+54}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 9: 76.6% accurate, 5.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{-183}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 0.047:\\ \;\;\;\;x \cdot \frac{z}{\frac{z + \frac{a}{\frac{z}{t}} \cdot -0.5}{y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

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 -2.3e-183)
   (* x (- y))
   (if (<= z 0.047) (* x (/ z (/ (+ z (* (/ a (/ z t)) -0.5)) y))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.3e-183) {
		tmp = x * -y;
	} else if (z <= 0.047) {
		tmp = x * (z / ((z + ((a / (z / t)) * -0.5)) / 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 <= (-2.3d-183)) then
        tmp = x * -y
    else if (z <= 0.047d0) then
        tmp = x * (z / ((z + ((a / (z / t)) * (-0.5d0))) / 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 <= -2.3e-183) {
		tmp = x * -y;
	} else if (z <= 0.047) {
		tmp = x * (z / ((z + ((a / (z / t)) * -0.5)) / y));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.3e-183:
		tmp = x * -y
	elif z <= 0.047:
		tmp = x * (z / ((z + ((a / (z / t)) * -0.5)) / 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 <= -2.3e-183)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 0.047)
		tmp = Float64(x * Float64(z / Float64(Float64(z + Float64(Float64(a / Float64(z / t)) * -0.5)) / 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 <= -2.3e-183)
		tmp = x * -y;
	elseif (z <= 0.047)
		tmp = x * (z / ((z + ((a / (z / t)) * -0.5)) / 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, -2.3e-183], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 0.047], N[(x * N[(z / N[(N[(z + N[(N[(a / N[(z / t), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.30000000000000016e-183

   1. Initial program 57.9%

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

      1. associate-*l* 55.8%

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

      2. associate-*r/ 60.7%

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

      3. *-commutative 60.7%

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

      4. associate-/l* 61.9%

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

   3. Simplified 61.9%

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

   4. Taylor expanded in z around -inf 79.5%

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

      1. neg-mul-1 79.5%

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

   6. Simplified 79.5%

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

   if -2.30000000000000016e-183 < z < 0.047

   1. Initial program 86.4%

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

      1. associate-*l* 86.4%

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

      2. associate-*r/ 84.9%

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

      3. *-commutative 84.9%

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

      4. associate-/l* 84.6%

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

   3. Simplified 84.6%

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

   4. Taylor expanded in z around inf 59.3%

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

      1. associate-/l* 59.3%

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

   6. Simplified 59.3%

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

   if 0.047 < z

   1. Initial program 57.4%

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

      1. associate-*l* 55.9%

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

      2. associate-*r/ 59.7%

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

      3. *-commutative 59.7%

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

      4. associate-/l* 54.3%

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

   3. Simplified 54.3%

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

   4. Taylor expanded in z around inf 92.6%

      \[\leadsto \color{blue}{x \cdot y} \]
   5. Step-by-step derivation

      1. *-commutative 92.6%

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

   6. Simplified 92.6%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{-183}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 0.047:\\ \;\;\;\;x \cdot \frac{z}{\frac{z + \frac{a}{\frac{z}{t}} \cdot -0.5}{y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 10: 76.6% accurate, 5.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{-183}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 0.052:\\ \;\;\;\;x \cdot \frac{z \cdot y}{z + -0.5 \cdot \frac{t \cdot a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

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 -2.6e-183)
   (* x (- y))
   (if (<= z 0.052) (* x (/ (* z y) (+ z (* -0.5 (/ (* t a) z))))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.6e-183) {
		tmp = x * -y;
	} else if (z <= 0.052) {
		tmp = x * ((z * y) / (z + (-0.5 * ((t * a) / 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 <= (-2.6d-183)) then
        tmp = x * -y
    else if (z <= 0.052d0) then
        tmp = x * ((z * y) / (z + ((-0.5d0) * ((t * a) / 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 <= -2.6e-183) {
		tmp = x * -y;
	} else if (z <= 0.052) {
		tmp = x * ((z * y) / (z + (-0.5 * ((t * a) / z))));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.6e-183:
		tmp = x * -y
	elif z <= 0.052:
		tmp = x * ((z * y) / (z + (-0.5 * ((t * a) / 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 <= -2.6e-183)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 0.052)
		tmp = Float64(x * Float64(Float64(z * y) / Float64(z + Float64(-0.5 * Float64(Float64(t * a) / 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 <= -2.6e-183)
		tmp = x * -y;
	elseif (z <= 0.052)
		tmp = x * ((z * y) / (z + (-0.5 * ((t * a) / 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, -2.6e-183], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 0.052], N[(x * N[(N[(z * y), $MachinePrecision] / N[(z + N[(-0.5 * N[(N[(t * a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.5999999999999999e-183

   1. Initial program 57.9%

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

      1. associate-*l* 55.8%

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

      2. associate-*r/ 60.7%

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

      3. *-commutative 60.7%

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

      4. associate-/l* 61.9%

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

   3. Simplified 61.9%

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

   4. Taylor expanded in z around -inf 79.5%

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

      1. neg-mul-1 79.5%

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

   6. Simplified 79.5%

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

   if -2.5999999999999999e-183 < z < 0.0519999999999999976

   1. Initial program 86.4%

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

      1. associate-*l* 86.4%

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

      2. associate-*r/ 84.9%

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

   3. Simplified 84.9%

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

   4. Taylor expanded in z around inf 59.9%

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

   if 0.0519999999999999976 < z

   1. Initial program 57.4%

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

      1. associate-*l* 55.9%

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

      2. associate-*r/ 59.7%

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

      3. *-commutative 59.7%

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

      4. associate-/l* 54.3%

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

   3. Simplified 54.3%

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

   4. Taylor expanded in z around inf 92.6%

      \[\leadsto \color{blue}{x \cdot y} \]
   5. Step-by-step derivation

      1. *-commutative 92.6%

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

   6. Simplified 92.6%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{-183}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 0.052:\\ \;\;\;\;x \cdot \frac{z \cdot y}{z + -0.5 \cdot \frac{t \cdot a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 11: 75.1% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{-183}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-121}:\\ \;\;\;\;-2 \cdot \left(\frac{x}{t} \cdot \frac{y \cdot \left(z \cdot z\right)}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

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 -3.6e-183)
   (* x (- y))
   (if (<= z 2.4e-121) (* -2.0 (* (/ x t) (/ (* y (* z z)) a))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.6e-183) {
		tmp = x * -y;
	} else if (z <= 2.4e-121) {
		tmp = -2.0 * ((x / t) * ((y * (z * z)) / a));
	} 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 <= (-3.6d-183)) then
        tmp = x * -y
    else if (z <= 2.4d-121) then
        tmp = (-2.0d0) * ((x / t) * ((y * (z * z)) / a))
    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 <= -3.6e-183) {
		tmp = x * -y;
	} else if (z <= 2.4e-121) {
		tmp = -2.0 * ((x / t) * ((y * (z * z)) / a));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.6e-183:
		tmp = x * -y
	elif z <= 2.4e-121:
		tmp = -2.0 * ((x / t) * ((y * (z * z)) / a))
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.6e-183)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 2.4e-121)
		tmp = Float64(-2.0 * Float64(Float64(x / t) * Float64(Float64(y * Float64(z * z)) / a)));
	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 <= -3.6e-183)
		tmp = x * -y;
	elseif (z <= 2.4e-121)
		tmp = -2.0 * ((x / t) * ((y * (z * z)) / a));
	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, -3.6e-183], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 2.4e-121], N[(-2.0 * N[(N[(x / t), $MachinePrecision] * N[(N[(y * N[(z * z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.6000000000000001e-183

   1. Initial program 57.9%

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

      1. associate-*l* 55.8%

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

      2. associate-*r/ 60.7%

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

      3. *-commutative 60.7%

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

      4. associate-/l* 61.9%

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

   3. Simplified 61.9%

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

   4. Taylor expanded in z around -inf 79.5%

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

      1. neg-mul-1 79.5%

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

   6. Simplified 79.5%

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

   if -3.6000000000000001e-183 < z < 2.40000000000000003e-121

   1. Initial program 84.4%

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

      1. associate-*l/ 84.2%

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

   3. Simplified 84.2%

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

   4. Taylor expanded in z around inf 55.8%

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

   5. Taylor expanded in z around 0 53.0%

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

      1. *-commutative 53.0%

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

      2. times-frac 52.9%

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

      3. unpow2 52.9%

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

   7. Simplified 52.9%

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

   if 2.40000000000000003e-121 < z

   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. associate-*l* 63.4%

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

      2. associate-*r/ 67.4%

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

      3. *-commutative 67.4%

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

      4. associate-/l* 62.8%

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

   3. Simplified 62.8%

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

   4. Taylor expanded in z around inf 84.4%

      \[\leadsto \color{blue}{x \cdot y} \]
   5. Step-by-step derivation

      1. *-commutative 84.4%

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

   6. Simplified 84.4%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{-183}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-121}:\\ \;\;\;\;-2 \cdot \left(\frac{x}{t} \cdot \frac{y \cdot \left(z \cdot z\right)}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 12: 75.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{-183}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-122}:\\ \;\;\;\;-2 \cdot \frac{x}{\frac{t}{\frac{y}{\frac{a}{z \cdot z}}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

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 -3.7e-183)
   (* x (- y))
   (if (<= z 2.5e-122) (* -2.0 (/ x (/ t (/ y (/ a (* z z)))))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.7e-183) {
		tmp = x * -y;
	} else if (z <= 2.5e-122) {
		tmp = -2.0 * (x / (t / (y / (a / (z * 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 <= (-3.7d-183)) then
        tmp = x * -y
    else if (z <= 2.5d-122) then
        tmp = (-2.0d0) * (x / (t / (y / (a / (z * 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 <= -3.7e-183) {
		tmp = x * -y;
	} else if (z <= 2.5e-122) {
		tmp = -2.0 * (x / (t / (y / (a / (z * z)))));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.7e-183:
		tmp = x * -y
	elif z <= 2.5e-122:
		tmp = -2.0 * (x / (t / (y / (a / (z * 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 <= -3.7e-183)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 2.5e-122)
		tmp = Float64(-2.0 * Float64(x / Float64(t / Float64(y / Float64(a / Float64(z * 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 <= -3.7e-183)
		tmp = x * -y;
	elseif (z <= 2.5e-122)
		tmp = -2.0 * (x / (t / (y / (a / (z * 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, -3.7e-183], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 2.5e-122], N[(-2.0 * N[(x / N[(t / N[(y / N[(a / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.6999999999999999e-183

   1. Initial program 57.9%

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

      1. associate-*l* 55.8%

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

      2. associate-*r/ 60.7%

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

      3. *-commutative 60.7%

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

      4. associate-/l* 61.9%

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

   3. Simplified 61.9%

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

   4. Taylor expanded in z around -inf 79.5%

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

      1. neg-mul-1 79.5%

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

   6. Simplified 79.5%

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

   if -3.6999999999999999e-183 < z < 2.4999999999999999e-122

   1. Initial program 84.4%

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

      1. associate-*l/ 84.2%

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

   3. Simplified 84.2%

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

   4. Taylor expanded in z around inf 55.8%

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

   5. Taylor expanded in z around 0 53.0%

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

      1. *-commutative 53.0%

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

      2. times-frac 52.9%

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

      3. unpow2 52.9%

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

   7. Simplified 52.9%

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

   8. Taylor expanded in x around 0 53.0%

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

      1. associate-/l* 53.0%

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

      2. *-commutative 53.0%

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

      3. *-commutative 53.0%

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

      4. unpow2 53.0%

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

      5. associate-/l* 55.5%

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

      6. unpow2 55.5%

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

      7. *-commutative 55.5%

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

      8. associate-/l* 55.5%

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

      9. unpow2 55.5%

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

   10. Simplified 55.5%

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

   if 2.4999999999999999e-122 < z

   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. associate-*l* 63.4%

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

      2. associate-*r/ 67.4%

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

      3. *-commutative 67.4%

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

      4. associate-/l* 62.8%

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

   3. Simplified 62.8%

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

   4. Taylor expanded in z around inf 84.4%

      \[\leadsto \color{blue}{x \cdot y} \]
   5. Step-by-step derivation

      1. *-commutative 84.4%

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

   6. Simplified 84.4%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{-183}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-122}:\\ \;\;\;\;-2 \cdot \frac{x}{\frac{t}{\frac{y}{\frac{a}{z \cdot z}}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 13: 75.4% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{-183}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-122}:\\ \;\;\;\;z \cdot \left(-2 \cdot \left(\frac{z}{t} \cdot \frac{x \cdot y}{a}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

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 -3.3e-183)
   (* x (- y))
   (if (<= z 1.15e-122) (* z (* -2.0 (* (/ z t) (/ (* x y) a)))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.3e-183) {
		tmp = x * -y;
	} else if (z <= 1.15e-122) {
		tmp = z * (-2.0 * ((z / t) * ((x * y) / a)));
	} 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 <= (-3.3d-183)) then
        tmp = x * -y
    else if (z <= 1.15d-122) then
        tmp = z * ((-2.0d0) * ((z / t) * ((x * y) / a)))
    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 <= -3.3e-183) {
		tmp = x * -y;
	} else if (z <= 1.15e-122) {
		tmp = z * (-2.0 * ((z / t) * ((x * y) / a)));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.3e-183:
		tmp = x * -y
	elif z <= 1.15e-122:
		tmp = z * (-2.0 * ((z / t) * ((x * y) / a)))
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.3e-183)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 1.15e-122)
		tmp = Float64(z * Float64(-2.0 * Float64(Float64(z / t) * Float64(Float64(x * y) / a))));
	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 <= -3.3e-183)
		tmp = x * -y;
	elseif (z <= 1.15e-122)
		tmp = z * (-2.0 * ((z / t) * ((x * y) / a)));
	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, -3.3e-183], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 1.15e-122], N[(z * N[(-2.0 * N[(N[(z / t), $MachinePrecision] * N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.3e-183

   1. Initial program 57.9%

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

      1. associate-*l* 55.8%

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

      2. associate-*r/ 60.7%

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

      3. *-commutative 60.7%

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

      4. associate-/l* 61.9%

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

   3. Simplified 61.9%

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

   4. Taylor expanded in z around -inf 79.5%

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

      1. neg-mul-1 79.5%

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

   6. Simplified 79.5%

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

   if -3.3e-183 < z < 1.15000000000000003e-122

   1. Initial program 84.4%

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

      1. associate-*l/ 84.2%

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

   3. Simplified 84.2%

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

   4. Taylor expanded in z around inf 55.8%

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

   5. Taylor expanded in z around 0 53.1%

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

      1. associate-*r* 53.2%

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

      2. *-commutative 53.2%

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

      3. *-commutative 53.2%

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

      4. times-frac 55.6%

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

   7. Simplified 55.6%

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

   if 1.15000000000000003e-122 < z

   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. associate-*l* 63.4%

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

      2. associate-*r/ 67.4%

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

      3. *-commutative 67.4%

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

      4. associate-/l* 62.8%

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

   3. Simplified 62.8%

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

   4. Taylor expanded in z around inf 84.4%

      \[\leadsto \color{blue}{x \cdot y} \]
   5. Step-by-step derivation

      1. *-commutative 84.4%

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

   6. Simplified 84.4%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{-183}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-122}:\\ \;\;\;\;z \cdot \left(-2 \cdot \left(\frac{z}{t} \cdot \frac{x \cdot y}{a}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 14: 75.9% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{-183}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{z + -0.5 \cdot \frac{t \cdot a}{z}}\right)\\ \end{array} \end{array} \]

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 -2.7e-183)
   (* x (- y))
   (* y (* x (/ z (+ z (* -0.5 (/ (* t a) z))))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.7e-183) {
		tmp = x * -y;
	} else {
		tmp = y * (x * (z / (z + (-0.5 * ((t * a) / z)))));
	}
	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 <= (-2.7d-183)) then
        tmp = x * -y
    else
        tmp = y * (x * (z / (z + ((-0.5d0) * ((t * a) / z)))))
    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 <= -2.7e-183) {
		tmp = x * -y;
	} else {
		tmp = y * (x * (z / (z + (-0.5 * ((t * a) / z)))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.7e-183:
		tmp = x * -y
	else:
		tmp = y * (x * (z / (z + (-0.5 * ((t * a) / z)))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.7e-183)
		tmp = Float64(x * Float64(-y));
	else
		tmp = Float64(y * Float64(x * Float64(z / Float64(z + Float64(-0.5 * Float64(Float64(t * a) / z))))));
	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 <= -2.7e-183)
		tmp = x * -y;
	else
		tmp = y * (x * (z / (z + (-0.5 * ((t * a) / z)))));
	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, -2.7e-183], N[(x * (-y)), $MachinePrecision], N[(y * N[(x * N[(z / N[(z + N[(-0.5 * N[(N[(t * a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.70000000000000008e-183

   1. Initial program 57.9%

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

      1. associate-*l* 55.8%

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

      2. associate-*r/ 60.7%

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

      3. *-commutative 60.7%

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

      4. associate-/l* 61.9%

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

   3. Simplified 61.9%

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

   4. Taylor expanded in z around -inf 79.5%

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

      1. neg-mul-1 79.5%

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

   6. Simplified 79.5%

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

   if -2.70000000000000008e-183 < z

   1. Initial program 70.7%

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

      1. associate-*l* 69.9%

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

      2. associate-*r/ 71.3%

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

      3. *-commutative 71.3%

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

      4. associate-*l/ 71.7%

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

      5. associate-*r* 71.7%

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

   3. Applied egg-rr 71.7%

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

   4. Taylor expanded in z around inf 76.7%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{-183}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{z + -0.5 \cdot \frac{t \cdot a}{z}}\right)\\ \end{array} \]

Alternative 15: 77.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{-207}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{0.5 \cdot \frac{a}{\frac{z}{t}} - z}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{z + -0.5 \cdot \frac{t \cdot a}{z}}\right)\\ \end{array} \end{array} \]

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 -1.45e-207)
   (* y (* x (/ z (- (* 0.5 (/ a (/ z t))) z))))
   (* y (* x (/ z (+ z (* -0.5 (/ (* t a) z))))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.45e-207) {
		tmp = y * (x * (z / ((0.5 * (a / (z / t))) - z)));
	} else {
		tmp = y * (x * (z / (z + (-0.5 * ((t * a) / z)))));
	}
	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 <= (-1.45d-207)) then
        tmp = y * (x * (z / ((0.5d0 * (a / (z / t))) - z)))
    else
        tmp = y * (x * (z / (z + ((-0.5d0) * ((t * a) / z)))))
    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 <= -1.45e-207) {
		tmp = y * (x * (z / ((0.5 * (a / (z / t))) - z)));
	} else {
		tmp = y * (x * (z / (z + (-0.5 * ((t * a) / z)))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.45e-207:
		tmp = y * (x * (z / ((0.5 * (a / (z / t))) - z)))
	else:
		tmp = y * (x * (z / (z + (-0.5 * ((t * a) / z)))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.45e-207)
		tmp = Float64(y * Float64(x * Float64(z / Float64(Float64(0.5 * Float64(a / Float64(z / t))) - z))));
	else
		tmp = Float64(y * Float64(x * Float64(z / Float64(z + Float64(-0.5 * Float64(Float64(t * a) / z))))));
	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 <= -1.45e-207)
		tmp = y * (x * (z / ((0.5 * (a / (z / t))) - z)));
	else
		tmp = y * (x * (z / (z + (-0.5 * ((t * a) / z)))));
	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, -1.45e-207], N[(y * N[(x * N[(z / N[(N[(0.5 * N[(a / N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(x * N[(z / N[(z + N[(-0.5 * N[(N[(t * a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.45000000000000006e-207

   1. Initial program 59.2%

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

      1. associate-*l* 57.2%

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

      2. associate-*r/ 61.9%

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

      3. *-commutative 61.9%

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

      4. associate-*l/ 63.8%

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

      5. associate-*r* 63.1%

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

   3. Applied egg-rr 63.1%

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

   4. Taylor expanded in z around -inf 78.7%

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

      1. neg-mul-1 78.7%

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

      2. +-commutative 78.7%

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

      3. unsub-neg 78.7%

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

      4. associate-/l* 80.3%

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

   6. Simplified 80.3%

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

   if -1.45000000000000006e-207 < z

   1. Initial program 70.1%

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

      1. associate-*l* 69.3%

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

      2. associate-*r/ 70.7%

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

      3. *-commutative 70.7%

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

      4. associate-*l/ 71.1%

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

      5. associate-*r* 71.1%

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

   3. Applied egg-rr 71.1%

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

   4. Taylor expanded in z around inf 77.8%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{-207}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{0.5 \cdot \frac{a}{\frac{z}{t}} - z}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{z + -0.5 \cdot \frac{t \cdot a}{z}}\right)\\ \end{array} \]

Alternative 16: 77.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{-207}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{t \cdot \frac{a \cdot 0.5}{z} - z}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{z + -0.5 \cdot \frac{t \cdot a}{z}}\right)\\ \end{array} \end{array} \]

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 -1.45e-207)
   (* y (* x (/ z (- (* t (/ (* a 0.5) z)) z))))
   (* y (* x (/ z (+ z (* -0.5 (/ (* t a) z))))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.45e-207) {
		tmp = y * (x * (z / ((t * ((a * 0.5) / z)) - z)));
	} else {
		tmp = y * (x * (z / (z + (-0.5 * ((t * a) / z)))));
	}
	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 <= (-1.45d-207)) then
        tmp = y * (x * (z / ((t * ((a * 0.5d0) / z)) - z)))
    else
        tmp = y * (x * (z / (z + ((-0.5d0) * ((t * a) / z)))))
    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 <= -1.45e-207) {
		tmp = y * (x * (z / ((t * ((a * 0.5) / z)) - z)));
	} else {
		tmp = y * (x * (z / (z + (-0.5 * ((t * a) / z)))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.45e-207:
		tmp = y * (x * (z / ((t * ((a * 0.5) / z)) - z)))
	else:
		tmp = y * (x * (z / (z + (-0.5 * ((t * a) / z)))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.45e-207)
		tmp = Float64(y * Float64(x * Float64(z / Float64(Float64(t * Float64(Float64(a * 0.5) / z)) - z))));
	else
		tmp = Float64(y * Float64(x * Float64(z / Float64(z + Float64(-0.5 * Float64(Float64(t * a) / z))))));
	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 <= -1.45e-207)
		tmp = y * (x * (z / ((t * ((a * 0.5) / z)) - z)));
	else
		tmp = y * (x * (z / (z + (-0.5 * ((t * a) / z)))));
	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, -1.45e-207], N[(y * N[(x * N[(z / N[(N[(t * N[(N[(a * 0.5), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(x * N[(z / N[(z + N[(-0.5 * N[(N[(t * a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.45000000000000006e-207

   1. Initial program 59.2%

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

      1. associate-*l* 57.2%

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

      2. associate-*r/ 61.9%

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

      3. *-commutative 61.9%

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

      4. associate-*l/ 63.8%

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

      5. associate-*r* 63.1%

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

   3. Applied egg-rr 63.1%

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

   4. Taylor expanded in z around -inf 78.7%

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

      1. neg-mul-1 78.7%

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

      2. +-commutative 78.7%

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

      3. unsub-neg 78.7%

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

      4. associate-/l* 80.3%

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

   6. Simplified 80.3%

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

      1. expm1-log1p-u 78.8%

         \[\leadsto \left(x \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{z}{0.5 \cdot \frac{a}{\frac{z}{t}} - z}\right)\right)}\right) \cdot y \]

      2. expm1-udef 77.6%

         \[\leadsto \left(x \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{z}{0.5 \cdot \frac{a}{\frac{z}{t}} - z}\right)} - 1\right)}\right) \cdot y \]

      3. associate-*r/ 77.6%

         \[\leadsto \left(x \cdot \left(e^{\mathsf{log1p}\left(\frac{z}{\color{blue}{\frac{0.5 \cdot a}{\frac{z}{t}}} - z}\right)} - 1\right)\right) \cdot y \]

   8. Applied egg-rr 77.6%

      \[\leadsto \left(x \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{z}{\frac{0.5 \cdot a}{\frac{z}{t}} - z}\right)} - 1\right)}\right) \cdot y \]
   9. Step-by-step derivation

      1. expm1-def 78.8%

         \[\leadsto \left(x \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{z}{\frac{0.5 \cdot a}{\frac{z}{t}} - z}\right)\right)}\right) \cdot y \]

      2. expm1-log1p 80.3%

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

      3. associate-/r/ 80.2%

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

      4. *-commutative 80.2%

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

   10. Simplified 80.2%

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

   if -1.45000000000000006e-207 < z

   1. Initial program 70.1%

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

      1. associate-*l* 69.3%

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

      2. associate-*r/ 70.7%

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

      3. *-commutative 70.7%

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

      4. associate-*l/ 71.1%

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

      5. associate-*r* 71.1%

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

   3. Applied egg-rr 71.1%

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

   4. Taylor expanded in z around inf 77.8%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{-207}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{t \cdot \frac{a \cdot 0.5}{z} - z}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{z + -0.5 \cdot \frac{t \cdot a}{z}}\right)\\ \end{array} \]

Alternative 17: 74.2% accurate, 10.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{-212}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 10^{-104}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

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 -5.8e-212)
   (* x (- y))
   (if (<= z 1e-104) (* x (/ (* z y) z)) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.8e-212) {
		tmp = x * -y;
	} else if (z <= 1e-104) {
		tmp = x * ((z * y) / 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 <= (-5.8d-212)) then
        tmp = x * -y
    else if (z <= 1d-104) then
        tmp = x * ((z * y) / 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 <= -5.8e-212) {
		tmp = x * -y;
	} else if (z <= 1e-104) {
		tmp = x * ((z * y) / z);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -5.8e-212:
		tmp = x * -y
	elif z <= 1e-104:
		tmp = x * ((z * y) / 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 <= -5.8e-212)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 1e-104)
		tmp = Float64(x * Float64(Float64(z * y) / 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 <= -5.8e-212)
		tmp = x * -y;
	elseif (z <= 1e-104)
		tmp = x * ((z * y) / 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, -5.8e-212], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 1e-104], N[(x * N[(N[(z * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.7999999999999999e-212

   1. Initial program 59.8%

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

      1. associate-*l* 57.9%

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

      2. associate-*r/ 62.5%

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

      3. *-commutative 62.5%

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

      4. associate-/l* 63.6%

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

   3. Simplified 63.6%

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

   4. Taylor expanded in z around -inf 76.2%

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

      1. neg-mul-1 76.2%

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

   6. Simplified 76.2%

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

   if -5.7999999999999999e-212 < z < 9.99999999999999927e-105

   1. Initial program 82.5%

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

      1. associate-*l* 82.5%

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

      2. associate-*r/ 79.9%

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

   3. Simplified 79.9%

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

   4. Taylor expanded in z around inf 42.1%

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

   if 9.99999999999999927e-105 < z

   1. Initial program 64.5%

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

      1. associate-*l* 63.2%

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

      2. associate-*r/ 66.4%

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

      3. *-commutative 66.4%

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

      4. associate-/l* 61.6%

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

   3. Simplified 61.6%

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

   4. Taylor expanded in z around inf 85.9%

      \[\leadsto \color{blue}{x \cdot y} \]
   5. Step-by-step derivation

      1. *-commutative 85.9%

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

   6. Simplified 85.9%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{-212}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 10^{-104}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 18: 73.1% accurate, 18.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 10^{-303}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

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 1e-303) (* 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-303) {
		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 <= 1d-303) 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 <= 1e-303) {
		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 <= 1e-303:
		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 <= 1e-303)
		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 <= 1e-303)
		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, 1e-303], N[(x * (-y)), $MachinePrecision], N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 9.99999999999999931e-304

   1. Initial program 63.4%

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

      1. associate-*l* 61.0%

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

      2. associate-*r/ 65.1%

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

      3. *-commutative 65.1%

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

      4. associate-/l* 66.1%

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

   3. Simplified 66.1%

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

   4. Taylor expanded in z around -inf 72.1%

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

      1. neg-mul-1 72.1%

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

   6. Simplified 72.1%

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

   if 9.99999999999999931e-304 < z

   1. Initial program 66.7%

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

      1. associate-*l* 66.5%

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

      2. associate-*r/ 68.2%

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

      3. *-commutative 68.2%

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

      4. associate-/l* 64.4%

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

   3. Simplified 64.4%

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

   4. Taylor expanded in z around inf 73.1%

      \[\leadsto \color{blue}{x \cdot y} \]
   5. Step-by-step derivation

      1. *-commutative 73.1%

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

   6. Simplified 73.1%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 10^{-303}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 19: 43.6% 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 64.9%

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

   1. associate-*l* 63.5%

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

   2. associate-*r/ 66.5%

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

   3. *-commutative 66.5%

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

   4. associate-/l* 65.3%

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

3. Simplified 65.3%

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

4. Taylor expanded in z around inf 43.4%

   \[\leadsto \color{blue}{x \cdot y} \]
5. Step-by-step derivation

   1. *-commutative 43.4%

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

6. Simplified 43.4%

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

7. Final simplification 43.4%

   \[\leadsto x \cdot y \]

Developer target: 89.2% 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 2023290 
(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)))))