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

Percentage Accurate: 60.9% → 89.5%

Time: 20.6s

Alternatives: 15

Speedup: 18.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 60.9% 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: 89.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := y \cdot \left(x \cdot \frac{z}{\sqrt{z \cdot z - a \cdot t}}\right)\\ \mathbf{if}\;z \leq -1 \cdot 10^{+154}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq -2.9 \cdot 10^{-289}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-86}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{x}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{+55}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (* x (/ z (sqrt (- (* z z) (* a t))))))))
   (if (<= z -1e+154)
     (* x (- y))
     (if (<= z -2.9e-289)
       t_1
       (if (<= z 9.5e-86)
         (* (* z y) (/ x (sqrt (* a (- t)))))
         (if (<= z 4.4e+55) t_1 (* x y)))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (x * (z / sqrt(((z * z) - (a * t)))));
	double tmp;
	if (z <= -1e+154) {
		tmp = x * -y;
	} else if (z <= -2.9e-289) {
		tmp = t_1;
	} else if (z <= 9.5e-86) {
		tmp = (z * y) * (x / sqrt((a * -t)));
	} else if (z <= 4.4e+55) {
		tmp = t_1;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = y * (x * (z / sqrt(((z * z) - (a * t)))))
    if (z <= (-1d+154)) then
        tmp = x * -y
    else if (z <= (-2.9d-289)) then
        tmp = t_1
    else if (z <= 9.5d-86) then
        tmp = (z * y) * (x / sqrt((a * -t)))
    else if (z <= 4.4d+55) then
        tmp = t_1
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (x * (z / Math.sqrt(((z * z) - (a * t)))));
	double tmp;
	if (z <= -1e+154) {
		tmp = x * -y;
	} else if (z <= -2.9e-289) {
		tmp = t_1;
	} else if (z <= 9.5e-86) {
		tmp = (z * y) * (x / Math.sqrt((a * -t)));
	} else if (z <= 4.4e+55) {
		tmp = t_1;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	t_1 = y * (x * (z / math.sqrt(((z * z) - (a * t)))))
	tmp = 0
	if z <= -1e+154:
		tmp = x * -y
	elif z <= -2.9e-289:
		tmp = t_1
	elif z <= 9.5e-86:
		tmp = (z * y) * (x / math.sqrt((a * -t)))
	elif z <= 4.4e+55:
		tmp = t_1
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(x * Float64(z / sqrt(Float64(Float64(z * z) - Float64(a * t))))))
	tmp = 0.0
	if (z <= -1e+154)
		tmp = Float64(x * Float64(-y));
	elseif (z <= -2.9e-289)
		tmp = t_1;
	elseif (z <= 9.5e-86)
		tmp = Float64(Float64(z * y) * Float64(x / sqrt(Float64(a * Float64(-t)))));
	elseif (z <= 4.4e+55)
		tmp = t_1;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = y * (x * (z / sqrt(((z * z) - (a * t)))));
	tmp = 0.0;
	if (z <= -1e+154)
		tmp = x * -y;
	elseif (z <= -2.9e-289)
		tmp = t_1;
	elseif (z <= 9.5e-86)
		tmp = (z * y) * (x / sqrt((a * -t)));
	elseif (z <= 4.4e+55)
		tmp = t_1;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(x * N[(z / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1e+154], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, -2.9e-289], t$95$1, If[LessEqual[z, 9.5e-86], N[(N[(z * y), $MachinePrecision] * N[(x / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.4e+55], t$95$1, N[(x * y), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.00000000000000004e154

   1. Initial program 13.6%

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

      1. *-commutative 13.6%

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

      2. associate-*l* 13.0%

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

      3. associate-*r/ 13.3%

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

   3. Simplified 13.3%

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

   4. Taylor expanded in z around -inf 100.0%

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

      1. neg-mul-1 100.0%

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

   6. Simplified 100.0%

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

   if -1.00000000000000004e154 < z < -2.90000000000000006e-289 or 9.4999999999999996e-86 < z < 4.40000000000000021e55

   1. Initial program 83.3%

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

      1. associate-/l* 84.3%

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

   3. Simplified 84.3%

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

      1. associate-*l/ 88.0%

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

      2. associate-/l* 86.2%

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

      3. div-inv 86.1%

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

      4. associate-*l* 88.6%

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

      5. div-inv 88.8%

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

   5. Applied egg-rr 88.8%

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

   if -2.90000000000000006e-289 < z < 9.4999999999999996e-86

   1. Initial program 66.8%

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

      1. *-commutative 66.8%

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

      2. associate-*l* 75.2%

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

      3. associate-*r/ 72.7%

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

   3. Simplified 72.7%

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

      1. *-commutative 72.7%

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

      2. associate-/l* 64.3%

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

      3. associate-/r/ 77.4%

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

      4. associate-/l/ 86.0%

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

      5. *-commutative 86.0%

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

   5. Applied egg-rr 86.0%

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

      1. associate-/r/ 85.9%

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

   7. Applied egg-rr 85.9%

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

   8. Taylor expanded in z around 0 85.9%

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

      1. associate-*r* 85.9%

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

      2. neg-mul-1 85.9%

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

   10. Simplified 85.9%

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

   if 4.40000000000000021e55 < z

   1. Initial program 45.4%

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

      1. *-commutative 45.4%

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

      2. associate-*l* 43.8%

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

      3. associate-*r/ 47.8%

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

   3. Simplified 47.8%

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

   4. Taylor expanded in z around inf 100.0%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+154}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq -2.9 \cdot 10^{-289}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{\sqrt{z \cdot z - a \cdot t}}\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-86}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{x}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{+55}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{\sqrt{z \cdot z - a \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 2: 89.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+110}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+55}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.6e+110)
   (* x (- y))
   (if (<= z 3.6e+55) (* y (/ (* z x) (sqrt (- (* z z) (* a t))))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.6e+110) {
		tmp = x * -y;
	} else if (z <= 3.6e+55) {
		tmp = y * ((z * x) / sqrt(((z * z) - (a * t))));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-2.6d+110)) then
        tmp = x * -y
    else if (z <= 3.6d+55) then
        tmp = y * ((z * x) / sqrt(((z * z) - (a * t))))
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.6e+110) {
		tmp = x * -y;
	} else if (z <= 3.6e+55) {
		tmp = y * ((z * x) / Math.sqrt(((z * z) - (a * t))));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.6e+110:
		tmp = x * -y
	elif z <= 3.6e+55:
		tmp = y * ((z * x) / math.sqrt(((z * z) - (a * t))))
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.6e+110)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 3.6e+55)
		tmp = Float64(y * Float64(Float64(z * x) / sqrt(Float64(Float64(z * z) - Float64(a * t)))));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.6e+110)
		tmp = x * -y;
	elseif (z <= 3.6e+55)
		tmp = y * ((z * x) / sqrt(((z * z) - (a * t))));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.6e+110], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 3.6e+55], N[(y * N[(N[(z * x), $MachinePrecision] / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.6e110

   1. Initial program 29.7%

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

      1. *-commutative 29.7%

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

      2. associate-*l* 29.1%

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

      3. associate-*r/ 29.4%

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

   3. Simplified 29.4%

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

   4. Taylor expanded in z around -inf 100.0%

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

      1. neg-mul-1 100.0%

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

   6. Simplified 100.0%

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

   if -2.6e110 < z < 3.59999999999999987e55

   1. Initial program 79.0%

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

      1. *-commutative 79.0%

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

      2. associate-*l* 81.0%

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

      3. associate-*r/ 82.9%

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

   3. Simplified 82.9%

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

   if 3.59999999999999987e55 < z

   1. Initial program 45.4%

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

      1. *-commutative 45.4%

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

      2. associate-*l* 43.8%

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

      3. associate-*r/ 47.8%

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

   3. Simplified 47.8%

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

   4. Taylor expanded in z around inf 100.0%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+110}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+55}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 3: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4.05 \cdot 10^{+61}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(0.5, \frac{a}{z} \cdot \frac{t}{z}, -1\right)} \cdot y\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{+54}:\\ \;\;\;\;\frac{x}{\sqrt{z \cdot z - a \cdot t}} \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4.05e+61)
   (* (/ x (fma 0.5 (* (/ a z) (/ t z)) -1.0)) y)
   (if (<= z 8.4e+54) (* (/ x (sqrt (- (* z z) (* a t)))) (* z y)) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.05e+61) {
		tmp = (x / fma(0.5, ((a / z) * (t / z)), -1.0)) * y;
	} else if (z <= 8.4e+54) {
		tmp = (x / sqrt(((z * z) - (a * t)))) * (z * 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 <= -4.05e+61)
		tmp = Float64(Float64(x / fma(0.5, Float64(Float64(a / z) * Float64(t / z)), -1.0)) * y);
	elseif (z <= 8.4e+54)
		tmp = Float64(Float64(x / sqrt(Float64(Float64(z * z) - Float64(a * t)))) * Float64(z * y));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4.05e+61], N[(N[(x / N[(0.5 * N[(N[(a / z), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, 8.4e+54], N[(N[(x / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.0499999999999998e61

   1. Initial program 37.9%

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

      1. associate-/l* 39.7%

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

   3. Simplified 39.7%

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

   4. Taylor expanded in z around -inf 93.4%

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

      1. associate-*r/ 93.4%

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

      2. *-commutative 93.4%

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

   6. Applied egg-rr 93.4%

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

      1. div-inv 93.4%

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

      2. +-commutative 93.4%

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

      3. *-commutative 93.4%

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

      4. fma-def 93.4%

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

      5. associate-/l* 93.4%

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

   8. Applied egg-rr 93.4%

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

      1. associate-*r/ 93.4%

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

      2. *-commutative 93.4%

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

      3. *-rgt-identity 93.4%

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

      4. *-commutative 93.4%

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

      5. rem-log-exp 14.9%

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

      6. fma-udef 14.9%

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

      7. *-commutative 14.9%

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

      8. neg-mul-1 14.9%

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

      9. prod-exp 14.9%

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

      10. *-commutative 14.9%

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

      11. prod-exp 14.9%

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

      12. rem-log-exp 93.4%

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

      13. unsub-neg 93.4%

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

      14. associate-/r/ 93.4%

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

      15. *-commutative 93.4%

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

   10. Simplified 93.4%

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

       1. expm1-log1p-u 64.3%

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

       2. expm1-udef 37.4%

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

       3. associate-/l* 37.5%

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

       4. div-sub 37.5%

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

       5. *-commutative 37.5%

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

       6. *-commutative 37.5%

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

       7. pow1 37.5%

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

       8. pow1 37.5%

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

       9. pow-div 37.5%

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

       10. metadata-eval 37.5%

           \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{\frac{\frac{\left(t \cdot a\right) \cdot \frac{0.5}{z}}{z} - {z}^{\color{blue}{0}}}{y}}\right)} - 1 \]

       11. metadata-eval 37.5%

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

   12. Applied egg-rr 37.5%

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

       1. expm1-def 64.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\frac{\frac{\left(t \cdot a\right) \cdot \frac{0.5}{z}}{z} - 1}{y}}\right)\right)} \]

       2. expm1-log1p 93.2%

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

       3. associate-/r/ 93.5%

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

       4. associate-*r/ 93.5%

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

       5. *-commutative 93.5%

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

       6. *-commutative 93.5%

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

       7. associate-/r* 93.3%

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

       8. unpow2 93.3%

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

       9. associate-*r/ 93.3%

          \[\leadsto \frac{x}{\color{blue}{0.5 \cdot \frac{a \cdot t}{{z}^{2}}} - 1} \cdot y \]

       10. fma-neg 93.3%

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

       11. unpow2 93.3%

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

       12. times-frac 98.7%

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

       13. metadata-eval 98.7%

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

   14. Simplified 98.7%

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

   if -4.0499999999999998e61 < z < 8.39999999999999943e54

   1. Initial program 77.9%

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

      1. *-commutative 77.9%

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

      2. associate-*l* 80.2%

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

      3. associate-*r/ 82.2%

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

   3. Simplified 82.2%

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

      1. *-commutative 82.2%

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

      2. associate-/l* 81.2%

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

      3. associate-/r/ 83.0%

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

      4. associate-/l/ 83.0%

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

      5. *-commutative 83.0%

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

   5. Applied egg-rr 83.0%

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

      1. associate-/r/ 83.7%

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

   7. Applied egg-rr 83.7%

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

   if 8.39999999999999943e54 < z

   1. Initial program 45.4%

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

      1. *-commutative 45.4%

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

      2. associate-*l* 43.8%

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

      3. associate-*r/ 47.8%

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

   3. Simplified 47.8%

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

   4. Taylor expanded in z around inf 100.0%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.05 \cdot 10^{+61}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(0.5, \frac{a}{z} \cdot \frac{t}{z}, -1\right)} \cdot y\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{+54}:\\ \;\;\;\;\frac{x}{\sqrt{z \cdot z - a \cdot t}} \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 4: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+81}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{+54}:\\ \;\;\;\;\frac{x}{\sqrt{z \cdot z - a \cdot t}} \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4e+81)
   (* x (- y))
   (if (<= z 1.32e+54) (* (/ x (sqrt (- (* z z) (* a t)))) (* z y)) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4e+81) {
		tmp = x * -y;
	} else if (z <= 1.32e+54) {
		tmp = (x / sqrt(((z * z) - (a * t)))) * (z * 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 <= (-4d+81)) then
        tmp = x * -y
    else if (z <= 1.32d+54) then
        tmp = (x / sqrt(((z * z) - (a * t)))) * (z * 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 <= -4e+81) {
		tmp = x * -y;
	} else if (z <= 1.32e+54) {
		tmp = (x / Math.sqrt(((z * z) - (a * t)))) * (z * y);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -4e+81:
		tmp = x * -y
	elif z <= 1.32e+54:
		tmp = (x / math.sqrt(((z * z) - (a * t)))) * (z * 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 <= -4e+81)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 1.32e+54)
		tmp = Float64(Float64(x / sqrt(Float64(Float64(z * z) - Float64(a * t)))) * Float64(z * 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 <= -4e+81)
		tmp = x * -y;
	elseif (z <= 1.32e+54)
		tmp = (x / sqrt(((z * z) - (a * t)))) * (z * 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, -4e+81], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 1.32e+54], N[(N[(x / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.99999999999999969e81

   1. Initial program 36.6%

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

      1. *-commutative 36.6%

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

      2. associate-*l* 36.1%

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

      3. associate-*r/ 36.3%

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

   3. Simplified 36.3%

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

   4. Taylor expanded in z around -inf 99.6%

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

      1. neg-mul-1 99.6%

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

   6. Simplified 99.6%

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

   if -3.99999999999999969e81 < z < 1.3200000000000001e54

   1. Initial program 77.6%

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

      1. *-commutative 77.6%

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

      2. associate-*l* 79.8%

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

      3. associate-*r/ 81.8%

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

   3. Simplified 81.8%

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

      1. *-commutative 81.8%

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

      2. associate-/l* 80.9%

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

      3. associate-/r/ 82.6%

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

      4. associate-/l/ 82.6%

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

      5. *-commutative 82.6%

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

   5. Applied egg-rr 82.6%

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

      1. associate-/r/ 83.3%

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

   7. Applied egg-rr 83.3%

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

   if 1.3200000000000001e54 < z

   1. Initial program 45.4%

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

      1. *-commutative 45.4%

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

      2. associate-*l* 43.8%

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

      3. associate-*r/ 47.8%

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

   3. Simplified 47.8%

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

   4. Taylor expanded in z around inf 100.0%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+81}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{+54}:\\ \;\;\;\;\frac{x}{\sqrt{z \cdot z - a \cdot t}} \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 5: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-125}:\\ \;\;\;\;\frac{x \cdot y}{\frac{-0.5 \cdot \left(\frac{a}{z} \cdot t\right) - z}{z}}\\ \mathbf{elif}\;z \leq 240000:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.25e-125)
   (/ (* x y) (/ (- (* -0.5 (* (/ a z) t)) z) z))
   (if (<= z 240000.0)
     (* y (/ (* z x) (sqrt (* a (- t)))))
     (/ (* x y) (+ 1.0 (* -0.5 (/ a (/ (* z z) t))))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.25e-125) {
		tmp = (x * y) / (((-0.5 * ((a / z) * t)) - z) / z);
	} else if (z <= 240000.0) {
		tmp = y * ((z * x) / sqrt((a * -t)));
	} else {
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.25d-125)) then
        tmp = (x * y) / ((((-0.5d0) * ((a / z) * t)) - z) / z)
    else if (z <= 240000.0d0) then
        tmp = y * ((z * x) / sqrt((a * -t)))
    else
        tmp = (x * y) / (1.0d0 + ((-0.5d0) * (a / ((z * z) / t))))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.25e-125) {
		tmp = (x * y) / (((-0.5 * ((a / z) * t)) - z) / z);
	} else if (z <= 240000.0) {
		tmp = y * ((z * x) / Math.sqrt((a * -t)));
	} else {
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.25e-125:
		tmp = (x * y) / (((-0.5 * ((a / z) * t)) - z) / z)
	elif z <= 240000.0:
		tmp = y * ((z * x) / math.sqrt((a * -t)))
	else:
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.25e-125)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(-0.5 * Float64(Float64(a / z) * t)) - z) / z));
	elseif (z <= 240000.0)
		tmp = Float64(y * Float64(Float64(z * x) / sqrt(Float64(a * Float64(-t)))));
	else
		tmp = Float64(Float64(x * y) / Float64(1.0 + Float64(-0.5 * Float64(a / Float64(Float64(z * z) / t)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.25e-125)
		tmp = (x * y) / (((-0.5 * ((a / z) * t)) - z) / z);
	elseif (z <= 240000.0)
		tmp = y * ((z * x) / sqrt((a * -t)));
	else
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.25e-125], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(-0.5 * N[(N[(a / z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 240000.0], N[(y * N[(N[(z * x), $MachinePrecision] / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(a / N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.24999999999999992e-125

   1. Initial program 52.5%

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

      1. associate-/l* 53.8%

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

   3. Simplified 53.8%

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

   4. Taylor expanded in z around -inf 90.3%

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

      1. associate-*r/ 90.3%

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

      2. frac-2neg 90.3%

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

      3. *-commutative 90.3%

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

      4. mul-1-neg 90.3%

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

      5. add-sqr-sqrt 90.3%

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

      6. sqrt-unprod 90.2%

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

      7. mul-1-neg 90.2%

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

      8. mul-1-neg 90.2%

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

      9. sqr-neg 90.2%

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

      10. sqrt-prod 0.0%

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

      11. add-sqr-sqrt 89.8%

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

   6. Applied egg-rr 89.8%

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

      1. distribute-frac-neg 89.8%

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

      2. *-commutative 89.8%

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

      3. associate-*r/ 89.8%

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

      4. associate-/l* 93.5%

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

      5. distribute-lft-neg-in 93.5%

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

      6. metadata-eval 93.5%

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

      7. associate-/r/ 93.5%

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

      8. *-commutative 93.5%

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

   8. Simplified 93.5%

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

   if -1.24999999999999992e-125 < z < 2.4e5

   1. Initial program 72.9%

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

      1. *-commutative 72.9%

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

      2. associate-*l* 79.5%

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

      3. associate-*r/ 78.3%

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

   3. Simplified 78.3%

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

   4. Taylor expanded in z around 0 67.2%

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

      1. associate-*r* 70.1%

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

      2. neg-mul-1 70.1%

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

   6. Simplified 67.2%

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

   if 2.4e5 < z

   1. Initial program 49.6%

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

      1. associate-/l* 55.5%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in z around inf 94.8%

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

      1. unpow2 94.8%

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

      2. associate-/l* 98.4%

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

   6. Simplified 98.4%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-125}:\\ \;\;\;\;\frac{x \cdot y}{\frac{-0.5 \cdot \left(\frac{a}{z} \cdot t\right) - z}{z}}\\ \mathbf{elif}\;z \leq 240000:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}\\ \end{array} \]

Alternative 6: 81.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{-121}:\\ \;\;\;\;\frac{x \cdot y}{\frac{-0.5 \cdot \left(\frac{a}{z} \cdot t\right) - z}{z}}\\ \mathbf{elif}\;z \leq 240000:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{x}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.45e-121)
   (/ (* x y) (/ (- (* -0.5 (* (/ a z) t)) z) z))
   (if (<= z 240000.0)
     (* (* z y) (/ x (sqrt (* a (- t)))))
     (/ (* x y) (+ 1.0 (* -0.5 (/ a (/ (* z z) t))))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.45e-121) {
		tmp = (x * y) / (((-0.5 * ((a / z) * t)) - z) / z);
	} else if (z <= 240000.0) {
		tmp = (z * y) * (x / sqrt((a * -t)));
	} else {
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.45d-121)) then
        tmp = (x * y) / ((((-0.5d0) * ((a / z) * t)) - z) / z)
    else if (z <= 240000.0d0) then
        tmp = (z * y) * (x / sqrt((a * -t)))
    else
        tmp = (x * y) / (1.0d0 + ((-0.5d0) * (a / ((z * z) / t))))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.45e-121) {
		tmp = (x * y) / (((-0.5 * ((a / z) * t)) - z) / z);
	} else if (z <= 240000.0) {
		tmp = (z * y) * (x / Math.sqrt((a * -t)));
	} else {
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.45e-121:
		tmp = (x * y) / (((-0.5 * ((a / z) * t)) - z) / z)
	elif z <= 240000.0:
		tmp = (z * y) * (x / math.sqrt((a * -t)))
	else:
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.45e-121)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(-0.5 * Float64(Float64(a / z) * t)) - z) / z));
	elseif (z <= 240000.0)
		tmp = Float64(Float64(z * y) * Float64(x / sqrt(Float64(a * Float64(-t)))));
	else
		tmp = Float64(Float64(x * y) / Float64(1.0 + Float64(-0.5 * Float64(a / Float64(Float64(z * z) / t)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.45e-121)
		tmp = (x * y) / (((-0.5 * ((a / z) * t)) - z) / z);
	elseif (z <= 240000.0)
		tmp = (z * y) * (x / sqrt((a * -t)));
	else
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.45e-121], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(-0.5 * N[(N[(a / z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 240000.0], N[(N[(z * y), $MachinePrecision] * N[(x / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(a / N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.45e-121

   1. Initial program 52.5%

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

      1. associate-/l* 53.8%

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

   3. Simplified 53.8%

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

   4. Taylor expanded in z around -inf 90.3%

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

      1. associate-*r/ 90.3%

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

      2. frac-2neg 90.3%

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

      3. *-commutative 90.3%

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

      4. mul-1-neg 90.3%

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

      5. add-sqr-sqrt 90.3%

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

      6. sqrt-unprod 90.2%

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

      7. mul-1-neg 90.2%

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

      8. mul-1-neg 90.2%

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

      9. sqr-neg 90.2%

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

      10. sqrt-prod 0.0%

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

      11. add-sqr-sqrt 89.8%

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

   6. Applied egg-rr 89.8%

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

      1. distribute-frac-neg 89.8%

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

      2. *-commutative 89.8%

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

      3. associate-*r/ 89.8%

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

      4. associate-/l* 93.5%

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

      5. distribute-lft-neg-in 93.5%

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

      6. metadata-eval 93.5%

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

      7. associate-/r/ 93.5%

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

      8. *-commutative 93.5%

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

   8. Simplified 93.5%

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

   if -1.45e-121 < z < 2.4e5

   1. Initial program 72.9%

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

      1. *-commutative 72.9%

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

      2. associate-*l* 79.5%

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

      3. associate-*r/ 78.3%

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

   3. Simplified 78.3%

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

      1. *-commutative 78.3%

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

      2. associate-/l* 74.0%

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

      3. associate-/r/ 77.0%

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

      4. associate-/l/ 79.9%

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

      5. *-commutative 79.9%

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

   5. Applied egg-rr 79.9%

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

      1. associate-/r/ 81.3%

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

   7. Applied egg-rr 81.3%

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

   8. Taylor expanded in z around 0 70.1%

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

      1. associate-*r* 70.1%

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

      2. neg-mul-1 70.1%

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

   10. Simplified 70.1%

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

   if 2.4e5 < z

   1. Initial program 49.6%

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

      1. associate-/l* 55.5%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in z around inf 94.8%

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

      1. unpow2 94.8%

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

      2. associate-/l* 98.4%

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

   6. Simplified 98.4%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{-121}:\\ \;\;\;\;\frac{x \cdot y}{\frac{-0.5 \cdot \left(\frac{a}{z} \cdot t\right) - z}{z}}\\ \mathbf{elif}\;z \leq 240000:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{x}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}\\ \end{array} \]

Alternative 7: 83.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-121}:\\ \;\;\;\;\frac{x \cdot y}{\frac{-0.5 \cdot \left(\frac{a}{z} \cdot t\right) - z}{z}}\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-88}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{a \cdot \left(-t\right)}}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.55e-121)
   (/ (* x y) (/ (- (* -0.5 (* (/ a z) t)) z) z))
   (if (<= z 9.5e-88)
     (/ x (/ (sqrt (* a (- t))) (* z y)))
     (/ (* x y) (+ 1.0 (* -0.5 (/ a (/ (* z z) t))))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.55e-121) {
		tmp = (x * y) / (((-0.5 * ((a / z) * t)) - z) / z);
	} else if (z <= 9.5e-88) {
		tmp = x / (sqrt((a * -t)) / (z * y));
	} else {
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.55d-121)) then
        tmp = (x * y) / ((((-0.5d0) * ((a / z) * t)) - z) / z)
    else if (z <= 9.5d-88) then
        tmp = x / (sqrt((a * -t)) / (z * y))
    else
        tmp = (x * y) / (1.0d0 + ((-0.5d0) * (a / ((z * z) / t))))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.55e-121) {
		tmp = (x * y) / (((-0.5 * ((a / z) * t)) - z) / z);
	} else if (z <= 9.5e-88) {
		tmp = x / (Math.sqrt((a * -t)) / (z * y));
	} else {
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.55e-121:
		tmp = (x * y) / (((-0.5 * ((a / z) * t)) - z) / z)
	elif z <= 9.5e-88:
		tmp = x / (math.sqrt((a * -t)) / (z * y))
	else:
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.55e-121)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(-0.5 * Float64(Float64(a / z) * t)) - z) / z));
	elseif (z <= 9.5e-88)
		tmp = Float64(x / Float64(sqrt(Float64(a * Float64(-t))) / Float64(z * y)));
	else
		tmp = Float64(Float64(x * y) / Float64(1.0 + Float64(-0.5 * Float64(a / Float64(Float64(z * z) / t)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.55e-121)
		tmp = (x * y) / (((-0.5 * ((a / z) * t)) - z) / z);
	elseif (z <= 9.5e-88)
		tmp = x / (sqrt((a * -t)) / (z * y));
	else
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.55e-121], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(-0.5 * N[(N[(a / z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9.5e-88], N[(x / N[(N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(a / N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.5499999999999999e-121

   1. Initial program 52.5%

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

      1. associate-/l* 53.8%

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

   3. Simplified 53.8%

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

   4. Taylor expanded in z around -inf 90.3%

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

      1. associate-*r/ 90.3%

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

      2. frac-2neg 90.3%

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

      3. *-commutative 90.3%

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

      4. mul-1-neg 90.3%

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

      5. add-sqr-sqrt 90.3%

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

      6. sqrt-unprod 90.2%

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

      7. mul-1-neg 90.2%

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

      8. mul-1-neg 90.2%

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

      9. sqr-neg 90.2%

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

      10. sqrt-prod 0.0%

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

      11. add-sqr-sqrt 89.8%

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

   6. Applied egg-rr 89.8%

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

      1. distribute-frac-neg 89.8%

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

      2. *-commutative 89.8%

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

      3. associate-*r/ 89.8%

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

      4. associate-/l* 93.5%

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

      5. distribute-lft-neg-in 93.5%

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

      6. metadata-eval 93.5%

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

      7. associate-/r/ 93.5%

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

      8. *-commutative 93.5%

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

   8. Simplified 93.5%

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

   if -1.5499999999999999e-121 < z < 9.5e-88

   1. Initial program 69.1%

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

      1. *-commutative 69.1%

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

      2. associate-*l* 76.2%

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

      3. associate-*r/ 74.2%

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

   3. Simplified 74.2%

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

      1. *-commutative 74.2%

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

      2. associate-/l* 68.4%

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

      3. associate-/r/ 74.4%

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

      4. associate-/l/ 78.4%

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

      5. *-commutative 78.4%

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

   5. Applied egg-rr 78.4%

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

   6. Taylor expanded in z around 0 76.3%

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

      1. associate-*r* 76.3%

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

      2. neg-mul-1 76.3%

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

   8. Simplified 76.3%

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

   if 9.5e-88 < z

   1. Initial program 55.5%

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

      1. associate-/l* 60.5%

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

   3. Simplified 60.5%

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

   4. Taylor expanded in z around inf 87.6%

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

      1. unpow2 87.6%

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

      2. associate-/l* 90.5%

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

   6. Simplified 90.5%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-121}:\\ \;\;\;\;\frac{x \cdot y}{\frac{-0.5 \cdot \left(\frac{a}{z} \cdot t\right) - z}{z}}\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-88}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{a \cdot \left(-t\right)}}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}\\ \end{array} \]

Alternative 8: 76.1% accurate, 5.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{+81}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-86}:\\ \;\;\;\;z \cdot \frac{x \cdot y}{\frac{0.5}{z} \cdot \left(a \cdot t\right) - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.7e+81)
   (* x (- y))
   (if (<= z 9.5e-86) (* z (/ (* x y) (- (* (/ 0.5 z) (* a t)) 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+81) {
		tmp = x * -y;
	} else if (z <= 9.5e-86) {
		tmp = z * ((x * y) / (((0.5 / z) * (a * t)) - z));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-3.7d+81)) then
        tmp = x * -y
    else if (z <= 9.5d-86) then
        tmp = z * ((x * y) / (((0.5d0 / z) * (a * t)) - z))
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.7e+81) {
		tmp = x * -y;
	} else if (z <= 9.5e-86) {
		tmp = z * ((x * y) / (((0.5 / z) * (a * t)) - 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+81:
		tmp = x * -y
	elif z <= 9.5e-86:
		tmp = z * ((x * y) / (((0.5 / z) * (a * t)) - 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+81)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 9.5e-86)
		tmp = Float64(z * Float64(Float64(x * y) / Float64(Float64(Float64(0.5 / z) * Float64(a * t)) - z)));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -3.7e+81)
		tmp = x * -y;
	elseif (z <= 9.5e-86)
		tmp = z * ((x * y) / (((0.5 / z) * (a * t)) - z));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.7e+81], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 9.5e-86], N[(z * N[(N[(x * y), $MachinePrecision] / N[(N[(N[(0.5 / z), $MachinePrecision] * N[(a * t), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.7000000000000001e81

   1. Initial program 36.6%

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

      1. *-commutative 36.6%

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

      2. associate-*l* 36.1%

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

      3. associate-*r/ 36.3%

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

   3. Simplified 36.3%

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

   4. Taylor expanded in z around -inf 99.6%

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

      1. neg-mul-1 99.6%

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

   6. Simplified 99.6%

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

   if -3.7000000000000001e81 < z < 9.4999999999999996e-86

   1. Initial program 75.1%

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

      1. associate-/l* 76.2%

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

   3. Simplified 76.2%

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

   4. Taylor expanded in z around -inf 56.2%

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

      1. associate-*r/ 56.2%

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

      2. *-commutative 56.2%

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

   6. Applied egg-rr 56.2%

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

      1. expm1-log1p-u 51.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot y}{\frac{\frac{0.5 \cdot \left(t \cdot a\right)}{z} + -1 \cdot z}{z}}\right)\right)} \]

      2. expm1-udef 36.8%

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

      3. associate-/r/ 36.8%

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

      4. +-commutative 36.8%

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

      5. *-commutative 36.8%

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

      6. fma-def 36.8%

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

      7. associate-/l* 36.8%

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

   8. Applied egg-rr 36.8%

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

      1. expm1-def 51.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot y}{\mathsf{fma}\left(z, -1, \frac{0.5}{\frac{z}{t \cdot a}}\right)} \cdot z\right)\right)} \]

      2. expm1-log1p 56.3%

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

      3. *-commutative 56.3%

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

      4. rem-log-exp 30.7%

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

      5. fma-udef 30.7%

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

      6. *-commutative 30.7%

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

      7. neg-mul-1 30.7%

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

      8. prod-exp 30.0%

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

      9. *-commutative 30.0%

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

      10. prod-exp 30.7%

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

      11. rem-log-exp 56.3%

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

      12. unsub-neg 56.3%

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

      13. associate-/r/ 56.3%

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

      14. *-commutative 56.3%

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

   10. Simplified 56.3%

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

   if 9.4999999999999996e-86 < z

   1. Initial program 56.0%

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

      1. *-commutative 56.0%

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

      2. associate-*l* 55.7%

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

      3. associate-*r/ 59.7%

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

   3. Simplified 59.7%

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

   4. Taylor expanded in z around inf 91.1%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{+81}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-86}:\\ \;\;\;\;z \cdot \frac{x \cdot y}{\frac{0.5}{z} \cdot \left(a \cdot t\right) - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 9: 78.4% accurate, 5.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+81}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{-232}:\\ \;\;\;\;z \cdot \frac{x \cdot y}{\frac{0.5}{z} \cdot \left(a \cdot t\right) - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4e+81)
   (* x (- y))
   (if (<= z -8.2e-232)
     (* z (/ (* x y) (- (* (/ 0.5 z) (* a t)) z)))
     (/ (* x y) (+ 1.0 (* -0.5 (/ a (/ (* z z) t))))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4e+81) {
		tmp = x * -y;
	} else if (z <= -8.2e-232) {
		tmp = z * ((x * y) / (((0.5 / z) * (a * t)) - z));
	} else {
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-4d+81)) then
        tmp = x * -y
    else if (z <= (-8.2d-232)) then
        tmp = z * ((x * y) / (((0.5d0 / z) * (a * t)) - z))
    else
        tmp = (x * y) / (1.0d0 + ((-0.5d0) * (a / ((z * z) / t))))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4e+81) {
		tmp = x * -y;
	} else if (z <= -8.2e-232) {
		tmp = z * ((x * y) / (((0.5 / z) * (a * t)) - z));
	} else {
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -4e+81:
		tmp = x * -y
	elif z <= -8.2e-232:
		tmp = z * ((x * y) / (((0.5 / z) * (a * t)) - z))
	else:
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4e+81)
		tmp = Float64(x * Float64(-y));
	elseif (z <= -8.2e-232)
		tmp = Float64(z * Float64(Float64(x * y) / Float64(Float64(Float64(0.5 / z) * Float64(a * t)) - z)));
	else
		tmp = Float64(Float64(x * y) / Float64(1.0 + Float64(-0.5 * Float64(a / Float64(Float64(z * z) / t)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -4e+81)
		tmp = x * -y;
	elseif (z <= -8.2e-232)
		tmp = z * ((x * y) / (((0.5 / z) * (a * t)) - z));
	else
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4e+81], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, -8.2e-232], N[(z * N[(N[(x * y), $MachinePrecision] / N[(N[(N[(0.5 / z), $MachinePrecision] * N[(a * t), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(a / N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.99999999999999969e81

   1. Initial program 36.6%

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

      1. *-commutative 36.6%

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

      2. associate-*l* 36.1%

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

      3. associate-*r/ 36.3%

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

   3. Simplified 36.3%

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

   4. Taylor expanded in z around -inf 99.6%

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

      1. neg-mul-1 99.6%

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

   6. Simplified 99.6%

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

   if -3.99999999999999969e81 < z < -8.19999999999999945e-232

   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* 82.2%

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

   3. Simplified 82.2%

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

   4. Taylor expanded in z around -inf 69.2%

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

      1. associate-*r/ 69.2%

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

      2. *-commutative 69.2%

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

   6. Applied egg-rr 69.2%

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

      1. expm1-log1p-u 61.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot y}{\frac{\frac{0.5 \cdot \left(t \cdot a\right)}{z} + -1 \cdot z}{z}}\right)\right)} \]

      2. expm1-udef 38.5%

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

      3. associate-/r/ 38.4%

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

      4. +-commutative 38.4%

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

      5. *-commutative 38.4%

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

      6. fma-def 38.4%

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

      7. associate-/l* 38.4%

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

   8. Applied egg-rr 38.4%

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

      1. expm1-def 61.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot y}{\mathsf{fma}\left(z, -1, \frac{0.5}{\frac{z}{t \cdot a}}\right)} \cdot z\right)\right)} \]

      2. expm1-log1p 69.1%

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

      3. *-commutative 69.1%

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

      4. rem-log-exp 30.1%

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

      5. fma-udef 30.1%

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

      6. *-commutative 30.1%

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

      7. neg-mul-1 30.1%

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

      8. prod-exp 29.1%

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

      9. *-commutative 29.1%

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

      10. prod-exp 30.1%

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

      11. rem-log-exp 69.1%

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

      12. unsub-neg 69.1%

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

      13. associate-/r/ 69.1%

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

      14. *-commutative 69.1%

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

   10. Simplified 69.1%

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

   if -8.19999999999999945e-232 < z

   1. Initial program 56.7%

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

      1. associate-/l* 61.9%

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

   3. Simplified 61.9%

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

   4. Taylor expanded in z around inf 75.9%

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

      1. unpow2 75.9%

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

      2. associate-/l* 78.6%

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

   6. Simplified 78.6%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+81}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{-232}:\\ \;\;\;\;z \cdot \frac{x \cdot y}{\frac{0.5}{z} \cdot \left(a \cdot t\right) - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}\\ \end{array} \]

Alternative 10: 74.3% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{-254}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-86}:\\ \;\;\;\;-2 \cdot \frac{x \cdot \left(y \cdot \left(z \cdot z\right)\right)}{a \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.3e-254)
   (* x (- y))
   (if (<= z 9.5e-86) (* -2.0 (/ (* x (* y (* z z))) (* a t))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.3e-254) {
		tmp = x * -y;
	} else if (z <= 9.5e-86) {
		tmp = -2.0 * ((x * (y * (z * z))) / (a * t));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-2.3d-254)) then
        tmp = x * -y
    else if (z <= 9.5d-86) then
        tmp = (-2.0d0) * ((x * (y * (z * z))) / (a * t))
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.3e-254) {
		tmp = x * -y;
	} else if (z <= 9.5e-86) {
		tmp = -2.0 * ((x * (y * (z * z))) / (a * t));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.3e-254:
		tmp = x * -y
	elif z <= 9.5e-86:
		tmp = -2.0 * ((x * (y * (z * z))) / (a * t))
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.3e-254)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 9.5e-86)
		tmp = Float64(-2.0 * Float64(Float64(x * Float64(y * Float64(z * z))) / Float64(a * t)));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.3e-254)
		tmp = x * -y;
	elseif (z <= 9.5e-86)
		tmp = -2.0 * ((x * (y * (z * z))) / (a * t));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.3e-254], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 9.5e-86], N[(-2.0 * N[(N[(x * N[(y * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.2999999999999999e-254

   1. Initial program 55.8%

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

      1. *-commutative 55.8%

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

      2. associate-*l* 54.5%

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

      3. associate-*r/ 55.8%

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

   3. Simplified 55.8%

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

   4. Taylor expanded in z around -inf 80.7%

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

      1. neg-mul-1 80.7%

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

   6. Simplified 80.7%

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

   if -2.2999999999999999e-254 < z < 9.4999999999999996e-86

   1. Initial program 65.6%

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

      1. associate-*l/ 73.1%

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

   3. Simplified 73.1%

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

   4. Taylor expanded in z around inf 37.6%

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

   5. Taylor expanded in z around 0 37.2%

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

      1. associate-*r* 37.4%

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

      2. *-commutative 37.4%

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

      3. unpow2 37.4%

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

   7. Simplified 37.4%

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

   if 9.4999999999999996e-86 < z

   1. Initial program 56.0%

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

      1. *-commutative 56.0%

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

      2. associate-*l* 55.7%

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

      3. associate-*r/ 59.7%

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

   3. Simplified 59.7%

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

   4. Taylor expanded in z around inf 91.1%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{-254}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-86}:\\ \;\;\;\;-2 \cdot \frac{x \cdot \left(y \cdot \left(z \cdot z\right)\right)}{a \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 11: 75.1% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{-124}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-86}:\\ \;\;\;\;2 \cdot \left(\frac{y}{a} \cdot \frac{z \cdot \left(z \cdot x\right)}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

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 -1.75e-124)
   (* x (- y))
   (if (<= z 9.5e-86) (* 2.0 (* (/ y a) (/ (* z (* z x)) t))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.75e-124) {
		tmp = x * -y;
	} else if (z <= 9.5e-86) {
		tmp = 2.0 * ((y / a) * ((z * (z * x)) / t));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.75d-124)) then
        tmp = x * -y
    else if (z <= 9.5d-86) then
        tmp = 2.0d0 * ((y / a) * ((z * (z * x)) / t))
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.75e-124) {
		tmp = x * -y;
	} else if (z <= 9.5e-86) {
		tmp = 2.0 * ((y / a) * ((z * (z * x)) / t));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.75e-124:
		tmp = x * -y
	elif z <= 9.5e-86:
		tmp = 2.0 * ((y / a) * ((z * (z * x)) / t))
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.75e-124)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 9.5e-86)
		tmp = Float64(2.0 * Float64(Float64(y / a) * Float64(Float64(z * Float64(z * x)) / t)));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.75e-124)
		tmp = x * -y;
	elseif (z <= 9.5e-86)
		tmp = 2.0 * ((y / a) * ((z * (z * x)) / t));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.75e-124], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 9.5e-86], N[(2.0 * N[(N[(y / a), $MachinePrecision] * N[(N[(z * N[(z * x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.7499999999999999e-124

   1. Initial program 52.5%

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

      1. *-commutative 52.5%

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

      2. associate-*l* 50.4%

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

      3. associate-*r/ 52.4%

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

   3. Simplified 52.4%

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

   4. Taylor expanded in z around -inf 92.8%

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

      1. neg-mul-1 92.8%

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

   6. Simplified 92.8%

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

   if -1.7499999999999999e-124 < z < 9.4999999999999996e-86

   1. Initial program 67.7%

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

      1. associate-/l* 69.5%

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

   3. Simplified 69.5%

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

   4. Taylor expanded in z around -inf 38.6%

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

   5. Taylor expanded in a around inf 30.1%

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

      1. times-frac 31.2%

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

      2. unpow2 31.2%

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

      3. associate-*l* 33.6%

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

   7. Simplified 33.6%

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

   if 9.4999999999999996e-86 < z

   1. Initial program 56.0%

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

      1. *-commutative 56.0%

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

      2. associate-*l* 55.7%

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

      3. associate-*r/ 59.7%

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

   3. Simplified 59.7%

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

   4. Taylor expanded in z around inf 91.1%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{-124}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-86}:\\ \;\;\;\;2 \cdot \left(\frac{y}{a} \cdot \frac{z \cdot \left(z \cdot x\right)}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 12: 77.7% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{-232}:\\ \;\;\;\;\frac{x \cdot y}{\frac{\frac{0.5}{z} \cdot \left(a \cdot t\right) - z}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -8.2e-232)
   (/ (* x y) (/ (- (* (/ 0.5 z) (* a t)) z) z))
   (/ (* x y) (+ 1.0 (* -0.5 (/ a (/ (* z z) t)))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.2e-232) {
		tmp = (x * y) / ((((0.5 / z) * (a * t)) - z) / z);
	} else {
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-8.2d-232)) then
        tmp = (x * y) / ((((0.5d0 / z) * (a * t)) - z) / z)
    else
        tmp = (x * y) / (1.0d0 + ((-0.5d0) * (a / ((z * z) / t))))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.2e-232) {
		tmp = (x * y) / ((((0.5 / z) * (a * t)) - z) / z);
	} else {
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -8.2e-232:
		tmp = (x * y) / ((((0.5 / z) * (a * t)) - z) / z)
	else:
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -8.2e-232)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(Float64(0.5 / z) * Float64(a * t)) - z) / z));
	else
		tmp = Float64(Float64(x * y) / Float64(1.0 + Float64(-0.5 * Float64(a / Float64(Float64(z * z) / t)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -8.2e-232)
		tmp = (x * y) / ((((0.5 / z) * (a * t)) - z) / z);
	else
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -8.2e-232], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(N[(0.5 / z), $MachinePrecision] * N[(a * t), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(a / N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.19999999999999945e-232

   1. Initial program 57.0%

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

      1. associate-/l* 57.4%

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

   3. Simplified 57.4%

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

   4. Taylor expanded in z around -inf 83.5%

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

      1. associate-*r/ 83.5%

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

      2. *-commutative 83.5%

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

   6. Applied egg-rr 83.5%

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

      1. div-inv 83.5%

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

      2. +-commutative 83.5%

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

      3. *-commutative 83.5%

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

      4. fma-def 83.5%

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

      5. associate-/l* 83.5%

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

   8. Applied egg-rr 83.5%

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

      1. associate-*r/ 83.5%

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

      2. *-commutative 83.5%

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

      3. *-rgt-identity 83.5%

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

      4. *-commutative 83.5%

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

      5. rem-log-exp 20.9%

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

      6. fma-udef 20.9%

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

      7. *-commutative 20.9%

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

      8. neg-mul-1 20.9%

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

      9. prod-exp 20.5%

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

      10. *-commutative 20.5%

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

      11. prod-exp 20.9%

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

      12. rem-log-exp 83.5%

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

      13. unsub-neg 83.5%

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

      14. associate-/r/ 83.5%

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

      15. *-commutative 83.5%

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

   10. Simplified 83.5%

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

   if -8.19999999999999945e-232 < z

   1. Initial program 56.7%

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

      1. associate-/l* 61.9%

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

   3. Simplified 61.9%

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

   4. Taylor expanded in z around inf 75.9%

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

      1. unpow2 75.9%

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

      2. associate-/l* 78.6%

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

   6. Simplified 78.6%

      \[\leadsto \frac{x \cdot y}{\color{blue}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{-232}:\\ \;\;\;\;\frac{x \cdot y}{\frac{\frac{0.5}{z} \cdot \left(a \cdot t\right) - z}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}\\ \end{array} \]

Alternative 13: 77.3% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 3 \cdot 10^{-107}:\\ \;\;\;\;\frac{x \cdot y}{\frac{-0.5 \cdot \left(\frac{a}{z} \cdot t\right) - z}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z 3e-107)
   (/ (* x y) (/ (- (* -0.5 (* (/ a z) t)) z) z))
   (/ (* x y) (+ 1.0 (* -0.5 (/ a (/ (* z z) t)))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= 3e-107) {
		tmp = (x * y) / (((-0.5 * ((a / z) * t)) - z) / z);
	} else {
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= 3d-107) then
        tmp = (x * y) / ((((-0.5d0) * ((a / z) * t)) - z) / z)
    else
        tmp = (x * y) / (1.0d0 + ((-0.5d0) * (a / ((z * z) / t))))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= 3e-107) {
		tmp = (x * y) / (((-0.5 * ((a / z) * t)) - z) / z);
	} else {
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= 3e-107:
		tmp = (x * y) / (((-0.5 * ((a / z) * t)) - z) / z)
	else:
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= 3e-107)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(-0.5 * Float64(Float64(a / z) * t)) - z) / z));
	else
		tmp = Float64(Float64(x * y) / Float64(1.0 + Float64(-0.5 * Float64(a / Float64(Float64(z * z) / t)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= 3e-107)
		tmp = (x * y) / (((-0.5 * ((a / z) * t)) - z) / z);
	else
		tmp = (x * y) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, 3e-107], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(-0.5 * N[(N[(a / z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(a / N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.9999999999999997e-107

   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* 58.9%

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

   3. Simplified 58.9%

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

   4. Taylor expanded in z around -inf 74.3%

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

      1. associate-*r/ 74.3%

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

      2. frac-2neg 74.3%

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

      3. *-commutative 74.3%

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

      4. mul-1-neg 74.3%

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

      5. add-sqr-sqrt 69.5%

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

      6. sqrt-unprod 72.2%

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

      7. mul-1-neg 72.2%

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

      8. mul-1-neg 72.2%

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

      9. sqr-neg 72.2%

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

      10. sqrt-prod 5.0%

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

      11. add-sqr-sqrt 73.8%

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

   6. Applied egg-rr 73.8%

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

      1. distribute-frac-neg 73.8%

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

      2. *-commutative 73.8%

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

      3. associate-*r/ 73.8%

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

      4. associate-/l* 76.7%

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

      5. distribute-lft-neg-in 76.7%

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

      6. metadata-eval 76.7%

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

      7. associate-/r/ 76.7%

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

      8. *-commutative 76.7%

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

   8. Simplified 76.7%

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

   if 2.9999999999999997e-107 < z

   1. Initial program 55.9%

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

      1. associate-/l* 60.8%

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

   3. Simplified 60.8%

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

   4. Taylor expanded in z around inf 87.4%

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

      1. unpow2 87.4%

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

      2. associate-/l* 90.3%

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

   6. Simplified 90.3%

      \[\leadsto \frac{x \cdot y}{\color{blue}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3 \cdot 10^{-107}:\\ \;\;\;\;\frac{x \cdot y}{\frac{-0.5 \cdot \left(\frac{a}{z} \cdot t\right) - z}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}\\ \end{array} \]

Alternative 14: 73.0% accurate, 18.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

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 -2e-310) (* x (- y)) (* x y)))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2e-310) {
		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 <= (-2d-310)) 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 <= -2e-310) {
		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 <= -2e-310:
		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 <= -2e-310)
		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 <= -2e-310)
		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, -2e-310], N[(x * (-y)), $MachinePrecision], N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.999999999999994e-310

   1. Initial program 55.3%

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

      1. *-commutative 55.3%

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

      2. associate-*l* 55.5%

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

      3. associate-*r/ 56.8%

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

   3. Simplified 56.8%

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

   4. Taylor expanded in z around -inf 78.9%

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

      1. neg-mul-1 78.9%

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

   6. Simplified 78.9%

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

   if -1.999999999999994e-310 < z

   1. Initial program 58.4%

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

      1. *-commutative 58.4%

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

      2. associate-*l* 58.9%

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

      3. associate-*r/ 61.8%

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

   3. Simplified 61.8%

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

   4. Taylor expanded in z around inf 78.9%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 15: 43.5% 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 56.8%

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

   1. *-commutative 56.8%

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

   2. associate-*l* 57.2%

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

   3. associate-*r/ 59.2%

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

3. Simplified 59.2%

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

4. Taylor expanded in z around inf 45.4%

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

5. Final simplification 45.4%

   \[\leadsto x \cdot y \]

Developer target: 89.1% 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 2023221 
(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)))))