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

Percentage Accurate: 61.4% → 91.2%

Time: 17.6s

Alternatives: 16

Speedup: 18.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.4% 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: 91.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := \sqrt{z \cdot z - a \cdot t}\\ \mathbf{if}\;z \leq -1.25 \cdot 10^{+104}:\\ \;\;\;\;\left(\frac{z}{\mathsf{fma}\left(0.5, \frac{a}{\frac{z}{t}}, -z\right)} \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq 10^{-302}:\\ \;\;\;\;\frac{z}{\frac{t_1}{x \cdot y}}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-127}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+118}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (sqrt (- (* z z) (* a t)))))
   (if (<= z -1.25e+104)
     (* (* (/ z (fma 0.5 (/ a (/ z t)) (- z))) x) y)
     (if (<= z 1e-302)
       (/ z (/ t_1 (* x y)))
       (if (<= z 1.1e-127)
         (/ (* x (* z y)) (sqrt (* a (- t))))
         (if (<= z 2e+118) (* y (* x (/ z t_1))) (* x y)))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = sqrt(((z * z) - (a * t)));
	double tmp;
	if (z <= -1.25e+104) {
		tmp = ((z / fma(0.5, (a / (z / t)), -z)) * x) * y;
	} else if (z <= 1e-302) {
		tmp = z / (t_1 / (x * y));
	} else if (z <= 1.1e-127) {
		tmp = (x * (z * y)) / sqrt((a * -t));
	} else if (z <= 2e+118) {
		tmp = y * (x * (z / t_1));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = sqrt(Float64(Float64(z * z) - Float64(a * t)))
	tmp = 0.0
	if (z <= -1.25e+104)
		tmp = Float64(Float64(Float64(z / fma(0.5, Float64(a / Float64(z / t)), Float64(-z))) * x) * y);
	elseif (z <= 1e-302)
		tmp = Float64(z / Float64(t_1 / Float64(x * y)));
	elseif (z <= 1.1e-127)
		tmp = Float64(Float64(x * Float64(z * y)) / sqrt(Float64(a * Float64(-t))));
	elseif (z <= 2e+118)
		tmp = Float64(y * Float64(x * Float64(z / t_1)));
	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_] := Block[{t$95$1 = N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -1.25e+104], N[(N[(N[(z / N[(0.5 * N[(a / N[(z / t), $MachinePrecision]), $MachinePrecision] + (-z)), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, 1e-302], N[(z / N[(t$95$1 / N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.1e-127], N[(N[(x * N[(z * y), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2e+118], N[(y * N[(x * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -1.2499999999999999e104

   1. Initial program 32.1%

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

      1. *-commutative 32.1%

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

      2. associate-/l* 28.3%

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

   3. Simplified 28.3%

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

      1. associate-/r/ 32.5%

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

      2. associate-*r* 32.6%

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

   5. Applied egg-rr 32.6%

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

   6. Taylor expanded in z around -inf 93.5%

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

      1. fma-def 93.5%

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

      2. associate-/l* 98.3%

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

      3. neg-mul-1 98.3%

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

   8. Simplified 98.3%

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

   if -1.2499999999999999e104 < z < 9.9999999999999996e-303

   1. Initial program 82.0%

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

      1. *-commutative 82.0%

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

      2. associate-/l* 86.1%

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

   3. Simplified 86.1%

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

   if 9.9999999999999996e-303 < z < 1.1000000000000001e-127

   1. Initial program 69.9%

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

      1. associate-*l* 83.5%

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

   3. Simplified 83.5%

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

   4. Taylor expanded in z around 0 83.5%

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

      1. associate-*r* 83.5%

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

      2. neg-mul-1 83.5%

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

      3. *-commutative 83.5%

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

   6. Simplified 83.5%

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

   if 1.1000000000000001e-127 < z < 1.99999999999999993e118

   1. Initial program 94.3%

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

      1. *-commutative 94.3%

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

      2. associate-/l* 91.3%

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

   3. Simplified 91.3%

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

      1. associate-/r/ 96.3%

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

      2. associate-*r* 96.3%

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

   5. Applied egg-rr 96.3%

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

   if 1.99999999999999993e118 < z

   1. Initial program 17.4%

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

      1. associate-*l* 17.0%

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

      2. associate-*r/ 17.3%

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

   3. Simplified 17.3%

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

   4. Taylor expanded in z around inf 100.0%

      \[\leadsto \color{blue}{y \cdot x} \]
3. Recombined 5 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+104}:\\ \;\;\;\;\left(\frac{z}{\mathsf{fma}\left(0.5, \frac{a}{\frac{z}{t}}, -z\right)} \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq 10^{-302}:\\ \;\;\;\;\frac{z}{\frac{\sqrt{z \cdot z - a \cdot t}}{x \cdot y}}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-127}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+118}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{\sqrt{z \cdot z - a \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 2: 89.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := x \cdot \frac{z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{if}\;z \leq -1.05 \cdot 10^{+117}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq -1.18 \cdot 10^{-116}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-305}:\\ \;\;\;\;\frac{z}{\frac{\sqrt{a \cdot \left(-t\right)}}{x \cdot y}}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+71}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{z + \frac{a}{\frac{z}{t}} \cdot -0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* x (/ z (/ (sqrt (- (* z z) (* a t))) y)))))
   (if (<= z -1.05e+117)
     (* x (- y))
     (if (<= z -1.18e-116)
       t_1
       (if (<= z 1.35e-305)
         (/ z (/ (sqrt (* a (- t))) (* x y)))
         (if (<= z 2e+71)
           t_1
           (* y (* x (/ z (+ z (* (/ a (/ z t)) -0.5)))))))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = x * (z / (sqrt(((z * z) - (a * t))) / y));
	double tmp;
	if (z <= -1.05e+117) {
		tmp = x * -y;
	} else if (z <= -1.18e-116) {
		tmp = t_1;
	} else if (z <= 1.35e-305) {
		tmp = z / (sqrt((a * -t)) / (x * y));
	} else if (z <= 2e+71) {
		tmp = t_1;
	} else {
		tmp = y * (x * (z / (z + ((a / (z / t)) * -0.5))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * (z / (sqrt(((z * z) - (a * t))) / y))
    if (z <= (-1.05d+117)) then
        tmp = x * -y
    else if (z <= (-1.18d-116)) then
        tmp = t_1
    else if (z <= 1.35d-305) then
        tmp = z / (sqrt((a * -t)) / (x * y))
    else if (z <= 2d+71) then
        tmp = t_1
    else
        tmp = y * (x * (z / (z + ((a / (z / t)) * (-0.5d0)))))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x * (z / (Math.sqrt(((z * z) - (a * t))) / y));
	double tmp;
	if (z <= -1.05e+117) {
		tmp = x * -y;
	} else if (z <= -1.18e-116) {
		tmp = t_1;
	} else if (z <= 1.35e-305) {
		tmp = z / (Math.sqrt((a * -t)) / (x * y));
	} else if (z <= 2e+71) {
		tmp = t_1;
	} else {
		tmp = y * (x * (z / (z + ((a / (z / t)) * -0.5))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	t_1 = x * (z / (math.sqrt(((z * z) - (a * t))) / y))
	tmp = 0
	if z <= -1.05e+117:
		tmp = x * -y
	elif z <= -1.18e-116:
		tmp = t_1
	elif z <= 1.35e-305:
		tmp = z / (math.sqrt((a * -t)) / (x * y))
	elif z <= 2e+71:
		tmp = t_1
	else:
		tmp = y * (x * (z / (z + ((a / (z / t)) * -0.5))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = Float64(x * Float64(z / Float64(sqrt(Float64(Float64(z * z) - Float64(a * t))) / y)))
	tmp = 0.0
	if (z <= -1.05e+117)
		tmp = Float64(x * Float64(-y));
	elseif (z <= -1.18e-116)
		tmp = t_1;
	elseif (z <= 1.35e-305)
		tmp = Float64(z / Float64(sqrt(Float64(a * Float64(-t))) / Float64(x * y)));
	elseif (z <= 2e+71)
		tmp = t_1;
	else
		tmp = Float64(y * Float64(x * Float64(z / Float64(z + Float64(Float64(a / Float64(z / t)) * -0.5)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = x * (z / (sqrt(((z * z) - (a * t))) / y));
	tmp = 0.0;
	if (z <= -1.05e+117)
		tmp = x * -y;
	elseif (z <= -1.18e-116)
		tmp = t_1;
	elseif (z <= 1.35e-305)
		tmp = z / (sqrt((a * -t)) / (x * y));
	elseif (z <= 2e+71)
		tmp = t_1;
	else
		tmp = y * (x * (z / (z + ((a / (z / t)) * -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x * N[(z / N[(N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.05e+117], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, -1.18e-116], t$95$1, If[LessEqual[z, 1.35e-305], N[(z / N[(N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision] / N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2e+71], t$95$1, N[(y * N[(x * N[(z / N[(z + N[(N[(a / N[(z / t), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.0500000000000001e117

   1. Initial program 26.9%

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

      1. associate-*l* 24.9%

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

      2. associate-*r/ 25.0%

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

   3. Simplified 25.0%

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

   4. Taylor expanded in z around -inf 97.9%

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

      1. mul-1-neg 97.9%

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

      2. distribute-rgt-neg-out 97.9%

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

   6. Simplified 97.9%

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

   if -1.0500000000000001e117 < z < -1.1800000000000001e-116 or 1.35e-305 < z < 2.0000000000000001e71

   1. Initial program 85.6%

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

      1. associate-*l* 89.3%

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

      2. associate-*r/ 91.9%

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

      3. *-commutative 91.9%

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

      4. associate-/l* 91.1%

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

   3. Simplified 91.1%

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

   if -1.1800000000000001e-116 < z < 1.35e-305

   1. Initial program 77.7%

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

      1. *-commutative 77.7%

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

      2. associate-/l* 78.1%

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

   3. Simplified 78.1%

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

   4. Taylor expanded in z around 0 78.0%

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

      1. associate-*r* 59.3%

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

      2. neg-mul-1 59.3%

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

      3. *-commutative 59.3%

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

   6. Simplified 78.0%

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

   if 2.0000000000000001e71 < z

   1. Initial program 33.7%

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

      1. *-commutative 33.7%

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

      2. associate-/l* 31.1%

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

   3. Simplified 31.1%

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

      1. associate-/r/ 35.8%

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

      2. associate-*r* 36.0%

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

   5. Applied egg-rr 36.0%

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

   6. Taylor expanded in z around inf 90.4%

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

      1. associate-/l* 96.8%

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

   8. Simplified 96.8%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+117}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq -1.18 \cdot 10^{-116}:\\ \;\;\;\;x \cdot \frac{z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-305}:\\ \;\;\;\;\frac{z}{\frac{\sqrt{a \cdot \left(-t\right)}}{x \cdot y}}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+71}:\\ \;\;\;\;x \cdot \frac{z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{z + \frac{a}{\frac{z}{t}} \cdot -0.5}\right)\\ \end{array} \]

Alternative 3: 90.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := x \cdot \frac{z \cdot y}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{if}\;z \leq -2.05 \cdot 10^{+123}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{-125}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-305}:\\ \;\;\;\;\frac{z}{\frac{\sqrt{a \cdot \left(-t\right)}}{x \cdot y}}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+71}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{z + \frac{a}{\frac{z}{t}} \cdot -0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* x (/ (* z y) (sqrt (- (* z z) (* a t)))))))
   (if (<= z -2.05e+123)
     (* x (- y))
     (if (<= z -2.4e-125)
       t_1
       (if (<= z 4e-305)
         (/ z (/ (sqrt (* a (- t))) (* x y)))
         (if (<= z 4e+71)
           t_1
           (* y (* x (/ z (+ z (* (/ a (/ z t)) -0.5)))))))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = x * ((z * y) / sqrt(((z * z) - (a * t))));
	double tmp;
	if (z <= -2.05e+123) {
		tmp = x * -y;
	} else if (z <= -2.4e-125) {
		tmp = t_1;
	} else if (z <= 4e-305) {
		tmp = z / (sqrt((a * -t)) / (x * y));
	} else if (z <= 4e+71) {
		tmp = t_1;
	} else {
		tmp = y * (x * (z / (z + ((a / (z / t)) * -0.5))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * ((z * y) / sqrt(((z * z) - (a * t))))
    if (z <= (-2.05d+123)) then
        tmp = x * -y
    else if (z <= (-2.4d-125)) then
        tmp = t_1
    else if (z <= 4d-305) then
        tmp = z / (sqrt((a * -t)) / (x * y))
    else if (z <= 4d+71) then
        tmp = t_1
    else
        tmp = y * (x * (z / (z + ((a / (z / t)) * (-0.5d0)))))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x * ((z * y) / Math.sqrt(((z * z) - (a * t))));
	double tmp;
	if (z <= -2.05e+123) {
		tmp = x * -y;
	} else if (z <= -2.4e-125) {
		tmp = t_1;
	} else if (z <= 4e-305) {
		tmp = z / (Math.sqrt((a * -t)) / (x * y));
	} else if (z <= 4e+71) {
		tmp = t_1;
	} else {
		tmp = y * (x * (z / (z + ((a / (z / t)) * -0.5))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	t_1 = x * ((z * y) / math.sqrt(((z * z) - (a * t))))
	tmp = 0
	if z <= -2.05e+123:
		tmp = x * -y
	elif z <= -2.4e-125:
		tmp = t_1
	elif z <= 4e-305:
		tmp = z / (math.sqrt((a * -t)) / (x * y))
	elif z <= 4e+71:
		tmp = t_1
	else:
		tmp = y * (x * (z / (z + ((a / (z / t)) * -0.5))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = Float64(x * Float64(Float64(z * y) / sqrt(Float64(Float64(z * z) - Float64(a * t)))))
	tmp = 0.0
	if (z <= -2.05e+123)
		tmp = Float64(x * Float64(-y));
	elseif (z <= -2.4e-125)
		tmp = t_1;
	elseif (z <= 4e-305)
		tmp = Float64(z / Float64(sqrt(Float64(a * Float64(-t))) / Float64(x * y)));
	elseif (z <= 4e+71)
		tmp = t_1;
	else
		tmp = Float64(y * Float64(x * Float64(z / Float64(z + Float64(Float64(a / Float64(z / t)) * -0.5)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = x * ((z * y) / sqrt(((z * z) - (a * t))));
	tmp = 0.0;
	if (z <= -2.05e+123)
		tmp = x * -y;
	elseif (z <= -2.4e-125)
		tmp = t_1;
	elseif (z <= 4e-305)
		tmp = z / (sqrt((a * -t)) / (x * y));
	elseif (z <= 4e+71)
		tmp = t_1;
	else
		tmp = y * (x * (z / (z + ((a / (z / t)) * -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x * N[(N[(z * y), $MachinePrecision] / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.05e+123], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, -2.4e-125], t$95$1, If[LessEqual[z, 4e-305], N[(z / N[(N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision] / N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4e+71], t$95$1, N[(y * N[(x * N[(z / N[(z + N[(N[(a / N[(z / t), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -2.04999999999999995e123

   1. Initial program 25.5%

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

      1. associate-*l* 25.4%

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

      2. associate-*r/ 25.5%

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

   3. Simplified 25.5%

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

   4. Taylor expanded in z around -inf 97.9%

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

      1. mul-1-neg 97.9%

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

      2. distribute-rgt-neg-out 97.9%

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

   6. Simplified 97.9%

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

   if -2.04999999999999995e123 < z < -2.4000000000000001e-125 or 3.99999999999999999e-305 < z < 4.0000000000000002e71

   1. Initial program 85.8%

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

      1. associate-*l* 88.7%

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

      2. associate-*r/ 91.2%

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

   3. Simplified 91.2%

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

   if -2.4000000000000001e-125 < z < 3.99999999999999999e-305

   1. Initial program 77.0%

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

      1. *-commutative 77.0%

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

      2. associate-/l* 80.5%

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

   3. Simplified 80.5%

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

   4. Taylor expanded in z around 0 80.4%

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

      1. associate-*r* 61.0%

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

      2. neg-mul-1 61.0%

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

      3. *-commutative 61.0%

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

   6. Simplified 80.4%

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

   if 4.0000000000000002e71 < z

   1. Initial program 33.7%

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

      1. *-commutative 33.7%

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

      2. associate-/l* 31.1%

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

   3. Simplified 31.1%

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

      1. associate-/r/ 35.8%

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

      2. associate-*r* 36.0%

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

   5. Applied egg-rr 36.0%

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

   6. Taylor expanded in z around inf 90.4%

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

      1. associate-/l* 96.8%

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

   8. Simplified 96.8%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.05 \cdot 10^{+123}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{-125}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-305}:\\ \;\;\;\;\frac{z}{\frac{\sqrt{a \cdot \left(-t\right)}}{x \cdot y}}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+71}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{z + \frac{a}{\frac{z}{t}} \cdot -0.5}\right)\\ \end{array} \]

Alternative 4: 91.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := \sqrt{z \cdot z - a \cdot t}\\ \mathbf{if}\;z \leq -4.5 \cdot 10^{+104}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-306}:\\ \;\;\;\;z \cdot \frac{x \cdot y}{t_1}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-127}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+118}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (sqrt (- (* z z) (* a t)))))
   (if (<= z -4.5e+104)
     (* x (- y))
     (if (<= z 2.7e-306)
       (* z (/ (* x y) t_1))
       (if (<= z 1.1e-127)
         (/ (* x (* z y)) (sqrt (* a (- t))))
         (if (<= z 2e+118) (* y (* x (/ z t_1))) (* x y)))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = sqrt(((z * z) - (a * t)));
	double tmp;
	if (z <= -4.5e+104) {
		tmp = x * -y;
	} else if (z <= 2.7e-306) {
		tmp = z * ((x * y) / t_1);
	} else if (z <= 1.1e-127) {
		tmp = (x * (z * y)) / sqrt((a * -t));
	} else if (z <= 2e+118) {
		tmp = y * (x * (z / 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 = sqrt(((z * z) - (a * t)))
    if (z <= (-4.5d+104)) then
        tmp = x * -y
    else if (z <= 2.7d-306) then
        tmp = z * ((x * y) / t_1)
    else if (z <= 1.1d-127) then
        tmp = (x * (z * y)) / sqrt((a * -t))
    else if (z <= 2d+118) then
        tmp = y * (x * (z / t_1))
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.sqrt(((z * z) - (a * t)));
	double tmp;
	if (z <= -4.5e+104) {
		tmp = x * -y;
	} else if (z <= 2.7e-306) {
		tmp = z * ((x * y) / t_1);
	} else if (z <= 1.1e-127) {
		tmp = (x * (z * y)) / Math.sqrt((a * -t));
	} else if (z <= 2e+118) {
		tmp = y * (x * (z / t_1));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	t_1 = math.sqrt(((z * z) - (a * t)))
	tmp = 0
	if z <= -4.5e+104:
		tmp = x * -y
	elif z <= 2.7e-306:
		tmp = z * ((x * y) / t_1)
	elif z <= 1.1e-127:
		tmp = (x * (z * y)) / math.sqrt((a * -t))
	elif z <= 2e+118:
		tmp = y * (x * (z / t_1))
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = sqrt(Float64(Float64(z * z) - Float64(a * t)))
	tmp = 0.0
	if (z <= -4.5e+104)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 2.7e-306)
		tmp = Float64(z * Float64(Float64(x * y) / t_1));
	elseif (z <= 1.1e-127)
		tmp = Float64(Float64(x * Float64(z * y)) / sqrt(Float64(a * Float64(-t))));
	elseif (z <= 2e+118)
		tmp = Float64(y * Float64(x * Float64(z / t_1)));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = sqrt(((z * z) - (a * t)));
	tmp = 0.0;
	if (z <= -4.5e+104)
		tmp = x * -y;
	elseif (z <= 2.7e-306)
		tmp = z * ((x * y) / t_1);
	elseif (z <= 1.1e-127)
		tmp = (x * (z * y)) / sqrt((a * -t));
	elseif (z <= 2e+118)
		tmp = y * (x * (z / t_1));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -4.5e+104], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 2.7e-306], N[(z * N[(N[(x * y), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.1e-127], N[(N[(x * N[(z * y), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2e+118], N[(y * N[(x * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -4.4999999999999998e104

   1. Initial program 32.1%

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

      1. associate-*l* 30.3%

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

      2. associate-*r/ 30.4%

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

   3. Simplified 30.4%

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

   4. Taylor expanded in z around -inf 98.1%

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

      1. mul-1-neg 98.1%

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

      2. distribute-rgt-neg-out 98.1%

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

   6. Simplified 98.1%

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

   if -4.4999999999999998e104 < z < 2.70000000000000009e-306

   1. Initial program 82.0%

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

      1. associate-*l/ 84.8%

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

   3. Simplified 84.8%

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

   if 2.70000000000000009e-306 < z < 1.1000000000000001e-127

   1. Initial program 69.9%

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

      1. associate-*l* 83.5%

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

   3. Simplified 83.5%

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

   4. Taylor expanded in z around 0 83.5%

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

      1. associate-*r* 83.5%

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

      2. neg-mul-1 83.5%

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

      3. *-commutative 83.5%

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

   6. Simplified 83.5%

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

   if 1.1000000000000001e-127 < z < 1.99999999999999993e118

   1. Initial program 94.3%

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

      1. *-commutative 94.3%

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

      2. associate-/l* 91.3%

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

   3. Simplified 91.3%

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

      1. associate-/r/ 96.3%

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

      2. associate-*r* 96.3%

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

   5. Applied egg-rr 96.3%

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

   if 1.99999999999999993e118 < z

   1. Initial program 17.4%

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

      1. associate-*l* 17.0%

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

      2. associate-*r/ 17.3%

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

   3. Simplified 17.3%

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

   4. Taylor expanded in z around inf 100.0%

      \[\leadsto \color{blue}{y \cdot x} \]
3. Recombined 5 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+104}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-306}:\\ \;\;\;\;z \cdot \frac{x \cdot y}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-127}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+118}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{\sqrt{z \cdot z - a \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 5: 91.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := \sqrt{z \cdot z - a \cdot t}\\ \mathbf{if}\;z \leq -1.25 \cdot 10^{+104}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-307}:\\ \;\;\;\;\frac{z}{\frac{t_1}{x \cdot y}}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-127}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+118}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (sqrt (- (* z z) (* a t)))))
   (if (<= z -1.25e+104)
     (* x (- y))
     (if (<= z 9e-307)
       (/ z (/ t_1 (* x y)))
       (if (<= z 1.1e-127)
         (/ (* x (* z y)) (sqrt (* a (- t))))
         (if (<= z 2e+118) (* y (* x (/ z t_1))) (* x y)))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = sqrt(((z * z) - (a * t)));
	double tmp;
	if (z <= -1.25e+104) {
		tmp = x * -y;
	} else if (z <= 9e-307) {
		tmp = z / (t_1 / (x * y));
	} else if (z <= 1.1e-127) {
		tmp = (x * (z * y)) / sqrt((a * -t));
	} else if (z <= 2e+118) {
		tmp = y * (x * (z / 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 = sqrt(((z * z) - (a * t)))
    if (z <= (-1.25d+104)) then
        tmp = x * -y
    else if (z <= 9d-307) then
        tmp = z / (t_1 / (x * y))
    else if (z <= 1.1d-127) then
        tmp = (x * (z * y)) / sqrt((a * -t))
    else if (z <= 2d+118) then
        tmp = y * (x * (z / t_1))
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.sqrt(((z * z) - (a * t)));
	double tmp;
	if (z <= -1.25e+104) {
		tmp = x * -y;
	} else if (z <= 9e-307) {
		tmp = z / (t_1 / (x * y));
	} else if (z <= 1.1e-127) {
		tmp = (x * (z * y)) / Math.sqrt((a * -t));
	} else if (z <= 2e+118) {
		tmp = y * (x * (z / t_1));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	t_1 = math.sqrt(((z * z) - (a * t)))
	tmp = 0
	if z <= -1.25e+104:
		tmp = x * -y
	elif z <= 9e-307:
		tmp = z / (t_1 / (x * y))
	elif z <= 1.1e-127:
		tmp = (x * (z * y)) / math.sqrt((a * -t))
	elif z <= 2e+118:
		tmp = y * (x * (z / t_1))
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = sqrt(Float64(Float64(z * z) - Float64(a * t)))
	tmp = 0.0
	if (z <= -1.25e+104)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 9e-307)
		tmp = Float64(z / Float64(t_1 / Float64(x * y)));
	elseif (z <= 1.1e-127)
		tmp = Float64(Float64(x * Float64(z * y)) / sqrt(Float64(a * Float64(-t))));
	elseif (z <= 2e+118)
		tmp = Float64(y * Float64(x * Float64(z / t_1)));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = sqrt(((z * z) - (a * t)));
	tmp = 0.0;
	if (z <= -1.25e+104)
		tmp = x * -y;
	elseif (z <= 9e-307)
		tmp = z / (t_1 / (x * y));
	elseif (z <= 1.1e-127)
		tmp = (x * (z * y)) / sqrt((a * -t));
	elseif (z <= 2e+118)
		tmp = y * (x * (z / t_1));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -1.25e+104], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 9e-307], N[(z / N[(t$95$1 / N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.1e-127], N[(N[(x * N[(z * y), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2e+118], N[(y * N[(x * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -1.2499999999999999e104

   1. Initial program 32.1%

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

      1. associate-*l* 30.3%

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

      2. associate-*r/ 30.4%

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

   3. Simplified 30.4%

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

   4. Taylor expanded in z around -inf 98.1%

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

      1. mul-1-neg 98.1%

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

      2. distribute-rgt-neg-out 98.1%

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

   6. Simplified 98.1%

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

   if -1.2499999999999999e104 < z < 8.99999999999999978e-307

   1. Initial program 82.0%

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

      1. *-commutative 82.0%

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

      2. associate-/l* 86.1%

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

   3. Simplified 86.1%

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

   if 8.99999999999999978e-307 < z < 1.1000000000000001e-127

   1. Initial program 69.9%

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

      1. associate-*l* 83.5%

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

   3. Simplified 83.5%

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

   4. Taylor expanded in z around 0 83.5%

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

      1. associate-*r* 83.5%

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

      2. neg-mul-1 83.5%

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

      3. *-commutative 83.5%

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

   6. Simplified 83.5%

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

   if 1.1000000000000001e-127 < z < 1.99999999999999993e118

   1. Initial program 94.3%

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

      1. *-commutative 94.3%

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

      2. associate-/l* 91.3%

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

   3. Simplified 91.3%

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

      1. associate-/r/ 96.3%

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

      2. associate-*r* 96.3%

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

   5. Applied egg-rr 96.3%

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

   if 1.99999999999999993e118 < z

   1. Initial program 17.4%

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

      1. associate-*l* 17.0%

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

      2. associate-*r/ 17.3%

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

   3. Simplified 17.3%

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

   4. Taylor expanded in z around inf 100.0%

      \[\leadsto \color{blue}{y \cdot x} \]
3. Recombined 5 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+104}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-307}:\\ \;\;\;\;\frac{z}{\frac{\sqrt{z \cdot z - a \cdot t}}{x \cdot y}}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-127}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+118}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{\sqrt{z \cdot z - a \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 6: 91.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := \sqrt{z \cdot z - a \cdot t}\\ \mathbf{if}\;z \leq -6 \cdot 10^{+104}:\\ \;\;\;\;\frac{x \cdot y}{\mathsf{fma}\left(0.5, \frac{a}{\frac{z \cdot z}{t}}, -1\right)}\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-305}:\\ \;\;\;\;\frac{z}{\frac{t_1}{x \cdot y}}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-127}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{elif}\;z \leq 10^{+123}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (sqrt (- (* z z) (* a t)))))
   (if (<= z -6e+104)
     (/ (* x y) (fma 0.5 (/ a (/ (* z z) t)) -1.0))
     (if (<= z 2.9e-305)
       (/ z (/ t_1 (* x y)))
       (if (<= z 1.1e-127)
         (/ (* x (* z y)) (sqrt (* a (- t))))
         (if (<= z 1e+123) (* y (* x (/ z t_1))) (* x y)))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = sqrt(((z * z) - (a * t)));
	double tmp;
	if (z <= -6e+104) {
		tmp = (x * y) / fma(0.5, (a / ((z * z) / t)), -1.0);
	} else if (z <= 2.9e-305) {
		tmp = z / (t_1 / (x * y));
	} else if (z <= 1.1e-127) {
		tmp = (x * (z * y)) / sqrt((a * -t));
	} else if (z <= 1e+123) {
		tmp = y * (x * (z / t_1));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = sqrt(Float64(Float64(z * z) - Float64(a * t)))
	tmp = 0.0
	if (z <= -6e+104)
		tmp = Float64(Float64(x * y) / fma(0.5, Float64(a / Float64(Float64(z * z) / t)), -1.0));
	elseif (z <= 2.9e-305)
		tmp = Float64(z / Float64(t_1 / Float64(x * y)));
	elseif (z <= 1.1e-127)
		tmp = Float64(Float64(x * Float64(z * y)) / sqrt(Float64(a * Float64(-t))));
	elseif (z <= 1e+123)
		tmp = Float64(y * Float64(x * Float64(z / t_1)));
	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_] := Block[{t$95$1 = N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -6e+104], N[(N[(x * y), $MachinePrecision] / N[(0.5 * N[(a / N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.9e-305], N[(z / N[(t$95$1 / N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.1e-127], N[(N[(x * N[(z * y), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1e+123], N[(y * N[(x * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -5.99999999999999937e104

   1. Initial program 32.1%

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

      1. associate-*l* 30.3%

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

      2. associate-*r/ 30.4%

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

   3. Simplified 30.4%

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

      1. associate-/l* 32.6%

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

      2. associate-*r/ 32.5%

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

      3. *-commutative 32.5%

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

   5. Applied egg-rr 32.5%

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

   6. Taylor expanded in z around -inf 89.4%

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

      1. fma-neg 89.4%

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

      2. associate-/l* 98.1%

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

      3. unpow2 98.1%

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

      4. metadata-eval 98.1%

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

   8. Simplified 98.1%

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

   if -5.99999999999999937e104 < z < 2.89999999999999988e-305

   1. Initial program 82.0%

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

      1. *-commutative 82.0%

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

      2. associate-/l* 86.1%

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

   3. Simplified 86.1%

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

   if 2.89999999999999988e-305 < z < 1.1000000000000001e-127

   1. Initial program 69.9%

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

      1. associate-*l* 83.5%

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

   3. Simplified 83.5%

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

   4. Taylor expanded in z around 0 83.5%

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

      1. associate-*r* 83.5%

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

      2. neg-mul-1 83.5%

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

      3. *-commutative 83.5%

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

   6. Simplified 83.5%

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

   if 1.1000000000000001e-127 < z < 9.99999999999999978e122

   1. Initial program 94.3%

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

      1. *-commutative 94.3%

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

      2. associate-/l* 91.3%

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

   3. Simplified 91.3%

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

      1. associate-/r/ 96.3%

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

      2. associate-*r* 96.3%

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

   5. Applied egg-rr 96.3%

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

   if 9.99999999999999978e122 < z

   1. Initial program 17.4%

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

      1. associate-*l* 17.0%

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

      2. associate-*r/ 17.3%

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

   3. Simplified 17.3%

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

   4. Taylor expanded in z around inf 100.0%

      \[\leadsto \color{blue}{y \cdot x} \]
3. Recombined 5 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+104}:\\ \;\;\;\;\frac{x \cdot y}{\mathsf{fma}\left(0.5, \frac{a}{\frac{z \cdot z}{t}}, -1\right)}\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-305}:\\ \;\;\;\;\frac{z}{\frac{\sqrt{z \cdot z - a \cdot t}}{x \cdot y}}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-127}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{elif}\;z \leq 10^{+123}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{\sqrt{z \cdot z - a \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 7: 91.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+129}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+118}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{\sqrt{z \cdot z - a \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

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 -4e+129)
   (* x (- y))
   (if (<= z 2e+118) (* y (* x (/ z (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 <= -4e+129) {
		tmp = x * -y;
	} else if (z <= 2e+118) {
		tmp = y * (x * (z / 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 <= (-4d+129)) then
        tmp = x * -y
    else if (z <= 2d+118) then
        tmp = y * (x * (z / 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 <= -4e+129) {
		tmp = x * -y;
	} else if (z <= 2e+118) {
		tmp = y * (x * (z / 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 <= -4e+129:
		tmp = x * -y
	elif z <= 2e+118:
		tmp = y * (x * (z / 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 <= -4e+129)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 2e+118)
		tmp = Float64(y * Float64(x * Float64(z / 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 <= -4e+129)
		tmp = x * -y;
	elseif (z <= 2e+118)
		tmp = y * (x * (z / 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, -4e+129], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 2e+118], N[(y * N[(x * N[(z / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4e129

   1. Initial program 25.5%

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

      1. associate-*l* 25.4%

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

      2. associate-*r/ 25.5%

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

   3. Simplified 25.5%

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

   4. Taylor expanded in z around -inf 97.9%

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

      1. mul-1-neg 97.9%

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

      2. distribute-rgt-neg-out 97.9%

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

   6. Simplified 97.9%

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

   if -4e129 < z < 1.99999999999999993e118

   1. Initial program 84.2%

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

      1. *-commutative 84.2%

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

      2. associate-/l* 84.0%

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

   3. Simplified 84.0%

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

      1. associate-/r/ 86.0%

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

      2. associate-*r* 87.2%

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

   5. Applied egg-rr 87.2%

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

   if 1.99999999999999993e118 < z

   1. Initial program 17.4%

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

      1. associate-*l* 17.0%

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

      2. associate-*r/ 17.3%

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

   3. Simplified 17.3%

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

   4. Taylor expanded in z around inf 100.0%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+129}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+118}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{\sqrt{z \cdot z - a \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 8: 83.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := \sqrt{a \cdot \left(-t\right)}\\ \mathbf{if}\;z \leq -1.3 \cdot 10^{-28}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 3.35 \cdot 10^{-304}:\\ \;\;\;\;\frac{z}{\frac{t_1}{x \cdot y}}\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-123}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\frac{z + \frac{a}{\frac{z}{t}} \cdot -0.5}{z}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (sqrt (* a (- t)))))
   (if (<= z -1.3e-28)
     (* x (- y))
     (if (<= z 3.35e-304)
       (/ z (/ t_1 (* x y)))
       (if (<= z 2.8e-123)
         (* x (/ (* z y) t_1))
         (/ (* x y) (/ (+ z (* (/ a (/ z t)) -0.5)) z)))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = sqrt((a * -t));
	double tmp;
	if (z <= -1.3e-28) {
		tmp = x * -y;
	} else if (z <= 3.35e-304) {
		tmp = z / (t_1 / (x * y));
	} else if (z <= 2.8e-123) {
		tmp = x * ((z * y) / t_1);
	} else {
		tmp = (x * y) / ((z + ((a / (z / t)) * -0.5)) / z);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((a * -t))
    if (z <= (-1.3d-28)) then
        tmp = x * -y
    else if (z <= 3.35d-304) then
        tmp = z / (t_1 / (x * y))
    else if (z <= 2.8d-123) then
        tmp = x * ((z * y) / t_1)
    else
        tmp = (x * y) / ((z + ((a / (z / t)) * (-0.5d0))) / z)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.sqrt((a * -t));
	double tmp;
	if (z <= -1.3e-28) {
		tmp = x * -y;
	} else if (z <= 3.35e-304) {
		tmp = z / (t_1 / (x * y));
	} else if (z <= 2.8e-123) {
		tmp = x * ((z * y) / t_1);
	} else {
		tmp = (x * y) / ((z + ((a / (z / t)) * -0.5)) / z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	t_1 = math.sqrt((a * -t))
	tmp = 0
	if z <= -1.3e-28:
		tmp = x * -y
	elif z <= 3.35e-304:
		tmp = z / (t_1 / (x * y))
	elif z <= 2.8e-123:
		tmp = x * ((z * y) / t_1)
	else:
		tmp = (x * y) / ((z + ((a / (z / t)) * -0.5)) / z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = sqrt(Float64(a * Float64(-t)))
	tmp = 0.0
	if (z <= -1.3e-28)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 3.35e-304)
		tmp = Float64(z / Float64(t_1 / Float64(x * y)));
	elseif (z <= 2.8e-123)
		tmp = Float64(x * Float64(Float64(z * y) / t_1));
	else
		tmp = Float64(Float64(x * y) / Float64(Float64(z + Float64(Float64(a / Float64(z / t)) * -0.5)) / z));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = sqrt((a * -t));
	tmp = 0.0;
	if (z <= -1.3e-28)
		tmp = x * -y;
	elseif (z <= 3.35e-304)
		tmp = z / (t_1 / (x * y));
	elseif (z <= 2.8e-123)
		tmp = x * ((z * y) / t_1);
	else
		tmp = (x * y) / ((z + ((a / (z / t)) * -0.5)) / z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -1.3e-28], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 3.35e-304], N[(z / N[(t$95$1 / N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.8e-123], N[(x * N[(N[(z * y), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(N[(z + N[(N[(a / N[(z / t), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.3e-28

   1. Initial program 50.7%

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

      1. associate-*l* 48.3%

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

      2. associate-*r/ 50.7%

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

   3. Simplified 50.7%

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

   4. Taylor expanded in z around -inf 94.2%

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

      1. mul-1-neg 94.2%

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

      2. distribute-rgt-neg-out 94.2%

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

   6. Simplified 94.2%

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

   if -1.3e-28 < z < 3.3500000000000002e-304

   1. Initial program 77.5%

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

      1. *-commutative 77.5%

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

      2. associate-/l* 79.6%

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

   3. Simplified 79.6%

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

   4. Taylor expanded in z around 0 72.9%

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

      1. associate-*r* 61.9%

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

      2. neg-mul-1 61.9%

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

      3. *-commutative 61.9%

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

   6. Simplified 72.9%

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

   if 3.3500000000000002e-304 < z < 2.7999999999999999e-123

   1. Initial program 69.9%

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

      1. associate-*l* 83.5%

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

      2. associate-*r/ 80.6%

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

   3. Simplified 80.6%

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

   4. Taylor expanded in z around 0 80.6%

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

      1. associate-*r* 83.5%

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

      2. neg-mul-1 83.5%

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

      3. *-commutative 83.5%

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

   6. Simplified 80.6%

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

   if 2.7999999999999999e-123 < z

   1. Initial program 59.0%

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

      1. associate-*l* 57.7%

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

      2. associate-*r/ 59.9%

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

   3. Simplified 59.9%

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

      1. associate-/l* 61.3%

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

      2. associate-*r/ 60.4%

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

      3. *-commutative 60.4%

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

   5. Applied egg-rr 60.4%

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

   6. Taylor expanded in z around inf 81.1%

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

      1. associate-/l* 85.0%

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

   8. Simplified 85.0%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{-28}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 3.35 \cdot 10^{-304}:\\ \;\;\;\;\frac{z}{\frac{\sqrt{a \cdot \left(-t\right)}}{x \cdot y}}\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-123}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\frac{z + \frac{a}{\frac{z}{t}} \cdot -0.5}{z}}\\ \end{array} \]

Alternative 9: 83.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := \sqrt{a \cdot \left(-t\right)}\\ \mathbf{if}\;z \leq -7 \cdot 10^{-31}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-302}:\\ \;\;\;\;\frac{z}{\frac{t_1}{x \cdot y}}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-122}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\frac{z + \frac{a}{\frac{z}{t}} \cdot -0.5}{z}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (sqrt (* a (- t)))))
   (if (<= z -7e-31)
     (* x (- y))
     (if (<= z 3.3e-302)
       (/ z (/ t_1 (* x y)))
       (if (<= z 1.1e-122)
         (/ (* x (* z y)) t_1)
         (/ (* x y) (/ (+ z (* (/ a (/ z t)) -0.5)) z)))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = sqrt((a * -t));
	double tmp;
	if (z <= -7e-31) {
		tmp = x * -y;
	} else if (z <= 3.3e-302) {
		tmp = z / (t_1 / (x * y));
	} else if (z <= 1.1e-122) {
		tmp = (x * (z * y)) / t_1;
	} else {
		tmp = (x * y) / ((z + ((a / (z / t)) * -0.5)) / z);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((a * -t))
    if (z <= (-7d-31)) then
        tmp = x * -y
    else if (z <= 3.3d-302) then
        tmp = z / (t_1 / (x * y))
    else if (z <= 1.1d-122) then
        tmp = (x * (z * y)) / t_1
    else
        tmp = (x * y) / ((z + ((a / (z / t)) * (-0.5d0))) / z)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.sqrt((a * -t));
	double tmp;
	if (z <= -7e-31) {
		tmp = x * -y;
	} else if (z <= 3.3e-302) {
		tmp = z / (t_1 / (x * y));
	} else if (z <= 1.1e-122) {
		tmp = (x * (z * y)) / t_1;
	} else {
		tmp = (x * y) / ((z + ((a / (z / t)) * -0.5)) / z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	t_1 = math.sqrt((a * -t))
	tmp = 0
	if z <= -7e-31:
		tmp = x * -y
	elif z <= 3.3e-302:
		tmp = z / (t_1 / (x * y))
	elif z <= 1.1e-122:
		tmp = (x * (z * y)) / t_1
	else:
		tmp = (x * y) / ((z + ((a / (z / t)) * -0.5)) / z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = sqrt(Float64(a * Float64(-t)))
	tmp = 0.0
	if (z <= -7e-31)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 3.3e-302)
		tmp = Float64(z / Float64(t_1 / Float64(x * y)));
	elseif (z <= 1.1e-122)
		tmp = Float64(Float64(x * Float64(z * y)) / t_1);
	else
		tmp = Float64(Float64(x * y) / Float64(Float64(z + Float64(Float64(a / Float64(z / t)) * -0.5)) / z));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = sqrt((a * -t));
	tmp = 0.0;
	if (z <= -7e-31)
		tmp = x * -y;
	elseif (z <= 3.3e-302)
		tmp = z / (t_1 / (x * y));
	elseif (z <= 1.1e-122)
		tmp = (x * (z * y)) / t_1;
	else
		tmp = (x * y) / ((z + ((a / (z / t)) * -0.5)) / z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -7e-31], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 3.3e-302], N[(z / N[(t$95$1 / N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.1e-122], N[(N[(x * N[(z * y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(N[(z + N[(N[(a / N[(z / t), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -6.99999999999999971e-31

   1. Initial program 50.7%

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

      1. associate-*l* 48.3%

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

      2. associate-*r/ 50.7%

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

   3. Simplified 50.7%

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

   4. Taylor expanded in z around -inf 94.2%

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

      1. mul-1-neg 94.2%

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

      2. distribute-rgt-neg-out 94.2%

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

   6. Simplified 94.2%

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

   if -6.99999999999999971e-31 < z < 3.3000000000000002e-302

   1. Initial program 77.5%

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

      1. *-commutative 77.5%

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

      2. associate-/l* 79.6%

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

   3. Simplified 79.6%

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

   4. Taylor expanded in z around 0 72.9%

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

      1. associate-*r* 61.9%

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

      2. neg-mul-1 61.9%

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

      3. *-commutative 61.9%

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

   6. Simplified 72.9%

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

   if 3.3000000000000002e-302 < z < 1.1e-122

   1. Initial program 69.9%

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

      1. associate-*l* 83.5%

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

   3. Simplified 83.5%

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

   4. Taylor expanded in z around 0 83.5%

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

      1. associate-*r* 83.5%

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

      2. neg-mul-1 83.5%

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

      3. *-commutative 83.5%

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

   6. Simplified 83.5%

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

   if 1.1e-122 < z

   1. Initial program 59.0%

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

      1. associate-*l* 57.7%

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

      2. associate-*r/ 59.9%

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

   3. Simplified 59.9%

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

      1. associate-/l* 61.3%

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

      2. associate-*r/ 60.4%

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

      3. *-commutative 60.4%

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

   5. Applied egg-rr 60.4%

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

   6. Taylor expanded in z around inf 81.1%

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

      1. associate-/l* 85.0%

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

   8. Simplified 85.0%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-31}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-302}:\\ \;\;\;\;\frac{z}{\frac{\sqrt{a \cdot \left(-t\right)}}{x \cdot y}}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-122}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\frac{z + \frac{a}{\frac{z}{t}} \cdot -0.5}{z}}\\ \end{array} \]

Alternative 10: 83.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{-31}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{-123}:\\ \;\;\;\;x \cdot \frac{z}{\frac{\sqrt{a \cdot \left(-t\right)}}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\frac{z + \frac{a}{\frac{z}{t}} \cdot -0.5}{z}}\\ \end{array} \end{array} \]

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 -7.5e-31)
   (* x (- y))
   (if (<= z 6.4e-123)
     (* x (/ z (/ (sqrt (* a (- t))) y)))
     (/ (* x y) (/ (+ z (* (/ a (/ z t)) -0.5)) z)))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7.5e-31) {
		tmp = x * -y;
	} else if (z <= 6.4e-123) {
		tmp = x * (z / (sqrt((a * -t)) / y));
	} else {
		tmp = (x * y) / ((z + ((a / (z / t)) * -0.5)) / z);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-7.5d-31)) then
        tmp = x * -y
    else if (z <= 6.4d-123) then
        tmp = x * (z / (sqrt((a * -t)) / y))
    else
        tmp = (x * y) / ((z + ((a / (z / t)) * (-0.5d0))) / z)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7.5e-31) {
		tmp = x * -y;
	} else if (z <= 6.4e-123) {
		tmp = x * (z / (Math.sqrt((a * -t)) / y));
	} else {
		tmp = (x * y) / ((z + ((a / (z / t)) * -0.5)) / z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -7.5e-31:
		tmp = x * -y
	elif z <= 6.4e-123:
		tmp = x * (z / (math.sqrt((a * -t)) / y))
	else:
		tmp = (x * y) / ((z + ((a / (z / t)) * -0.5)) / z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -7.5e-31)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 6.4e-123)
		tmp = Float64(x * Float64(z / Float64(sqrt(Float64(a * Float64(-t))) / y)));
	else
		tmp = Float64(Float64(x * y) / Float64(Float64(z + Float64(Float64(a / Float64(z / t)) * -0.5)) / z));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -7.5e-31)
		tmp = x * -y;
	elseif (z <= 6.4e-123)
		tmp = x * (z / (sqrt((a * -t)) / y));
	else
		tmp = (x * y) / ((z + ((a / (z / t)) * -0.5)) / z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -7.5e-31], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 6.4e-123], N[(x * N[(z / N[(N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(N[(z + N[(N[(a / N[(z / t), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.49999999999999975e-31

   1. Initial program 50.7%

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

      1. associate-*l* 48.3%

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

      2. associate-*r/ 50.7%

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

   3. Simplified 50.7%

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

   4. Taylor expanded in z around -inf 94.2%

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

      1. mul-1-neg 94.2%

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

      2. distribute-rgt-neg-out 94.2%

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

   6. Simplified 94.2%

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

   if -7.49999999999999975e-31 < z < 6.39999999999999957e-123

   1. Initial program 74.2%

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

      1. associate-*l* 74.9%

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

      2. associate-*r/ 76.3%

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

      3. *-commutative 76.3%

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

      4. associate-/l* 73.8%

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

   3. Simplified 73.8%

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

   4. Taylor expanded in z around 0 70.0%

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

      1. associate-*r* 71.1%

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

      2. neg-mul-1 71.1%

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

      3. *-commutative 71.1%

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

   6. Simplified 70.0%

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

   if 6.39999999999999957e-123 < z

   1. Initial program 59.0%

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

      1. associate-*l* 57.7%

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

      2. associate-*r/ 59.9%

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

   3. Simplified 59.9%

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

      1. associate-/l* 61.3%

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

      2. associate-*r/ 60.4%

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

      3. *-commutative 60.4%

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

   5. Applied egg-rr 60.4%

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

   6. Taylor expanded in z around inf 81.1%

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

      1. associate-/l* 85.0%

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

   8. Simplified 85.0%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{-31}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{-123}:\\ \;\;\;\;x \cdot \frac{z}{\frac{\sqrt{a \cdot \left(-t\right)}}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\frac{z + \frac{a}{\frac{z}{t}} \cdot -0.5}{z}}\\ \end{array} \]

Alternative 11: 83.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{-27}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-123}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\frac{z + \frac{a}{\frac{z}{t}} \cdot -0.5}{z}}\\ \end{array} \end{array} \]

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 -1.02e-27)
   (* x (- y))
   (if (<= z 2.9e-123)
     (* x (/ (* z y) (sqrt (* a (- t)))))
     (/ (* x y) (/ (+ z (* (/ a (/ z t)) -0.5)) z)))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.02e-27) {
		tmp = x * -y;
	} else if (z <= 2.9e-123) {
		tmp = x * ((z * y) / sqrt((a * -t)));
	} else {
		tmp = (x * y) / ((z + ((a / (z / t)) * -0.5)) / z);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.02d-27)) then
        tmp = x * -y
    else if (z <= 2.9d-123) then
        tmp = x * ((z * y) / sqrt((a * -t)))
    else
        tmp = (x * y) / ((z + ((a / (z / t)) * (-0.5d0))) / z)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.02e-27) {
		tmp = x * -y;
	} else if (z <= 2.9e-123) {
		tmp = x * ((z * y) / Math.sqrt((a * -t)));
	} else {
		tmp = (x * y) / ((z + ((a / (z / t)) * -0.5)) / z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.02e-27:
		tmp = x * -y
	elif z <= 2.9e-123:
		tmp = x * ((z * y) / math.sqrt((a * -t)))
	else:
		tmp = (x * y) / ((z + ((a / (z / t)) * -0.5)) / z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.02e-27)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 2.9e-123)
		tmp = Float64(x * Float64(Float64(z * y) / sqrt(Float64(a * Float64(-t)))));
	else
		tmp = Float64(Float64(x * y) / Float64(Float64(z + Float64(Float64(a / Float64(z / t)) * -0.5)) / z));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.02e-27)
		tmp = x * -y;
	elseif (z <= 2.9e-123)
		tmp = x * ((z * y) / sqrt((a * -t)));
	else
		tmp = (x * y) / ((z + ((a / (z / t)) * -0.5)) / z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.02e-27], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 2.9e-123], N[(x * N[(N[(z * y), $MachinePrecision] / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(N[(z + N[(N[(a / N[(z / t), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.02000000000000002e-27

   1. Initial program 50.7%

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

      1. associate-*l* 48.3%

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

      2. associate-*r/ 50.7%

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

   3. Simplified 50.7%

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

   4. Taylor expanded in z around -inf 94.2%

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

      1. mul-1-neg 94.2%

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

      2. distribute-rgt-neg-out 94.2%

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

   6. Simplified 94.2%

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

   if -1.02000000000000002e-27 < z < 2.90000000000000004e-123

   1. Initial program 74.2%

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

      1. associate-*l* 74.9%

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

      2. associate-*r/ 76.3%

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

   3. Simplified 76.3%

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

   4. Taylor expanded in z around 0 71.2%

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

      1. associate-*r* 71.1%

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

      2. neg-mul-1 71.1%

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

      3. *-commutative 71.1%

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

   6. Simplified 71.2%

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

   if 2.90000000000000004e-123 < z

   1. Initial program 59.0%

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

      1. associate-*l* 57.7%

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

      2. associate-*r/ 59.9%

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

   3. Simplified 59.9%

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

      1. associate-/l* 61.3%

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

      2. associate-*r/ 60.4%

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

      3. *-commutative 60.4%

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

   5. Applied egg-rr 60.4%

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

   6. Taylor expanded in z around inf 81.1%

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

      1. associate-/l* 85.0%

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

   8. Simplified 85.0%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{-27}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-123}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\frac{z + \frac{a}{\frac{z}{t}} \cdot -0.5}{z}}\\ \end{array} \]

Alternative 12: 78.1% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -6.7 \cdot 10^{-158}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{z + \frac{a}{\frac{z}{t}} \cdot -0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -6.7e-158)
   (* x (- y))
   (* y (* x (/ z (+ z (* (/ a (/ z t)) -0.5)))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6.7e-158) {
		tmp = x * -y;
	} else {
		tmp = y * (x * (z / (z + ((a / (z / t)) * -0.5))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-6.7d-158)) then
        tmp = x * -y
    else
        tmp = y * (x * (z / (z + ((a / (z / t)) * (-0.5d0)))))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6.7e-158) {
		tmp = x * -y;
	} else {
		tmp = y * (x * (z / (z + ((a / (z / t)) * -0.5))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -6.7e-158:
		tmp = x * -y
	else:
		tmp = y * (x * (z / (z + ((a / (z / t)) * -0.5))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -6.7e-158)
		tmp = Float64(x * Float64(-y));
	else
		tmp = Float64(y * Float64(x * Float64(z / Float64(z + Float64(Float64(a / Float64(z / t)) * -0.5)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -6.7e-158)
		tmp = x * -y;
	else
		tmp = y * (x * (z / (z + ((a / (z / t)) * -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -6.7e-158], N[(x * (-y)), $MachinePrecision], N[(y * N[(x * N[(z / N[(z + N[(N[(a / N[(z / t), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.7000000000000001e-158

   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* 53.8%

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

      2. associate-*r/ 56.6%

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

   3. Simplified 56.6%

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

   4. Taylor expanded in z around -inf 83.4%

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

      1. mul-1-neg 83.4%

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

      2. distribute-rgt-neg-out 83.4%

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

   6. Simplified 83.4%

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

   if -6.7000000000000001e-158 < z

   1. Initial program 63.4%

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

      1. *-commutative 63.4%

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

      2. associate-/l* 61.6%

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

   3. Simplified 61.6%

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

      1. associate-/r/ 63.6%

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

      2. associate-*r* 64.5%

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

   5. Applied egg-rr 64.5%

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

   6. Taylor expanded in z around inf 65.6%

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

      1. associate-/l* 68.0%

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

   8. Simplified 68.0%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.7 \cdot 10^{-158}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{z + \frac{a}{\frac{z}{t}} \cdot -0.5}\right)\\ \end{array} \]

Alternative 13: 78.3% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.14 \cdot 10^{-157}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\frac{z + \frac{a}{\frac{z}{t}} \cdot -0.5}{z}}\\ \end{array} \end{array} \]

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 -1.14e-157)
   (* x (- y))
   (/ (* x y) (/ (+ z (* (/ a (/ z t)) -0.5)) z))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.14e-157) {
		tmp = x * -y;
	} else {
		tmp = (x * y) / ((z + ((a / (z / t)) * -0.5)) / z);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.14d-157)) then
        tmp = x * -y
    else
        tmp = (x * y) / ((z + ((a / (z / t)) * (-0.5d0))) / z)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.14e-157) {
		tmp = x * -y;
	} else {
		tmp = (x * y) / ((z + ((a / (z / t)) * -0.5)) / z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.14e-157:
		tmp = x * -y
	else:
		tmp = (x * y) / ((z + ((a / (z / t)) * -0.5)) / z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.14e-157)
		tmp = Float64(x * Float64(-y));
	else
		tmp = Float64(Float64(x * y) / Float64(Float64(z + Float64(Float64(a / Float64(z / t)) * -0.5)) / z));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.14e-157)
		tmp = x * -y;
	else
		tmp = (x * y) / ((z + ((a / (z / t)) * -0.5)) / z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.14e-157], N[(x * (-y)), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(N[(z + N[(N[(a / N[(z / t), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.13999999999999996e-157

   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* 53.8%

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

      2. associate-*r/ 56.6%

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

   3. Simplified 56.6%

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

   4. Taylor expanded in z around -inf 83.4%

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

      1. mul-1-neg 83.4%

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

      2. distribute-rgt-neg-out 83.4%

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

   6. Simplified 83.4%

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

   if -1.13999999999999996e-157 < z

   1. Initial program 63.4%

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

      1. associate-*l* 63.6%

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

      2. associate-*r/ 65.0%

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

   3. Simplified 65.0%

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

      1. associate-/l* 64.1%

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

      2. associate-*r/ 63.0%

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

      3. *-commutative 63.0%

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

   5. Applied egg-rr 63.0%

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

   6. Taylor expanded in z around inf 66.1%

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

      1. associate-/l* 68.0%

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

   8. Simplified 68.6%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.14 \cdot 10^{-157}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\frac{z + \frac{a}{\frac{z}{t}} \cdot -0.5}{z}}\\ \end{array} \]

Alternative 14: 76.2% accurate, 10.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-158}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-68}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

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 -2.9e-158)
   (* x (- y))
   (if (<= z 3.6e-68) (/ (* x (* z y)) z) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.9e-158) {
		tmp = x * -y;
	} else if (z <= 3.6e-68) {
		tmp = (x * (z * y)) / z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-2.9d-158)) then
        tmp = x * -y
    else if (z <= 3.6d-68) then
        tmp = (x * (z * y)) / z
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.9e-158) {
		tmp = x * -y;
	} else if (z <= 3.6e-68) {
		tmp = (x * (z * y)) / z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.9e-158:
		tmp = x * -y
	elif z <= 3.6e-68:
		tmp = (x * (z * y)) / z
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.9e-158)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 3.6e-68)
		tmp = Float64(Float64(x * Float64(z * y)) / z);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.9e-158)
		tmp = x * -y;
	elseif (z <= 3.6e-68)
		tmp = (x * (z * y)) / z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.9e-158], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 3.6e-68], N[(N[(x * N[(z * y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.8999999999999998e-158

   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* 53.8%

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

      2. associate-*r/ 56.6%

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

   3. Simplified 56.6%

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

   4. Taylor expanded in z around -inf 83.4%

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

      1. mul-1-neg 83.4%

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

      2. distribute-rgt-neg-out 83.4%

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

   6. Simplified 83.4%

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

   if -2.8999999999999998e-158 < z < 3.60000000000000007e-68

   1. Initial program 74.4%

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

      1. associate-*l* 76.8%

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

   3. Simplified 76.8%

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

   4. Taylor expanded in z around inf 43.3%

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

   if 3.60000000000000007e-68 < 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* 54.0%

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

      2. associate-*r/ 56.4%

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

   3. Simplified 56.4%

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

   4. Taylor expanded in z around inf 85.8%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-158}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-68}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 15: 73.4% 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 59.5%

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

      1. associate-*l* 55.6%

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

      2. associate-*r/ 58.7%

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

   3. Simplified 58.7%

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

   4. Taylor expanded in z around -inf 74.4%

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

      1. mul-1-neg 74.4%

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

      2. distribute-rgt-neg-out 74.4%

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

   6. Simplified 74.4%

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

   if -1.999999999999994e-310 < z

   1. Initial program 61.9%

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

      1. associate-*l* 63.6%

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

      2. associate-*r/ 64.5%

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

   3. Simplified 64.5%

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

   4. Taylor expanded in z around inf 68.2%

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

4. Final simplification 71.2%

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

Alternative 16: 43.8% accurate, 37.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.8%

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

   1. associate-*l* 59.7%

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

   2. associate-*r/ 61.7%

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

3. Simplified 61.7%

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

4. Taylor expanded in z around inf 43.5%

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

5. Final simplification 43.5%

   \[\leadsto x \cdot y \]

Developer target: 89.5% 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 2023268 
(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)))))