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

Percentage Accurate: 61.1% → 90.5%

Time: 22.7s

Alternatives: 21

Speedup: 18.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := z \cdot z - a \cdot t\\ t_2 := \sqrt{t_1}\\ t_3 := \frac{a}{\frac{z \cdot z}{t}}\\ \mathbf{if}\;z \leq -4 \cdot 10^{+110}:\\ \;\;\;\;\frac{x \cdot y}{\mathsf{fma}\left(0.5, t_3, -1\right)}\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-150}:\\ \;\;\;\;\frac{z}{\frac{t_2}{x \cdot y}}\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-193}:\\ \;\;\;\;\left(x \cdot \left(z \cdot y\right)\right) \cdot {t_1}^{-0.5}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-90}:\\ \;\;\;\;y \cdot \left(z \cdot \frac{x}{t_2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\sqrt{1 - t_3}}\\ \end{array} \end{array} \]

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 (- (* z z) (* a t))) (t_2 (sqrt t_1)) (t_3 (/ a (/ (* z z) t))))
   (if (<= z -4e+110)
     (/ (* x y) (fma 0.5 t_3 -1.0))
     (if (<= z -1.8e-150)
       (/ z (/ t_2 (* x y)))
       (if (<= z 5.8e-193)
         (* (* x (* z y)) (pow t_1 -0.5))
         (if (<= z 1.6e-90)
           (* y (* z (/ x t_2)))
           (/ (* x y) (sqrt (- 1.0 t_3)))))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * z) - (a * t);
	double t_2 = sqrt(t_1);
	double t_3 = a / ((z * z) / t);
	double tmp;
	if (z <= -4e+110) {
		tmp = (x * y) / fma(0.5, t_3, -1.0);
	} else if (z <= -1.8e-150) {
		tmp = z / (t_2 / (x * y));
	} else if (z <= 5.8e-193) {
		tmp = (x * (z * y)) * pow(t_1, -0.5);
	} else if (z <= 1.6e-90) {
		tmp = y * (z * (x / t_2));
	} else {
		tmp = (x * y) / sqrt((1.0 - t_3));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * z) - Float64(a * t))
	t_2 = sqrt(t_1)
	t_3 = Float64(a / Float64(Float64(z * z) / t))
	tmp = 0.0
	if (z <= -4e+110)
		tmp = Float64(Float64(x * y) / fma(0.5, t_3, -1.0));
	elseif (z <= -1.8e-150)
		tmp = Float64(z / Float64(t_2 / Float64(x * y)));
	elseif (z <= 5.8e-193)
		tmp = Float64(Float64(x * Float64(z * y)) * (t_1 ^ -0.5));
	elseif (z <= 1.6e-90)
		tmp = Float64(y * Float64(z * Float64(x / t_2)));
	else
		tmp = Float64(Float64(x * y) / sqrt(Float64(1.0 - t_3)));
	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[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[(a / N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4e+110], N[(N[(x * y), $MachinePrecision] / N[(0.5 * t$95$3 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -1.8e-150], N[(z / N[(t$95$2 / N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.8e-193], N[(N[(x * N[(z * y), $MachinePrecision]), $MachinePrecision] * N[Power[t$95$1, -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.6e-90], N[(y * N[(z * N[(x / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -4.0000000000000001e110

   1. Initial program 29.3%

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

      1. associate-/l* 33.5%

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

   3. Simplified 33.5%

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

   4. Taylor expanded in z around -inf 93.9%

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

      1. fma-neg 93.9%

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

      2. unpow2 93.9%

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

      3. associate-/l* 94.3%

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

      4. metadata-eval 94.3%

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

   6. Simplified 94.3%

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

   if -4.0000000000000001e110 < z < -1.8000000000000001e-150

   1. Initial program 89.4%

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

      1. *-commutative 89.4%

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

      2. associate-*l* 84.0%

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

      3. associate-*r/ 86.4%

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

   3. Simplified 86.4%

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

      1. *-commutative 86.4%

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

      2. associate-*l/ 84.0%

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

      3. *-commutative 84.0%

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

      4. associate-*r* 89.4%

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

      5. associate-/l* 94.5%

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

   5. Applied egg-rr 94.5%

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

   if -1.8000000000000001e-150 < z < 5.80000000000000013e-193

   1. Initial program 72.3%

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

      1. associate-/l* 69.5%

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

   3. Simplified 69.5%

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

      1. associate-/l* 72.3%

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

      2. div-inv 72.2%

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

      3. associate-*l* 76.3%

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

      4. associate-*l* 76.3%

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

      5. *-commutative 76.3%

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

      6. pow1/2 76.3%

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

      7. pow-flip 76.4%

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

      8. metadata-eval 76.4%

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

   5. Applied egg-rr 76.4%

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

      1. associate-*r* 76.4%

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

      2. *-commutative 76.4%

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

   7. Simplified 76.4%

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

   if 5.80000000000000013e-193 < z < 1.60000000000000004e-90

   1. Initial program 88.0%

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

      1. *-commutative 88.0%

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

      2. associate-*l* 88.2%

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

      3. associate-*r/ 88.3%

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

   3. Simplified 88.3%

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

      1. associate-/l* 88.4%

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

      2. associate-/r/ 83.4%

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

   5. Applied egg-rr 83.4%

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

   if 1.60000000000000004e-90 < z

   1. Initial program 53.8%

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

      1. associate-/l* 61.0%

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

   3. Simplified 61.0%

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

      1. add-sqr-sqrt 61.0%

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

      2. sqrt-unprod 61.0%

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

      3. frac-times 55.3%

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

      4. add-sqr-sqrt 55.3%

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

   5. Applied egg-rr 55.3%

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

      1. div-sub 55.3%

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

      2. *-inverses 93.8%

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

      3. *-commutative 93.8%

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

      4. associate-/l* 98.2%

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

   7. Simplified 98.2%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\sqrt{1 - \frac{a}{\frac{z \cdot z}{t}}}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+110}:\\ \;\;\;\;\frac{x \cdot y}{\mathsf{fma}\left(0.5, \frac{a}{\frac{z \cdot z}{t}}, -1\right)}\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-150}:\\ \;\;\;\;\frac{z}{\frac{\sqrt{z \cdot z - a \cdot t}}{x \cdot y}}\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-193}:\\ \;\;\;\;\left(x \cdot \left(z \cdot y\right)\right) \cdot {\left(z \cdot z - a \cdot t\right)}^{-0.5}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-90}:\\ \;\;\;\;y \cdot \left(z \cdot \frac{x}{\sqrt{z \cdot z - a \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\sqrt{1 - \frac{a}{\frac{z \cdot z}{t}}}}\\ \end{array} \]

Alternative 2: 88.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -3.15 \cdot 10^{+71}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-91}:\\ \;\;\;\;y \cdot \left(z \cdot \frac{x}{\sqrt{z \cdot z - a \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}\\ \end{array} \end{array} \]

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 -3.15e+71)
   (* x (- y))
   (if (<= z 2e-91)
     (* y (* z (/ x (sqrt (- (* z z) (* a t))))))
     (/ (* x y) (/ (+ z (* -0.5 (/ a (/ z t)))) z)))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.15e+71) {
		tmp = x * -y;
	} else if (z <= 2e-91) {
		tmp = y * (z * (x / sqrt(((z * z) - (a * t)))));
	} else {
		tmp = (x * y) / ((z + (-0.5 * (a / (z / t)))) / 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 <= (-3.15d+71)) then
        tmp = x * -y
    else if (z <= 2d-91) then
        tmp = y * (z * (x / sqrt(((z * z) - (a * t)))))
    else
        tmp = (x * y) / ((z + ((-0.5d0) * (a / (z / t)))) / 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 <= -3.15e+71) {
		tmp = x * -y;
	} else if (z <= 2e-91) {
		tmp = y * (z * (x / Math.sqrt(((z * z) - (a * t)))));
	} else {
		tmp = (x * y) / ((z + (-0.5 * (a / (z / t)))) / z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.15e+71:
		tmp = x * -y
	elif z <= 2e-91:
		tmp = y * (z * (x / math.sqrt(((z * z) - (a * t)))))
	else:
		tmp = (x * y) / ((z + (-0.5 * (a / (z / t)))) / z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.15e+71)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 2e-91)
		tmp = Float64(y * Float64(z * Float64(x / sqrt(Float64(Float64(z * z) - Float64(a * t))))));
	else
		tmp = Float64(Float64(x * y) / Float64(Float64(z + Float64(-0.5 * Float64(a / Float64(z / t)))) / 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 <= -3.15e+71)
		tmp = x * -y;
	elseif (z <= 2e-91)
		tmp = y * (z * (x / sqrt(((z * z) - (a * t)))));
	else
		tmp = (x * y) / ((z + (-0.5 * (a / (z / t)))) / 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, -3.15e+71], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 2e-91], N[(y * N[(z * N[(x / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(N[(z + N[(-0.5 * N[(a / N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.15e71

   1. Initial program 36.6%

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

      1. *-commutative 36.6%

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

      2. associate-*l* 32.8%

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

      3. associate-*r/ 33.1%

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

   3. Simplified 33.1%

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

   4. Taylor expanded in z around -inf 95.1%

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

      1. neg-mul-1 95.1%

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

   6. Simplified 95.1%

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

   if -3.15e71 < z < 2.00000000000000004e-91

   1. Initial program 82.9%

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

      1. *-commutative 82.9%

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

      2. associate-*l* 83.6%

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

      3. associate-*r/ 82.3%

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

   3. Simplified 82.3%

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

      1. associate-/l* 83.8%

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

      2. associate-/r/ 81.3%

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

   5. Applied egg-rr 81.3%

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

   if 2.00000000000000004e-91 < z

   1. Initial program 53.8%

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

      1. associate-/l* 61.0%

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

   3. Simplified 61.0%

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

      1. pow1/2 61.0%

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

      2. pow-to-exp 57.9%

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

   5. Applied egg-rr 57.9%

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

   6. Taylor expanded in z around inf 92.3%

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

      1. associate-/l* 95.6%

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

   8. Simplified 95.6%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.15 \cdot 10^{+71}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-91}:\\ \;\;\;\;y \cdot \left(z \cdot \frac{x}{\sqrt{z \cdot z - a \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}\\ \end{array} \]

Alternative 3: 91.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+135}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+142}:\\ \;\;\;\;\frac{z}{\sqrt{z \cdot z - a \cdot t}} \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}\\ \end{array} \end{array} \]

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 -1e+135)
   (* x (- y))
   (if (<= z 4e+142)
     (* (/ z (sqrt (- (* z z) (* a t)))) (* x y))
     (/ (* x y) (/ (+ z (* -0.5 (/ a (/ z t)))) z)))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1e+135) {
		tmp = x * -y;
	} else if (z <= 4e+142) {
		tmp = (z / sqrt(((z * z) - (a * t)))) * (x * y);
	} else {
		tmp = (x * y) / ((z + (-0.5 * (a / (z / t)))) / 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 <= (-1d+135)) then
        tmp = x * -y
    else if (z <= 4d+142) then
        tmp = (z / sqrt(((z * z) - (a * t)))) * (x * y)
    else
        tmp = (x * y) / ((z + ((-0.5d0) * (a / (z / t)))) / 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 <= -1e+135) {
		tmp = x * -y;
	} else if (z <= 4e+142) {
		tmp = (z / Math.sqrt(((z * z) - (a * t)))) * (x * y);
	} else {
		tmp = (x * y) / ((z + (-0.5 * (a / (z / t)))) / z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1e+135:
		tmp = x * -y
	elif z <= 4e+142:
		tmp = (z / math.sqrt(((z * z) - (a * t)))) * (x * y)
	else:
		tmp = (x * y) / ((z + (-0.5 * (a / (z / t)))) / z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1e+135)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 4e+142)
		tmp = Float64(Float64(z / sqrt(Float64(Float64(z * z) - Float64(a * t)))) * Float64(x * y));
	else
		tmp = Float64(Float64(x * y) / Float64(Float64(z + Float64(-0.5 * Float64(a / Float64(z / t)))) / 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 <= -1e+135)
		tmp = x * -y;
	elseif (z <= 4e+142)
		tmp = (z / sqrt(((z * z) - (a * t)))) * (x * y);
	else
		tmp = (x * y) / ((z + (-0.5 * (a / (z / t)))) / 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, -1e+135], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 4e+142], N[(N[(z / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(N[(z + N[(-0.5 * N[(a / N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.99999999999999962e134

   1. Initial program 19.7%

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

      1. *-commutative 19.7%

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

      2. associate-*l* 19.3%

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

      3. associate-*r/ 19.6%

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

   3. Simplified 19.6%

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

   4. Taylor expanded in z around -inf 93.2%

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

      1. neg-mul-1 93.2%

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

   6. Simplified 93.2%

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

   if -9.99999999999999962e134 < z < 4.0000000000000002e142

   1. Initial program 84.6%

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

      1. associate-/l* 89.4%

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

   3. Simplified 89.4%

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

      1. clear-num 89.3%

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

      2. associate-/r/ 89.3%

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

      3. clear-num 89.4%

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

   5. Applied egg-rr 89.4%

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

   if 4.0000000000000002e142 < z

   1. Initial program 15.8%

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

      1. associate-/l* 20.6%

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

   3. Simplified 20.6%

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

      1. pow1/2 20.6%

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

      2. pow-to-exp 20.0%

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

   5. Applied egg-rr 20.0%

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

   6. Taylor expanded in z around inf 89.5%

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

      1. associate-/l* 96.4%

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

   8. Simplified 96.4%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+135}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+142}:\\ \;\;\;\;\frac{z}{\sqrt{z \cdot z - a \cdot t}} \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}\\ \end{array} \]

Alternative 4: 91.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+142}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+29}:\\ \;\;\;\;\frac{z}{\sqrt{z \cdot z - a \cdot t}} \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\sqrt{1 - \frac{a}{\frac{z \cdot z}{t}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2e+142)
   (* x (- y))
   (if (<= z 4e+29)
     (* (/ z (sqrt (- (* z z) (* a t)))) (* x y))
     (/ (* x y) (sqrt (- 1.0 (/ a (/ (* z z) t))))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2e+142) {
		tmp = x * -y;
	} else if (z <= 4e+29) {
		tmp = (z / sqrt(((z * z) - (a * t)))) * (x * y);
	} else {
		tmp = (x * y) / sqrt((1.0 - (a / ((z * z) / t))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-2d+142)) then
        tmp = x * -y
    else if (z <= 4d+29) then
        tmp = (z / sqrt(((z * z) - (a * t)))) * (x * y)
    else
        tmp = (x * y) / sqrt((1.0d0 - (a / ((z * z) / t))))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2e+142) {
		tmp = x * -y;
	} else if (z <= 4e+29) {
		tmp = (z / Math.sqrt(((z * z) - (a * t)))) * (x * y);
	} else {
		tmp = (x * y) / Math.sqrt((1.0 - (a / ((z * z) / t))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -2e+142:
		tmp = x * -y
	elif z <= 4e+29:
		tmp = (z / math.sqrt(((z * z) - (a * t)))) * (x * y)
	else:
		tmp = (x * y) / math.sqrt((1.0 - (a / ((z * z) / t))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2e+142)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 4e+29)
		tmp = Float64(Float64(z / sqrt(Float64(Float64(z * z) - Float64(a * t)))) * Float64(x * y));
	else
		tmp = Float64(Float64(x * y) / sqrt(Float64(1.0 - Float64(a / Float64(Float64(z * z) / t)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2e+142)
		tmp = x * -y;
	elseif (z <= 4e+29)
		tmp = (z / sqrt(((z * z) - (a * t)))) * (x * y);
	else
		tmp = (x * y) / sqrt((1.0 - (a / ((z * z) / t))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2e+142], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 4e+29], N[(N[(z / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[Sqrt[N[(1.0 - N[(a / N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.0000000000000001e142

   1. Initial program 19.7%

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

      1. *-commutative 19.7%

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

      2. associate-*l* 19.3%

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

      3. associate-*r/ 19.6%

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

   3. Simplified 19.6%

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

   4. Taylor expanded in z around -inf 93.2%

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

      1. neg-mul-1 93.2%

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

   6. Simplified 93.2%

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

   if -2.0000000000000001e142 < z < 3.99999999999999966e29

   1. Initial program 84.8%

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

      1. associate-/l* 87.4%

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

   3. Simplified 87.4%

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

      1. clear-num 87.3%

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

      2. associate-/r/ 87.4%

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

      3. clear-num 87.4%

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

   5. Applied egg-rr 87.4%

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

   if 3.99999999999999966e29 < z

   1. Initial program 44.6%

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

      1. associate-/l* 53.2%

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

   3. Simplified 53.2%

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

      1. add-sqr-sqrt 53.2%

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

      2. sqrt-unprod 53.2%

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

      3. frac-times 46.4%

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

      4. add-sqr-sqrt 46.4%

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

   5. Applied egg-rr 46.4%

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

      1. div-sub 46.4%

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

      2. *-inverses 92.6%

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

      3. *-commutative 92.6%

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

      4. associate-/l* 97.9%

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

   7. Simplified 97.9%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+142}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+29}:\\ \;\;\;\;\frac{z}{\sqrt{z \cdot z - a \cdot t}} \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\sqrt{1 - \frac{a}{\frac{z \cdot z}{t}}}}\\ \end{array} \]

Alternative 5: 91.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := \frac{a}{\frac{z \cdot z}{t}}\\ \mathbf{if}\;z \leq -2 \cdot 10^{+142}:\\ \;\;\;\;\frac{x \cdot y}{\mathsf{fma}\left(0.5, t_1, -1\right)}\\ \mathbf{elif}\;z \leq 10^{+30}:\\ \;\;\;\;\frac{z}{\sqrt{z \cdot z - a \cdot t}} \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\sqrt{1 - t_1}}\\ \end{array} \end{array} \]

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 (/ a (/ (* z z) t))))
   (if (<= z -2e+142)
     (/ (* x y) (fma 0.5 t_1 -1.0))
     (if (<= z 1e+30)
       (* (/ z (sqrt (- (* z z) (* a t)))) (* x y))
       (/ (* x y) (sqrt (- 1.0 t_1)))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = a / ((z * z) / t);
	double tmp;
	if (z <= -2e+142) {
		tmp = (x * y) / fma(0.5, t_1, -1.0);
	} else if (z <= 1e+30) {
		tmp = (z / sqrt(((z * z) - (a * t)))) * (x * y);
	} else {
		tmp = (x * y) / sqrt((1.0 - t_1));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = Float64(a / Float64(Float64(z * z) / t))
	tmp = 0.0
	if (z <= -2e+142)
		tmp = Float64(Float64(x * y) / fma(0.5, t_1, -1.0));
	elseif (z <= 1e+30)
		tmp = Float64(Float64(z / sqrt(Float64(Float64(z * z) - Float64(a * t)))) * Float64(x * y));
	else
		tmp = Float64(Float64(x * y) / sqrt(Float64(1.0 - t_1)));
	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[(a / N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2e+142], N[(N[(x * y), $MachinePrecision] / N[(0.5 * t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1e+30], N[(N[(z / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.0000000000000001e142

   1. Initial program 19.7%

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

      1. associate-/l* 20.2%

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

   3. Simplified 20.2%

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

   4. Taylor expanded in z around -inf 92.6%

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

      1. fma-neg 92.6%

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

      2. unpow2 92.6%

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

      3. associate-/l* 93.2%

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

      4. metadata-eval 93.2%

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

   6. Simplified 93.2%

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

   if -2.0000000000000001e142 < z < 1e30

   1. Initial program 84.8%

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

      1. associate-/l* 87.4%

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

   3. Simplified 87.4%

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

      1. clear-num 87.3%

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

      2. associate-/r/ 87.4%

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

      3. clear-num 87.4%

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

   5. Applied egg-rr 87.4%

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

   if 1e30 < z

   1. Initial program 44.6%

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

      1. associate-/l* 53.2%

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

   3. Simplified 53.2%

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

      1. add-sqr-sqrt 53.2%

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

      2. sqrt-unprod 53.2%

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

      3. frac-times 46.4%

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

      4. add-sqr-sqrt 46.4%

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

   5. Applied egg-rr 46.4%

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

      1. div-sub 46.4%

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

      2. *-inverses 92.6%

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

      3. *-commutative 92.6%

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

      4. associate-/l* 97.9%

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

   7. Simplified 97.9%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+142}:\\ \;\;\;\;\frac{x \cdot y}{\mathsf{fma}\left(0.5, \frac{a}{\frac{z \cdot z}{t}}, -1\right)}\\ \mathbf{elif}\;z \leq 10^{+30}:\\ \;\;\;\;\frac{z}{\sqrt{z \cdot z - a \cdot t}} \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\sqrt{1 - \frac{a}{\frac{z \cdot z}{t}}}}\\ \end{array} \]

Alternative 6: 82.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.18 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.65 \cdot 10^{-127}:\\ \;\;\;\;y \cdot \left(z \cdot \frac{x}{\sqrt{a \cdot \left(-t\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}\\ \end{array} \end{array} \]

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 -1.18e-15)
   (* x (- y))
   (if (<= z 2.65e-127)
     (* y (* z (/ x (sqrt (* a (- t))))))
     (/ (* x y) (/ (+ z (* -0.5 (/ a (/ z t)))) z)))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.18e-15) {
		tmp = x * -y;
	} else if (z <= 2.65e-127) {
		tmp = y * (z * (x / sqrt((a * -t))));
	} else {
		tmp = (x * y) / ((z + (-0.5 * (a / (z / t)))) / 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 <= (-1.18d-15)) then
        tmp = x * -y
    else if (z <= 2.65d-127) then
        tmp = y * (z * (x / sqrt((a * -t))))
    else
        tmp = (x * y) / ((z + ((-0.5d0) * (a / (z / t)))) / 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 <= -1.18e-15) {
		tmp = x * -y;
	} else if (z <= 2.65e-127) {
		tmp = y * (z * (x / Math.sqrt((a * -t))));
	} else {
		tmp = (x * y) / ((z + (-0.5 * (a / (z / t)))) / z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.18e-15:
		tmp = x * -y
	elif z <= 2.65e-127:
		tmp = y * (z * (x / math.sqrt((a * -t))))
	else:
		tmp = (x * y) / ((z + (-0.5 * (a / (z / t)))) / z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.18e-15)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 2.65e-127)
		tmp = Float64(y * Float64(z * Float64(x / sqrt(Float64(a * Float64(-t))))));
	else
		tmp = Float64(Float64(x * y) / Float64(Float64(z + Float64(-0.5 * Float64(a / Float64(z / t)))) / 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 <= -1.18e-15)
		tmp = x * -y;
	elseif (z <= 2.65e-127)
		tmp = y * (z * (x / sqrt((a * -t))));
	else
		tmp = (x * y) / ((z + (-0.5 * (a / (z / t)))) / 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, -1.18e-15], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 2.65e-127], N[(y * N[(z * N[(x / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(N[(z + N[(-0.5 * N[(a / N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.18000000000000004e-15

   1. Initial program 53.4%

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

      1. *-commutative 53.4%

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

      2. associate-*l* 50.6%

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

      3. associate-*r/ 50.9%

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

   3. Simplified 50.9%

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

   4. Taylor expanded in z around -inf 87.2%

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

      1. neg-mul-1 87.2%

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

   6. Simplified 87.2%

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

   if -1.18000000000000004e-15 < z < 2.6500000000000001e-127

   1. Initial program 80.1%

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

      1. *-commutative 80.1%

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

      2. associate-*l* 81.2%

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

      3. associate-*r/ 79.2%

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

   3. Simplified 79.2%

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

      1. associate-/l* 81.3%

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

      2. associate-/r/ 77.7%

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

   5. Applied egg-rr 77.7%

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

   6. Taylor expanded in z around 0 69.0%

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

      1. associate-*r* 69.0%

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

      2. neg-mul-1 69.0%

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

   8. Simplified 69.0%

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

   if 2.6500000000000001e-127 < z

   1. Initial program 55.6%

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

      1. associate-/l* 62.3%

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

   3. Simplified 62.3%

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

      1. pow1/2 62.3%

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

      2. pow-to-exp 59.2%

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

   5. Applied egg-rr 59.2%

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

   6. Taylor expanded in z around inf 89.9%

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

      1. associate-/l* 93.0%

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

   8. Simplified 93.0%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.18 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.65 \cdot 10^{-127}:\\ \;\;\;\;y \cdot \left(z \cdot \frac{x}{\sqrt{a \cdot \left(-t\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}\\ \end{array} \]

Alternative 7: 82.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{-16}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{-127}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot x\right)}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}\\ \end{array} \end{array} \]

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 -5.5e-16)
   (* x (- y))
   (if (<= z 1.32e-127)
     (/ (* y (* z x)) (sqrt (* a (- t))))
     (/ (* x y) (/ (+ z (* -0.5 (/ a (/ z t)))) z)))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.5e-16) {
		tmp = x * -y;
	} else if (z <= 1.32e-127) {
		tmp = (y * (z * x)) / sqrt((a * -t));
	} else {
		tmp = (x * y) / ((z + (-0.5 * (a / (z / t)))) / 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 <= (-5.5d-16)) then
        tmp = x * -y
    else if (z <= 1.32d-127) then
        tmp = (y * (z * x)) / sqrt((a * -t))
    else
        tmp = (x * y) / ((z + ((-0.5d0) * (a / (z / t)))) / 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 <= -5.5e-16) {
		tmp = x * -y;
	} else if (z <= 1.32e-127) {
		tmp = (y * (z * x)) / Math.sqrt((a * -t));
	} else {
		tmp = (x * y) / ((z + (-0.5 * (a / (z / t)))) / z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -5.5e-16:
		tmp = x * -y
	elif z <= 1.32e-127:
		tmp = (y * (z * x)) / math.sqrt((a * -t))
	else:
		tmp = (x * y) / ((z + (-0.5 * (a / (z / t)))) / z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.5e-16)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 1.32e-127)
		tmp = Float64(Float64(y * Float64(z * x)) / sqrt(Float64(a * Float64(-t))));
	else
		tmp = Float64(Float64(x * y) / Float64(Float64(z + Float64(-0.5 * Float64(a / Float64(z / t)))) / 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 <= -5.5e-16)
		tmp = x * -y;
	elseif (z <= 1.32e-127)
		tmp = (y * (z * x)) / sqrt((a * -t));
	else
		tmp = (x * y) / ((z + (-0.5 * (a / (z / t)))) / 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, -5.5e-16], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 1.32e-127], N[(N[(y * N[(z * x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(N[(z + N[(-0.5 * N[(a / N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.49999999999999964e-16

   1. Initial program 53.4%

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

      1. *-commutative 53.4%

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

      2. associate-*l* 50.6%

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

      3. associate-*r/ 50.9%

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

   3. Simplified 50.9%

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

   4. Taylor expanded in z around -inf 87.2%

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

      1. neg-mul-1 87.2%

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

   6. Simplified 87.2%

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

   if -5.49999999999999964e-16 < z < 1.32e-127

   1. Initial program 80.1%

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

   2. Taylor expanded in z around 0 72.8%

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

      1. associate-*r* 69.0%

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

      2. neg-mul-1 69.0%

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

   4. Simplified 72.8%

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

   5. Taylor expanded in x around 0 75.1%

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

   if 1.32e-127 < z

   1. Initial program 55.6%

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

      1. associate-/l* 62.3%

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

   3. Simplified 62.3%

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

      1. pow1/2 62.3%

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

      2. pow-to-exp 59.2%

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

   5. Applied egg-rr 59.2%

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

   6. Taylor expanded in z around inf 89.9%

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

      1. associate-/l* 93.0%

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

   8. Simplified 93.0%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{-16}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{-127}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot x\right)}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}\\ \end{array} \]

Alternative 8: 76.6% accurate, 4.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -0.002:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq -6.2 \cdot 10^{-81}:\\ \;\;\;\;z \cdot \frac{x \cdot y}{\frac{-0.5}{\frac{\frac{z}{t}}{a}} - z}\\ \mathbf{elif}\;z \leq -3.6 \cdot 10^{-220}:\\ \;\;\;\;\frac{z \cdot \left(x \cdot y\right)}{-z}\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-136}:\\ \;\;\;\;z \cdot \frac{x \cdot y}{z + -0.5 \cdot \frac{a \cdot t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -0.002)
   (* x (- y))
   (if (<= z -6.2e-81)
     (* z (/ (* x y) (- (/ -0.5 (/ (/ z t) a)) z)))
     (if (<= z -3.6e-220)
       (/ (* z (* x y)) (- z))
       (if (<= z 2.1e-136)
         (* z (/ (* x y) (+ z (* -0.5 (/ (* a t) z)))))
         (* x y))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -0.002) {
		tmp = x * -y;
	} else if (z <= -6.2e-81) {
		tmp = z * ((x * y) / ((-0.5 / ((z / t) / a)) - z));
	} else if (z <= -3.6e-220) {
		tmp = (z * (x * y)) / -z;
	} else if (z <= 2.1e-136) {
		tmp = z * ((x * y) / (z + (-0.5 * ((a * t) / z))));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-0.002d0)) then
        tmp = x * -y
    else if (z <= (-6.2d-81)) then
        tmp = z * ((x * y) / (((-0.5d0) / ((z / t) / a)) - z))
    else if (z <= (-3.6d-220)) then
        tmp = (z * (x * y)) / -z
    else if (z <= 2.1d-136) then
        tmp = z * ((x * y) / (z + ((-0.5d0) * ((a * t) / z))))
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -0.002) {
		tmp = x * -y;
	} else if (z <= -6.2e-81) {
		tmp = z * ((x * y) / ((-0.5 / ((z / t) / a)) - z));
	} else if (z <= -3.6e-220) {
		tmp = (z * (x * y)) / -z;
	} else if (z <= 2.1e-136) {
		tmp = z * ((x * y) / (z + (-0.5 * ((a * t) / z))));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -0.002:
		tmp = x * -y
	elif z <= -6.2e-81:
		tmp = z * ((x * y) / ((-0.5 / ((z / t) / a)) - z))
	elif z <= -3.6e-220:
		tmp = (z * (x * y)) / -z
	elif z <= 2.1e-136:
		tmp = z * ((x * y) / (z + (-0.5 * ((a * t) / z))))
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -0.002)
		tmp = Float64(x * Float64(-y));
	elseif (z <= -6.2e-81)
		tmp = Float64(z * Float64(Float64(x * y) / Float64(Float64(-0.5 / Float64(Float64(z / t) / a)) - z)));
	elseif (z <= -3.6e-220)
		tmp = Float64(Float64(z * Float64(x * y)) / Float64(-z));
	elseif (z <= 2.1e-136)
		tmp = Float64(z * Float64(Float64(x * y) / Float64(z + Float64(-0.5 * Float64(Float64(a * t) / z)))));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -0.002)
		tmp = x * -y;
	elseif (z <= -6.2e-81)
		tmp = z * ((x * y) / ((-0.5 / ((z / t) / a)) - z));
	elseif (z <= -3.6e-220)
		tmp = (z * (x * y)) / -z;
	elseif (z <= 2.1e-136)
		tmp = z * ((x * y) / (z + (-0.5 * ((a * t) / z))));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -0.002], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, -6.2e-81], N[(z * N[(N[(x * y), $MachinePrecision] / N[(N[(-0.5 / N[(N[(z / t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -3.6e-220], N[(N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision] / (-z)), $MachinePrecision], If[LessEqual[z, 2.1e-136], N[(z * N[(N[(x * y), $MachinePrecision] / N[(z + N[(-0.5 * N[(N[(a * t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -2e-3

   1. Initial program 49.5%

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

      1. *-commutative 49.5%

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

      2. associate-*l* 46.4%

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

      3. associate-*r/ 46.7%

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

   3. Simplified 46.7%

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

   4. Taylor expanded in z around -inf 89.7%

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

      1. neg-mul-1 89.7%

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

   6. Simplified 89.7%

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

   if -2e-3 < z < -6.19999999999999976e-81

   1. Initial program 79.0%

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

      1. associate-*l/ 89.0%

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

   3. Simplified 89.0%

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

   4. Taylor expanded in z around -inf 79.1%

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

      1. frac-2neg 79.1%

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

      2. distribute-rgt-neg-out 79.1%

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

      3. div-inv 79.0%

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

      4. add-sqr-sqrt 55.5%

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

      5. sqrt-unprod 61.9%

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

      6. sqr-neg 61.9%

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

      7. sqrt-unprod 16.8%

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

      8. add-sqr-sqrt 40.3%

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

      9. fma-def 40.3%

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

      10. mul-1-neg 40.3%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-prod 78.0%

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

      13. sqr-neg 78.0%

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

      14. mul-1-neg 78.0%

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

      15. mul-1-neg 78.0%

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

      16. sqrt-unprod 78.0%

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

      17. add-sqr-sqrt 78.0%

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

      18. fma-neg 78.0%

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

   6. Applied egg-rr 78.0%

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

      1. associate-*r/ 78.1%

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

      2. *-rgt-identity 78.1%

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

      3. *-commutative 78.1%

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

      4. neg-sub0 78.1%

         \[\leadsto \frac{y \cdot x}{\color{blue}{0 - \mathsf{fma}\left(0.5, \frac{a}{\frac{z}{t}}, z\right)}} \cdot z \]

      5. fma-udef 78.1%

         \[\leadsto \frac{y \cdot x}{0 - \color{blue}{\left(0.5 \cdot \frac{a}{\frac{z}{t}} + z\right)}} \cdot z \]

      6. associate-/l* 78.1%

         \[\leadsto \frac{y \cdot x}{0 - \left(0.5 \cdot \color{blue}{\frac{a \cdot t}{z}} + z\right)} \cdot z \]

      7. associate-*r/ 78.1%

         \[\leadsto \frac{y \cdot x}{0 - \left(\color{blue}{\frac{0.5 \cdot \left(a \cdot t\right)}{z}} + z\right)} \cdot z \]

      8. *-commutative 78.1%

         \[\leadsto \frac{y \cdot x}{0 - \left(\frac{0.5 \cdot \color{blue}{\left(t \cdot a\right)}}{z} + z\right)} \cdot z \]

      9. associate--r+ 78.1%

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

      10. neg-sub0 78.1%

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

      11. associate-/l* 78.1%

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

      12. distribute-neg-frac 78.1%

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

      13. metadata-eval 78.1%

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

      14. associate-/r* 78.1%

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

   8. Simplified 78.1%

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

   if -6.19999999999999976e-81 < z < -3.60000000000000021e-220

   1. Initial program 79.8%

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

   2. Taylor expanded in z around -inf 54.9%

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

      1. neg-mul-1 54.9%

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

   4. Simplified 54.9%

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

   if -3.60000000000000021e-220 < z < 2.0999999999999999e-136

   1. Initial program 84.3%

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

      1. associate-*l/ 77.9%

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

   3. Simplified 77.9%

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

   4. Taylor expanded in z around inf 41.2%

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

   if 2.0999999999999999e-136 < z

   1. Initial program 55.1%

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

      1. *-commutative 55.1%

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

      2. associate-*l* 53.8%

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

      3. associate-*r/ 58.5%

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

   3. Simplified 58.5%

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

   4. Taylor expanded in z around inf 92.1%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.002:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq -6.2 \cdot 10^{-81}:\\ \;\;\;\;z \cdot \frac{x \cdot y}{\frac{-0.5}{\frac{\frac{z}{t}}{a}} - z}\\ \mathbf{elif}\;z \leq -3.6 \cdot 10^{-220}:\\ \;\;\;\;\frac{z \cdot \left(x \cdot y\right)}{-z}\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-136}:\\ \;\;\;\;z \cdot \frac{x \cdot y}{z + -0.5 \cdot \frac{a \cdot t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 9: 77.6% accurate, 5.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -0.058:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq -5 \cdot 10^{-79}:\\ \;\;\;\;z \cdot \frac{x \cdot y}{\frac{-0.5}{\frac{\frac{z}{t}}{a}} - z}\\ \mathbf{elif}\;z \leq -2.7 \cdot 10^{-224}:\\ \;\;\;\;\frac{z \cdot \left(x \cdot y\right)}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}\\ \end{array} \end{array} \]

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 -0.058)
   (* x (- y))
   (if (<= z -5e-79)
     (* z (/ (* x y) (- (/ -0.5 (/ (/ z t) a)) z)))
     (if (<= z -2.7e-224)
       (/ (* z (* x y)) (- z))
       (/ (* x y) (/ (+ z (* -0.5 (/ a (/ z t)))) z))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -0.058) {
		tmp = x * -y;
	} else if (z <= -5e-79) {
		tmp = z * ((x * y) / ((-0.5 / ((z / t) / a)) - z));
	} else if (z <= -2.7e-224) {
		tmp = (z * (x * y)) / -z;
	} else {
		tmp = (x * y) / ((z + (-0.5 * (a / (z / t)))) / 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 <= (-0.058d0)) then
        tmp = x * -y
    else if (z <= (-5d-79)) then
        tmp = z * ((x * y) / (((-0.5d0) / ((z / t) / a)) - z))
    else if (z <= (-2.7d-224)) then
        tmp = (z * (x * y)) / -z
    else
        tmp = (x * y) / ((z + ((-0.5d0) * (a / (z / t)))) / 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 <= -0.058) {
		tmp = x * -y;
	} else if (z <= -5e-79) {
		tmp = z * ((x * y) / ((-0.5 / ((z / t) / a)) - z));
	} else if (z <= -2.7e-224) {
		tmp = (z * (x * y)) / -z;
	} else {
		tmp = (x * y) / ((z + (-0.5 * (a / (z / t)))) / z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -0.058:
		tmp = x * -y
	elif z <= -5e-79:
		tmp = z * ((x * y) / ((-0.5 / ((z / t) / a)) - z))
	elif z <= -2.7e-224:
		tmp = (z * (x * y)) / -z
	else:
		tmp = (x * y) / ((z + (-0.5 * (a / (z / t)))) / z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -0.058)
		tmp = Float64(x * Float64(-y));
	elseif (z <= -5e-79)
		tmp = Float64(z * Float64(Float64(x * y) / Float64(Float64(-0.5 / Float64(Float64(z / t) / a)) - z)));
	elseif (z <= -2.7e-224)
		tmp = Float64(Float64(z * Float64(x * y)) / Float64(-z));
	else
		tmp = Float64(Float64(x * y) / Float64(Float64(z + Float64(-0.5 * Float64(a / Float64(z / t)))) / 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 <= -0.058)
		tmp = x * -y;
	elseif (z <= -5e-79)
		tmp = z * ((x * y) / ((-0.5 / ((z / t) / a)) - z));
	elseif (z <= -2.7e-224)
		tmp = (z * (x * y)) / -z;
	else
		tmp = (x * y) / ((z + (-0.5 * (a / (z / t)))) / 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, -0.058], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, -5e-79], N[(z * N[(N[(x * y), $MachinePrecision] / N[(N[(-0.5 / N[(N[(z / t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -2.7e-224], N[(N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision] / (-z)), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(N[(z + N[(-0.5 * N[(a / N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -0.0580000000000000029

   1. Initial program 49.5%

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

      1. *-commutative 49.5%

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

      2. associate-*l* 46.4%

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

      3. associate-*r/ 46.7%

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

   3. Simplified 46.7%

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

   4. Taylor expanded in z around -inf 89.7%

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

      1. neg-mul-1 89.7%

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

   6. Simplified 89.7%

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

   if -0.0580000000000000029 < z < -4.99999999999999999e-79

   1. Initial program 79.0%

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

      1. associate-*l/ 89.0%

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

   3. Simplified 89.0%

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

   4. Taylor expanded in z around -inf 79.1%

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

      1. frac-2neg 79.1%

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

      2. distribute-rgt-neg-out 79.1%

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

      3. div-inv 79.0%

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

      4. add-sqr-sqrt 55.5%

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

      5. sqrt-unprod 61.9%

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

      6. sqr-neg 61.9%

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

      7. sqrt-unprod 16.8%

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

      8. add-sqr-sqrt 40.3%

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

      9. fma-def 40.3%

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

      10. mul-1-neg 40.3%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-prod 78.0%

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

      13. sqr-neg 78.0%

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

      14. mul-1-neg 78.0%

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

      15. mul-1-neg 78.0%

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

      16. sqrt-unprod 78.0%

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

      17. add-sqr-sqrt 78.0%

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

      18. fma-neg 78.0%

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

   6. Applied egg-rr 78.0%

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

      1. associate-*r/ 78.1%

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

      2. *-rgt-identity 78.1%

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

      3. *-commutative 78.1%

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

      4. neg-sub0 78.1%

         \[\leadsto \frac{y \cdot x}{\color{blue}{0 - \mathsf{fma}\left(0.5, \frac{a}{\frac{z}{t}}, z\right)}} \cdot z \]

      5. fma-udef 78.1%

         \[\leadsto \frac{y \cdot x}{0 - \color{blue}{\left(0.5 \cdot \frac{a}{\frac{z}{t}} + z\right)}} \cdot z \]

      6. associate-/l* 78.1%

         \[\leadsto \frac{y \cdot x}{0 - \left(0.5 \cdot \color{blue}{\frac{a \cdot t}{z}} + z\right)} \cdot z \]

      7. associate-*r/ 78.1%

         \[\leadsto \frac{y \cdot x}{0 - \left(\color{blue}{\frac{0.5 \cdot \left(a \cdot t\right)}{z}} + z\right)} \cdot z \]

      8. *-commutative 78.1%

         \[\leadsto \frac{y \cdot x}{0 - \left(\frac{0.5 \cdot \color{blue}{\left(t \cdot a\right)}}{z} + z\right)} \cdot z \]

      9. associate--r+ 78.1%

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

      10. neg-sub0 78.1%

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

      11. associate-/l* 78.1%

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

      12. distribute-neg-frac 78.1%

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

      13. metadata-eval 78.1%

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

      14. associate-/r* 78.1%

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

   8. Simplified 78.1%

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

   if -4.99999999999999999e-79 < z < -2.69999999999999998e-224

   1. Initial program 79.8%

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

   2. Taylor expanded in z around -inf 54.9%

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

      1. neg-mul-1 54.9%

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

   4. Simplified 54.9%

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

   if -2.69999999999999998e-224 < z

   1. Initial program 63.1%

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

      1. associate-/l* 67.2%

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

   3. Simplified 67.2%

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

      1. pow1/2 67.2%

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

      2. pow-to-exp 64.3%

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

   5. Applied egg-rr 64.3%

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

   6. Taylor expanded in z around inf 75.8%

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

      1. associate-/l* 78.0%

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

   8. Simplified 78.0%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.058:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq -5 \cdot 10^{-79}:\\ \;\;\;\;z \cdot \frac{x \cdot y}{\frac{-0.5}{\frac{\frac{z}{t}}{a}} - z}\\ \mathbf{elif}\;z \leq -2.7 \cdot 10^{-224}:\\ \;\;\;\;\frac{z \cdot \left(x \cdot y\right)}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}\\ \end{array} \]

Alternative 10: 77.9% accurate, 5.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -26500:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq -9.6 \cdot 10^{-94}:\\ \;\;\;\;y \cdot \left(z \cdot \frac{x}{0.5 \cdot \frac{a \cdot t}{z} - z}\right)\\ \mathbf{elif}\;z \leq -3 \cdot 10^{-222}:\\ \;\;\;\;\frac{z \cdot \left(x \cdot y\right)}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}\\ \end{array} \end{array} \]

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 -26500.0)
   (* x (- y))
   (if (<= z -9.6e-94)
     (* y (* z (/ x (- (* 0.5 (/ (* a t) z)) z))))
     (if (<= z -3e-222)
       (/ (* z (* x y)) (- z))
       (/ (* x y) (/ (+ z (* -0.5 (/ a (/ z t)))) z))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -26500.0) {
		tmp = x * -y;
	} else if (z <= -9.6e-94) {
		tmp = y * (z * (x / ((0.5 * ((a * t) / z)) - z)));
	} else if (z <= -3e-222) {
		tmp = (z * (x * y)) / -z;
	} else {
		tmp = (x * y) / ((z + (-0.5 * (a / (z / t)))) / 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 <= (-26500.0d0)) then
        tmp = x * -y
    else if (z <= (-9.6d-94)) then
        tmp = y * (z * (x / ((0.5d0 * ((a * t) / z)) - z)))
    else if (z <= (-3d-222)) then
        tmp = (z * (x * y)) / -z
    else
        tmp = (x * y) / ((z + ((-0.5d0) * (a / (z / t)))) / 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 <= -26500.0) {
		tmp = x * -y;
	} else if (z <= -9.6e-94) {
		tmp = y * (z * (x / ((0.5 * ((a * t) / z)) - z)));
	} else if (z <= -3e-222) {
		tmp = (z * (x * y)) / -z;
	} else {
		tmp = (x * y) / ((z + (-0.5 * (a / (z / t)))) / z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -26500.0:
		tmp = x * -y
	elif z <= -9.6e-94:
		tmp = y * (z * (x / ((0.5 * ((a * t) / z)) - z)))
	elif z <= -3e-222:
		tmp = (z * (x * y)) / -z
	else:
		tmp = (x * y) / ((z + (-0.5 * (a / (z / t)))) / z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -26500.0)
		tmp = Float64(x * Float64(-y));
	elseif (z <= -9.6e-94)
		tmp = Float64(y * Float64(z * Float64(x / Float64(Float64(0.5 * Float64(Float64(a * t) / z)) - z))));
	elseif (z <= -3e-222)
		tmp = Float64(Float64(z * Float64(x * y)) / Float64(-z));
	else
		tmp = Float64(Float64(x * y) / Float64(Float64(z + Float64(-0.5 * Float64(a / Float64(z / t)))) / 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 <= -26500.0)
		tmp = x * -y;
	elseif (z <= -9.6e-94)
		tmp = y * (z * (x / ((0.5 * ((a * t) / z)) - z)));
	elseif (z <= -3e-222)
		tmp = (z * (x * y)) / -z;
	else
		tmp = (x * y) / ((z + (-0.5 * (a / (z / t)))) / 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, -26500.0], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, -9.6e-94], N[(y * N[(z * N[(x / N[(N[(0.5 * N[(N[(a * t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -3e-222], N[(N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision] / (-z)), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(N[(z + N[(-0.5 * N[(a / N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -26500

   1. Initial program 48.8%

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

      1. *-commutative 48.8%

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

      2. associate-*l* 45.7%

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

      3. associate-*r/ 45.9%

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

   3. Simplified 45.9%

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

   4. Taylor expanded in z around -inf 90.9%

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

      1. neg-mul-1 90.9%

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

   6. Simplified 90.9%

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

   if -26500 < z < -9.6e-94

   1. Initial program 82.7%

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

      1. *-commutative 82.7%

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

      2. associate-*l* 82.7%

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

      3. associate-*r/ 88.7%

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

   3. Simplified 88.7%

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

      1. associate-/l* 91.1%

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

      2. associate-/r/ 91.1%

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

   5. Applied egg-rr 91.1%

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

   6. Taylor expanded in z around -inf 75.0%

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

   if -9.6e-94 < z < -3.0000000000000003e-222

   1. Initial program 77.2%

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

   2. Taylor expanded in z around -inf 52.6%

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

      1. neg-mul-1 52.6%

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

   4. Simplified 52.6%

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

   if -3.0000000000000003e-222 < z

   1. Initial program 63.1%

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

      1. associate-/l* 67.2%

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

   3. Simplified 67.2%

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

      1. pow1/2 67.2%

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

      2. pow-to-exp 64.3%

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

   5. Applied egg-rr 64.3%

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

   6. Taylor expanded in z around inf 75.8%

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

      1. associate-/l* 78.0%

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

   8. Simplified 78.0%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -26500:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq -9.6 \cdot 10^{-94}:\\ \;\;\;\;y \cdot \left(z \cdot \frac{x}{0.5 \cdot \frac{a \cdot t}{z} - z}\right)\\ \mathbf{elif}\;z \leq -3 \cdot 10^{-222}:\\ \;\;\;\;\frac{z \cdot \left(x \cdot y\right)}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}\\ \end{array} \]

Alternative 11: 75.9% accurate, 5.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{-218}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{-191}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z + -0.5 \cdot \frac{a \cdot t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.75e-218)
   (* x (- y))
   (if (<= z 2.55e-191)
     (* y (/ (* z x) (+ z (* -0.5 (/ (* a t) z)))))
     (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.75e-218) {
		tmp = x * -y;
	} else if (z <= 2.55e-191) {
		tmp = y * ((z * x) / (z + (-0.5 * ((a * t) / z))));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.75d-218)) then
        tmp = x * -y
    else if (z <= 2.55d-191) then
        tmp = y * ((z * x) / (z + ((-0.5d0) * ((a * t) / z))))
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.75e-218) {
		tmp = x * -y;
	} else if (z <= 2.55e-191) {
		tmp = y * ((z * x) / (z + (-0.5 * ((a * t) / z))));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.75e-218:
		tmp = x * -y
	elif z <= 2.55e-191:
		tmp = y * ((z * x) / (z + (-0.5 * ((a * t) / z))))
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.75e-218)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 2.55e-191)
		tmp = Float64(y * Float64(Float64(z * x) / Float64(z + Float64(-0.5 * Float64(Float64(a * t) / z)))));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.75e-218)
		tmp = x * -y;
	elseif (z <= 2.55e-191)
		tmp = y * ((z * x) / (z + (-0.5 * ((a * t) / z))));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.75e-218], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 2.55e-191], N[(y * N[(N[(z * x), $MachinePrecision] / N[(z + N[(-0.5 * N[(N[(a * t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.75e-218

   1. Initial program 60.8%

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

      1. *-commutative 60.8%

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

      2. associate-*l* 57.3%

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

      3. associate-*r/ 57.8%

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

   3. Simplified 57.8%

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

   4. Taylor expanded in z around -inf 75.3%

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

      1. neg-mul-1 75.3%

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

   6. Simplified 75.3%

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

   if -1.75e-218 < z < 2.5500000000000001e-191

   1. Initial program 79.3%

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

      1. *-commutative 79.3%

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

      2. associate-*l* 88.4%

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

      3. associate-*r/ 81.8%

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

   3. Simplified 81.8%

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

   4. Taylor expanded in z around inf 44.7%

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

   if 2.5500000000000001e-191 < z

   1. Initial program 59.1%

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

      1. *-commutative 59.1%

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

      2. associate-*l* 57.2%

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

      3. associate-*r/ 61.4%

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

   3. Simplified 61.4%

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

   4. Taylor expanded in z around inf 86.2%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{-218}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{-191}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z + -0.5 \cdot \frac{a \cdot t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 12: 76.3% accurate, 5.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-216}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-135}:\\ \;\;\;\;z \cdot \frac{x \cdot y}{z + -0.5 \cdot \frac{a \cdot t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.9e-216)
   (* x (- y))
   (if (<= z 6.2e-135)
     (* z (/ (* x y) (+ z (* -0.5 (/ (* a t) z)))))
     (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.9e-216) {
		tmp = x * -y;
	} else if (z <= 6.2e-135) {
		tmp = z * ((x * y) / (z + (-0.5 * ((a * t) / z))));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-2.9d-216)) then
        tmp = x * -y
    else if (z <= 6.2d-135) then
        tmp = z * ((x * y) / (z + ((-0.5d0) * ((a * t) / z))))
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.9e-216) {
		tmp = x * -y;
	} else if (z <= 6.2e-135) {
		tmp = z * ((x * y) / (z + (-0.5 * ((a * t) / z))));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.9e-216:
		tmp = x * -y
	elif z <= 6.2e-135:
		tmp = z * ((x * y) / (z + (-0.5 * ((a * t) / z))))
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.9e-216)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 6.2e-135)
		tmp = Float64(z * Float64(Float64(x * y) / Float64(z + Float64(-0.5 * Float64(Float64(a * t) / z)))));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.9e-216)
		tmp = x * -y;
	elseif (z <= 6.2e-135)
		tmp = z * ((x * y) / (z + (-0.5 * ((a * t) / z))));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.9e-216], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 6.2e-135], N[(z * N[(N[(x * y), $MachinePrecision] / N[(z + N[(-0.5 * N[(N[(a * t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.9000000000000001e-216

   1. Initial program 60.8%

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

      1. *-commutative 60.8%

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

      2. associate-*l* 57.3%

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

      3. associate-*r/ 57.8%

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

   3. Simplified 57.8%

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

   4. Taylor expanded in z around -inf 75.3%

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

      1. neg-mul-1 75.3%

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

   6. Simplified 75.3%

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

   if -2.9000000000000001e-216 < z < 6.2000000000000001e-135

   1. Initial program 84.3%

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

      1. associate-*l/ 77.9%

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

   3. Simplified 77.9%

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

   4. Taylor expanded in z around inf 41.2%

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

   if 6.2000000000000001e-135 < z

   1. Initial program 55.1%

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

      1. *-commutative 55.1%

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

      2. associate-*l* 53.8%

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

      3. associate-*r/ 58.5%

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

   3. Simplified 58.5%

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

   4. Taylor expanded in z around inf 92.1%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-216}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-135}:\\ \;\;\;\;z \cdot \frac{x \cdot y}{z + -0.5 \cdot \frac{a \cdot t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 13: 76.9% accurate, 5.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -380000:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{-192}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{0.5 \cdot \frac{a \cdot t}{z} - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

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 -380000.0)
   (* x (- y))
   (if (<= z 2.55e-192)
     (* y (/ (* z x) (- (* 0.5 (/ (* a t) z)) z)))
     (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -380000.0) {
		tmp = x * -y;
	} else if (z <= 2.55e-192) {
		tmp = y * ((z * x) / ((0.5 * ((a * t) / z)) - z));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-380000.0d0)) then
        tmp = x * -y
    else if (z <= 2.55d-192) then
        tmp = y * ((z * x) / ((0.5d0 * ((a * t) / z)) - z))
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -380000.0) {
		tmp = x * -y;
	} else if (z <= 2.55e-192) {
		tmp = y * ((z * x) / ((0.5 * ((a * t) / z)) - z));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -380000.0:
		tmp = x * -y
	elif z <= 2.55e-192:
		tmp = y * ((z * x) / ((0.5 * ((a * t) / z)) - z))
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -380000.0)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 2.55e-192)
		tmp = Float64(y * Float64(Float64(z * x) / Float64(Float64(0.5 * Float64(Float64(a * t) / z)) - z)));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -380000.0)
		tmp = x * -y;
	elseif (z <= 2.55e-192)
		tmp = y * ((z * x) / ((0.5 * ((a * t) / z)) - z));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -380000.0], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 2.55e-192], N[(y * N[(N[(z * x), $MachinePrecision] / N[(N[(0.5 * N[(N[(a * t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.8e5

   1. Initial program 48.8%

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

      1. *-commutative 48.8%

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

      2. associate-*l* 45.7%

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

      3. associate-*r/ 45.9%

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

   3. Simplified 45.9%

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

   4. Taylor expanded in z around -inf 90.9%

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

      1. neg-mul-1 90.9%

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

   6. Simplified 90.9%

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

   if -3.8e5 < z < 2.5500000000000001e-192

   1. Initial program 79.7%

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

      1. *-commutative 79.7%

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

      2. associate-*l* 80.8%

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

      3. associate-*r/ 78.7%

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

   3. Simplified 78.7%

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

   4. Taylor expanded in z around -inf 53.0%

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

   if 2.5500000000000001e-192 < z

   1. Initial program 59.1%

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

      1. *-commutative 59.1%

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

      2. associate-*l* 57.2%

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

      3. associate-*r/ 61.4%

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

   3. Simplified 61.4%

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

   4. Taylor expanded in z around inf 86.2%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -380000:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{-192}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{0.5 \cdot \frac{a \cdot t}{z} - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 14: 75.5% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-217}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-192}:\\ \;\;\;\;2 \cdot \left(\frac{y}{a} \cdot \frac{x \cdot \left(z \cdot z\right)}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.5e-217)
   (* x (- y))
   (if (<= z 2.1e-192) (* 2.0 (* (/ y a) (/ (* x (* z z)) t))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.5e-217) {
		tmp = x * -y;
	} else if (z <= 2.1e-192) {
		tmp = 2.0 * ((y / a) * ((x * (z * z)) / t));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.5d-217)) then
        tmp = x * -y
    else if (z <= 2.1d-192) then
        tmp = 2.0d0 * ((y / a) * ((x * (z * z)) / t))
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.5e-217) {
		tmp = x * -y;
	} else if (z <= 2.1e-192) {
		tmp = 2.0 * ((y / a) * ((x * (z * z)) / t));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.5e-217:
		tmp = x * -y
	elif z <= 2.1e-192:
		tmp = 2.0 * ((y / a) * ((x * (z * z)) / t))
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.5e-217)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 2.1e-192)
		tmp = Float64(2.0 * Float64(Float64(y / a) * Float64(Float64(x * Float64(z * z)) / t)));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.5e-217)
		tmp = x * -y;
	elseif (z <= 2.1e-192)
		tmp = 2.0 * ((y / a) * ((x * (z * z)) / t));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.5e-217], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 2.1e-192], N[(2.0 * N[(N[(y / a), $MachinePrecision] * N[(N[(x * N[(z * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.50000000000000002e-217

   1. Initial program 60.8%

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

      1. *-commutative 60.8%

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

      2. associate-*l* 57.3%

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

      3. associate-*r/ 57.8%

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

   3. Simplified 57.8%

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

   4. Taylor expanded in z around -inf 75.3%

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

      1. neg-mul-1 75.3%

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

   6. Simplified 75.3%

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

   if -1.50000000000000002e-217 < z < 2.09999999999999993e-192

   1. Initial program 79.3%

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

      1. associate-*l/ 76.0%

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

   3. Simplified 76.0%

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

   4. Taylor expanded in z around -inf 44.4%

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

   5. Taylor expanded in a around inf 44.4%

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

      1. times-frac 43.7%

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

      2. *-commutative 43.7%

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

      3. unpow2 43.7%

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

   7. Simplified 43.7%

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

   if 2.09999999999999993e-192 < z

   1. Initial program 59.1%

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

      1. *-commutative 59.1%

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

      2. associate-*l* 57.2%

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

      3. associate-*r/ 61.4%

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

   3. Simplified 61.4%

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

   4. Taylor expanded in z around inf 86.2%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-217}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-192}:\\ \;\;\;\;2 \cdot \left(\frac{y}{a} \cdot \frac{x \cdot \left(z \cdot z\right)}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 15: 75.8% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-216}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 6.9 \cdot 10^{-192}:\\ \;\;\;\;z \cdot \left(2 \cdot \left(\frac{z}{t} \cdot \frac{x \cdot y}{a}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.9e-216)
   (* x (- y))
   (if (<= z 6.9e-192) (* z (* 2.0 (* (/ z t) (/ (* x y) a)))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.9e-216) {
		tmp = x * -y;
	} else if (z <= 6.9e-192) {
		tmp = z * (2.0 * ((z / t) * ((x * y) / a)));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-2.9d-216)) then
        tmp = x * -y
    else if (z <= 6.9d-192) then
        tmp = z * (2.0d0 * ((z / t) * ((x * y) / a)))
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.9e-216) {
		tmp = x * -y;
	} else if (z <= 6.9e-192) {
		tmp = z * (2.0 * ((z / t) * ((x * y) / a)));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.9e-216:
		tmp = x * -y
	elif z <= 6.9e-192:
		tmp = z * (2.0 * ((z / t) * ((x * y) / a)))
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.9e-216)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 6.9e-192)
		tmp = Float64(z * Float64(2.0 * Float64(Float64(z / t) * Float64(Float64(x * y) / a))));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.9e-216)
		tmp = x * -y;
	elseif (z <= 6.9e-192)
		tmp = z * (2.0 * ((z / t) * ((x * y) / a)));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.9e-216], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 6.9e-192], N[(z * N[(2.0 * N[(N[(z / t), $MachinePrecision] * N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.9000000000000001e-216

   1. Initial program 60.8%

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

      1. *-commutative 60.8%

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

      2. associate-*l* 57.3%

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

      3. associate-*r/ 57.8%

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

   3. Simplified 57.8%

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

   4. Taylor expanded in z around -inf 75.3%

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

      1. neg-mul-1 75.3%

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

   6. Simplified 75.3%

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

   if -2.9000000000000001e-216 < z < 6.90000000000000016e-192

   1. Initial program 79.3%

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

      1. associate-*l/ 76.0%

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

   3. Simplified 76.0%

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

   4. Taylor expanded in z around -inf 44.4%

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

   5. Taylor expanded in a around inf 44.2%

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

      1. *-commutative 44.2%

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

      2. associate-*r* 44.1%

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

      3. *-commutative 44.1%

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

      4. times-frac 44.0%

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

      5. *-commutative 44.0%

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

   7. Simplified 44.0%

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

   if 6.90000000000000016e-192 < z

   1. Initial program 59.1%

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

      1. *-commutative 59.1%

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

      2. associate-*l* 57.2%

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

      3. associate-*r/ 61.4%

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

   3. Simplified 61.4%

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

   4. Taylor expanded in z around inf 86.2%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-216}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 6.9 \cdot 10^{-192}:\\ \;\;\;\;z \cdot \left(2 \cdot \left(\frac{z}{t} \cdot \frac{x \cdot y}{a}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 16: 75.6% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{-217}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-192}:\\ \;\;\;\;z \cdot \left(2 \cdot \frac{x \cdot \left(z \cdot y\right)}{a \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

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 -2.3e-217)
   (* x (- y))
   (if (<= z 5.8e-192) (* z (* 2.0 (/ (* x (* z y)) (* a t)))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.3e-217) {
		tmp = x * -y;
	} else if (z <= 5.8e-192) {
		tmp = z * (2.0 * ((x * (z * y)) / (a * t)));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-2.3d-217)) then
        tmp = x * -y
    else if (z <= 5.8d-192) then
        tmp = z * (2.0d0 * ((x * (z * y)) / (a * t)))
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.3e-217) {
		tmp = x * -y;
	} else if (z <= 5.8e-192) {
		tmp = z * (2.0 * ((x * (z * y)) / (a * t)));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.3e-217:
		tmp = x * -y
	elif z <= 5.8e-192:
		tmp = z * (2.0 * ((x * (z * y)) / (a * t)))
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.3e-217)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 5.8e-192)
		tmp = Float64(z * Float64(2.0 * Float64(Float64(x * Float64(z * y)) / Float64(a * t))));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.3e-217)
		tmp = x * -y;
	elseif (z <= 5.8e-192)
		tmp = z * (2.0 * ((x * (z * y)) / (a * t)));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.3e-217], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 5.8e-192], N[(z * N[(2.0 * N[(N[(x * N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.30000000000000005e-217

   1. Initial program 60.8%

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

      1. *-commutative 60.8%

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

      2. associate-*l* 57.3%

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

      3. associate-*r/ 57.8%

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

   3. Simplified 57.8%

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

   4. Taylor expanded in z around -inf 75.3%

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

      1. neg-mul-1 75.3%

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

   6. Simplified 75.3%

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

   if -2.30000000000000005e-217 < z < 5.80000000000000033e-192

   1. Initial program 79.3%

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

      1. associate-*l/ 76.0%

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

   3. Simplified 76.0%

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

   4. Taylor expanded in z around -inf 44.4%

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

   5. Taylor expanded in a around inf 44.2%

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

      1. *-commutative 44.2%

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

      2. associate-*r* 44.1%

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

      3. *-commutative 44.1%

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

      4. associate-*l* 44.2%

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

   7. Simplified 44.2%

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

   if 5.80000000000000033e-192 < z

   1. Initial program 59.1%

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

      1. *-commutative 59.1%

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

      2. associate-*l* 57.2%

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

      3. associate-*r/ 61.4%

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

   3. Simplified 61.4%

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

   4. Taylor expanded in z around inf 86.2%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{-217}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-192}:\\ \;\;\;\;z \cdot \left(2 \cdot \frac{x \cdot \left(z \cdot y\right)}{a \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 17: 75.4% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{-216}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-192}:\\ \;\;\;\;\frac{-2 \cdot \left(y \cdot \left(x \cdot \left(z \cdot z\right)\right)\right)}{a \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.2e-216)
   (* x (- y))
   (if (<= z 2.6e-192) (/ (* -2.0 (* y (* x (* z z)))) (* a t)) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.2e-216) {
		tmp = x * -y;
	} else if (z <= 2.6e-192) {
		tmp = (-2.0 * (y * (x * (z * z)))) / (a * t);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-2.2d-216)) then
        tmp = x * -y
    else if (z <= 2.6d-192) then
        tmp = ((-2.0d0) * (y * (x * (z * z)))) / (a * t)
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.2e-216) {
		tmp = x * -y;
	} else if (z <= 2.6e-192) {
		tmp = (-2.0 * (y * (x * (z * z)))) / (a * t);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.2e-216:
		tmp = x * -y
	elif z <= 2.6e-192:
		tmp = (-2.0 * (y * (x * (z * z)))) / (a * t)
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.2e-216)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 2.6e-192)
		tmp = Float64(Float64(-2.0 * Float64(y * Float64(x * Float64(z * z)))) / Float64(a * t));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.2e-216)
		tmp = x * -y;
	elseif (z <= 2.6e-192)
		tmp = (-2.0 * (y * (x * (z * z)))) / (a * t);
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.2e-216], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 2.6e-192], N[(N[(-2.0 * N[(y * N[(x * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * t), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.1999999999999999e-216

   1. Initial program 60.8%

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

      1. *-commutative 60.8%

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

      2. associate-*l* 57.3%

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

      3. associate-*r/ 57.8%

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

   3. Simplified 57.8%

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

   4. Taylor expanded in z around -inf 75.3%

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

      1. neg-mul-1 75.3%

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

   6. Simplified 75.3%

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

   if -2.1999999999999999e-216 < z < 2.6000000000000002e-192

   1. Initial program 79.3%

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

   2. Taylor expanded in x around 0 88.4%

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

   3. Taylor expanded in z around inf 44.8%

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

   4. Taylor expanded in z around 0 44.4%

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

      1. associate-*r/ 44.4%

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

      2. *-commutative 44.4%

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

      3. unpow2 44.4%

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

   6. Simplified 44.4%

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

   if 2.6000000000000002e-192 < z

   1. Initial program 59.1%

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

      1. *-commutative 59.1%

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

      2. associate-*l* 57.2%

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

      3. associate-*r/ 61.4%

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

   3. Simplified 61.4%

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

   4. Taylor expanded in z around inf 86.2%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{-216}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-192}:\\ \;\;\;\;\frac{-2 \cdot \left(y \cdot \left(x \cdot \left(z \cdot z\right)\right)\right)}{a \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 18: 75.4% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-219}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-192}:\\ \;\;\;\;\frac{\left(y \cdot -2\right) \cdot \left(z \cdot \left(z \cdot x\right)\right)}{a \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -9e-219)
   (* x (- y))
   (if (<= z 1.6e-192) (/ (* (* y -2.0) (* z (* z x))) (* a t)) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -9e-219) {
		tmp = x * -y;
	} else if (z <= 1.6e-192) {
		tmp = ((y * -2.0) * (z * (z * x))) / (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 <= (-9d-219)) then
        tmp = x * -y
    else if (z <= 1.6d-192) then
        tmp = ((y * (-2.0d0)) * (z * (z * x))) / (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 <= -9e-219) {
		tmp = x * -y;
	} else if (z <= 1.6e-192) {
		tmp = ((y * -2.0) * (z * (z * x))) / (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 <= -9e-219:
		tmp = x * -y
	elif z <= 1.6e-192:
		tmp = ((y * -2.0) * (z * (z * x))) / (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 <= -9e-219)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 1.6e-192)
		tmp = Float64(Float64(Float64(y * -2.0) * Float64(z * Float64(z * x))) / 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 <= -9e-219)
		tmp = x * -y;
	elseif (z <= 1.6e-192)
		tmp = ((y * -2.0) * (z * (z * x))) / (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, -9e-219], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 1.6e-192], N[(N[(N[(y * -2.0), $MachinePrecision] * N[(z * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * t), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.00000000000000029e-219

   1. Initial program 60.8%

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

      1. *-commutative 60.8%

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

      2. associate-*l* 57.3%

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

      3. associate-*r/ 57.8%

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

   3. Simplified 57.8%

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

   4. Taylor expanded in z around -inf 75.3%

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

      1. neg-mul-1 75.3%

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

   6. Simplified 75.3%

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

   if -9.00000000000000029e-219 < z < 1.6000000000000001e-192

   1. Initial program 79.3%

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

   2. Taylor expanded in x around 0 88.4%

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

   3. Taylor expanded in z around inf 44.8%

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

   4. Taylor expanded in z around 0 44.4%

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

      1. associate-*r/ 44.4%

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

      2. associate-*r* 44.4%

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

      3. unpow2 44.4%

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

      4. associate-*l* 44.5%

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

   6. Simplified 44.5%

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

   if 1.6000000000000001e-192 < z

   1. Initial program 59.1%

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

      1. *-commutative 59.1%

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

      2. associate-*l* 57.2%

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

      3. associate-*r/ 61.4%

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

   3. Simplified 61.4%

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

   4. Taylor expanded in z around inf 86.2%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-219}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-192}:\\ \;\;\;\;\frac{\left(y \cdot -2\right) \cdot \left(z \cdot \left(z \cdot x\right)\right)}{a \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 19: 74.4% accurate, 10.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{-240}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-156}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

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 -3e-240)
   (* x (- y))
   (if (<= z 5e-156) (* y (/ (* z x) z)) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3e-240) {
		tmp = x * -y;
	} else if (z <= 5e-156) {
		tmp = y * ((z * x) / z);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-3d-240)) then
        tmp = x * -y
    else if (z <= 5d-156) then
        tmp = y * ((z * x) / z)
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3e-240) {
		tmp = x * -y;
	} else if (z <= 5e-156) {
		tmp = y * ((z * x) / z);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -3e-240:
		tmp = x * -y
	elif z <= 5e-156:
		tmp = y * ((z * x) / z)
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3e-240)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 5e-156)
		tmp = Float64(y * Float64(Float64(z * x) / 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 <= -3e-240)
		tmp = x * -y;
	elseif (z <= 5e-156)
		tmp = y * ((z * x) / z);
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3e-240], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 5e-156], N[(y * N[(N[(z * x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.99999999999999991e-240

   1. Initial program 61.8%

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

      1. *-commutative 61.8%

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

      2. associate-*l* 57.6%

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

      3. associate-*r/ 57.3%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in z around -inf 73.4%

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

      1. neg-mul-1 73.4%

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

   6. Simplified 73.4%

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

   if -2.99999999999999991e-240 < z < 5.00000000000000007e-156

   1. Initial program 80.5%

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

      1. *-commutative 80.5%

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

      2. associate-*l* 91.9%

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

      3. associate-*r/ 88.9%

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

   3. Simplified 88.9%

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

   4. Taylor expanded in z around inf 35.1%

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

   if 5.00000000000000007e-156 < z

   1. Initial program 57.1%

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

      1. *-commutative 57.1%

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

      2. associate-*l* 55.0%

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

      3. associate-*r/ 59.5%

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

   3. Simplified 59.5%

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

   4. Taylor expanded in z around inf 88.9%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{-240}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-156}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 20: 72.8% accurate, 18.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.22 \cdot 10^{-294}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

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 -1.22e-294) (* x (- y)) (* x y)))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.22e-294) {
		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 <= (-1.22d-294)) 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 <= -1.22e-294) {
		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 <= -1.22e-294:
		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 <= -1.22e-294)
		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 <= -1.22e-294)
		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, -1.22e-294], N[(x * (-y)), $MachinePrecision], N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.21999999999999995e-294

   1. Initial program 62.1%

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

      1. *-commutative 62.1%

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

      2. associate-*l* 58.0%

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

      3. associate-*r/ 57.7%

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

   3. Simplified 57.7%

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

   4. Taylor expanded in z around -inf 70.1%

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

      1. neg-mul-1 70.1%

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

   6. Simplified 70.1%

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

   if -1.21999999999999995e-294 < z

   1. Initial program 62.1%

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

      1. *-commutative 62.1%

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

      2. associate-*l* 63.1%

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

      3. associate-*r/ 66.0%

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

   3. Simplified 66.0%

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

   4. Taylor expanded in z around inf 76.0%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.22 \cdot 10^{-294}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 21: 43.0% 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 62.1%

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

   1. *-commutative 62.1%

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

   2. associate-*l* 60.6%

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

   3. associate-*r/ 62.1%

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

3. Simplified 62.1%

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

4. Taylor expanded in z around inf 46.2%

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

5. Final simplification 46.2%

   \[\leadsto x \cdot y \]

Developer target: 89.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023257 
(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)))))