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

Percentage Accurate: 61.5% → 91.8%

Time: 31.7s

Alternatives: 23

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: 61.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} t_1 := \frac{\sqrt{z \cdot z - a \cdot t}}{z}\\ \mathbf{if}\;z \leq -2 \cdot 10^{+153}:\\ \;\;\;\;\frac{y}{-1 - \frac{\frac{a}{z}}{\frac{z}{\frac{t}{-2}}}} \cdot x\\ \mathbf{elif}\;z \leq -2.9 \cdot 10^{-201}:\\ \;\;\;\;y \cdot \frac{x}{t_1}\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-223}:\\ \;\;\;\;x \cdot \left(\left(z \cdot y\right) \cdot {\left(e^{-0.5}\right)}^{\left(\log a - \log \left(\frac{-1}{t}\right)\right)}\right)\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+85}:\\ \;\;\;\;\frac{x}{\frac{t_1}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \left(\frac{a}{z} \cdot \frac{t}{z}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (sqrt (- (* z z) (* a t))) z)))
   (if (<= z -2e+153)
     (* (/ y (- -1.0 (/ (/ a z) (/ z (/ t -2.0))))) x)
     (if (<= z -2.9e-201)
       (* y (/ x t_1))
       (if (<= z 8.5e-223)
         (* x (* (* z y) (pow (exp -0.5) (- (log a) (log (/ -1.0 t))))))
         (if (<= z 4.6e+85)
           (/ x (/ t_1 y))
           (/ (* y x) (+ 1.0 (* -0.5 (* (/ a z) (/ t z)))))))))))

C

assert(x < y);
assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = sqrt(((z * z) - (a * t))) / z;
	double tmp;
	if (z <= -2e+153) {
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x;
	} else if (z <= -2.9e-201) {
		tmp = y * (x / t_1);
	} else if (z <= 8.5e-223) {
		tmp = x * ((z * y) * pow(exp(-0.5), (log(a) - log((-1.0 / t)))));
	} else if (z <= 4.6e+85) {
		tmp = x / (t_1 / y);
	} else {
		tmp = (y * x) / (1.0 + (-0.5 * ((a / z) * (t / z))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(((z * z) - (a * t))) / z
    if (z <= (-2d+153)) then
        tmp = (y / ((-1.0d0) - ((a / z) / (z / (t / (-2.0d0)))))) * x
    else if (z <= (-2.9d-201)) then
        tmp = y * (x / t_1)
    else if (z <= 8.5d-223) then
        tmp = x * ((z * y) * (exp((-0.5d0)) ** (log(a) - log(((-1.0d0) / t)))))
    else if (z <= 4.6d+85) then
        tmp = x / (t_1 / y)
    else
        tmp = (y * x) / (1.0d0 + ((-0.5d0) * ((a / z) * (t / z))))
    end if
    code = tmp
end function

Java

assert x < y;
assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.sqrt(((z * z) - (a * t))) / z;
	double tmp;
	if (z <= -2e+153) {
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x;
	} else if (z <= -2.9e-201) {
		tmp = y * (x / t_1);
	} else if (z <= 8.5e-223) {
		tmp = x * ((z * y) * Math.pow(Math.exp(-0.5), (Math.log(a) - Math.log((-1.0 / t)))));
	} else if (z <= 4.6e+85) {
		tmp = x / (t_1 / y);
	} else {
		tmp = (y * x) / (1.0 + (-0.5 * ((a / z) * (t / z))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[t, a] = sort([t, a])
def code(x, y, z, t, a):
	t_1 = math.sqrt(((z * z) - (a * t))) / z
	tmp = 0
	if z <= -2e+153:
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x
	elif z <= -2.9e-201:
		tmp = y * (x / t_1)
	elif z <= 8.5e-223:
		tmp = x * ((z * y) * math.pow(math.exp(-0.5), (math.log(a) - math.log((-1.0 / t)))))
	elif z <= 4.6e+85:
		tmp = x / (t_1 / y)
	else:
		tmp = (y * x) / (1.0 + (-0.5 * ((a / z) * (t / z))))
	return tmp

Julia

x, y = sort([x, y])
t, a = sort([t, a])
function code(x, y, z, t, a)
	t_1 = Float64(sqrt(Float64(Float64(z * z) - Float64(a * t))) / z)
	tmp = 0.0
	if (z <= -2e+153)
		tmp = Float64(Float64(y / Float64(-1.0 - Float64(Float64(a / z) / Float64(z / Float64(t / -2.0))))) * x);
	elseif (z <= -2.9e-201)
		tmp = Float64(y * Float64(x / t_1));
	elseif (z <= 8.5e-223)
		tmp = Float64(x * Float64(Float64(z * y) * (exp(-0.5) ^ Float64(log(a) - log(Float64(-1.0 / t))))));
	elseif (z <= 4.6e+85)
		tmp = Float64(x / Float64(t_1 / y));
	else
		tmp = Float64(Float64(y * x) / Float64(1.0 + Float64(-0.5 * Float64(Float64(a / z) * Float64(t / z)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = sqrt(((z * z) - (a * t))) / z;
	tmp = 0.0;
	if (z <= -2e+153)
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x;
	elseif (z <= -2.9e-201)
		tmp = y * (x / t_1);
	elseif (z <= 8.5e-223)
		tmp = x * ((z * y) * (exp(-0.5) ^ (log(a) - log((-1.0 / t)))));
	elseif (z <= 4.6e+85)
		tmp = x / (t_1 / y);
	else
		tmp = (y * x) / (1.0 + (-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.
NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[z, -2e+153], N[(N[(y / N[(-1.0 - N[(N[(a / z), $MachinePrecision] / N[(z / N[(t / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, -2.9e-201], N[(y * N[(x / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 8.5e-223], N[(x * N[(N[(z * y), $MachinePrecision] * N[Power[N[Exp[-0.5], $MachinePrecision], N[(N[Log[a], $MachinePrecision] - N[Log[N[(-1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.6e+85], N[(x / N[(t$95$1 / y), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(N[(a / z), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -2e153

   1. Initial program 22.9%

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

      1. associate-/l* 23.6%

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

      2. associate-/l* 23.7%

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

   3. Simplified 23.7%

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

   4. Taylor expanded in z around -inf 79.7%

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

      1. *-commutative 79.7%

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

      2. associate-*r/ 79.7%

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

      3. *-commutative 79.7%

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

      4. associate-*r* 79.7%

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

      5. unpow2 79.7%

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

   6. Simplified 79.7%

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

      1. associate-*l* 79.7%

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

      2. *-commutative 79.7%

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

      3. associate-*r* 79.8%

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

      4. times-frac 83.2%

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

      5. *-commutative 83.2%

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

      6. associate-*l/ 93.2%

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

      7. associate-/r/ 93.2%

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

      8. div-inv 93.2%

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

      9. clear-num 93.2%

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

   8. Applied egg-rr 93.2%

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

   9. Taylor expanded in y around -inf 80.0%

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

       1. associate-*r/ 80.0%

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

       2. mul-1-neg 80.0%

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

       3. distribute-rgt-neg-in 80.0%

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

       4. +-commutative 80.0%

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

       5. associate-/l* 93.5%

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

       6. unpow2 93.5%

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

       7. associate-*r/ 93.5%

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

   11. Simplified 93.5%

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

       1. div-inv 93.5%

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

       2. associate-*l* 93.5%

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

       3. *-commutative 93.5%

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

       4. div-inv 93.5%

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

   13. Applied egg-rr 93.5%

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

   if -2e153 < z < -2.9000000000000002e-201

   1. Initial program 83.4%

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

      1. associate-/l* 86.4%

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

      2. associate-*l/ 89.5%

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

   3. Simplified 89.5%

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

   if -2.9000000000000002e-201 < z < 8.5000000000000003e-223

   1. Initial program 78.5%

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

   2. Taylor expanded in z around 0 78.5%

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

      1. associate-*r* 78.5%

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

      2. neg-mul-1 78.5%

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

      3. *-commutative 78.5%

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

   4. Simplified 78.5%

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

      1. div-inv 78.6%

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

      2. associate-*l* 77.5%

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

      3. associate-*l* 73.3%

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

      4. distribute-rgt-neg-out 73.3%

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

      5. pow1/2 73.3%

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

      6. pow-flip 73.3%

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

      7. distribute-rgt-neg-out 73.3%

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

      8. metadata-eval 73.3%

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

   6. Applied egg-rr 73.3%

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

   7. Taylor expanded in t around -inf 40.8%

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

      1. exp-prod 40.8%

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

      2. mul-1-neg 40.8%

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

   9. Simplified 40.8%

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

   if 8.5000000000000003e-223 < z < 4.5999999999999998e85

   1. Initial program 88.1%

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

      1. associate-/l* 89.6%

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

      2. associate-/l* 89.5%

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

   3. Simplified 89.5%

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

   if 4.5999999999999998e85 < z

   1. Initial program 38.5%

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

      1. associate-/l* 42.0%

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

   3. Simplified 42.0%

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

   4. Taylor expanded in z around inf 90.3%

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

      1. associate-/l* 98.5%

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

   6. Simplified 98.5%

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

   7. Taylor expanded in z around 0 88.3%

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

      1. unpow2 88.3%

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

      2. *-commutative 88.3%

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

      3. times-frac 98.5%

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

   9. Simplified 98.5%

      \[\leadsto \frac{x \cdot y}{\color{blue}{1 + -0.5 \cdot \left(\frac{t}{z} \cdot \frac{a}{z}\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+153}:\\ \;\;\;\;\frac{y}{-1 - \frac{\frac{a}{z}}{\frac{z}{\frac{t}{-2}}}} \cdot x\\ \mathbf{elif}\;z \leq -2.9 \cdot 10^{-201}:\\ \;\;\;\;y \cdot \frac{x}{\frac{\sqrt{z \cdot z - a \cdot t}}{z}}\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-223}:\\ \;\;\;\;x \cdot \left(\left(z \cdot y\right) \cdot {\left(e^{-0.5}\right)}^{\left(\log a - \log \left(\frac{-1}{t}\right)\right)}\right)\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+85}:\\ \;\;\;\;\frac{x}{\frac{\frac{\sqrt{z \cdot z - a \cdot t}}{z}}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \left(\frac{a}{z} \cdot \frac{t}{z}\right)}\\ \end{array} \]

Alternative 2: 91.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} t_1 := \frac{\sqrt{z \cdot z - a \cdot t}}{z}\\ \mathbf{if}\;z \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\frac{y}{-1 - \frac{\frac{a}{z}}{\frac{z}{\frac{t}{-2}}}} \cdot x\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{-196}:\\ \;\;\;\;y \cdot \frac{x}{t_1}\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-223}:\\ \;\;\;\;x \cdot \left(\left(z \cdot y\right) \cdot e^{-0.5 \cdot \left(\log a + \log \left(-t\right)\right)}\right)\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+85}:\\ \;\;\;\;\frac{x}{\frac{t_1}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \left(\frac{a}{z} \cdot \frac{t}{z}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (sqrt (- (* z z) (* a t))) z)))
   (if (<= z -1e+154)
     (* (/ y (- -1.0 (/ (/ a z) (/ z (/ t -2.0))))) x)
     (if (<= z -8.2e-196)
       (* y (/ x t_1))
       (if (<= z 9e-223)
         (* x (* (* z y) (exp (* -0.5 (+ (log a) (log (- t)))))))
         (if (<= z 2.6e+85)
           (/ x (/ t_1 y))
           (/ (* y x) (+ 1.0 (* -0.5 (* (/ a z) (/ t z)))))))))))

C

assert(x < y);
assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = sqrt(((z * z) - (a * t))) / z;
	double tmp;
	if (z <= -1e+154) {
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x;
	} else if (z <= -8.2e-196) {
		tmp = y * (x / t_1);
	} else if (z <= 9e-223) {
		tmp = x * ((z * y) * exp((-0.5 * (log(a) + log(-t)))));
	} else if (z <= 2.6e+85) {
		tmp = x / (t_1 / y);
	} else {
		tmp = (y * x) / (1.0 + (-0.5 * ((a / z) * (t / z))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(((z * z) - (a * t))) / z
    if (z <= (-1d+154)) then
        tmp = (y / ((-1.0d0) - ((a / z) / (z / (t / (-2.0d0)))))) * x
    else if (z <= (-8.2d-196)) then
        tmp = y * (x / t_1)
    else if (z <= 9d-223) then
        tmp = x * ((z * y) * exp(((-0.5d0) * (log(a) + log(-t)))))
    else if (z <= 2.6d+85) then
        tmp = x / (t_1 / y)
    else
        tmp = (y * x) / (1.0d0 + ((-0.5d0) * ((a / z) * (t / z))))
    end if
    code = tmp
end function

Java

assert x < y;
assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.sqrt(((z * z) - (a * t))) / z;
	double tmp;
	if (z <= -1e+154) {
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x;
	} else if (z <= -8.2e-196) {
		tmp = y * (x / t_1);
	} else if (z <= 9e-223) {
		tmp = x * ((z * y) * Math.exp((-0.5 * (Math.log(a) + Math.log(-t)))));
	} else if (z <= 2.6e+85) {
		tmp = x / (t_1 / y);
	} else {
		tmp = (y * x) / (1.0 + (-0.5 * ((a / z) * (t / z))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[t, a] = sort([t, a])
def code(x, y, z, t, a):
	t_1 = math.sqrt(((z * z) - (a * t))) / z
	tmp = 0
	if z <= -1e+154:
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x
	elif z <= -8.2e-196:
		tmp = y * (x / t_1)
	elif z <= 9e-223:
		tmp = x * ((z * y) * math.exp((-0.5 * (math.log(a) + math.log(-t)))))
	elif z <= 2.6e+85:
		tmp = x / (t_1 / y)
	else:
		tmp = (y * x) / (1.0 + (-0.5 * ((a / z) * (t / z))))
	return tmp

Julia

x, y = sort([x, y])
t, a = sort([t, a])
function code(x, y, z, t, a)
	t_1 = Float64(sqrt(Float64(Float64(z * z) - Float64(a * t))) / z)
	tmp = 0.0
	if (z <= -1e+154)
		tmp = Float64(Float64(y / Float64(-1.0 - Float64(Float64(a / z) / Float64(z / Float64(t / -2.0))))) * x);
	elseif (z <= -8.2e-196)
		tmp = Float64(y * Float64(x / t_1));
	elseif (z <= 9e-223)
		tmp = Float64(x * Float64(Float64(z * y) * exp(Float64(-0.5 * Float64(log(a) + log(Float64(-t)))))));
	elseif (z <= 2.6e+85)
		tmp = Float64(x / Float64(t_1 / y));
	else
		tmp = Float64(Float64(y * x) / Float64(1.0 + Float64(-0.5 * Float64(Float64(a / z) * Float64(t / z)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = sqrt(((z * z) - (a * t))) / z;
	tmp = 0.0;
	if (z <= -1e+154)
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x;
	elseif (z <= -8.2e-196)
		tmp = y * (x / t_1);
	elseif (z <= 9e-223)
		tmp = x * ((z * y) * exp((-0.5 * (log(a) + log(-t)))));
	elseif (z <= 2.6e+85)
		tmp = x / (t_1 / y);
	else
		tmp = (y * x) / (1.0 + (-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.
NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[z, -1e+154], N[(N[(y / N[(-1.0 - N[(N[(a / z), $MachinePrecision] / N[(z / N[(t / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, -8.2e-196], N[(y * N[(x / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9e-223], N[(x * N[(N[(z * y), $MachinePrecision] * N[Exp[N[(-0.5 * N[(N[Log[a], $MachinePrecision] + N[Log[(-t)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.6e+85], N[(x / N[(t$95$1 / y), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(N[(a / z), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -1.00000000000000004e154

   1. Initial program 22.9%

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

      1. associate-/l* 23.6%

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

      2. associate-/l* 23.7%

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

   3. Simplified 23.7%

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

   4. Taylor expanded in z around -inf 79.7%

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

      1. *-commutative 79.7%

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

      2. associate-*r/ 79.7%

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

      3. *-commutative 79.7%

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

      4. associate-*r* 79.7%

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

      5. unpow2 79.7%

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

   6. Simplified 79.7%

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

      1. associate-*l* 79.7%

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

      2. *-commutative 79.7%

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

      3. associate-*r* 79.8%

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

      4. times-frac 83.2%

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

      5. *-commutative 83.2%

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

      6. associate-*l/ 93.2%

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

      7. associate-/r/ 93.2%

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

      8. div-inv 93.2%

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

      9. clear-num 93.2%

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

   8. Applied egg-rr 93.2%

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

   9. Taylor expanded in y around -inf 80.0%

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

       1. associate-*r/ 80.0%

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

       2. mul-1-neg 80.0%

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

       3. distribute-rgt-neg-in 80.0%

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

       4. +-commutative 80.0%

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

       5. associate-/l* 93.5%

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

       6. unpow2 93.5%

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

       7. associate-*r/ 93.5%

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

   11. Simplified 93.5%

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

       1. div-inv 93.5%

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

       2. associate-*l* 93.5%

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

       3. *-commutative 93.5%

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

       4. div-inv 93.5%

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

   13. Applied egg-rr 93.5%

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

   if -1.00000000000000004e154 < z < -8.20000000000000043e-196

   1. Initial program 83.4%

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

      1. associate-/l* 86.4%

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

      2. associate-*l/ 89.5%

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

   3. Simplified 89.5%

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

   if -8.20000000000000043e-196 < z < 8.99999999999999935e-223

   1. Initial program 78.5%

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

   2. Taylor expanded in z around 0 78.5%

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

      1. associate-*r* 78.5%

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

      2. neg-mul-1 78.5%

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

      3. *-commutative 78.5%

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

   4. Simplified 78.5%

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

      1. div-inv 78.6%

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

      2. associate-*l* 77.5%

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

      3. associate-*l* 73.3%

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

      4. distribute-rgt-neg-out 73.3%

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

      5. pow1/2 73.3%

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

      6. pow-flip 73.3%

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

      7. distribute-rgt-neg-out 73.3%

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

      8. metadata-eval 73.3%

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

   6. Applied egg-rr 73.3%

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

   7. Taylor expanded in a around 0 40.8%

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

   if 8.99999999999999935e-223 < z < 2.60000000000000011e85

   1. Initial program 88.1%

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

      1. associate-/l* 89.6%

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

      2. associate-/l* 89.5%

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

   3. Simplified 89.5%

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

   if 2.60000000000000011e85 < z

   1. Initial program 38.5%

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

      1. associate-/l* 42.0%

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

   3. Simplified 42.0%

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

   4. Taylor expanded in z around inf 90.3%

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

      1. associate-/l* 98.5%

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

   6. Simplified 98.5%

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

   7. Taylor expanded in z around 0 88.3%

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

      1. unpow2 88.3%

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

      2. *-commutative 88.3%

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

      3. times-frac 98.5%

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

   9. Simplified 98.5%

      \[\leadsto \frac{x \cdot y}{\color{blue}{1 + -0.5 \cdot \left(\frac{t}{z} \cdot \frac{a}{z}\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\frac{y}{-1 - \frac{\frac{a}{z}}{\frac{z}{\frac{t}{-2}}}} \cdot x\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{-196}:\\ \;\;\;\;y \cdot \frac{x}{\frac{\sqrt{z \cdot z - a \cdot t}}{z}}\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-223}:\\ \;\;\;\;x \cdot \left(\left(z \cdot y\right) \cdot e^{-0.5 \cdot \left(\log a + \log \left(-t\right)\right)}\right)\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+85}:\\ \;\;\;\;\frac{x}{\frac{\frac{\sqrt{z \cdot z - a \cdot t}}{z}}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \left(\frac{a}{z} \cdot \frac{t}{z}\right)}\\ \end{array} \]

Alternative 3: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+34}:\\ \;\;\;\;\frac{y \cdot \left(-x\right)}{1 + -0.5 \cdot \frac{a}{z \cdot \frac{z}{t}}}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+86}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \left(\frac{a}{z} \cdot \frac{t}{z}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.5e+34)
   (/ (* y (- x)) (+ 1.0 (* -0.5 (/ a (* z (/ z t))))))
   (if (<= z 1.1e+86)
     (* y (/ (* z x) (sqrt (- (* z z) (* a t)))))
     (/ (* y x) (+ 1.0 (* -0.5 (* (/ a z) (/ t z))))))))

C

assert(x < y);
assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.5e+34) {
		tmp = (y * -x) / (1.0 + (-0.5 * (a / (z * (z / t)))));
	} else if (z <= 1.1e+86) {
		tmp = y * ((z * x) / sqrt(((z * z) - (a * t))));
	} else {
		tmp = (y * x) / (1.0 + (-0.5 * ((a / z) * (t / z))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a 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+34)) then
        tmp = (y * -x) / (1.0d0 + ((-0.5d0) * (a / (z * (z / t)))))
    else if (z <= 1.1d+86) then
        tmp = y * ((z * x) / sqrt(((z * z) - (a * t))))
    else
        tmp = (y * x) / (1.0d0 + ((-0.5d0) * ((a / z) * (t / z))))
    end if
    code = tmp
end function

Java

assert x < y;
assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.5e+34) {
		tmp = (y * -x) / (1.0 + (-0.5 * (a / (z * (z / t)))));
	} else if (z <= 1.1e+86) {
		tmp = y * ((z * x) / Math.sqrt(((z * z) - (a * t))));
	} else {
		tmp = (y * x) / (1.0 + (-0.5 * ((a / z) * (t / z))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -5.5e+34:
		tmp = (y * -x) / (1.0 + (-0.5 * (a / (z * (z / t)))))
	elif z <= 1.1e+86:
		tmp = y * ((z * x) / math.sqrt(((z * z) - (a * t))))
	else:
		tmp = (y * x) / (1.0 + (-0.5 * ((a / z) * (t / z))))
	return tmp

Julia

x, y = sort([x, y])
t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.5e+34)
		tmp = Float64(Float64(y * Float64(-x)) / Float64(1.0 + Float64(-0.5 * Float64(a / Float64(z * Float64(z / t))))));
	elseif (z <= 1.1e+86)
		tmp = Float64(y * Float64(Float64(z * x) / sqrt(Float64(Float64(z * z) - Float64(a * t)))));
	else
		tmp = Float64(Float64(y * x) / Float64(1.0 + Float64(-0.5 * Float64(Float64(a / z) * Float64(t / z)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -5.5e+34)
		tmp = (y * -x) / (1.0 + (-0.5 * (a / (z * (z / t)))));
	elseif (z <= 1.1e+86)
		tmp = y * ((z * x) / sqrt(((z * z) - (a * t))));
	else
		tmp = (y * x) / (1.0 + (-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.
NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.5e+34], N[(N[(y * (-x)), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(a / N[(z * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.1e+86], N[(y * N[(N[(z * x), $MachinePrecision] / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(N[(a / z), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.4999999999999996e34

   1. Initial program 47.3%

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

      1. associate-/l* 50.8%

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

      2. associate-/l* 50.7%

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

   3. Simplified 50.7%

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

   4. Taylor expanded in z around -inf 83.2%

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

      1. *-commutative 83.2%

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

      2. associate-*r/ 83.2%

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

      3. *-commutative 83.2%

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

      4. associate-*r* 83.2%

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

      5. unpow2 83.2%

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

   6. Simplified 83.2%

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

      1. associate-*l* 83.2%

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

      2. *-commutative 83.2%

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

      3. associate-*r* 83.3%

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

      4. times-frac 85.4%

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

      5. *-commutative 85.4%

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

      6. associate-*l/ 91.5%

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

      7. associate-/r/ 91.5%

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

      8. div-inv 91.5%

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

      9. clear-num 91.5%

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

   8. Applied egg-rr 91.5%

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

   9. Taylor expanded in y around -inf 83.6%

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

       1. associate-*r/ 83.6%

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

       2. mul-1-neg 83.6%

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

       3. distribute-rgt-neg-in 83.6%

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

       4. +-commutative 83.6%

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

       5. associate-/l* 91.8%

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

       6. unpow2 91.8%

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

       7. associate-*r/ 91.8%

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

   11. Simplified 91.8%

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

   if -5.4999999999999996e34 < z < 1.10000000000000002e86

   1. Initial program 84.4%

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

      1. associate-/l* 83.2%

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

      2. associate-*l/ 86.1%

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

      3. associate-/l* 86.6%

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

      4. *-commutative 86.6%

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

   3. Simplified 86.6%

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

   if 1.10000000000000002e86 < z

   1. Initial program 38.5%

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

      1. associate-/l* 42.0%

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

   3. Simplified 42.0%

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

   4. Taylor expanded in z around inf 90.3%

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

      1. associate-/l* 98.5%

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

   6. Simplified 98.5%

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

   7. Taylor expanded in z around 0 88.3%

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

      1. unpow2 88.3%

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

      2. *-commutative 88.3%

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

      3. times-frac 98.5%

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

   9. Simplified 98.5%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+34}:\\ \;\;\;\;\frac{y \cdot \left(-x\right)}{1 + -0.5 \cdot \frac{a}{z \cdot \frac{z}{t}}}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+86}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \left(\frac{a}{z} \cdot \frac{t}{z}\right)}\\ \end{array} \]

Alternative 4: 90.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+153}:\\ \;\;\;\;\frac{y}{-1 - \frac{\frac{a}{z}}{\frac{z}{\frac{t}{-2}}}} \cdot x\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+86}:\\ \;\;\;\;y \cdot \frac{x}{\frac{\sqrt{z \cdot z - a \cdot t}}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \left(\frac{a}{z} \cdot \frac{t}{z}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2e+153)
   (* (/ y (- -1.0 (/ (/ a z) (/ z (/ t -2.0))))) x)
   (if (<= z 1.25e+86)
     (* y (/ x (/ (sqrt (- (* z z) (* a t))) z)))
     (/ (* y x) (+ 1.0 (* -0.5 (* (/ a z) (/ t z))))))))

C

assert(x < y);
assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2e+153) {
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x;
	} else if (z <= 1.25e+86) {
		tmp = y * (x / (sqrt(((z * z) - (a * t))) / z));
	} else {
		tmp = (y * x) / (1.0 + (-0.5 * ((a / z) * (t / z))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a 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+153)) then
        tmp = (y / ((-1.0d0) - ((a / z) / (z / (t / (-2.0d0)))))) * x
    else if (z <= 1.25d+86) then
        tmp = y * (x / (sqrt(((z * z) - (a * t))) / z))
    else
        tmp = (y * x) / (1.0d0 + ((-0.5d0) * ((a / z) * (t / z))))
    end if
    code = tmp
end function

Java

assert x < y;
assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2e+153) {
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x;
	} else if (z <= 1.25e+86) {
		tmp = y * (x / (Math.sqrt(((z * z) - (a * t))) / z));
	} else {
		tmp = (y * x) / (1.0 + (-0.5 * ((a / z) * (t / z))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -2e+153:
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x
	elif z <= 1.25e+86:
		tmp = y * (x / (math.sqrt(((z * z) - (a * t))) / z))
	else:
		tmp = (y * x) / (1.0 + (-0.5 * ((a / z) * (t / z))))
	return tmp

Julia

x, y = sort([x, y])
t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2e+153)
		tmp = Float64(Float64(y / Float64(-1.0 - Float64(Float64(a / z) / Float64(z / Float64(t / -2.0))))) * x);
	elseif (z <= 1.25e+86)
		tmp = Float64(y * Float64(x / Float64(sqrt(Float64(Float64(z * z) - Float64(a * t))) / z)));
	else
		tmp = Float64(Float64(y * x) / Float64(1.0 + Float64(-0.5 * Float64(Float64(a / z) * Float64(t / z)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2e+153)
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x;
	elseif (z <= 1.25e+86)
		tmp = y * (x / (sqrt(((z * z) - (a * t))) / z));
	else
		tmp = (y * x) / (1.0 + (-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.
NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2e+153], N[(N[(y / N[(-1.0 - N[(N[(a / z), $MachinePrecision] / N[(z / N[(t / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 1.25e+86], N[(y * N[(x / N[(N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(N[(a / z), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2e153

   1. Initial program 22.9%

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

      1. associate-/l* 23.6%

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

      2. associate-/l* 23.7%

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

   3. Simplified 23.7%

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

   4. Taylor expanded in z around -inf 79.7%

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

      1. *-commutative 79.7%

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

      2. associate-*r/ 79.7%

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

      3. *-commutative 79.7%

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

      4. associate-*r* 79.7%

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

      5. unpow2 79.7%

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

   6. Simplified 79.7%

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

      1. associate-*l* 79.7%

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

      2. *-commutative 79.7%

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

      3. associate-*r* 79.8%

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

      4. times-frac 83.2%

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

      5. *-commutative 83.2%

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

      6. associate-*l/ 93.2%

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

      7. associate-/r/ 93.2%

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

      8. div-inv 93.2%

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

      9. clear-num 93.2%

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

   8. Applied egg-rr 93.2%

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

   9. Taylor expanded in y around -inf 80.0%

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

       1. associate-*r/ 80.0%

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

       2. mul-1-neg 80.0%

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

       3. distribute-rgt-neg-in 80.0%

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

       4. +-commutative 80.0%

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

       5. associate-/l* 93.5%

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

       6. unpow2 93.5%

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

       7. associate-*r/ 93.5%

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

   11. Simplified 93.5%

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

       1. div-inv 93.5%

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

       2. associate-*l* 93.5%

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

       3. *-commutative 93.5%

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

       4. div-inv 93.5%

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

   13. Applied egg-rr 93.5%

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

   if -2e153 < z < 1.2499999999999999e86

   1. Initial program 84.5%

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

      1. associate-/l* 84.7%

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

      2. associate-*l/ 87.2%

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

   3. Simplified 87.2%

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

   if 1.2499999999999999e86 < z

   1. Initial program 38.5%

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

      1. associate-/l* 42.0%

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

   3. Simplified 42.0%

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

   4. Taylor expanded in z around inf 90.3%

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

      1. associate-/l* 98.5%

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

   6. Simplified 98.5%

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

   7. Taylor expanded in z around 0 88.3%

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

      1. unpow2 88.3%

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

      2. *-commutative 88.3%

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

      3. times-frac 98.5%

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

   9. Simplified 98.5%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+153}:\\ \;\;\;\;\frac{y}{-1 - \frac{\frac{a}{z}}{\frac{z}{\frac{t}{-2}}}} \cdot x\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+86}:\\ \;\;\;\;y \cdot \frac{x}{\frac{\sqrt{z \cdot z - a \cdot t}}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \left(\frac{a}{z} \cdot \frac{t}{z}\right)}\\ \end{array} \]

Alternative 5: 90.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+115}:\\ \;\;\;\;\frac{y}{-1 - \frac{\frac{a}{z}}{\frac{z}{\frac{t}{-2}}}} \cdot x\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{+85}:\\ \;\;\;\;\frac{x}{\frac{\frac{\sqrt{z \cdot z - a \cdot t}}{z}}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \left(\frac{a}{z} \cdot \frac{t}{z}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.8e+115)
   (* (/ y (- -1.0 (/ (/ a z) (/ z (/ t -2.0))))) x)
   (if (<= z 9.8e+85)
     (/ x (/ (/ (sqrt (- (* z z) (* a t))) z) y))
     (/ (* y x) (+ 1.0 (* -0.5 (* (/ a z) (/ t z))))))))

C

assert(x < y);
assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.8e+115) {
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x;
	} else if (z <= 9.8e+85) {
		tmp = x / ((sqrt(((z * z) - (a * t))) / z) / y);
	} else {
		tmp = (y * x) / (1.0 + (-0.5 * ((a / z) * (t / z))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a 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 <= (-3.8d+115)) then
        tmp = (y / ((-1.0d0) - ((a / z) / (z / (t / (-2.0d0)))))) * x
    else if (z <= 9.8d+85) then
        tmp = x / ((sqrt(((z * z) - (a * t))) / z) / y)
    else
        tmp = (y * x) / (1.0d0 + ((-0.5d0) * ((a / z) * (t / z))))
    end if
    code = tmp
end function

Java

assert x < y;
assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.8e+115) {
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x;
	} else if (z <= 9.8e+85) {
		tmp = x / ((Math.sqrt(((z * z) - (a * t))) / z) / y);
	} else {
		tmp = (y * x) / (1.0 + (-0.5 * ((a / z) * (t / z))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.8e+115:
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x
	elif z <= 9.8e+85:
		tmp = x / ((math.sqrt(((z * z) - (a * t))) / z) / y)
	else:
		tmp = (y * x) / (1.0 + (-0.5 * ((a / z) * (t / z))))
	return tmp

Julia

x, y = sort([x, y])
t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.8e+115)
		tmp = Float64(Float64(y / Float64(-1.0 - Float64(Float64(a / z) / Float64(z / Float64(t / -2.0))))) * x);
	elseif (z <= 9.8e+85)
		tmp = Float64(x / Float64(Float64(sqrt(Float64(Float64(z * z) - Float64(a * t))) / z) / y));
	else
		tmp = Float64(Float64(y * x) / Float64(1.0 + Float64(-0.5 * Float64(Float64(a / z) * Float64(t / z)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -3.8e+115)
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x;
	elseif (z <= 9.8e+85)
		tmp = x / ((sqrt(((z * z) - (a * t))) / z) / y);
	else
		tmp = (y * x) / (1.0 + (-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.
NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.8e+115], N[(N[(y / N[(-1.0 - N[(N[(a / z), $MachinePrecision] / N[(z / N[(t / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 9.8e+85], N[(x / N[(N[(N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(N[(a / z), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.8000000000000001e115

   1. Initial program 33.1%

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

      1. associate-/l* 37.6%

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

      2. associate-/l* 37.6%

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

   3. Simplified 37.6%

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

   4. Taylor expanded in z around -inf 83.4%

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

      1. *-commutative 83.4%

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

      2. associate-*r/ 83.4%

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

      3. *-commutative 83.4%

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

      4. associate-*r* 83.4%

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

      5. unpow2 83.4%

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

   6. Simplified 83.4%

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

      1. associate-*l* 83.4%

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

      2. *-commutative 83.4%

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

      3. associate-*r* 83.4%

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

      4. times-frac 86.2%

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

      5. *-commutative 86.2%

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

      6. associate-*l/ 94.4%

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

      7. associate-/r/ 94.4%

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

      8. div-inv 94.4%

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

      9. clear-num 94.4%

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

   8. Applied egg-rr 94.4%

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

   9. Taylor expanded in y around -inf 83.7%

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

       1. associate-*r/ 83.7%

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

       2. mul-1-neg 83.7%

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

       3. distribute-rgt-neg-in 83.7%

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

       4. +-commutative 83.7%

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

       5. associate-/l* 94.7%

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

       6. unpow2 94.7%

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

       7. associate-*r/ 94.7%

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

   11. Simplified 94.7%

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

       1. div-inv 94.7%

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

       2. associate-*l* 94.7%

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

       3. *-commutative 94.7%

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

       4. div-inv 94.7%

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

   13. Applied egg-rr 94.7%

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

   if -3.8000000000000001e115 < z < 9.7999999999999993e85

   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* 83.8%

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

      2. associate-/l* 85.0%

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

   3. Simplified 85.0%

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

   if 9.7999999999999993e85 < z

   1. Initial program 38.5%

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

      1. associate-/l* 42.0%

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

   3. Simplified 42.0%

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

   4. Taylor expanded in z around inf 90.3%

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

      1. associate-/l* 98.5%

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

   6. Simplified 98.5%

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

   7. Taylor expanded in z around 0 88.3%

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

      1. unpow2 88.3%

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

      2. *-commutative 88.3%

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

      3. times-frac 98.5%

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

   9. Simplified 98.5%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+115}:\\ \;\;\;\;\frac{y}{-1 - \frac{\frac{a}{z}}{\frac{z}{\frac{t}{-2}}}} \cdot x\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{+85}:\\ \;\;\;\;\frac{x}{\frac{\frac{\sqrt{z \cdot z - a \cdot t}}{z}}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \left(\frac{a}{z} \cdot \frac{t}{z}\right)}\\ \end{array} \]

Alternative 6: 83.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.16 \cdot 10^{-82}:\\ \;\;\;\;\frac{y}{-1 - \frac{\frac{a}{z}}{\frac{z}{\frac{t}{-2}}}} \cdot x\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-124}:\\ \;\;\;\;x \cdot \left(\left(z \cdot y\right) \cdot {\left(a \cdot \left(-t\right)\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \left(\frac{a}{z} \cdot \frac{t}{z}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.16e-82)
   (* (/ y (- -1.0 (/ (/ a z) (/ z (/ t -2.0))))) x)
   (if (<= z 3e-124)
     (* x (* (* z y) (pow (* a (- t)) -0.5)))
     (/ (* y x) (+ 1.0 (* -0.5 (* (/ a z) (/ t z))))))))

C

assert(x < y);
assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.16e-82) {
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x;
	} else if (z <= 3e-124) {
		tmp = x * ((z * y) * pow((a * -t), -0.5));
	} else {
		tmp = (y * x) / (1.0 + (-0.5 * ((a / z) * (t / z))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.16d-82)) then
        tmp = (y / ((-1.0d0) - ((a / z) / (z / (t / (-2.0d0)))))) * x
    else if (z <= 3d-124) then
        tmp = x * ((z * y) * ((a * -t) ** (-0.5d0)))
    else
        tmp = (y * x) / (1.0d0 + ((-0.5d0) * ((a / z) * (t / z))))
    end if
    code = tmp
end function

Java

assert x < y;
assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.16e-82) {
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x;
	} else if (z <= 3e-124) {
		tmp = x * ((z * y) * Math.pow((a * -t), -0.5));
	} else {
		tmp = (y * x) / (1.0 + (-0.5 * ((a / z) * (t / z))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.16e-82:
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x
	elif z <= 3e-124:
		tmp = x * ((z * y) * math.pow((a * -t), -0.5))
	else:
		tmp = (y * x) / (1.0 + (-0.5 * ((a / z) * (t / z))))
	return tmp

Julia

x, y = sort([x, y])
t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.16e-82)
		tmp = Float64(Float64(y / Float64(-1.0 - Float64(Float64(a / z) / Float64(z / Float64(t / -2.0))))) * x);
	elseif (z <= 3e-124)
		tmp = Float64(x * Float64(Float64(z * y) * (Float64(a * Float64(-t)) ^ -0.5)));
	else
		tmp = Float64(Float64(y * x) / Float64(1.0 + Float64(-0.5 * Float64(Float64(a / z) * Float64(t / z)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.16e-82)
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x;
	elseif (z <= 3e-124)
		tmp = x * ((z * y) * ((a * -t) ^ -0.5));
	else
		tmp = (y * x) / (1.0 + (-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.
NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.16e-82], N[(N[(y / N[(-1.0 - N[(N[(a / z), $MachinePrecision] / N[(z / N[(t / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 3e-124], N[(x * N[(N[(z * y), $MachinePrecision] * N[Power[N[(a * (-t)), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(N[(a / z), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.16e-82

   1. Initial program 57.3%

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

      1. associate-/l* 59.9%

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

      2. associate-/l* 59.8%

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

   3. Simplified 59.8%

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

   4. Taylor expanded in z around -inf 77.8%

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

      1. *-commutative 77.8%

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

      2. associate-*r/ 77.8%

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

      3. *-commutative 77.8%

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

      4. associate-*r* 77.8%

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

      5. unpow2 77.8%

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

   6. Simplified 77.8%

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

      1. associate-*l* 77.8%

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

      2. *-commutative 77.8%

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

      3. associate-*r* 77.8%

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

      4. times-frac 79.4%

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

      5. *-commutative 79.4%

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

      6. associate-*l/ 84.1%

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

      7. associate-/r/ 84.1%

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

      8. div-inv 84.1%

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

      9. clear-num 84.1%

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

   8. Applied egg-rr 84.1%

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

   9. Taylor expanded in y around -inf 78.1%

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

       1. associate-*r/ 78.1%

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

       2. mul-1-neg 78.1%

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

       3. distribute-rgt-neg-in 78.1%

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

       4. +-commutative 78.1%

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

       5. associate-/l* 84.4%

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

       6. unpow2 84.4%

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

       7. associate-*r/ 84.4%

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

   11. Simplified 84.4%

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

       1. div-inv 84.4%

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

       2. associate-*l* 84.4%

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

       3. *-commutative 84.4%

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

       4. div-inv 84.4%

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

   13. Applied egg-rr 84.4%

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

   if -1.16e-82 < z < 3e-124

   1. Initial program 78.3%

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

   2. Taylor expanded in z around 0 76.6%

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

      1. associate-*r* 76.6%

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

      2. neg-mul-1 76.6%

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

      3. *-commutative 76.6%

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

   4. Simplified 76.6%

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

      1. div-inv 76.7%

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

      2. associate-*l* 81.0%

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

      3. associate-*l* 79.1%

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

      4. distribute-rgt-neg-out 79.1%

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

      5. pow1/2 79.2%

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

      6. pow-flip 79.1%

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

      7. distribute-rgt-neg-out 79.1%

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

      8. metadata-eval 79.1%

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

   6. Applied egg-rr 79.1%

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

   if 3e-124 < z

   1. Initial program 61.9%

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

      1. associate-/l* 64.7%

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

   3. Simplified 64.7%

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

   4. Taylor expanded in z around inf 85.5%

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

      1. associate-/l* 89.9%

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

   6. Simplified 89.9%

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

   7. Taylor expanded in z around 0 84.5%

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

      1. unpow2 84.5%

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

      2. *-commutative 84.5%

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

      3. times-frac 89.9%

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

   9. Simplified 89.9%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.16 \cdot 10^{-82}:\\ \;\;\;\;\frac{y}{-1 - \frac{\frac{a}{z}}{\frac{z}{\frac{t}{-2}}}} \cdot x\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-124}:\\ \;\;\;\;x \cdot \left(\left(z \cdot y\right) \cdot {\left(a \cdot \left(-t\right)\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \left(\frac{a}{z} \cdot \frac{t}{z}\right)}\\ \end{array} \]

Alternative 7: 83.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-87}:\\ \;\;\;\;\frac{y}{-1 - \frac{\frac{a}{z}}{\frac{z}{\frac{t}{-2}}}} \cdot x\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-127}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \left(\frac{a}{z} \cdot \frac{t}{z}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.4e-87)
   (* (/ y (- -1.0 (/ (/ a z) (/ z (/ t -2.0))))) x)
   (if (<= z 1.8e-127)
     (* y (/ (* z x) (sqrt (* a (- t)))))
     (/ (* y x) (+ 1.0 (* -0.5 (* (/ a z) (/ t z))))))))

C

assert(x < y);
assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.4e-87) {
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x;
	} else if (z <= 1.8e-127) {
		tmp = y * ((z * x) / sqrt((a * -t)));
	} else {
		tmp = (y * x) / (1.0 + (-0.5 * ((a / z) * (t / z))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a 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 <= (-2.4d-87)) then
        tmp = (y / ((-1.0d0) - ((a / z) / (z / (t / (-2.0d0)))))) * x
    else if (z <= 1.8d-127) then
        tmp = y * ((z * x) / sqrt((a * -t)))
    else
        tmp = (y * x) / (1.0d0 + ((-0.5d0) * ((a / z) * (t / z))))
    end if
    code = tmp
end function

Java

assert x < y;
assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.4e-87) {
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x;
	} else if (z <= 1.8e-127) {
		tmp = y * ((z * x) / Math.sqrt((a * -t)));
	} else {
		tmp = (y * x) / (1.0 + (-0.5 * ((a / z) * (t / z))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.4e-87:
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x
	elif z <= 1.8e-127:
		tmp = y * ((z * x) / math.sqrt((a * -t)))
	else:
		tmp = (y * x) / (1.0 + (-0.5 * ((a / z) * (t / z))))
	return tmp

Julia

x, y = sort([x, y])
t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.4e-87)
		tmp = Float64(Float64(y / Float64(-1.0 - Float64(Float64(a / z) / Float64(z / Float64(t / -2.0))))) * x);
	elseif (z <= 1.8e-127)
		tmp = Float64(y * Float64(Float64(z * x) / sqrt(Float64(a * Float64(-t)))));
	else
		tmp = Float64(Float64(y * x) / Float64(1.0 + Float64(-0.5 * Float64(Float64(a / z) * Float64(t / z)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.4e-87)
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x;
	elseif (z <= 1.8e-127)
		tmp = y * ((z * x) / sqrt((a * -t)));
	else
		tmp = (y * x) / (1.0 + (-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.
NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.4e-87], N[(N[(y / N[(-1.0 - N[(N[(a / z), $MachinePrecision] / N[(z / N[(t / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 1.8e-127], N[(y * N[(N[(z * x), $MachinePrecision] / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(N[(a / z), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.4e-87

   1. Initial program 57.3%

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

      1. associate-/l* 59.9%

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

      2. associate-/l* 59.8%

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

   3. Simplified 59.8%

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

   4. Taylor expanded in z around -inf 77.8%

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

      1. *-commutative 77.8%

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

      2. associate-*r/ 77.8%

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

      3. *-commutative 77.8%

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

      4. associate-*r* 77.8%

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

      5. unpow2 77.8%

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

   6. Simplified 77.8%

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

      1. associate-*l* 77.8%

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

      2. *-commutative 77.8%

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

      3. associate-*r* 77.8%

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

      4. times-frac 79.4%

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

      5. *-commutative 79.4%

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

      6. associate-*l/ 84.1%

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

      7. associate-/r/ 84.1%

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

      8. div-inv 84.1%

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

      9. clear-num 84.1%

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

   8. Applied egg-rr 84.1%

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

   9. Taylor expanded in y around -inf 78.1%

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

       1. associate-*r/ 78.1%

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

       2. mul-1-neg 78.1%

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

       3. distribute-rgt-neg-in 78.1%

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

       4. +-commutative 78.1%

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

       5. associate-/l* 84.4%

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

       6. unpow2 84.4%

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

       7. associate-*r/ 84.4%

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

   11. Simplified 84.4%

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

       1. div-inv 84.4%

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

       2. associate-*l* 84.4%

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

       3. *-commutative 84.4%

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

       4. div-inv 84.4%

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

   13. Applied egg-rr 84.4%

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

   if -2.4e-87 < z < 1.8e-127

   1. Initial program 78.3%

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

   2. Taylor expanded in z around 0 76.6%

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

      1. associate-*r* 76.6%

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

      2. neg-mul-1 76.6%

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

      3. *-commutative 76.6%

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

   4. Simplified 76.6%

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

   5. Taylor expanded in x around 0 81.0%

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

      1. *-commutative 81.0%

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

   7. Simplified 81.0%

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

      1. div-inv 81.0%

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

      2. associate-*l* 79.1%

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

      3. associate-*l* 72.5%

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

      4. un-div-inv 72.5%

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

      5. distribute-rgt-neg-out 72.5%

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

      6. *-commutative 72.5%

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

   9. Applied egg-rr 72.5%

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

       1. associate-*r* 78.1%

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

       2. *-commutative 78.1%

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

       3. associate-*l/ 81.3%

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

       4. associate-*r/ 81.5%

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

       5. distribute-rgt-neg-in 81.5%

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

   11. Simplified 81.5%

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

   if 1.8e-127 < z

   1. Initial program 61.9%

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

      1. associate-/l* 64.7%

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

   3. Simplified 64.7%

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

   4. Taylor expanded in z around inf 85.5%

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

      1. associate-/l* 89.9%

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

   6. Simplified 89.9%

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

   7. Taylor expanded in z around 0 84.5%

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

      1. unpow2 84.5%

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

      2. *-commutative 84.5%

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

      3. times-frac 89.9%

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

   9. Simplified 89.9%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-87}:\\ \;\;\;\;\frac{y}{-1 - \frac{\frac{a}{z}}{\frac{z}{\frac{t}{-2}}}} \cdot x\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-127}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \left(\frac{a}{z} \cdot \frac{t}{z}\right)}\\ \end{array} \]

Alternative 8: 83.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{-87}:\\ \;\;\;\;\frac{y}{-1 - \frac{\frac{a}{z}}{\frac{z}{\frac{t}{-2}}}} \cdot x\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-126}:\\ \;\;\;\;x \cdot \left(z \cdot \frac{y}{\sqrt{a \cdot \left(-t\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \left(\frac{a}{z} \cdot \frac{t}{z}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.2e-87)
   (* (/ y (- -1.0 (/ (/ a z) (/ z (/ t -2.0))))) x)
   (if (<= z 2.1e-126)
     (* x (* z (/ y (sqrt (* a (- t))))))
     (/ (* y x) (+ 1.0 (* -0.5 (* (/ a z) (/ t z))))))))

C

assert(x < y);
assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.2e-87) {
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x;
	} else if (z <= 2.1e-126) {
		tmp = x * (z * (y / sqrt((a * -t))));
	} else {
		tmp = (y * x) / (1.0 + (-0.5 * ((a / z) * (t / z))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a 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 <= (-3.2d-87)) then
        tmp = (y / ((-1.0d0) - ((a / z) / (z / (t / (-2.0d0)))))) * x
    else if (z <= 2.1d-126) then
        tmp = x * (z * (y / sqrt((a * -t))))
    else
        tmp = (y * x) / (1.0d0 + ((-0.5d0) * ((a / z) * (t / z))))
    end if
    code = tmp
end function

Java

assert x < y;
assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.2e-87) {
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x;
	} else if (z <= 2.1e-126) {
		tmp = x * (z * (y / Math.sqrt((a * -t))));
	} else {
		tmp = (y * x) / (1.0 + (-0.5 * ((a / z) * (t / z))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.2e-87:
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x
	elif z <= 2.1e-126:
		tmp = x * (z * (y / math.sqrt((a * -t))))
	else:
		tmp = (y * x) / (1.0 + (-0.5 * ((a / z) * (t / z))))
	return tmp

Julia

x, y = sort([x, y])
t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.2e-87)
		tmp = Float64(Float64(y / Float64(-1.0 - Float64(Float64(a / z) / Float64(z / Float64(t / -2.0))))) * x);
	elseif (z <= 2.1e-126)
		tmp = Float64(x * Float64(z * Float64(y / sqrt(Float64(a * Float64(-t))))));
	else
		tmp = Float64(Float64(y * x) / Float64(1.0 + Float64(-0.5 * Float64(Float64(a / z) * Float64(t / z)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -3.2e-87)
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x;
	elseif (z <= 2.1e-126)
		tmp = x * (z * (y / sqrt((a * -t))));
	else
		tmp = (y * x) / (1.0 + (-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.
NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.2e-87], N[(N[(y / N[(-1.0 - N[(N[(a / z), $MachinePrecision] / N[(z / N[(t / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 2.1e-126], N[(x * N[(z * N[(y / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(N[(a / z), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.19999999999999979e-87

   1. Initial program 57.3%

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

      1. associate-/l* 59.9%

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

      2. associate-/l* 59.8%

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

   3. Simplified 59.8%

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

   4. Taylor expanded in z around -inf 77.8%

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

      1. *-commutative 77.8%

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

      2. associate-*r/ 77.8%

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

      3. *-commutative 77.8%

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

      4. associate-*r* 77.8%

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

      5. unpow2 77.8%

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

   6. Simplified 77.8%

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

      1. associate-*l* 77.8%

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

      2. *-commutative 77.8%

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

      3. associate-*r* 77.8%

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

      4. times-frac 79.4%

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

      5. *-commutative 79.4%

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

      6. associate-*l/ 84.1%

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

      7. associate-/r/ 84.1%

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

      8. div-inv 84.1%

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

      9. clear-num 84.1%

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

   8. Applied egg-rr 84.1%

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

   9. Taylor expanded in y around -inf 78.1%

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

       1. associate-*r/ 78.1%

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

       2. mul-1-neg 78.1%

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

       3. distribute-rgt-neg-in 78.1%

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

       4. +-commutative 78.1%

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

       5. associate-/l* 84.4%

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

       6. unpow2 84.4%

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

       7. associate-*r/ 84.4%

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

   11. Simplified 84.4%

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

       1. div-inv 84.4%

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

       2. associate-*l* 84.4%

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

       3. *-commutative 84.4%

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

       4. div-inv 84.4%

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

   13. Applied egg-rr 84.4%

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

   if -3.19999999999999979e-87 < z < 2.0999999999999999e-126

   1. Initial program 78.3%

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

   2. Taylor expanded in z around 0 76.6%

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

      1. associate-*r* 76.6%

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

      2. neg-mul-1 76.6%

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

      3. *-commutative 76.6%

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

   4. Simplified 76.6%

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

   5. Taylor expanded in x around 0 81.0%

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

      1. *-commutative 81.0%

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

   7. Simplified 81.0%

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

      1. div-inv 81.0%

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

      2. associate-*l* 79.1%

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

      3. *-commutative 79.1%

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

      4. associate-*l* 72.5%

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

      5. un-div-inv 72.5%

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

      6. distribute-rgt-neg-out 72.5%

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

      7. *-commutative 72.5%

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

   9. Applied egg-rr 72.5%

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

   if 2.0999999999999999e-126 < z

   1. Initial program 61.9%

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

      1. associate-/l* 64.7%

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

   3. Simplified 64.7%

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

   4. Taylor expanded in z around inf 85.5%

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

      1. associate-/l* 89.9%

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

   6. Simplified 89.9%

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

   7. Taylor expanded in z around 0 84.5%

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

      1. unpow2 84.5%

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

      2. *-commutative 84.5%

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

      3. times-frac 89.9%

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

   9. Simplified 89.9%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{-87}:\\ \;\;\;\;\frac{y}{-1 - \frac{\frac{a}{z}}{\frac{z}{\frac{t}{-2}}}} \cdot x\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-126}:\\ \;\;\;\;x \cdot \left(z \cdot \frac{y}{\sqrt{a \cdot \left(-t\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \left(\frac{a}{z} \cdot \frac{t}{z}\right)}\\ \end{array} \]

Alternative 9: 83.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{-87}:\\ \;\;\;\;\frac{y}{-1 - \frac{\frac{a}{z}}{\frac{z}{\frac{t}{-2}}}} \cdot x\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-122}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{x}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \left(\frac{a}{z} \cdot \frac{t}{z}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.6e-87)
   (* (/ y (- -1.0 (/ (/ a z) (/ z (/ t -2.0))))) x)
   (if (<= z 3e-122)
     (* (* z y) (/ x (sqrt (* a (- t)))))
     (/ (* y x) (+ 1.0 (* -0.5 (* (/ a z) (/ t z))))))))

C

assert(x < y);
assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.6e-87) {
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x;
	} else if (z <= 3e-122) {
		tmp = (z * y) * (x / sqrt((a * -t)));
	} else {
		tmp = (y * x) / (1.0 + (-0.5 * ((a / z) * (t / z))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-2.6d-87)) then
        tmp = (y / ((-1.0d0) - ((a / z) / (z / (t / (-2.0d0)))))) * x
    else if (z <= 3d-122) then
        tmp = (z * y) * (x / sqrt((a * -t)))
    else
        tmp = (y * x) / (1.0d0 + ((-0.5d0) * ((a / z) * (t / z))))
    end if
    code = tmp
end function

Java

assert x < y;
assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.6e-87) {
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x;
	} else if (z <= 3e-122) {
		tmp = (z * y) * (x / Math.sqrt((a * -t)));
	} else {
		tmp = (y * x) / (1.0 + (-0.5 * ((a / z) * (t / z))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.6e-87:
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x
	elif z <= 3e-122:
		tmp = (z * y) * (x / math.sqrt((a * -t)))
	else:
		tmp = (y * x) / (1.0 + (-0.5 * ((a / z) * (t / z))))
	return tmp

Julia

x, y = sort([x, y])
t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.6e-87)
		tmp = Float64(Float64(y / Float64(-1.0 - Float64(Float64(a / z) / Float64(z / Float64(t / -2.0))))) * x);
	elseif (z <= 3e-122)
		tmp = Float64(Float64(z * y) * Float64(x / sqrt(Float64(a * Float64(-t)))));
	else
		tmp = Float64(Float64(y * x) / Float64(1.0 + Float64(-0.5 * Float64(Float64(a / z) * Float64(t / z)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.6e-87)
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x;
	elseif (z <= 3e-122)
		tmp = (z * y) * (x / sqrt((a * -t)));
	else
		tmp = (y * x) / (1.0 + (-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.
NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.6e-87], N[(N[(y / N[(-1.0 - N[(N[(a / z), $MachinePrecision] / N[(z / N[(t / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 3e-122], N[(N[(z * y), $MachinePrecision] * N[(x / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(N[(a / z), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.60000000000000002e-87

   1. Initial program 57.3%

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

      1. associate-/l* 59.9%

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

      2. associate-/l* 59.8%

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

   3. Simplified 59.8%

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

   4. Taylor expanded in z around -inf 77.8%

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

      1. *-commutative 77.8%

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

      2. associate-*r/ 77.8%

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

      3. *-commutative 77.8%

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

      4. associate-*r* 77.8%

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

      5. unpow2 77.8%

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

   6. Simplified 77.8%

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

      1. associate-*l* 77.8%

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

      2. *-commutative 77.8%

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

      3. associate-*r* 77.8%

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

      4. times-frac 79.4%

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

      5. *-commutative 79.4%

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

      6. associate-*l/ 84.1%

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

      7. associate-/r/ 84.1%

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

      8. div-inv 84.1%

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

      9. clear-num 84.1%

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

   8. Applied egg-rr 84.1%

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

   9. Taylor expanded in y around -inf 78.1%

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

       1. associate-*r/ 78.1%

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

       2. mul-1-neg 78.1%

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

       3. distribute-rgt-neg-in 78.1%

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

       4. +-commutative 78.1%

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

       5. associate-/l* 84.4%

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

       6. unpow2 84.4%

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

       7. associate-*r/ 84.4%

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

   11. Simplified 84.4%

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

       1. div-inv 84.4%

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

       2. associate-*l* 84.4%

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

       3. *-commutative 84.4%

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

       4. div-inv 84.4%

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

   13. Applied egg-rr 84.4%

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

   if -2.60000000000000002e-87 < z < 3.00000000000000004e-122

   1. Initial program 78.3%

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

   2. Taylor expanded in z around 0 76.6%

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

      1. associate-*r* 76.6%

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

      2. neg-mul-1 76.6%

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

      3. *-commutative 76.6%

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

   4. Simplified 76.6%

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

      1. clear-num 75.3%

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

      2. div-inv 75.3%

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

      3. clear-num 76.6%

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

      4. associate-*l* 81.0%

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

      5. associate-/l* 78.8%

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

   6. Applied egg-rr 78.8%

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

      1. *-lft-identity 78.8%

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

      2. associate-/r/ 77.8%

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

   8. Simplified 77.8%

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

   if 3.00000000000000004e-122 < z

   1. Initial program 61.9%

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

      1. associate-/l* 64.7%

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

   3. Simplified 64.7%

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

   4. Taylor expanded in z around inf 85.5%

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

      1. associate-/l* 89.9%

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

   6. Simplified 89.9%

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

   7. Taylor expanded in z around 0 84.5%

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

      1. unpow2 84.5%

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

      2. *-commutative 84.5%

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

      3. times-frac 89.9%

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

   9. Simplified 89.9%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{-87}:\\ \;\;\;\;\frac{y}{-1 - \frac{\frac{a}{z}}{\frac{z}{\frac{t}{-2}}}} \cdot x\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-122}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{x}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \left(\frac{a}{z} \cdot \frac{t}{z}\right)}\\ \end{array} \]

Alternative 10: 83.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-85}:\\ \;\;\;\;\frac{y}{-1 - \frac{\frac{a}{z}}{\frac{z}{\frac{t}{-2}}}} \cdot x\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-122}:\\ \;\;\;\;\frac{x}{\frac{\frac{\sqrt{a \cdot \left(-t\right)}}{z}}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \left(\frac{a}{z} \cdot \frac{t}{z}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5e-85)
   (* (/ y (- -1.0 (/ (/ a z) (/ z (/ t -2.0))))) x)
   (if (<= z 3.1e-122)
     (/ x (/ (/ (sqrt (* a (- t))) z) y))
     (/ (* y x) (+ 1.0 (* -0.5 (* (/ a z) (/ t z))))))))

C

assert(x < y);
assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5e-85) {
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x;
	} else if (z <= 3.1e-122) {
		tmp = x / ((sqrt((a * -t)) / z) / y);
	} else {
		tmp = (y * x) / (1.0 + (-0.5 * ((a / z) * (t / z))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-5d-85)) then
        tmp = (y / ((-1.0d0) - ((a / z) / (z / (t / (-2.0d0)))))) * x
    else if (z <= 3.1d-122) then
        tmp = x / ((sqrt((a * -t)) / z) / y)
    else
        tmp = (y * x) / (1.0d0 + ((-0.5d0) * ((a / z) * (t / z))))
    end if
    code = tmp
end function

Java

assert x < y;
assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5e-85) {
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x;
	} else if (z <= 3.1e-122) {
		tmp = x / ((Math.sqrt((a * -t)) / z) / y);
	} else {
		tmp = (y * x) / (1.0 + (-0.5 * ((a / z) * (t / z))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -5e-85:
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x
	elif z <= 3.1e-122:
		tmp = x / ((math.sqrt((a * -t)) / z) / y)
	else:
		tmp = (y * x) / (1.0 + (-0.5 * ((a / z) * (t / z))))
	return tmp

Julia

x, y = sort([x, y])
t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5e-85)
		tmp = Float64(Float64(y / Float64(-1.0 - Float64(Float64(a / z) / Float64(z / Float64(t / -2.0))))) * x);
	elseif (z <= 3.1e-122)
		tmp = Float64(x / Float64(Float64(sqrt(Float64(a * Float64(-t))) / z) / y));
	else
		tmp = Float64(Float64(y * x) / Float64(1.0 + Float64(-0.5 * Float64(Float64(a / z) * Float64(t / z)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -5e-85)
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x;
	elseif (z <= 3.1e-122)
		tmp = x / ((sqrt((a * -t)) / z) / y);
	else
		tmp = (y * x) / (1.0 + (-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.
NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5e-85], N[(N[(y / N[(-1.0 - N[(N[(a / z), $MachinePrecision] / N[(z / N[(t / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 3.1e-122], N[(x / N[(N[(N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision] / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(N[(a / z), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.0000000000000002e-85

   1. Initial program 57.3%

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

      1. associate-/l* 59.9%

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

      2. associate-/l* 59.8%

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

   3. Simplified 59.8%

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

   4. Taylor expanded in z around -inf 77.8%

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

      1. *-commutative 77.8%

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

      2. associate-*r/ 77.8%

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

      3. *-commutative 77.8%

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

      4. associate-*r* 77.8%

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

      5. unpow2 77.8%

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

   6. Simplified 77.8%

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

      1. associate-*l* 77.8%

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

      2. *-commutative 77.8%

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

      3. associate-*r* 77.8%

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

      4. times-frac 79.4%

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

      5. *-commutative 79.4%

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

      6. associate-*l/ 84.1%

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

      7. associate-/r/ 84.1%

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

      8. div-inv 84.1%

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

      9. clear-num 84.1%

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

   8. Applied egg-rr 84.1%

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

   9. Taylor expanded in y around -inf 78.1%

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

       1. associate-*r/ 78.1%

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

       2. mul-1-neg 78.1%

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

       3. distribute-rgt-neg-in 78.1%

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

       4. +-commutative 78.1%

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

       5. associate-/l* 84.4%

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

       6. unpow2 84.4%

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

       7. associate-*r/ 84.4%

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

   11. Simplified 84.4%

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

       1. div-inv 84.4%

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

       2. associate-*l* 84.4%

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

       3. *-commutative 84.4%

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

       4. div-inv 84.4%

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

   13. Applied egg-rr 84.4%

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

   if -5.0000000000000002e-85 < z < 3.0999999999999998e-122

   1. Initial program 78.3%

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

      1. associate-/l* 73.9%

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

      2. associate-/l* 78.9%

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

   3. Simplified 78.9%

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

   4. Taylor expanded in z around 0 77.2%

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

      1. associate-*r* 76.6%

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

      2. neg-mul-1 76.6%

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

      3. *-commutative 76.6%

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

   6. Simplified 77.2%

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

   if 3.0999999999999998e-122 < z

   1. Initial program 61.9%

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

      1. associate-/l* 64.7%

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

   3. Simplified 64.7%

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

   4. Taylor expanded in z around inf 85.5%

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

      1. associate-/l* 89.9%

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

   6. Simplified 89.9%

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

   7. Taylor expanded in z around 0 84.5%

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

      1. unpow2 84.5%

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

      2. *-commutative 84.5%

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

      3. times-frac 89.9%

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

   9. Simplified 89.9%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-85}:\\ \;\;\;\;\frac{y}{-1 - \frac{\frac{a}{z}}{\frac{z}{\frac{t}{-2}}}} \cdot x\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-122}:\\ \;\;\;\;\frac{x}{\frac{\frac{\sqrt{a \cdot \left(-t\right)}}{z}}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \left(\frac{a}{z} \cdot \frac{t}{z}\right)}\\ \end{array} \]

Alternative 11: 83.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{-87}:\\ \;\;\;\;\frac{y}{-1 - \frac{\frac{a}{z}}{\frac{z}{\frac{t}{-2}}}} \cdot x\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-123}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \left(\frac{a}{z} \cdot \frac{t}{z}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -7.6e-87)
   (* (/ y (- -1.0 (/ (/ a z) (/ z (/ t -2.0))))) x)
   (if (<= z 1.7e-123)
     (/ (* x (* z y)) (sqrt (* a (- t))))
     (/ (* y x) (+ 1.0 (* -0.5 (* (/ a z) (/ t z))))))))

C

assert(x < y);
assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7.6e-87) {
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x;
	} else if (z <= 1.7e-123) {
		tmp = (x * (z * y)) / sqrt((a * -t));
	} else {
		tmp = (y * x) / (1.0 + (-0.5 * ((a / z) * (t / z))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-7.6d-87)) then
        tmp = (y / ((-1.0d0) - ((a / z) / (z / (t / (-2.0d0)))))) * x
    else if (z <= 1.7d-123) then
        tmp = (x * (z * y)) / sqrt((a * -t))
    else
        tmp = (y * x) / (1.0d0 + ((-0.5d0) * ((a / z) * (t / z))))
    end if
    code = tmp
end function

Java

assert x < y;
assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7.6e-87) {
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x;
	} else if (z <= 1.7e-123) {
		tmp = (x * (z * y)) / Math.sqrt((a * -t));
	} else {
		tmp = (y * x) / (1.0 + (-0.5 * ((a / z) * (t / z))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -7.6e-87:
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x
	elif z <= 1.7e-123:
		tmp = (x * (z * y)) / math.sqrt((a * -t))
	else:
		tmp = (y * x) / (1.0 + (-0.5 * ((a / z) * (t / z))))
	return tmp

Julia

x, y = sort([x, y])
t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -7.6e-87)
		tmp = Float64(Float64(y / Float64(-1.0 - Float64(Float64(a / z) / Float64(z / Float64(t / -2.0))))) * x);
	elseif (z <= 1.7e-123)
		tmp = Float64(Float64(x * Float64(z * y)) / sqrt(Float64(a * Float64(-t))));
	else
		tmp = Float64(Float64(y * x) / Float64(1.0 + Float64(-0.5 * Float64(Float64(a / z) * Float64(t / z)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -7.6e-87)
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x;
	elseif (z <= 1.7e-123)
		tmp = (x * (z * y)) / sqrt((a * -t));
	else
		tmp = (y * x) / (1.0 + (-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.
NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -7.6e-87], N[(N[(y / N[(-1.0 - N[(N[(a / z), $MachinePrecision] / N[(z / N[(t / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 1.7e-123], N[(N[(x * N[(z * y), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(N[(a / z), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.6e-87

   1. Initial program 57.3%

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

      1. associate-/l* 59.9%

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

      2. associate-/l* 59.8%

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

   3. Simplified 59.8%

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

   4. Taylor expanded in z around -inf 77.8%

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

      1. *-commutative 77.8%

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

      2. associate-*r/ 77.8%

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

      3. *-commutative 77.8%

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

      4. associate-*r* 77.8%

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

      5. unpow2 77.8%

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

   6. Simplified 77.8%

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

      1. associate-*l* 77.8%

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

      2. *-commutative 77.8%

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

      3. associate-*r* 77.8%

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

      4. times-frac 79.4%

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

      5. *-commutative 79.4%

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

      6. associate-*l/ 84.1%

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

      7. associate-/r/ 84.1%

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

      8. div-inv 84.1%

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

      9. clear-num 84.1%

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

   8. Applied egg-rr 84.1%

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

   9. Taylor expanded in y around -inf 78.1%

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

       1. associate-*r/ 78.1%

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

       2. mul-1-neg 78.1%

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

       3. distribute-rgt-neg-in 78.1%

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

       4. +-commutative 78.1%

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

       5. associate-/l* 84.4%

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

       6. unpow2 84.4%

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

       7. associate-*r/ 84.4%

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

   11. Simplified 84.4%

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

       1. div-inv 84.4%

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

       2. associate-*l* 84.4%

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

       3. *-commutative 84.4%

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

       4. div-inv 84.4%

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

   13. Applied egg-rr 84.4%

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

   if -7.6e-87 < z < 1.7e-123

   1. Initial program 78.3%

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

   2. Taylor expanded in z around 0 76.6%

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

      1. associate-*r* 76.6%

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

      2. neg-mul-1 76.6%

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

      3. *-commutative 76.6%

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

   4. Simplified 76.6%

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

   5. Taylor expanded in x around 0 81.0%

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

      1. *-commutative 81.0%

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

   7. Simplified 81.0%

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

   if 1.7e-123 < z

   1. Initial program 61.9%

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

      1. associate-/l* 64.7%

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

   3. Simplified 64.7%

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

   4. Taylor expanded in z around inf 85.5%

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

      1. associate-/l* 89.9%

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

   6. Simplified 89.9%

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

   7. Taylor expanded in z around 0 84.5%

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

      1. unpow2 84.5%

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

      2. *-commutative 84.5%

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

      3. times-frac 89.9%

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

   9. Simplified 89.9%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{-87}:\\ \;\;\;\;\frac{y}{-1 - \frac{\frac{a}{z}}{\frac{z}{\frac{t}{-2}}}} \cdot x\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-123}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \left(\frac{a}{z} \cdot \frac{t}{z}\right)}\\ \end{array} \]

Alternative 12: 79.0% accurate, 6.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} t_1 := 1 + -0.5 \cdot \left(\frac{a}{z} \cdot \frac{t}{z}\right)\\ \mathbf{if}\;z \leq 5 \cdot 10^{-306}:\\ \;\;\;\;\frac{y \cdot \left(-x\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{t_1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ 1.0 (* -0.5 (* (/ a z) (/ t z))))))
   (if (<= z 5e-306) (/ (* y (- x)) t_1) (/ (* y x) t_1))))

C

assert(x < y);
assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = 1.0 + (-0.5 * ((a / z) * (t / z)));
	double tmp;
	if (z <= 5e-306) {
		tmp = (y * -x) / t_1;
	} else {
		tmp = (y * x) / t_1;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 + ((-0.5d0) * ((a / z) * (t / z)))
    if (z <= 5d-306) then
        tmp = (y * -x) / t_1
    else
        tmp = (y * x) / t_1
    end if
    code = tmp
end function

Java

assert x < y;
assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = 1.0 + (-0.5 * ((a / z) * (t / z)));
	double tmp;
	if (z <= 5e-306) {
		tmp = (y * -x) / t_1;
	} else {
		tmp = (y * x) / t_1;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[t, a] = sort([t, a])
def code(x, y, z, t, a):
	t_1 = 1.0 + (-0.5 * ((a / z) * (t / z)))
	tmp = 0
	if z <= 5e-306:
		tmp = (y * -x) / t_1
	else:
		tmp = (y * x) / t_1
	return tmp

Julia

x, y = sort([x, y])
t, a = sort([t, a])
function code(x, y, z, t, a)
	t_1 = Float64(1.0 + Float64(-0.5 * Float64(Float64(a / z) * Float64(t / z))))
	tmp = 0.0
	if (z <= 5e-306)
		tmp = Float64(Float64(y * Float64(-x)) / t_1);
	else
		tmp = Float64(Float64(y * x) / t_1);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = 1.0 + (-0.5 * ((a / z) * (t / z)));
	tmp = 0.0;
	if (z <= 5e-306)
		tmp = (y * -x) / t_1;
	else
		tmp = (y * x) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(1.0 + N[(-0.5 * N[(N[(a / z), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 5e-306], N[(N[(y * (-x)), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 4.99999999999999998e-306

   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* 64.2%

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

      2. associate-/l* 65.8%

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

   3. Simplified 65.8%

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

   4. Taylor expanded in z around -inf 68.2%

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

      1. *-commutative 68.2%

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

      2. associate-*r/ 68.2%

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

      3. *-commutative 68.2%

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

      4. associate-*r* 68.2%

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

      5. unpow2 68.2%

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

   6. Simplified 68.2%

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

   7. Taylor expanded in y around -inf 68.4%

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

      1. *-commutative 68.4%

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

      2. associate-*r/ 68.4%

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

      3. mul-1-neg 68.4%

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

      4. distribute-rgt-neg-out 68.4%

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

      5. unpow2 68.4%

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

      6. *-commutative 68.4%

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

      7. times-frac 76.5%

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

   9. Simplified 76.5%

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

   if 4.99999999999999998e-306 < z

   1. Initial program 64.9%

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

      1. associate-/l* 66.0%

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

   3. Simplified 66.0%

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

   4. Taylor expanded in z around inf 77.4%

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

      1. associate-/l* 80.9%

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

   6. Simplified 80.9%

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

   7. Taylor expanded in z around 0 76.3%

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

      1. unpow2 76.3%

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

      2. *-commutative 76.3%

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

      3. times-frac 80.9%

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

   9. Simplified 80.9%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{-306}:\\ \;\;\;\;\frac{y \cdot \left(-x\right)}{1 + -0.5 \cdot \left(\frac{a}{z} \cdot \frac{t}{z}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \left(\frac{a}{z} \cdot \frac{t}{z}\right)}\\ \end{array} \]

Alternative 13: 74.6% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -5.3 \cdot 10^{-107}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-137}:\\ \;\;\;\;2 \cdot \left(\frac{x}{a} \cdot \left(y \cdot \frac{z \cdot z}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.3e-107)
   (* y (- x))
   (if (<= z 1.45e-137) (* 2.0 (* (/ x a) (* y (/ (* z z) t)))) (* y x))))

C

assert(x < y);
assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.3e-107) {
		tmp = y * -x;
	} else if (z <= 1.45e-137) {
		tmp = 2.0 * ((x / a) * (y * ((z * z) / t)));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a 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 <= (-5.3d-107)) then
        tmp = y * -x
    else if (z <= 1.45d-137) then
        tmp = 2.0d0 * ((x / a) * (y * ((z * z) / t)))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.3e-107) {
		tmp = y * -x;
	} else if (z <= 1.45e-137) {
		tmp = 2.0 * ((x / a) * (y * ((z * z) / t)));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -5.3e-107:
		tmp = y * -x
	elif z <= 1.45e-137:
		tmp = 2.0 * ((x / a) * (y * ((z * z) / t)))
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.3e-107)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.45e-137)
		tmp = Float64(2.0 * Float64(Float64(x / a) * Float64(y * Float64(Float64(z * z) / t))));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -5.3e-107)
		tmp = y * -x;
	elseif (z <= 1.45e-137)
		tmp = 2.0 * ((x / a) * (y * ((z * z) / t)));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.3e-107], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.45e-137], N[(2.0 * N[(N[(x / a), $MachinePrecision] * N[(y * N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.3e-107

   1. Initial program 57.9%

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

      1. associate-/l* 60.3%

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

      2. associate-*l/ 60.4%

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

      3. associate-/l* 59.0%

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

      4. *-commutative 59.0%

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

   3. Simplified 59.0%

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

   4. Taylor expanded in z around -inf 79.6%

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

      1. mul-1-neg 79.6%

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

   6. Simplified 79.6%

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

   if -5.3e-107 < z < 1.44999999999999993e-137

   1. Initial program 79.2%

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

      1. associate-/l* 74.2%

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

      2. associate-/l* 79.9%

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

   3. Simplified 79.9%

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

   4. Taylor expanded in z around -inf 43.5%

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

      1. *-commutative 43.5%

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

      2. associate-*r/ 43.5%

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

      3. *-commutative 43.5%

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

      4. associate-*r* 43.5%

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

      5. unpow2 43.5%

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

   6. Simplified 43.5%

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

      1. times-frac 43.3%

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

      2. associate-*l/ 43.8%

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

      3. sub-div 43.8%

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

      4. *-commutative 43.8%

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

   8. Applied egg-rr 43.8%

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

   9. Taylor expanded in a around inf 43.5%

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

       1. times-frac 43.6%

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

       2. unpow2 43.6%

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

       3. associate-*r/ 43.5%

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

   11. Simplified 43.5%

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

   if 1.44999999999999993e-137 < z

   1. Initial program 62.3%

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

      1. associate-/l* 65.0%

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

      2. associate-/l* 64.1%

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

   3. Simplified 64.1%

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

   4. Taylor expanded in z around inf 86.2%

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

      1. *-commutative 86.2%

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

   6. Simplified 86.2%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.3 \cdot 10^{-107}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-137}:\\ \;\;\;\;2 \cdot \left(\frac{x}{a} \cdot \left(y \cdot \frac{z \cdot z}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 14: 74.8% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{-107}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{-136}:\\ \;\;\;\;y \cdot \left(-2 \cdot \frac{x}{t \cdot \frac{\frac{a}{z}}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.35e-107)
   (* y (- x))
   (if (<= z 8.2e-136) (* y (* -2.0 (/ x (* t (/ (/ a z) z))))) (* y x))))

C

assert(x < y);
assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.35e-107) {
		tmp = y * -x;
	} else if (z <= 8.2e-136) {
		tmp = y * (-2.0 * (x / (t * ((a / z) / z))));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a 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 <= (-1.35d-107)) then
        tmp = y * -x
    else if (z <= 8.2d-136) then
        tmp = y * ((-2.0d0) * (x / (t * ((a / z) / z))))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.35e-107) {
		tmp = y * -x;
	} else if (z <= 8.2e-136) {
		tmp = y * (-2.0 * (x / (t * ((a / z) / z))));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.35e-107:
		tmp = y * -x
	elif z <= 8.2e-136:
		tmp = y * (-2.0 * (x / (t * ((a / z) / z))))
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.35e-107)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 8.2e-136)
		tmp = Float64(y * Float64(-2.0 * Float64(x / Float64(t * Float64(Float64(a / z) / z)))));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.35e-107)
		tmp = y * -x;
	elseif (z <= 8.2e-136)
		tmp = y * (-2.0 * (x / (t * ((a / z) / z))));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.35e-107], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 8.2e-136], N[(y * N[(-2.0 * N[(x / N[(t * N[(N[(a / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.35e-107

   1. Initial program 57.9%

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

      1. associate-/l* 60.3%

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

      2. associate-*l/ 60.4%

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

      3. associate-/l* 59.0%

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

      4. *-commutative 59.0%

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

   3. Simplified 59.0%

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

   4. Taylor expanded in z around -inf 79.6%

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

      1. mul-1-neg 79.6%

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

   6. Simplified 79.6%

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

   if -1.35e-107 < z < 8.20000000000000051e-136

   1. Initial program 79.2%

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

      1. associate-/l* 74.2%

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

      2. associate-*l/ 81.7%

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

   3. Simplified 81.7%

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

   4. Taylor expanded in z around inf 43.9%

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

      1. associate-/l* 43.8%

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

   6. Simplified 44.0%

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

      1. associate-/r/ 44.0%

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

   8. Applied egg-rr 44.0%

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

   9. Taylor expanded in z around 0 43.4%

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

       1. associate-*r/ 43.4%

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

       2. unpow2 43.4%

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

       3. associate-/r* 43.8%

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

       4. associate-*r/ 43.8%

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

       5. *-commutative 43.8%

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

       6. associate-*r/ 43.9%

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

       7. *-commutative 43.9%

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

       8. associate-/r/ 44.0%

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

       9. associate-*l/ 44.0%

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

   11. Simplified 44.0%

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

   12. Taylor expanded in x around 0 43.5%

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

       1. times-frac 43.5%

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

       2. unpow2 43.5%

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

       3. associate-/l* 43.5%

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

       4. times-frac 44.0%

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

       5. associate-/l* 43.9%

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

       6. *-commutative 43.9%

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

       7. associate-*r/ 44.0%

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

       8. associate-*l/ 44.0%

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

       9. associate-*r/ 43.9%

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

   14. Simplified 43.9%

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

   if 8.20000000000000051e-136 < z

   1. Initial program 62.3%

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

      1. associate-/l* 65.0%

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

      2. associate-/l* 64.1%

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

   3. Simplified 64.1%

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

   4. Taylor expanded in z around inf 86.2%

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

      1. *-commutative 86.2%

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

   6. Simplified 86.2%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{-107}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{-136}:\\ \;\;\;\;y \cdot \left(-2 \cdot \frac{x}{t \cdot \frac{\frac{a}{z}}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 15: 74.6% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{-107}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-131}:\\ \;\;\;\;y \cdot \left(z \cdot \left(\frac{x}{a} \cdot \left(-2 \cdot \frac{z}{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4.1e-107)
   (* y (- x))
   (if (<= z 1.3e-131) (* y (* z (* (/ x a) (* -2.0 (/ z t))))) (* y x))))

C

assert(x < y);
assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.1e-107) {
		tmp = y * -x;
	} else if (z <= 1.3e-131) {
		tmp = y * (z * ((x / a) * (-2.0 * (z / t))));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a 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 <= (-4.1d-107)) then
        tmp = y * -x
    else if (z <= 1.3d-131) then
        tmp = y * (z * ((x / a) * ((-2.0d0) * (z / t))))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.1e-107) {
		tmp = y * -x;
	} else if (z <= 1.3e-131) {
		tmp = y * (z * ((x / a) * (-2.0 * (z / t))));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -4.1e-107:
		tmp = y * -x
	elif z <= 1.3e-131:
		tmp = y * (z * ((x / a) * (-2.0 * (z / t))))
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4.1e-107)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.3e-131)
		tmp = Float64(y * Float64(z * Float64(Float64(x / a) * Float64(-2.0 * Float64(z / t)))));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -4.1e-107)
		tmp = y * -x;
	elseif (z <= 1.3e-131)
		tmp = y * (z * ((x / a) * (-2.0 * (z / t))));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4.1e-107], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.3e-131], N[(y * N[(z * N[(N[(x / a), $MachinePrecision] * N[(-2.0 * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.0999999999999999e-107

   1. Initial program 57.9%

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

      1. associate-/l* 60.3%

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

      2. associate-*l/ 60.4%

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

      3. associate-/l* 59.0%

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

      4. *-commutative 59.0%

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

   3. Simplified 59.0%

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

   4. Taylor expanded in z around -inf 79.6%

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

      1. mul-1-neg 79.6%

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

   6. Simplified 79.6%

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

   if -4.0999999999999999e-107 < z < 1.29999999999999998e-131

   1. Initial program 79.2%

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

      1. associate-/l* 74.2%

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

      2. associate-*l/ 81.7%

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

   3. Simplified 81.7%

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

   4. Taylor expanded in z around inf 43.9%

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

      1. associate-/l* 43.8%

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

   6. Simplified 44.0%

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

      1. associate-/r/ 44.0%

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

   8. Applied egg-rr 44.0%

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

   9. Taylor expanded in z around 0 43.4%

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

       1. associate-*r/ 43.4%

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

       2. unpow2 43.4%

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

       3. associate-/r* 43.8%

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

       4. associate-*r/ 43.8%

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

       5. *-commutative 43.8%

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

       6. associate-*r/ 43.9%

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

       7. *-commutative 43.9%

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

       8. associate-/r/ 44.0%

          \[\leadsto \frac{x}{\frac{\color{blue}{\frac{t}{\frac{z}{a}}} \cdot -0.5}{z}} \cdot y \]

       9. associate-*l/ 44.0%

          \[\leadsto \frac{x}{\frac{\color{blue}{\frac{t \cdot -0.5}{\frac{z}{a}}}}{z}} \cdot y \]

   11. Simplified 44.0%

       \[\leadsto \frac{x}{\color{blue}{\frac{\frac{t \cdot -0.5}{\frac{z}{a}}}{z}}} \cdot y \]

   12. Taylor expanded in x around 0 43.5%

       \[\leadsto \color{blue}{\left(-2 \cdot \frac{x \cdot {z}^{2}}{a \cdot t}\right)} \cdot y \]
   13. Step-by-step derivation

       1. *-commutative 43.5%

          \[\leadsto \color{blue}{\left(\frac{x \cdot {z}^{2}}{a \cdot t} \cdot -2\right)} \cdot y \]

       2. associate-/r* 43.8%

          \[\leadsto \left(\color{blue}{\frac{\frac{x \cdot {z}^{2}}{a}}{t}} \cdot -2\right) \cdot y \]

       3. unpow2 43.8%

          \[\leadsto \left(\frac{\frac{x \cdot \color{blue}{\left(z \cdot z\right)}}{a}}{t} \cdot -2\right) \cdot y \]

       4. associate-*r/ 43.8%

          \[\leadsto \left(\frac{\color{blue}{x \cdot \frac{z \cdot z}{a}}}{t} \cdot -2\right) \cdot y \]

       5. associate-*r/ 44.0%

          \[\leadsto \left(\frac{x \cdot \color{blue}{\left(z \cdot \frac{z}{a}\right)}}{t} \cdot -2\right) \cdot y \]

       6. associate-/r/ 44.0%

          \[\leadsto \color{blue}{\frac{x \cdot \left(z \cdot \frac{z}{a}\right)}{\frac{t}{-2}}} \cdot y \]

       7. associate-*l/ 43.7%

          \[\leadsto \color{blue}{\left(\frac{x}{\frac{t}{-2}} \cdot \left(z \cdot \frac{z}{a}\right)\right)} \cdot y \]

       8. *-commutative 43.7%

          \[\leadsto \left(\frac{x}{\frac{t}{-2}} \cdot \color{blue}{\left(\frac{z}{a} \cdot z\right)}\right) \cdot y \]

       9. associate-/r/ 43.7%

          \[\leadsto \left(\frac{x}{\frac{t}{-2}} \cdot \color{blue}{\frac{z}{\frac{a}{z}}}\right) \cdot y \]

       10. associate-/r/ 43.9%

           \[\leadsto \color{blue}{\frac{x}{\frac{\frac{t}{-2}}{\frac{z}{\frac{a}{z}}}}} \cdot y \]

       11. associate-/l* 44.0%

           \[\leadsto \frac{x}{\color{blue}{\frac{\frac{t}{-2} \cdot \frac{a}{z}}{z}}} \cdot y \]

       12. associate-/r/ 44.0%

           \[\leadsto \color{blue}{\left(\frac{x}{\frac{t}{-2} \cdot \frac{a}{z}} \cdot z\right)} \cdot y \]

       13. *-commutative 44.0%

           \[\leadsto \color{blue}{\left(z \cdot \frac{x}{\frac{t}{-2} \cdot \frac{a}{z}}\right)} \cdot y \]

       14. *-commutative 44.0%

           \[\leadsto \left(z \cdot \frac{x}{\color{blue}{\frac{a}{z} \cdot \frac{t}{-2}}}\right) \cdot y \]

       15. associate-/r/ 44.0%

           \[\leadsto \left(z \cdot \frac{x}{\color{blue}{\frac{a}{\frac{z}{\frac{t}{-2}}}}}\right) \cdot y \]

       16. associate-/r/ 43.9%

           \[\leadsto \left(z \cdot \color{blue}{\left(\frac{x}{a} \cdot \frac{z}{\frac{t}{-2}}\right)}\right) \cdot y \]

       17. associate-/r/ 43.9%

           \[\leadsto \left(z \cdot \left(\frac{x}{a} \cdot \color{blue}{\left(\frac{z}{t} \cdot -2\right)}\right)\right) \cdot y \]

   14. Simplified 43.9%

       \[\leadsto \color{blue}{\left(z \cdot \left(\frac{x}{a} \cdot \left(\frac{z}{t} \cdot -2\right)\right)\right)} \cdot y \]

   if 1.29999999999999998e-131 < z

   1. Initial program 62.3%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. associate-/l* 65.0%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

      2. associate-/l* 64.1%

         \[\leadsto \color{blue}{\frac{x}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}{y}}} \]

   3. Simplified 64.1%

      \[\leadsto \color{blue}{\frac{x}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}{y}}} \]

   4. Taylor expanded in z around inf 86.2%

      \[\leadsto \color{blue}{x \cdot y} \]
   5. Step-by-step derivation

      1. *-commutative 86.2%

         \[\leadsto \color{blue}{y \cdot x} \]

   6. Simplified 86.2%

      \[\leadsto \color{blue}{y \cdot x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{-107}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-131}:\\ \;\;\;\;y \cdot \left(z \cdot \left(\frac{x}{a} \cdot \left(-2 \cdot \frac{z}{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 16: 74.8% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{-107}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-137}:\\ \;\;\;\;y \cdot \frac{x}{-0.5 \cdot \left(\frac{a}{z} \cdot \frac{t}{z}\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.35e-107)
   (* y (- x))
   (if (<= z 1.55e-137) (* y (/ x (* -0.5 (* (/ a z) (/ t z))))) (* y x))))

C

assert(x < y);
assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.35e-107) {
		tmp = y * -x;
	} else if (z <= 1.55e-137) {
		tmp = y * (x / (-0.5 * ((a / z) * (t / z))));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a 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 <= (-1.35d-107)) then
        tmp = y * -x
    else if (z <= 1.55d-137) then
        tmp = y * (x / ((-0.5d0) * ((a / z) * (t / z))))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.35e-107) {
		tmp = y * -x;
	} else if (z <= 1.55e-137) {
		tmp = y * (x / (-0.5 * ((a / z) * (t / z))));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.35e-107:
		tmp = y * -x
	elif z <= 1.55e-137:
		tmp = y * (x / (-0.5 * ((a / z) * (t / z))))
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.35e-107)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.55e-137)
		tmp = Float64(y * Float64(x / Float64(-0.5 * Float64(Float64(a / z) * Float64(t / z)))));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.35e-107)
		tmp = y * -x;
	elseif (z <= 1.55e-137)
		tmp = y * (x / (-0.5 * ((a / z) * (t / z))));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.35e-107], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.55e-137], N[(y * N[(x / N[(-0.5 * N[(N[(a / z), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.35e-107

   1. Initial program 57.9%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. associate-/l* 60.3%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

      2. associate-*l/ 60.4%

         \[\leadsto \color{blue}{\frac{x}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \cdot y} \]

      3. associate-/l* 59.0%

         \[\leadsto \color{blue}{\frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \cdot y \]

      4. *-commutative 59.0%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 59.0%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around -inf 79.6%

      \[\leadsto y \cdot \color{blue}{\left(-1 \cdot x\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 79.6%

         \[\leadsto y \cdot \color{blue}{\left(-x\right)} \]

   6. Simplified 79.6%

      \[\leadsto y \cdot \color{blue}{\left(-x\right)} \]

   if -1.35e-107 < z < 1.54999999999999989e-137

   1. Initial program 79.2%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. associate-/l* 74.2%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

      2. associate-*l/ 81.7%

         \[\leadsto \color{blue}{\frac{x}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \cdot y} \]

   3. Simplified 81.7%

      \[\leadsto \color{blue}{\frac{x}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \cdot y} \]

   4. Taylor expanded in z around inf 43.9%

      \[\leadsto \frac{x}{\frac{\color{blue}{z + -0.5 \cdot \frac{a \cdot t}{z}}}{z}} \cdot y \]
   5. Step-by-step derivation

      1. associate-/l* 43.8%

         \[\leadsto \frac{x \cdot y}{\frac{z + -0.5 \cdot \color{blue}{\frac{a}{\frac{z}{t}}}}{z}} \]

   6. Simplified 44.0%

      \[\leadsto \frac{x}{\frac{\color{blue}{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}}{z}} \cdot y \]

   7. Taylor expanded in z around 0 43.4%

      \[\leadsto \frac{x}{\color{blue}{-0.5 \cdot \frac{a \cdot t}{{z}^{2}}}} \cdot y \]
   8. Step-by-step derivation

      1. unpow2 43.4%

         \[\leadsto \frac{x}{-0.5 \cdot \frac{a \cdot t}{\color{blue}{z \cdot z}}} \cdot y \]

      2. *-commutative 43.4%

         \[\leadsto \frac{x}{-0.5 \cdot \frac{\color{blue}{t \cdot a}}{z \cdot z}} \cdot y \]

      3. times-frac 44.0%

         \[\leadsto \frac{x}{-0.5 \cdot \color{blue}{\left(\frac{t}{z} \cdot \frac{a}{z}\right)}} \cdot y \]

   9. Simplified 44.0%

      \[\leadsto \frac{x}{\color{blue}{-0.5 \cdot \left(\frac{t}{z} \cdot \frac{a}{z}\right)}} \cdot y \]

   if 1.54999999999999989e-137 < z

   1. Initial program 62.3%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. associate-/l* 65.0%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

      2. associate-/l* 64.1%

         \[\leadsto \color{blue}{\frac{x}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}{y}}} \]

   3. Simplified 64.1%

      \[\leadsto \color{blue}{\frac{x}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}{y}}} \]

   4. Taylor expanded in z around inf 86.2%

      \[\leadsto \color{blue}{x \cdot y} \]
   5. Step-by-step derivation

      1. *-commutative 86.2%

         \[\leadsto \color{blue}{y \cdot x} \]

   6. Simplified 86.2%

      \[\leadsto \color{blue}{y \cdot x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{-107}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-137}:\\ \;\;\;\;y \cdot \frac{x}{-0.5 \cdot \left(\frac{a}{z} \cdot \frac{t}{z}\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 17: 75.0% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{-105}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-137}:\\ \;\;\;\;\frac{x}{\frac{\frac{a}{z}}{\frac{\frac{z \cdot y}{0.5}}{t}}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.35e-105)
   (* y (- x))
   (if (<= z 1.9e-137) (/ x (/ (/ a z) (/ (/ (* z y) 0.5) t))) (* y x))))

C

assert(x < y);
assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.35e-105) {
		tmp = y * -x;
	} else if (z <= 1.9e-137) {
		tmp = x / ((a / z) / (((z * y) / 0.5) / t));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a 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 <= (-1.35d-105)) then
        tmp = y * -x
    else if (z <= 1.9d-137) then
        tmp = x / ((a / z) / (((z * y) / 0.5d0) / t))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.35e-105) {
		tmp = y * -x;
	} else if (z <= 1.9e-137) {
		tmp = x / ((a / z) / (((z * y) / 0.5) / t));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.35e-105:
		tmp = y * -x
	elif z <= 1.9e-137:
		tmp = x / ((a / z) / (((z * y) / 0.5) / t))
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.35e-105)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.9e-137)
		tmp = Float64(x / Float64(Float64(a / z) / Float64(Float64(Float64(z * y) / 0.5) / t)));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.35e-105)
		tmp = y * -x;
	elseif (z <= 1.9e-137)
		tmp = x / ((a / z) / (((z * y) / 0.5) / t));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.35e-105], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.9e-137], N[(x / N[(N[(a / z), $MachinePrecision] / N[(N[(N[(z * y), $MachinePrecision] / 0.5), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.34999999999999996e-105

   1. Initial program 57.9%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. associate-/l* 60.3%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

      2. associate-*l/ 60.4%

         \[\leadsto \color{blue}{\frac{x}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \cdot y} \]

      3. associate-/l* 59.0%

         \[\leadsto \color{blue}{\frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \cdot y \]

      4. *-commutative 59.0%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 59.0%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around -inf 79.6%

      \[\leadsto y \cdot \color{blue}{\left(-1 \cdot x\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 79.6%

         \[\leadsto y \cdot \color{blue}{\left(-x\right)} \]

   6. Simplified 79.6%

      \[\leadsto y \cdot \color{blue}{\left(-x\right)} \]

   if -1.34999999999999996e-105 < z < 1.89999999999999999e-137

   1. Initial program 79.2%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. associate-/l* 74.2%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

      2. associate-/l* 79.9%

         \[\leadsto \color{blue}{\frac{x}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}{y}}} \]

   3. Simplified 79.9%

      \[\leadsto \color{blue}{\frac{x}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}{y}}} \]

   4. Taylor expanded in z around -inf 43.5%

      \[\leadsto \frac{x}{\color{blue}{0.5 \cdot \frac{a \cdot t}{y \cdot {z}^{2}} - \frac{1}{y}}} \]
   5. Step-by-step derivation

      1. *-commutative 43.5%

         \[\leadsto \frac{x}{0.5 \cdot \frac{\color{blue}{t \cdot a}}{y \cdot {z}^{2}} - \frac{1}{y}} \]

      2. associate-*r/ 43.5%

         \[\leadsto \frac{x}{\color{blue}{\frac{0.5 \cdot \left(t \cdot a\right)}{y \cdot {z}^{2}}} - \frac{1}{y}} \]

      3. *-commutative 43.5%

         \[\leadsto \frac{x}{\frac{0.5 \cdot \color{blue}{\left(a \cdot t\right)}}{y \cdot {z}^{2}} - \frac{1}{y}} \]

      4. associate-*r* 43.5%

         \[\leadsto \frac{x}{\frac{\color{blue}{\left(0.5 \cdot a\right) \cdot t}}{y \cdot {z}^{2}} - \frac{1}{y}} \]

      5. unpow2 43.5%

         \[\leadsto \frac{x}{\frac{\left(0.5 \cdot a\right) \cdot t}{y \cdot \color{blue}{\left(z \cdot z\right)}} - \frac{1}{y}} \]

   6. Simplified 43.5%

      \[\leadsto \frac{x}{\color{blue}{\frac{\left(0.5 \cdot a\right) \cdot t}{y \cdot \left(z \cdot z\right)} - \frac{1}{y}}} \]
   7. Step-by-step derivation

      1. times-frac 43.3%

         \[\leadsto \frac{x}{\color{blue}{\frac{0.5 \cdot a}{y} \cdot \frac{t}{z \cdot z}} - \frac{1}{y}} \]

      2. associate-*l/ 43.8%

         \[\leadsto \frac{x}{\color{blue}{\frac{\left(0.5 \cdot a\right) \cdot \frac{t}{z \cdot z}}{y}} - \frac{1}{y}} \]

      3. sub-div 43.8%

         \[\leadsto \frac{x}{\color{blue}{\frac{\left(0.5 \cdot a\right) \cdot \frac{t}{z \cdot z} - 1}{y}}} \]

      4. *-commutative 43.8%

         \[\leadsto \frac{x}{\frac{\color{blue}{\left(a \cdot 0.5\right)} \cdot \frac{t}{z \cdot z} - 1}{y}} \]

   8. Applied egg-rr 43.8%

      \[\leadsto \frac{x}{\color{blue}{\frac{\left(a \cdot 0.5\right) \cdot \frac{t}{z \cdot z} - 1}{y}}} \]

   9. Taylor expanded in a around inf 43.5%

      \[\leadsto \frac{x}{\color{blue}{0.5 \cdot \frac{a \cdot t}{y \cdot {z}^{2}}}} \]
   10. Step-by-step derivation

       1. *-commutative 43.5%

          \[\leadsto \frac{x}{\color{blue}{\frac{a \cdot t}{y \cdot {z}^{2}} \cdot 0.5}} \]

       2. unpow2 43.5%

          \[\leadsto \frac{x}{\frac{a \cdot t}{y \cdot \color{blue}{\left(z \cdot z\right)}} \cdot 0.5} \]

       3. associate-*l/ 43.5%

          \[\leadsto \frac{x}{\color{blue}{\frac{\left(a \cdot t\right) \cdot 0.5}{y \cdot \left(z \cdot z\right)}}} \]

       4. associate-*r* 43.5%

          \[\leadsto \frac{x}{\frac{\color{blue}{a \cdot \left(t \cdot 0.5\right)}}{y \cdot \left(z \cdot z\right)}} \]

       5. *-commutative 43.5%

          \[\leadsto \frac{x}{\frac{a \cdot \left(t \cdot 0.5\right)}{\color{blue}{\left(z \cdot z\right) \cdot y}}} \]

       6. associate-*r* 47.8%

          \[\leadsto \frac{x}{\frac{a \cdot \left(t \cdot 0.5\right)}{\color{blue}{z \cdot \left(z \cdot y\right)}}} \]

       7. associate-/r* 47.8%

          \[\leadsto \frac{x}{\color{blue}{\frac{\frac{a \cdot \left(t \cdot 0.5\right)}{z}}{z \cdot y}}} \]

       8. associate-*l/ 47.9%

          \[\leadsto \frac{x}{\frac{\color{blue}{\frac{a}{z} \cdot \left(t \cdot 0.5\right)}}{z \cdot y}} \]

       9. associate-/l* 47.8%

          \[\leadsto \frac{x}{\color{blue}{\frac{\frac{a}{z}}{\frac{z \cdot y}{t \cdot 0.5}}}} \]

       10. *-commutative 47.8%

           \[\leadsto \frac{x}{\frac{\frac{a}{z}}{\frac{z \cdot y}{\color{blue}{0.5 \cdot t}}}} \]

       11. associate-/r* 47.8%

           \[\leadsto \frac{x}{\frac{\frac{a}{z}}{\color{blue}{\frac{\frac{z \cdot y}{0.5}}{t}}}} \]

   11. Simplified 47.8%

       \[\leadsto \frac{x}{\color{blue}{\frac{\frac{a}{z}}{\frac{\frac{z \cdot y}{0.5}}{t}}}} \]

   if 1.89999999999999999e-137 < z

   1. Initial program 62.3%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. associate-/l* 65.0%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

      2. associate-/l* 64.1%

         \[\leadsto \color{blue}{\frac{x}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}{y}}} \]

   3. Simplified 64.1%

      \[\leadsto \color{blue}{\frac{x}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}{y}}} \]

   4. Taylor expanded in z around inf 86.2%

      \[\leadsto \color{blue}{x \cdot y} \]
   5. Step-by-step derivation

      1. *-commutative 86.2%

         \[\leadsto \color{blue}{y \cdot x} \]

   6. Simplified 86.2%

      \[\leadsto \color{blue}{y \cdot x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{-105}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-137}:\\ \;\;\;\;\frac{x}{\frac{\frac{a}{z}}{\frac{\frac{z \cdot y}{0.5}}{t}}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 18: 76.4% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{-107}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\frac{z + -0.5 \cdot \left(\frac{a}{z} \cdot t\right)}{z}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.35e-107)
   (* y (- x))
   (* y (/ x (/ (+ z (* -0.5 (* (/ a z) t))) z)))))

C

assert(x < y);
assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.35e-107) {
		tmp = y * -x;
	} else {
		tmp = y * (x / ((z + (-0.5 * ((a / z) * t))) / z));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a 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 <= (-1.35d-107)) then
        tmp = y * -x
    else
        tmp = y * (x / ((z + ((-0.5d0) * ((a / z) * t))) / z))
    end if
    code = tmp
end function

Java

assert x < y;
assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.35e-107) {
		tmp = y * -x;
	} else {
		tmp = y * (x / ((z + (-0.5 * ((a / z) * t))) / z));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.35e-107:
		tmp = y * -x
	else:
		tmp = y * (x / ((z + (-0.5 * ((a / z) * t))) / z))
	return tmp

Julia

x, y = sort([x, y])
t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.35e-107)
		tmp = Float64(y * Float64(-x));
	else
		tmp = Float64(y * Float64(x / Float64(Float64(z + Float64(-0.5 * Float64(Float64(a / z) * t))) / z)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.35e-107)
		tmp = y * -x;
	else
		tmp = y * (x / ((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.
NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.35e-107], N[(y * (-x)), $MachinePrecision], N[(y * N[(x / N[(N[(z + N[(-0.5 * N[(N[(a / z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.35e-107

   1. Initial program 57.9%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. associate-/l* 60.3%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

      2. associate-*l/ 60.4%

         \[\leadsto \color{blue}{\frac{x}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \cdot y} \]

      3. associate-/l* 59.0%

         \[\leadsto \color{blue}{\frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \cdot y \]

      4. *-commutative 59.0%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 59.0%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around -inf 79.6%

      \[\leadsto y \cdot \color{blue}{\left(-1 \cdot x\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 79.6%

         \[\leadsto y \cdot \color{blue}{\left(-x\right)} \]

   6. Simplified 79.6%

      \[\leadsto y \cdot \color{blue}{\left(-x\right)} \]

   if -1.35e-107 < z

   1. Initial program 67.5%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. associate-/l* 67.9%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

      2. associate-*l/ 70.2%

         \[\leadsto \color{blue}{\frac{x}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \cdot y} \]

   3. Simplified 70.2%

      \[\leadsto \color{blue}{\frac{x}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \cdot y} \]

   4. Taylor expanded in z around inf 72.1%

      \[\leadsto \frac{x}{\frac{\color{blue}{z + -0.5 \cdot \frac{a \cdot t}{z}}}{z}} \cdot y \]
   5. Step-by-step derivation

      1. associate-/l* 75.0%

         \[\leadsto \frac{x \cdot y}{\frac{z + -0.5 \cdot \color{blue}{\frac{a}{\frac{z}{t}}}}{z}} \]

   6. Simplified 75.1%

      \[\leadsto \frac{x}{\frac{\color{blue}{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}}{z}} \cdot y \]
   7. Step-by-step derivation

      1. associate-/r/ 75.1%

         \[\leadsto \frac{x}{\frac{z + -0.5 \cdot \color{blue}{\left(\frac{a}{z} \cdot t\right)}}{z}} \cdot y \]

   8. Applied egg-rr 75.1%

      \[\leadsto \frac{x}{\frac{z + -0.5 \cdot \color{blue}{\left(\frac{a}{z} \cdot t\right)}}{z}} \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{-107}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\frac{z + -0.5 \cdot \left(\frac{a}{z} \cdot t\right)}{z}}\\ \end{array} \]

Alternative 19: 78.7% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 4.7 \cdot 10^{-306}:\\ \;\;\;\;\frac{y}{-1 - \frac{\frac{a}{z}}{\frac{z}{\frac{t}{-2}}}} \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\frac{z + -0.5 \cdot \left(\frac{a}{z} \cdot t\right)}{z}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z 4.7e-306)
   (* (/ y (- -1.0 (/ (/ a z) (/ z (/ t -2.0))))) x)
   (* y (/ x (/ (+ z (* -0.5 (* (/ a z) t))) z)))))

C

assert(x < y);
assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= 4.7e-306) {
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x;
	} else {
		tmp = y * (x / ((z + (-0.5 * ((a / z) * t))) / z));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a 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 <= 4.7d-306) then
        tmp = (y / ((-1.0d0) - ((a / z) / (z / (t / (-2.0d0)))))) * x
    else
        tmp = y * (x / ((z + ((-0.5d0) * ((a / z) * t))) / z))
    end if
    code = tmp
end function

Java

assert x < y;
assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= 4.7e-306) {
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x;
	} else {
		tmp = y * (x / ((z + (-0.5 * ((a / z) * t))) / z));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= 4.7e-306:
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x
	else:
		tmp = y * (x / ((z + (-0.5 * ((a / z) * t))) / z))
	return tmp

Julia

x, y = sort([x, y])
t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= 4.7e-306)
		tmp = Float64(Float64(y / Float64(-1.0 - Float64(Float64(a / z) / Float64(z / Float64(t / -2.0))))) * x);
	else
		tmp = Float64(y * Float64(x / Float64(Float64(z + Float64(-0.5 * Float64(Float64(a / z) * t))) / z)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= 4.7e-306)
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x;
	else
		tmp = y * (x / ((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.
NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, 4.7e-306], N[(N[(y / N[(-1.0 - N[(N[(a / z), $MachinePrecision] / N[(z / N[(t / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(y * N[(x / N[(N[(z + N[(-0.5 * N[(N[(a / z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.7000000000000001e-306

   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* 64.2%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

      2. associate-/l* 65.8%

         \[\leadsto \color{blue}{\frac{x}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}{y}}} \]

   3. Simplified 65.8%

      \[\leadsto \color{blue}{\frac{x}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}{y}}} \]

   4. Taylor expanded in z around -inf 68.2%

      \[\leadsto \frac{x}{\color{blue}{0.5 \cdot \frac{a \cdot t}{y \cdot {z}^{2}} - \frac{1}{y}}} \]
   5. Step-by-step derivation

      1. *-commutative 68.2%

         \[\leadsto \frac{x}{0.5 \cdot \frac{\color{blue}{t \cdot a}}{y \cdot {z}^{2}} - \frac{1}{y}} \]

      2. associate-*r/ 68.2%

         \[\leadsto \frac{x}{\color{blue}{\frac{0.5 \cdot \left(t \cdot a\right)}{y \cdot {z}^{2}}} - \frac{1}{y}} \]

      3. *-commutative 68.2%

         \[\leadsto \frac{x}{\frac{0.5 \cdot \color{blue}{\left(a \cdot t\right)}}{y \cdot {z}^{2}} - \frac{1}{y}} \]

      4. associate-*r* 68.2%

         \[\leadsto \frac{x}{\frac{\color{blue}{\left(0.5 \cdot a\right) \cdot t}}{y \cdot {z}^{2}} - \frac{1}{y}} \]

      5. unpow2 68.2%

         \[\leadsto \frac{x}{\frac{\left(0.5 \cdot a\right) \cdot t}{y \cdot \color{blue}{\left(z \cdot z\right)}} - \frac{1}{y}} \]

   6. Simplified 68.2%

      \[\leadsto \frac{x}{\color{blue}{\frac{\left(0.5 \cdot a\right) \cdot t}{y \cdot \left(z \cdot z\right)} - \frac{1}{y}}} \]
   7. Step-by-step derivation

      1. associate-*l* 68.2%

         \[\leadsto \frac{x}{\frac{\color{blue}{0.5 \cdot \left(a \cdot t\right)}}{y \cdot \left(z \cdot z\right)} - \frac{1}{y}} \]

      2. *-commutative 68.2%

         \[\leadsto \frac{x}{\frac{0.5 \cdot \color{blue}{\left(t \cdot a\right)}}{y \cdot \left(z \cdot z\right)} - \frac{1}{y}} \]

      3. associate-*r* 71.0%

         \[\leadsto \frac{x}{\frac{0.5 \cdot \left(t \cdot a\right)}{\color{blue}{\left(y \cdot z\right) \cdot z}} - \frac{1}{y}} \]

      4. times-frac 72.2%

         \[\leadsto \frac{x}{\color{blue}{\frac{0.5}{y \cdot z} \cdot \frac{t \cdot a}{z}} - \frac{1}{y}} \]

      5. *-commutative 72.2%

         \[\leadsto \frac{x}{\frac{0.5}{y \cdot z} \cdot \frac{\color{blue}{a \cdot t}}{z} - \frac{1}{y}} \]

      6. associate-*l/ 75.6%

         \[\leadsto \frac{x}{\frac{0.5}{y \cdot z} \cdot \color{blue}{\left(\frac{a}{z} \cdot t\right)} - \frac{1}{y}} \]

      7. associate-/r/ 75.6%

         \[\leadsto \frac{x}{\frac{0.5}{y \cdot z} \cdot \color{blue}{\frac{a}{\frac{z}{t}}} - \frac{1}{y}} \]

      8. div-inv 75.6%

         \[\leadsto \frac{x}{\frac{0.5}{y \cdot z} \cdot \color{blue}{\left(a \cdot \frac{1}{\frac{z}{t}}\right)} - \frac{1}{y}} \]

      9. clear-num 75.6%

         \[\leadsto \frac{x}{\frac{0.5}{y \cdot z} \cdot \left(a \cdot \color{blue}{\frac{t}{z}}\right) - \frac{1}{y}} \]

   8. Applied egg-rr 75.6%

      \[\leadsto \frac{x}{\color{blue}{\frac{0.5}{y \cdot z} \cdot \left(a \cdot \frac{t}{z}\right)} - \frac{1}{y}} \]

   9. Taylor expanded in y around -inf 68.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot y}{1 + -0.5 \cdot \frac{a \cdot t}{{z}^{2}}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 68.4%

          \[\leadsto \color{blue}{\frac{-1 \cdot \left(x \cdot y\right)}{1 + -0.5 \cdot \frac{a \cdot t}{{z}^{2}}}} \]

       2. mul-1-neg 68.4%

          \[\leadsto \frac{\color{blue}{-x \cdot y}}{1 + -0.5 \cdot \frac{a \cdot t}{{z}^{2}}} \]

       3. distribute-rgt-neg-in 68.4%

          \[\leadsto \frac{\color{blue}{x \cdot \left(-y\right)}}{1 + -0.5 \cdot \frac{a \cdot t}{{z}^{2}}} \]

       4. +-commutative 68.4%

          \[\leadsto \frac{x \cdot \left(-y\right)}{\color{blue}{-0.5 \cdot \frac{a \cdot t}{{z}^{2}} + 1}} \]

       5. associate-/l* 73.0%

          \[\leadsto \frac{x \cdot \left(-y\right)}{-0.5 \cdot \color{blue}{\frac{a}{\frac{{z}^{2}}{t}}} + 1} \]

       6. unpow2 73.0%

          \[\leadsto \frac{x \cdot \left(-y\right)}{-0.5 \cdot \frac{a}{\frac{\color{blue}{z \cdot z}}{t}} + 1} \]

       7. associate-*r/ 75.6%

          \[\leadsto \frac{x \cdot \left(-y\right)}{-0.5 \cdot \frac{a}{\color{blue}{z \cdot \frac{z}{t}}} + 1} \]

   11. Simplified 75.6%

       \[\leadsto \color{blue}{\frac{x \cdot \left(-y\right)}{-0.5 \cdot \frac{a}{z \cdot \frac{z}{t}} + 1}} \]
   12. Step-by-step derivation

       1. div-inv 75.6%

          \[\leadsto \color{blue}{\left(x \cdot \left(-y\right)\right) \cdot \frac{1}{-0.5 \cdot \frac{a}{z \cdot \frac{z}{t}} + 1}} \]

       2. associate-*l* 75.7%

          \[\leadsto \color{blue}{x \cdot \left(\left(-y\right) \cdot \frac{1}{-0.5 \cdot \frac{a}{z \cdot \frac{z}{t}} + 1}\right)} \]

       3. *-commutative 75.7%

          \[\leadsto \color{blue}{\left(\left(-y\right) \cdot \frac{1}{-0.5 \cdot \frac{a}{z \cdot \frac{z}{t}} + 1}\right) \cdot x} \]

       4. div-inv 75.7%

          \[\leadsto \color{blue}{\frac{-y}{-0.5 \cdot \frac{a}{z \cdot \frac{z}{t}} + 1}} \cdot x \]

   13. Applied egg-rr 75.8%

       \[\leadsto \color{blue}{\frac{y}{-1 - \frac{\frac{a}{z}}{\frac{z}{\frac{t}{-2}}}} \cdot x} \]

   if 4.7000000000000001e-306 < z

   1. Initial program 64.9%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. associate-/l* 66.0%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

      2. associate-*l/ 67.4%

         \[\leadsto \color{blue}{\frac{x}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \cdot y} \]

   3. Simplified 67.4%

      \[\leadsto \color{blue}{\frac{x}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \cdot y} \]

   4. Taylor expanded in z around inf 77.4%

      \[\leadsto \frac{x}{\frac{\color{blue}{z + -0.5 \cdot \frac{a \cdot t}{z}}}{z}} \cdot y \]
   5. Step-by-step derivation

      1. associate-/l* 80.9%

         \[\leadsto \frac{x \cdot y}{\frac{z + -0.5 \cdot \color{blue}{\frac{a}{\frac{z}{t}}}}{z}} \]

   6. Simplified 80.9%

      \[\leadsto \frac{x}{\frac{\color{blue}{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}}{z}} \cdot y \]
   7. Step-by-step derivation

      1. associate-/r/ 81.0%

         \[\leadsto \frac{x}{\frac{z + -0.5 \cdot \color{blue}{\left(\frac{a}{z} \cdot t\right)}}{z}} \cdot y \]

   8. Applied egg-rr 81.0%

      \[\leadsto \frac{x}{\frac{z + -0.5 \cdot \color{blue}{\left(\frac{a}{z} \cdot t\right)}}{z}} \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.7 \cdot 10^{-306}:\\ \;\;\;\;\frac{y}{-1 - \frac{\frac{a}{z}}{\frac{z}{\frac{t}{-2}}}} \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\frac{z + -0.5 \cdot \left(\frac{a}{z} \cdot t\right)}{z}}\\ \end{array} \]

Alternative 20: 78.9% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 8 \cdot 10^{-304}:\\ \;\;\;\;\frac{y}{-1 - \frac{\frac{a}{z}}{\frac{z}{\frac{t}{-2}}}} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \left(\frac{a}{z} \cdot \frac{t}{z}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z 8e-304)
   (* (/ y (- -1.0 (/ (/ a z) (/ z (/ t -2.0))))) x)
   (/ (* y x) (+ 1.0 (* -0.5 (* (/ a z) (/ t z)))))))

C

assert(x < y);
assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= 8e-304) {
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x;
	} else {
		tmp = (y * x) / (1.0 + (-0.5 * ((a / z) * (t / z))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a 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 <= 8d-304) then
        tmp = (y / ((-1.0d0) - ((a / z) / (z / (t / (-2.0d0)))))) * x
    else
        tmp = (y * x) / (1.0d0 + ((-0.5d0) * ((a / z) * (t / z))))
    end if
    code = tmp
end function

Java

assert x < y;
assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= 8e-304) {
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x;
	} else {
		tmp = (y * x) / (1.0 + (-0.5 * ((a / z) * (t / z))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= 8e-304:
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x
	else:
		tmp = (y * x) / (1.0 + (-0.5 * ((a / z) * (t / z))))
	return tmp

Julia

x, y = sort([x, y])
t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= 8e-304)
		tmp = Float64(Float64(y / Float64(-1.0 - Float64(Float64(a / z) / Float64(z / Float64(t / -2.0))))) * x);
	else
		tmp = Float64(Float64(y * x) / Float64(1.0 + Float64(-0.5 * Float64(Float64(a / z) * Float64(t / z)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= 8e-304)
		tmp = (y / (-1.0 - ((a / z) / (z / (t / -2.0))))) * x;
	else
		tmp = (y * x) / (1.0 + (-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.
NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, 8e-304], N[(N[(y / N[(-1.0 - N[(N[(a / z), $MachinePrecision] / N[(z / N[(t / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(N[(a / z), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 7.99999999999999977e-304

   1. Initial program 62.9%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. associate-/l* 64.0%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

      2. associate-/l* 66.4%

         \[\leadsto \color{blue}{\frac{x}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}{y}}} \]

   3. Simplified 66.4%

      \[\leadsto \color{blue}{\frac{x}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}{y}}} \]

   4. Taylor expanded in z around -inf 68.0%

      \[\leadsto \frac{x}{\color{blue}{0.5 \cdot \frac{a \cdot t}{y \cdot {z}^{2}} - \frac{1}{y}}} \]
   5. Step-by-step derivation

      1. *-commutative 68.0%

         \[\leadsto \frac{x}{0.5 \cdot \frac{\color{blue}{t \cdot a}}{y \cdot {z}^{2}} - \frac{1}{y}} \]

      2. associate-*r/ 68.0%

         \[\leadsto \frac{x}{\color{blue}{\frac{0.5 \cdot \left(t \cdot a\right)}{y \cdot {z}^{2}}} - \frac{1}{y}} \]

      3. *-commutative 68.0%

         \[\leadsto \frac{x}{\frac{0.5 \cdot \color{blue}{\left(a \cdot t\right)}}{y \cdot {z}^{2}} - \frac{1}{y}} \]

      4. associate-*r* 68.0%

         \[\leadsto \frac{x}{\frac{\color{blue}{\left(0.5 \cdot a\right) \cdot t}}{y \cdot {z}^{2}} - \frac{1}{y}} \]

      5. unpow2 68.0%

         \[\leadsto \frac{x}{\frac{\left(0.5 \cdot a\right) \cdot t}{y \cdot \color{blue}{\left(z \cdot z\right)}} - \frac{1}{y}} \]

   6. Simplified 68.0%

      \[\leadsto \frac{x}{\color{blue}{\frac{\left(0.5 \cdot a\right) \cdot t}{y \cdot \left(z \cdot z\right)} - \frac{1}{y}}} \]
   7. Step-by-step derivation

      1. associate-*l* 68.0%

         \[\leadsto \frac{x}{\frac{\color{blue}{0.5 \cdot \left(a \cdot t\right)}}{y \cdot \left(z \cdot z\right)} - \frac{1}{y}} \]

      2. *-commutative 68.0%

         \[\leadsto \frac{x}{\frac{0.5 \cdot \color{blue}{\left(t \cdot a\right)}}{y \cdot \left(z \cdot z\right)} - \frac{1}{y}} \]

      3. associate-*r* 70.7%

         \[\leadsto \frac{x}{\frac{0.5 \cdot \left(t \cdot a\right)}{\color{blue}{\left(y \cdot z\right) \cdot z}} - \frac{1}{y}} \]

      4. times-frac 71.8%

         \[\leadsto \frac{x}{\color{blue}{\frac{0.5}{y \cdot z} \cdot \frac{t \cdot a}{z}} - \frac{1}{y}} \]

      5. *-commutative 71.8%

         \[\leadsto \frac{x}{\frac{0.5}{y \cdot z} \cdot \frac{\color{blue}{a \cdot t}}{z} - \frac{1}{y}} \]

      6. associate-*l/ 75.2%

         \[\leadsto \frac{x}{\frac{0.5}{y \cdot z} \cdot \color{blue}{\left(\frac{a}{z} \cdot t\right)} - \frac{1}{y}} \]

      7. associate-/r/ 75.2%

         \[\leadsto \frac{x}{\frac{0.5}{y \cdot z} \cdot \color{blue}{\frac{a}{\frac{z}{t}}} - \frac{1}{y}} \]

      8. div-inv 75.2%

         \[\leadsto \frac{x}{\frac{0.5}{y \cdot z} \cdot \color{blue}{\left(a \cdot \frac{1}{\frac{z}{t}}\right)} - \frac{1}{y}} \]

      9. clear-num 75.2%

         \[\leadsto \frac{x}{\frac{0.5}{y \cdot z} \cdot \left(a \cdot \color{blue}{\frac{t}{z}}\right) - \frac{1}{y}} \]

   8. Applied egg-rr 75.2%

      \[\leadsto \frac{x}{\color{blue}{\frac{0.5}{y \cdot z} \cdot \left(a \cdot \frac{t}{z}\right)} - \frac{1}{y}} \]

   9. Taylor expanded in y around -inf 68.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot y}{1 + -0.5 \cdot \frac{a \cdot t}{{z}^{2}}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 68.1%

          \[\leadsto \color{blue}{\frac{-1 \cdot \left(x \cdot y\right)}{1 + -0.5 \cdot \frac{a \cdot t}{{z}^{2}}}} \]

       2. mul-1-neg 68.1%

          \[\leadsto \frac{\color{blue}{-x \cdot y}}{1 + -0.5 \cdot \frac{a \cdot t}{{z}^{2}}} \]

       3. distribute-rgt-neg-in 68.1%

          \[\leadsto \frac{\color{blue}{x \cdot \left(-y\right)}}{1 + -0.5 \cdot \frac{a \cdot t}{{z}^{2}}} \]

       4. +-commutative 68.1%

          \[\leadsto \frac{x \cdot \left(-y\right)}{\color{blue}{-0.5 \cdot \frac{a \cdot t}{{z}^{2}} + 1}} \]

       5. associate-/l* 72.6%

          \[\leadsto \frac{x \cdot \left(-y\right)}{-0.5 \cdot \color{blue}{\frac{a}{\frac{{z}^{2}}{t}}} + 1} \]

       6. unpow2 72.6%

          \[\leadsto \frac{x \cdot \left(-y\right)}{-0.5 \cdot \frac{a}{\frac{\color{blue}{z \cdot z}}{t}} + 1} \]

       7. associate-*r/ 75.2%

          \[\leadsto \frac{x \cdot \left(-y\right)}{-0.5 \cdot \frac{a}{\color{blue}{z \cdot \frac{z}{t}}} + 1} \]

   11. Simplified 75.2%

       \[\leadsto \color{blue}{\frac{x \cdot \left(-y\right)}{-0.5 \cdot \frac{a}{z \cdot \frac{z}{t}} + 1}} \]
   12. Step-by-step derivation

       1. div-inv 75.2%

          \[\leadsto \color{blue}{\left(x \cdot \left(-y\right)\right) \cdot \frac{1}{-0.5 \cdot \frac{a}{z \cdot \frac{z}{t}} + 1}} \]

       2. associate-*l* 75.3%

          \[\leadsto \color{blue}{x \cdot \left(\left(-y\right) \cdot \frac{1}{-0.5 \cdot \frac{a}{z \cdot \frac{z}{t}} + 1}\right)} \]

       3. *-commutative 75.3%

          \[\leadsto \color{blue}{\left(\left(-y\right) \cdot \frac{1}{-0.5 \cdot \frac{a}{z \cdot \frac{z}{t}} + 1}\right) \cdot x} \]

       4. div-inv 75.3%

          \[\leadsto \color{blue}{\frac{-y}{-0.5 \cdot \frac{a}{z \cdot \frac{z}{t}} + 1}} \cdot x \]

   13. Applied egg-rr 75.4%

       \[\leadsto \color{blue}{\frac{y}{-1 - \frac{\frac{a}{z}}{\frac{z}{\frac{t}{-2}}}} \cdot x} \]

   if 7.99999999999999977e-304 < z

   1. Initial program 65.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. associate-/l* 66.2%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

   3. Simplified 66.2%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

   4. Taylor expanded in z around inf 77.8%

      \[\leadsto \frac{x \cdot y}{\frac{\color{blue}{z + -0.5 \cdot \frac{a \cdot t}{z}}}{z}} \]
   5. Step-by-step derivation

      1. associate-/l* 81.4%

         \[\leadsto \frac{x \cdot y}{\frac{z + -0.5 \cdot \color{blue}{\frac{a}{\frac{z}{t}}}}{z}} \]

   6. Simplified 81.4%

      \[\leadsto \frac{x \cdot y}{\frac{\color{blue}{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}}{z}} \]

   7. Taylor expanded in z around 0 76.7%

      \[\leadsto \frac{x \cdot y}{\color{blue}{1 + -0.5 \cdot \frac{a \cdot t}{{z}^{2}}}} \]
   8. Step-by-step derivation

      1. unpow2 76.7%

         \[\leadsto \frac{x \cdot y}{1 + -0.5 \cdot \frac{a \cdot t}{\color{blue}{z \cdot z}}} \]

      2. *-commutative 76.7%

         \[\leadsto \frac{x \cdot y}{1 + -0.5 \cdot \frac{\color{blue}{t \cdot a}}{z \cdot z}} \]

      3. times-frac 81.4%

         \[\leadsto \frac{x \cdot y}{1 + -0.5 \cdot \color{blue}{\left(\frac{t}{z} \cdot \frac{a}{z}\right)}} \]

   9. Simplified 81.4%

      \[\leadsto \frac{x \cdot y}{\color{blue}{1 + -0.5 \cdot \left(\frac{t}{z} \cdot \frac{a}{z}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 8 \cdot 10^{-304}:\\ \;\;\;\;\frac{y}{-1 - \frac{\frac{a}{z}}{\frac{z}{\frac{t}{-2}}}} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \left(\frac{a}{z} \cdot \frac{t}{z}\right)}\\ \end{array} \]

Alternative 21: 74.3% accurate, 10.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4.7 \cdot 10^{-287}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-97}:\\ \;\;\;\;\frac{z \cdot \left(y \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4.7e-287)
   (* y (- x))
   (if (<= z 9.5e-97) (/ (* z (* y x)) z) (* y x))))

C

assert(x < y);
assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.7e-287) {
		tmp = y * -x;
	} else if (z <= 9.5e-97) {
		tmp = (z * (y * x)) / z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a 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 <= (-4.7d-287)) then
        tmp = y * -x
    else if (z <= 9.5d-97) then
        tmp = (z * (y * x)) / z
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.7e-287) {
		tmp = y * -x;
	} else if (z <= 9.5e-97) {
		tmp = (z * (y * x)) / z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -4.7e-287:
		tmp = y * -x
	elif z <= 9.5e-97:
		tmp = (z * (y * x)) / z
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4.7e-287)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 9.5e-97)
		tmp = Float64(Float64(z * Float64(y * x)) / z);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -4.7e-287)
		tmp = y * -x;
	elseif (z <= 9.5e-97)
		tmp = (z * (y * x)) / z;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4.7e-287], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 9.5e-97], N[(N[(z * N[(y * x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.6999999999999999e-287

   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. associate-/l* 63.3%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

      2. associate-*l/ 65.8%

         \[\leadsto \color{blue}{\frac{x}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \cdot y} \]

      3. associate-/l* 64.7%

         \[\leadsto \color{blue}{\frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \cdot y \]

      4. *-commutative 64.7%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 64.7%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around -inf 71.5%

      \[\leadsto y \cdot \color{blue}{\left(-1 \cdot x\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 71.5%

         \[\leadsto y \cdot \color{blue}{\left(-x\right)} \]

   6. Simplified 71.5%

      \[\leadsto y \cdot \color{blue}{\left(-x\right)} \]

   if -4.6999999999999999e-287 < z < 9.5000000000000001e-97

   1. Initial program 80.3%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]

   2. Taylor expanded in z around inf 41.0%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z}} \]

   if 9.5000000000000001e-97 < z

   1. Initial program 61.0%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. associate-/l* 63.9%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

      2. associate-/l* 63.0%

         \[\leadsto \color{blue}{\frac{x}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}{y}}} \]

   3. Simplified 63.0%

      \[\leadsto \color{blue}{\frac{x}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}{y}}} \]

   4. Taylor expanded in z around inf 88.1%

      \[\leadsto \color{blue}{x \cdot y} \]
   5. Step-by-step derivation

      1. *-commutative 88.1%

         \[\leadsto \color{blue}{y \cdot x} \]

   6. Simplified 88.1%

      \[\leadsto \color{blue}{y \cdot x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.7 \cdot 10^{-287}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-97}:\\ \;\;\;\;\frac{z \cdot \left(y \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 22: 72.5% accurate, 18.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{-297}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.3e-297) (* y (- x)) (* y x)))

C

assert(x < y);
assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.3e-297) {
		tmp = y * -x;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a 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-297)) then
        tmp = y * -x
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.3e-297) {
		tmp = y * -x;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.3e-297:
		tmp = y * -x
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.3e-297)
		tmp = Float64(y * Float64(-x));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.3e-297)
		tmp = y * -x;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.3e-297], N[(y * (-x)), $MachinePrecision], N[(y * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.2999999999999999e-297

   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. associate-/l* 63.3%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

      2. associate-*l/ 65.8%

         \[\leadsto \color{blue}{\frac{x}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \cdot y} \]

      3. associate-/l* 64.7%

         \[\leadsto \color{blue}{\frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \cdot y \]

      4. *-commutative 64.7%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 64.7%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around -inf 71.5%

      \[\leadsto y \cdot \color{blue}{\left(-1 \cdot x\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 71.5%

         \[\leadsto y \cdot \color{blue}{\left(-x\right)} \]

   6. Simplified 71.5%

      \[\leadsto y \cdot \color{blue}{\left(-x\right)} \]

   if -2.2999999999999999e-297 < z

   1. Initial program 65.7%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. associate-/l* 66.7%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

      2. associate-/l* 66.7%

         \[\leadsto \color{blue}{\frac{x}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}{y}}} \]

   3. Simplified 66.7%

      \[\leadsto \color{blue}{\frac{x}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}{y}}} \]

   4. Taylor expanded in z around inf 72.8%

      \[\leadsto \color{blue}{x \cdot y} \]
   5. Step-by-step derivation

      1. *-commutative 72.8%

         \[\leadsto \color{blue}{y \cdot x} \]

   6. Simplified 72.8%

      \[\leadsto \color{blue}{y \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{-297}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 23: 42.7% accurate, 37.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [t, a] = \mathsf{sort}([t, a])\\ \\ y \cdot x \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a) :precision binary64 (* y x))

C

assert(x < y);
assert(t < a);
double code(double x, double y, double z, double t, double a) {
	return y * x;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a 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 = y * x
end function

Java

assert x < y;
assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	return y * x;
}

Python

[x, y] = sort([x, y])
[t, a] = sort([t, a])
def code(x, y, z, t, a):
	return y * x

Julia

x, y = sort([x, y])
t, a = sort([t, a])
function code(x, y, z, t, a)
	return Float64(y * x)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
t, a = num2cell(sort([t, a])){:}
function tmp = code(x, y, z, t, a)
	tmp = y * x;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := N[(y * x), $MachinePrecision]

Derivation

1. Initial program 64.1%

   \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
2. Step-by-step derivation

   1. associate-/l* 65.2%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

   2. associate-/l* 65.9%

      \[\leadsto \color{blue}{\frac{x}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}{y}}} \]

3. Simplified 65.9%

   \[\leadsto \color{blue}{\frac{x}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}{y}}} \]

4. Taylor expanded in z around inf 47.6%

   \[\leadsto \color{blue}{x \cdot y} \]
5. Step-by-step derivation

   1. *-commutative 47.6%

      \[\leadsto \color{blue}{y \cdot x} \]

6. Simplified 47.6%

   \[\leadsto \color{blue}{y \cdot x} \]

7. Final simplification 47.6%

   \[\leadsto y \cdot x \]

Developer target: 89.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z < -3.1921305903852764 \cdot 10^{+46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z < 5.976268120920894 \cdot 10^{+90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (< z -3.1921305903852764e+46)
   (- (* y x))
   (if (< z 5.976268120920894e+90)
     (/ (* x z) (/ (sqrt (- (* z z) (* a t))) y))
     (* y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z < -3.1921305903852764e+46) {
		tmp = -(y * x);
	} else if (z < 5.976268120920894e+90) {
		tmp = (x * z) / (sqrt(((z * z) - (a * t))) / y);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z < (-3.1921305903852764d+46)) then
        tmp = -(y * x)
    else if (z < 5.976268120920894d+90) then
        tmp = (x * z) / (sqrt(((z * z) - (a * t))) / y)
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z < -3.1921305903852764e+46) {
		tmp = -(y * x);
	} else if (z < 5.976268120920894e+90) {
		tmp = (x * z) / (Math.sqrt(((z * z) - (a * t))) / y);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z < -3.1921305903852764e+46:
		tmp = -(y * x)
	elif z < 5.976268120920894e+90:
		tmp = (x * z) / (math.sqrt(((z * z) - (a * t))) / y)
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z < -3.1921305903852764e+46)
		tmp = Float64(-Float64(y * x));
	elseif (z < 5.976268120920894e+90)
		tmp = Float64(Float64(x * z) / Float64(sqrt(Float64(Float64(z * z) - Float64(a * t))) / y));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z < -3.1921305903852764e+46)
		tmp = -(y * x);
	elseif (z < 5.976268120920894e+90)
		tmp = (x * z) / (sqrt(((z * z) - (a * t))) / y);
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Less[z, -3.1921305903852764e+46], (-N[(y * x), $MachinePrecision]), If[Less[z, 5.976268120920894e+90], N[(N[(x * z), $MachinePrecision] / N[(N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Reproduce

herbie shell --seed 2023297 
(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)))))