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

Percentage Accurate: 61.4% → 92.1%

Time: 19.9s

Alternatives: 15

Speedup: 18.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+152}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{z}{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+53}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{\sqrt{1 - \frac{a}{\frac{z \cdot z}{t}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2e+152)
   (* (* y x) (/ z (- (* 0.5 (* t (/ a z))) z)))
   (if (<= z 2e+53)
     (* y (* x (/ z (sqrt (- (* z z) (* t a))))))
     (/ (* y x) (sqrt (- 1.0 (/ a (/ (* z z) t))))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2e+152) {
		tmp = (y * x) * (z / ((0.5 * (t * (a / z))) - z));
	} else if (z <= 2e+53) {
		tmp = y * (x * (z / sqrt(((z * z) - (t * a)))));
	} else {
		tmp = (y * x) / sqrt((1.0 - (a / ((z * z) / t))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-2d+152)) then
        tmp = (y * x) * (z / ((0.5d0 * (t * (a / z))) - z))
    else if (z <= 2d+53) then
        tmp = y * (x * (z / sqrt(((z * z) - (t * a)))))
    else
        tmp = (y * x) / sqrt((1.0d0 - (a / ((z * z) / t))))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2e+152) {
		tmp = (y * x) * (z / ((0.5 * (t * (a / z))) - z));
	} else if (z <= 2e+53) {
		tmp = y * (x * (z / Math.sqrt(((z * z) - (t * a)))));
	} else {
		tmp = (y * x) / Math.sqrt((1.0 - (a / ((z * z) / t))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -2e+152:
		tmp = (y * x) * (z / ((0.5 * (t * (a / z))) - z))
	elif z <= 2e+53:
		tmp = y * (x * (z / math.sqrt(((z * z) - (t * a)))))
	else:
		tmp = (y * x) / math.sqrt((1.0 - (a / ((z * z) / t))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2e+152)
		tmp = Float64(Float64(y * x) * Float64(z / Float64(Float64(0.5 * Float64(t * Float64(a / z))) - z)));
	elseif (z <= 2e+53)
		tmp = Float64(y * Float64(x * Float64(z / sqrt(Float64(Float64(z * z) - Float64(t * a))))));
	else
		tmp = Float64(Float64(y * x) / sqrt(Float64(1.0 - Float64(a / Float64(Float64(z * z) / t)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2e+152)
		tmp = (y * x) * (z / ((0.5 * (t * (a / z))) - z));
	elseif (z <= 2e+53)
		tmp = y * (x * (z / sqrt(((z * z) - (t * a)))));
	else
		tmp = (y * x) / sqrt((1.0 - (a / ((z * z) / t))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2e+152], N[(N[(y * x), $MachinePrecision] * N[(z / N[(N[(0.5 * N[(t * N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2e+53], N[(y * N[(x * N[(z / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / N[Sqrt[N[(1.0 - N[(a / N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.0000000000000001e152

   1. Initial program 19.6%

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

      1. associate-/l* 20.0%

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

   3. Simplified 20.0%

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

      1. associate-*l/ 20.2%

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

      2. associate-/l* 19.4%

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

      3. div-inv 19.4%

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

      4. associate-*l* 20.2%

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

      5. div-inv 20.2%

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

   5. Applied egg-rr 20.2%

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

   6. Taylor expanded in z around -inf 78.4%

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

      1. add-sqr-sqrt 58.2%

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

      2. sqrt-unprod 78.4%

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

      3. pow2 78.4%

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

      4. associate-/l* 85.6%

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

   8. Applied egg-rr 85.6%

      \[\leadsto \left(x \cdot \frac{z}{0.5 \cdot \color{blue}{\sqrt{{\left(\frac{a}{\frac{z}{t}}\right)}^{2}}} + -1 \cdot z}\right) \cdot y \]
   9. Step-by-step derivation

      1. unpow2 85.6%

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

      2. rem-sqrt-square 88.5%

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

      3. associate-/r/ 88.5%

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

   10. Simplified 88.5%

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

   11. Taylor expanded in x around 0 51.7%

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

       1. fma-def 51.7%

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

       2. *-commutative 51.7%

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

       3. associate-*r/ 54.7%

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

       4. neg-mul-1 54.7%

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

       5. *-commutative 54.7%

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

       6. associate-*r* 71.7%

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

       7. associate-*r/ 88.5%

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

       8. fma-udef 88.5%

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

       9. associate-*r/ 78.4%

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

       10. *-commutative 78.4%

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

       11. unsub-neg 78.4%

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

   13. Simplified 88.5%

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

   if -2.0000000000000001e152 < z < 2e53

   1. Initial program 85.3%

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

      1. associate-/l* 90.8%

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

   3. Simplified 90.8%

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

      1. associate-*l/ 91.3%

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

      2. associate-/l* 88.0%

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

      3. div-inv 87.8%

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

      4. associate-*l* 91.3%

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

      5. div-inv 91.3%

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

   5. Applied egg-rr 91.3%

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

   if 2e53 < z

   1. Initial program 49.7%

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

      1. associate-/l* 54.4%

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

   3. Simplified 54.4%

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

      1. add-sqr-sqrt 54.4%

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

      2. sqrt-unprod 54.4%

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

      3. frac-times 43.7%

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

      4. add-sqr-sqrt 43.7%

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

   5. Applied egg-rr 43.7%

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

      1. div-sub 43.7%

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

      2. *-inverses 87.2%

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

      3. *-commutative 87.2%

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

      4. associate-/l* 97.0%

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

   7. Simplified 97.0%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+152}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{z}{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+53}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{\sqrt{1 - \frac{a}{\frac{z \cdot z}{t}}}}\\ \end{array} \]

Alternative 2: 90.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{if}\;z \leq -8.5 \cdot 10^{+112}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{z}{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}\\ \mathbf{elif}\;z \leq -7.2 \cdot 10^{-252}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{-165}:\\ \;\;\;\;\frac{y \cdot x}{\frac{\sqrt{t \cdot \left(-a\right)}}{z}}\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+56}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (/ (* z x) (sqrt (- (* z z) (* t a)))))))
   (if (<= z -8.5e+112)
     (* (* y x) (/ z (- (* 0.5 (* t (/ a z))) z)))
     (if (<= z -7.2e-252)
       t_1
       (if (<= z 4.1e-165)
         (/ (* y x) (/ (sqrt (* t (- a))) z))
         (if (<= z 3.3e+56) t_1 (* y x)))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = y * ((z * x) / sqrt(((z * z) - (t * a))));
	double tmp;
	if (z <= -8.5e+112) {
		tmp = (y * x) * (z / ((0.5 * (t * (a / z))) - z));
	} else if (z <= -7.2e-252) {
		tmp = t_1;
	} else if (z <= 4.1e-165) {
		tmp = (y * x) / (sqrt((t * -a)) / z);
	} else if (z <= 3.3e+56) {
		tmp = t_1;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = y * ((z * x) / sqrt(((z * z) - (t * a))))
    if (z <= (-8.5d+112)) then
        tmp = (y * x) * (z / ((0.5d0 * (t * (a / z))) - z))
    else if (z <= (-7.2d-252)) then
        tmp = t_1
    else if (z <= 4.1d-165) then
        tmp = (y * x) / (sqrt((t * -a)) / z)
    else if (z <= 3.3d+56) then
        tmp = t_1
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = y * ((z * x) / Math.sqrt(((z * z) - (t * a))));
	double tmp;
	if (z <= -8.5e+112) {
		tmp = (y * x) * (z / ((0.5 * (t * (a / z))) - z));
	} else if (z <= -7.2e-252) {
		tmp = t_1;
	} else if (z <= 4.1e-165) {
		tmp = (y * x) / (Math.sqrt((t * -a)) / z);
	} else if (z <= 3.3e+56) {
		tmp = t_1;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	t_1 = y * ((z * x) / math.sqrt(((z * z) - (t * a))))
	tmp = 0
	if z <= -8.5e+112:
		tmp = (y * x) * (z / ((0.5 * (t * (a / z))) - z))
	elif z <= -7.2e-252:
		tmp = t_1
	elif z <= 4.1e-165:
		tmp = (y * x) / (math.sqrt((t * -a)) / z)
	elif z <= 3.3e+56:
		tmp = t_1
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(Float64(z * x) / sqrt(Float64(Float64(z * z) - Float64(t * a)))))
	tmp = 0.0
	if (z <= -8.5e+112)
		tmp = Float64(Float64(y * x) * Float64(z / Float64(Float64(0.5 * Float64(t * Float64(a / z))) - z)));
	elseif (z <= -7.2e-252)
		tmp = t_1;
	elseif (z <= 4.1e-165)
		tmp = Float64(Float64(y * x) / Float64(sqrt(Float64(t * Float64(-a))) / z));
	elseif (z <= 3.3e+56)
		tmp = t_1;
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = y * ((z * x) / sqrt(((z * z) - (t * a))));
	tmp = 0.0;
	if (z <= -8.5e+112)
		tmp = (y * x) * (z / ((0.5 * (t * (a / z))) - z));
	elseif (z <= -7.2e-252)
		tmp = t_1;
	elseif (z <= 4.1e-165)
		tmp = (y * x) / (sqrt((t * -a)) / z);
	elseif (z <= 3.3e+56)
		tmp = t_1;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(N[(z * x), $MachinePrecision] / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -8.5e+112], N[(N[(y * x), $MachinePrecision] * N[(z / N[(N[(0.5 * N[(t * N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -7.2e-252], t$95$1, If[LessEqual[z, 4.1e-165], N[(N[(y * x), $MachinePrecision] / N[(N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.3e+56], t$95$1, N[(y * x), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -8.50000000000000047e112

   1. Initial program 32.4%

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

      1. associate-/l* 34.7%

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

   3. Simplified 34.7%

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

      1. associate-*l/ 34.8%

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

      2. associate-/l* 32.2%

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

      3. div-inv 32.2%

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

      4. associate-*l* 34.8%

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

      5. div-inv 34.8%

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

   5. Applied egg-rr 34.8%

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

   6. Taylor expanded in z around -inf 80.6%

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

      1. add-sqr-sqrt 58.0%

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

      2. sqrt-unprod 80.6%

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

      3. pow2 80.6%

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

      4. associate-/l* 86.5%

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

   8. Applied egg-rr 86.5%

      \[\leadsto \left(x \cdot \frac{z}{0.5 \cdot \color{blue}{\sqrt{{\left(\frac{a}{\frac{z}{t}}\right)}^{2}}} + -1 \cdot z}\right) \cdot y \]
   9. Step-by-step derivation

      1. unpow2 86.5%

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

      2. rem-sqrt-square 88.8%

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

      3. associate-/r/ 88.8%

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

   10. Simplified 88.8%

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

   11. Taylor expanded in x around 0 56.7%

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

       1. fma-def 56.7%

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

       2. *-commutative 56.7%

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

       3. associate-*r/ 59.2%

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

       4. neg-mul-1 59.2%

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

       5. *-commutative 59.2%

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

       6. associate-*r* 73.2%

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

       7. associate-*r/ 88.8%

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

       8. fma-udef 88.8%

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

       9. associate-*r/ 80.6%

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

       10. *-commutative 80.6%

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

       11. unsub-neg 80.6%

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

   13. Simplified 88.8%

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

   if -8.50000000000000047e112 < z < -7.20000000000000046e-252 or 4.1000000000000002e-165 < z < 3.30000000000000002e56

   1. Initial program 87.0%

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

      1. *-commutative 87.0%

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

      2. associate-*l* 88.1%

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

      3. associate-*r/ 93.4%

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

   3. Simplified 93.4%

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

   if -7.20000000000000046e-252 < z < 4.1000000000000002e-165

   1. Initial program 75.5%

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

      1. associate-/l* 79.5%

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

   3. Simplified 79.5%

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

   4. Taylor expanded in z around 0 79.5%

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

      1. associate-*r* 59.9%

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

      2. neg-mul-1 59.9%

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

   6. Simplified 79.5%

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

   if 3.30000000000000002e56 < z

   1. Initial program 49.7%

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

      1. *-commutative 49.7%

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

      2. associate-*l* 47.9%

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

      3. associate-*r/ 50.9%

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

   3. Simplified 50.9%

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

   4. Taylor expanded in z around inf 95.5%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+112}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{z}{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}\\ \mathbf{elif}\;z \leq -7.2 \cdot 10^{-252}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{-165}:\\ \;\;\;\;\frac{y \cdot x}{\frac{\sqrt{t \cdot \left(-a\right)}}{z}}\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+56}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3: 90.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := \sqrt{z \cdot z - t \cdot a}\\ \mathbf{if}\;z \leq -1 \cdot 10^{+113}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{z}{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}\\ \mathbf{elif}\;z \leq -5.6 \cdot 10^{-254}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{t_1}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-165}:\\ \;\;\;\;\frac{y \cdot x}{\frac{\sqrt{t \cdot \left(-a\right)}}{z}}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+55}:\\ \;\;\;\;z \cdot \frac{y \cdot x}{t_1}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (sqrt (- (* z z) (* t a)))))
   (if (<= z -1e+113)
     (* (* y x) (/ z (- (* 0.5 (* t (/ a z))) z)))
     (if (<= z -5.6e-254)
       (* y (/ (* z x) t_1))
       (if (<= z 5e-165)
         (/ (* y x) (/ (sqrt (* t (- a))) z))
         (if (<= z 4e+55) (* z (/ (* y x) t_1)) (* y x)))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = sqrt(((z * z) - (t * a)));
	double tmp;
	if (z <= -1e+113) {
		tmp = (y * x) * (z / ((0.5 * (t * (a / z))) - z));
	} else if (z <= -5.6e-254) {
		tmp = y * ((z * x) / t_1);
	} else if (z <= 5e-165) {
		tmp = (y * x) / (sqrt((t * -a)) / z);
	} else if (z <= 4e+55) {
		tmp = z * ((y * x) / t_1);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(((z * z) - (t * a)))
    if (z <= (-1d+113)) then
        tmp = (y * x) * (z / ((0.5d0 * (t * (a / z))) - z))
    else if (z <= (-5.6d-254)) then
        tmp = y * ((z * x) / t_1)
    else if (z <= 5d-165) then
        tmp = (y * x) / (sqrt((t * -a)) / z)
    else if (z <= 4d+55) then
        tmp = z * ((y * x) / t_1)
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.sqrt(((z * z) - (t * a)));
	double tmp;
	if (z <= -1e+113) {
		tmp = (y * x) * (z / ((0.5 * (t * (a / z))) - z));
	} else if (z <= -5.6e-254) {
		tmp = y * ((z * x) / t_1);
	} else if (z <= 5e-165) {
		tmp = (y * x) / (Math.sqrt((t * -a)) / z);
	} else if (z <= 4e+55) {
		tmp = z * ((y * x) / t_1);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	t_1 = math.sqrt(((z * z) - (t * a)))
	tmp = 0
	if z <= -1e+113:
		tmp = (y * x) * (z / ((0.5 * (t * (a / z))) - z))
	elif z <= -5.6e-254:
		tmp = y * ((z * x) / t_1)
	elif z <= 5e-165:
		tmp = (y * x) / (math.sqrt((t * -a)) / z)
	elif z <= 4e+55:
		tmp = z * ((y * x) / t_1)
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = sqrt(Float64(Float64(z * z) - Float64(t * a)))
	tmp = 0.0
	if (z <= -1e+113)
		tmp = Float64(Float64(y * x) * Float64(z / Float64(Float64(0.5 * Float64(t * Float64(a / z))) - z)));
	elseif (z <= -5.6e-254)
		tmp = Float64(y * Float64(Float64(z * x) / t_1));
	elseif (z <= 5e-165)
		tmp = Float64(Float64(y * x) / Float64(sqrt(Float64(t * Float64(-a))) / z));
	elseif (z <= 4e+55)
		tmp = Float64(z * Float64(Float64(y * x) / t_1));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = sqrt(((z * z) - (t * a)));
	tmp = 0.0;
	if (z <= -1e+113)
		tmp = (y * x) * (z / ((0.5 * (t * (a / z))) - z));
	elseif (z <= -5.6e-254)
		tmp = y * ((z * x) / t_1);
	elseif (z <= 5e-165)
		tmp = (y * x) / (sqrt((t * -a)) / z);
	elseif (z <= 4e+55)
		tmp = z * ((y * x) / t_1);
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -1e+113], N[(N[(y * x), $MachinePrecision] * N[(z / N[(N[(0.5 * N[(t * N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -5.6e-254], N[(y * N[(N[(z * x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5e-165], N[(N[(y * x), $MachinePrecision] / N[(N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4e+55], N[(z * N[(N[(y * x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -1e113

   1. Initial program 32.4%

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

      1. associate-/l* 34.7%

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

   3. Simplified 34.7%

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

      1. associate-*l/ 34.8%

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

      2. associate-/l* 32.2%

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

      3. div-inv 32.2%

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

      4. associate-*l* 34.8%

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

      5. div-inv 34.8%

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

   5. Applied egg-rr 34.8%

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

   6. Taylor expanded in z around -inf 80.6%

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

      1. add-sqr-sqrt 58.0%

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

      2. sqrt-unprod 80.6%

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

      3. pow2 80.6%

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

      4. associate-/l* 86.5%

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

   8. Applied egg-rr 86.5%

      \[\leadsto \left(x \cdot \frac{z}{0.5 \cdot \color{blue}{\sqrt{{\left(\frac{a}{\frac{z}{t}}\right)}^{2}}} + -1 \cdot z}\right) \cdot y \]
   9. Step-by-step derivation

      1. unpow2 86.5%

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

      2. rem-sqrt-square 88.8%

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

      3. associate-/r/ 88.8%

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

   10. Simplified 88.8%

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

   11. Taylor expanded in x around 0 56.7%

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

       1. fma-def 56.7%

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

       2. *-commutative 56.7%

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

       3. associate-*r/ 59.2%

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

       4. neg-mul-1 59.2%

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

       5. *-commutative 59.2%

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

       6. associate-*r* 73.2%

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

       7. associate-*r/ 88.8%

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

       8. fma-udef 88.8%

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

       9. associate-*r/ 80.6%

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

       10. *-commutative 80.6%

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

       11. unsub-neg 80.6%

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

   13. Simplified 88.8%

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

   if -1e113 < z < -5.59999999999999966e-254

   1. Initial program 82.0%

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

      1. *-commutative 82.0%

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

      2. associate-*l* 86.0%

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

      3. associate-*r/ 95.4%

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

   3. Simplified 95.4%

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

   if -5.59999999999999966e-254 < z < 4.99999999999999981e-165

   1. Initial program 75.5%

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

      1. associate-/l* 79.5%

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

   3. Simplified 79.5%

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

   4. Taylor expanded in z around 0 79.5%

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

      1. associate-*r* 59.9%

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

      2. neg-mul-1 59.9%

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

   6. Simplified 79.5%

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

   if 4.99999999999999981e-165 < z < 4.00000000000000004e55

   1. Initial program 93.4%

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

      1. associate-/l* 93.4%

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

   3. Simplified 93.4%

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

      1. associate-/l* 93.4%

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

      2. div-inv 93.3%

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

      3. associate-*l* 87.9%

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

      4. associate-*l* 88.0%

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

      5. *-commutative 88.0%

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

      6. pow1/2 88.0%

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

      7. pow-flip 87.9%

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

      8. metadata-eval 87.9%

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

   5. Applied egg-rr 87.9%

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

      1. expm1-log1p-u 70.1%

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

      2. expm1-udef 38.1%

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

      3. *-commutative 38.1%

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

      4. associate-*l* 37.4%

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

      5. metadata-eval 37.4%

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

      6. pow-flip 37.4%

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

      7. pow1/2 37.4%

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

      8. div-inv 37.4%

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

   7. Applied egg-rr 37.4%

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

      1. expm1-def 69.5%

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

      2. expm1-log1p 89.2%

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

      3. associate-*l* 85.6%

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

      4. associate-*l/ 91.7%

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

      5. *-commutative 91.7%

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

   9. Simplified 91.7%

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

   if 4.00000000000000004e55 < z

   1. Initial program 49.7%

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

      1. *-commutative 49.7%

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

      2. associate-*l* 47.9%

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

      3. associate-*r/ 50.9%

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

   3. Simplified 50.9%

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

   4. Taylor expanded in z around inf 95.5%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+113}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{z}{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}\\ \mathbf{elif}\;z \leq -5.6 \cdot 10^{-254}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-165}:\\ \;\;\;\;\frac{y \cdot x}{\frac{\sqrt{t \cdot \left(-a\right)}}{z}}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+55}:\\ \;\;\;\;z \cdot \frac{y \cdot x}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 4: 91.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+152}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{z}{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+56}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4e+152)
   (* (* y x) (/ z (- (* 0.5 (* t (/ a z))) z)))
   (if (<= z 2e+56) (* y (* x (/ z (sqrt (- (* z z) (* t a)))))) (* y x))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4e+152) {
		tmp = (y * x) * (z / ((0.5 * (t * (a / z))) - z));
	} else if (z <= 2e+56) {
		tmp = y * (x * (z / sqrt(((z * z) - (t * a)))));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-4d+152)) then
        tmp = (y * x) * (z / ((0.5d0 * (t * (a / z))) - z))
    else if (z <= 2d+56) then
        tmp = y * (x * (z / sqrt(((z * z) - (t * a)))))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4e+152) {
		tmp = (y * x) * (z / ((0.5 * (t * (a / z))) - z));
	} else if (z <= 2e+56) {
		tmp = y * (x * (z / Math.sqrt(((z * z) - (t * a)))));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -4e+152:
		tmp = (y * x) * (z / ((0.5 * (t * (a / z))) - z))
	elif z <= 2e+56:
		tmp = y * (x * (z / math.sqrt(((z * z) - (t * a)))))
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4e+152)
		tmp = Float64(Float64(y * x) * Float64(z / Float64(Float64(0.5 * Float64(t * Float64(a / z))) - z)));
	elseif (z <= 2e+56)
		tmp = Float64(y * Float64(x * Float64(z / sqrt(Float64(Float64(z * z) - Float64(t * a))))));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -4e+152)
		tmp = (y * x) * (z / ((0.5 * (t * (a / z))) - z));
	elseif (z <= 2e+56)
		tmp = y * (x * (z / sqrt(((z * z) - (t * a)))));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4e+152], N[(N[(y * x), $MachinePrecision] * N[(z / N[(N[(0.5 * N[(t * N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2e+56], N[(y * N[(x * N[(z / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.0000000000000002e152

   1. Initial program 19.6%

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

      1. associate-/l* 20.0%

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

   3. Simplified 20.0%

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

      1. associate-*l/ 20.2%

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

      2. associate-/l* 19.4%

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

      3. div-inv 19.4%

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

      4. associate-*l* 20.2%

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

      5. div-inv 20.2%

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

   5. Applied egg-rr 20.2%

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

   6. Taylor expanded in z around -inf 78.4%

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

      1. add-sqr-sqrt 58.2%

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

      2. sqrt-unprod 78.4%

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

      3. pow2 78.4%

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

      4. associate-/l* 85.6%

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

   8. Applied egg-rr 85.6%

      \[\leadsto \left(x \cdot \frac{z}{0.5 \cdot \color{blue}{\sqrt{{\left(\frac{a}{\frac{z}{t}}\right)}^{2}}} + -1 \cdot z}\right) \cdot y \]
   9. Step-by-step derivation

      1. unpow2 85.6%

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

      2. rem-sqrt-square 88.5%

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

      3. associate-/r/ 88.5%

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

   10. Simplified 88.5%

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

   11. Taylor expanded in x around 0 51.7%

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

       1. fma-def 51.7%

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

       2. *-commutative 51.7%

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

       3. associate-*r/ 54.7%

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

       4. neg-mul-1 54.7%

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

       5. *-commutative 54.7%

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

       6. associate-*r* 71.7%

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

       7. associate-*r/ 88.5%

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

       8. fma-udef 88.5%

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

       9. associate-*r/ 78.4%

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

       10. *-commutative 78.4%

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

       11. unsub-neg 78.4%

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

   13. Simplified 88.5%

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

   if -4.0000000000000002e152 < z < 2.00000000000000018e56

   1. Initial program 85.3%

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

      1. associate-/l* 90.8%

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

   3. Simplified 90.8%

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

      1. associate-*l/ 91.3%

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

      2. associate-/l* 88.0%

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

      3. div-inv 87.8%

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

      4. associate-*l* 91.3%

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

      5. div-inv 91.3%

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

   5. Applied egg-rr 91.3%

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

   if 2.00000000000000018e56 < z

   1. Initial program 49.7%

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

      1. *-commutative 49.7%

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

      2. associate-*l* 47.9%

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

      3. associate-*r/ 50.9%

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

   3. Simplified 50.9%

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

   4. Taylor expanded in z around inf 95.5%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+152}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{z}{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+56}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 5: 83.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{-71}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{z}{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-70}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + \frac{a}{\frac{z \cdot z}{t}} \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.7e-71)
   (* (* y x) (/ z (- (* 0.5 (* t (/ a z))) z)))
   (if (<= z 1.1e-70)
     (* y (/ (* z x) (sqrt (* t (- a)))))
     (/ (* y x) (+ 1.0 (* (/ a (/ (* z z) t)) -0.5))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.7e-71) {
		tmp = (y * x) * (z / ((0.5 * (t * (a / z))) - z));
	} else if (z <= 1.1e-70) {
		tmp = y * ((z * x) / sqrt((t * -a)));
	} else {
		tmp = (y * x) / (1.0 + ((a / ((z * z) / t)) * -0.5));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-2.7d-71)) then
        tmp = (y * x) * (z / ((0.5d0 * (t * (a / z))) - z))
    else if (z <= 1.1d-70) then
        tmp = y * ((z * x) / sqrt((t * -a)))
    else
        tmp = (y * x) / (1.0d0 + ((a / ((z * z) / t)) * (-0.5d0)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.7e-71) {
		tmp = (y * x) * (z / ((0.5 * (t * (a / z))) - z));
	} else if (z <= 1.1e-70) {
		tmp = y * ((z * x) / Math.sqrt((t * -a)));
	} else {
		tmp = (y * x) / (1.0 + ((a / ((z * z) / t)) * -0.5));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.7e-71:
		tmp = (y * x) * (z / ((0.5 * (t * (a / z))) - z))
	elif z <= 1.1e-70:
		tmp = y * ((z * x) / math.sqrt((t * -a)))
	else:
		tmp = (y * x) / (1.0 + ((a / ((z * z) / t)) * -0.5))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.7e-71)
		tmp = Float64(Float64(y * x) * Float64(z / Float64(Float64(0.5 * Float64(t * Float64(a / z))) - z)));
	elseif (z <= 1.1e-70)
		tmp = Float64(y * Float64(Float64(z * x) / sqrt(Float64(t * Float64(-a)))));
	else
		tmp = Float64(Float64(y * x) / Float64(1.0 + Float64(Float64(a / Float64(Float64(z * z) / t)) * -0.5)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.7e-71)
		tmp = (y * x) * (z / ((0.5 * (t * (a / z))) - z));
	elseif (z <= 1.1e-70)
		tmp = y * ((z * x) / sqrt((t * -a)));
	else
		tmp = (y * x) / (1.0 + ((a / ((z * z) / t)) * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.7e-71], N[(N[(y * x), $MachinePrecision] * N[(z / N[(N[(0.5 * N[(t * N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.1e-70], N[(y * N[(N[(z * x), $MachinePrecision] / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / N[(1.0 + N[(N[(a / N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.7000000000000001e-71

   1. Initial program 55.9%

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

      1. associate-/l* 62.5%

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

   3. Simplified 62.5%

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

      1. associate-*l/ 62.5%

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

      2. associate-/l* 61.0%

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

      3. div-inv 60.9%

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

      4. associate-*l* 62.4%

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

      5. div-inv 62.5%

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

   5. Applied egg-rr 62.5%

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

   6. Taylor expanded in z around -inf 80.7%

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

      1. add-sqr-sqrt 53.4%

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

      2. sqrt-unprod 80.6%

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

      3. pow2 80.6%

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

      4. associate-/l* 83.9%

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

   8. Applied egg-rr 83.9%

      \[\leadsto \left(x \cdot \frac{z}{0.5 \cdot \color{blue}{\sqrt{{\left(\frac{a}{\frac{z}{t}}\right)}^{2}}} + -1 \cdot z}\right) \cdot y \]
   9. Step-by-step derivation

      1. unpow2 83.9%

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

      2. rem-sqrt-square 85.3%

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

      3. associate-/r/ 85.3%

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

   10. Simplified 85.3%

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

   11. Taylor expanded in x around 0 63.1%

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

       1. fma-def 63.1%

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

       2. *-commutative 63.1%

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

       3. associate-*r/ 64.5%

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

       4. neg-mul-1 64.5%

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

       5. *-commutative 64.5%

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

       6. associate-*r* 72.4%

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

       7. associate-*r/ 86.3%

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

       8. fma-udef 86.3%

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

       9. associate-*r/ 81.7%

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

       10. *-commutative 81.7%

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

       11. unsub-neg 81.7%

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

   13. Simplified 86.3%

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

   if -2.7000000000000001e-71 < z < 1.0999999999999999e-70

   1. Initial program 80.3%

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

      1. *-commutative 80.3%

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

      2. associate-*l* 77.2%

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

      3. associate-*r/ 79.7%

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

   3. Simplified 79.7%

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

   4. Taylor expanded in z around 0 73.2%

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

      1. associate-*r* 73.2%

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

      2. neg-mul-1 73.2%

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

   6. Simplified 73.2%

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

   if 1.0999999999999999e-70 < z

   1. Initial program 65.0%

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

      1. associate-/l* 68.2%

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

   3. Simplified 68.2%

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

   4. Taylor expanded in z around inf 86.3%

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

      1. unpow2 86.3%

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

      2. associate-/l* 91.9%

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

   6. Simplified 91.9%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{-71}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{z}{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-70}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + \frac{a}{\frac{z \cdot z}{t}} \cdot -0.5}\\ \end{array} \]

Alternative 6: 84.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-106}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{z}{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{-71}:\\ \;\;\;\;\frac{y \cdot x}{\frac{\sqrt{t \cdot \left(-a\right)}}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + \frac{a}{\frac{z \cdot z}{t}} \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5e-106)
   (* (* y x) (/ z (- (* 0.5 (* t (/ a z))) z)))
   (if (<= z 6.4e-71)
     (/ (* y x) (/ (sqrt (* t (- a))) z))
     (/ (* y x) (+ 1.0 (* (/ a (/ (* z z) t)) -0.5))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5e-106) {
		tmp = (y * x) * (z / ((0.5 * (t * (a / z))) - z));
	} else if (z <= 6.4e-71) {
		tmp = (y * x) / (sqrt((t * -a)) / z);
	} else {
		tmp = (y * x) / (1.0 + ((a / ((z * z) / t)) * -0.5));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-5d-106)) then
        tmp = (y * x) * (z / ((0.5d0 * (t * (a / z))) - z))
    else if (z <= 6.4d-71) then
        tmp = (y * x) / (sqrt((t * -a)) / z)
    else
        tmp = (y * x) / (1.0d0 + ((a / ((z * z) / t)) * (-0.5d0)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5e-106) {
		tmp = (y * x) * (z / ((0.5 * (t * (a / z))) - z));
	} else if (z <= 6.4e-71) {
		tmp = (y * x) / (Math.sqrt((t * -a)) / z);
	} else {
		tmp = (y * x) / (1.0 + ((a / ((z * z) / t)) * -0.5));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -5e-106:
		tmp = (y * x) * (z / ((0.5 * (t * (a / z))) - z))
	elif z <= 6.4e-71:
		tmp = (y * x) / (math.sqrt((t * -a)) / z)
	else:
		tmp = (y * x) / (1.0 + ((a / ((z * z) / t)) * -0.5))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5e-106)
		tmp = Float64(Float64(y * x) * Float64(z / Float64(Float64(0.5 * Float64(t * Float64(a / z))) - z)));
	elseif (z <= 6.4e-71)
		tmp = Float64(Float64(y * x) / Float64(sqrt(Float64(t * Float64(-a))) / z));
	else
		tmp = Float64(Float64(y * x) / Float64(1.0 + Float64(Float64(a / Float64(Float64(z * z) / t)) * -0.5)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -5e-106)
		tmp = (y * x) * (z / ((0.5 * (t * (a / z))) - z));
	elseif (z <= 6.4e-71)
		tmp = (y * x) / (sqrt((t * -a)) / z);
	else
		tmp = (y * x) / (1.0 + ((a / ((z * z) / t)) * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5e-106], N[(N[(y * x), $MachinePrecision] * N[(z / N[(N[(0.5 * N[(t * N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.4e-71], N[(N[(y * x), $MachinePrecision] / N[(N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / N[(1.0 + N[(N[(a / N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.99999999999999983e-106

   1. Initial program 56.4%

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

      1. associate-/l* 62.7%

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

   3. Simplified 62.7%

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

      1. associate-*l/ 62.6%

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

      2. associate-/l* 61.2%

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

      3. div-inv 61.2%

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

      4. associate-*l* 62.6%

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

      5. div-inv 62.6%

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

   5. Applied egg-rr 62.6%

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

   6. Taylor expanded in z around -inf 79.3%

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

      1. add-sqr-sqrt 52.8%

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

      2. sqrt-unprod 79.1%

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

      3. pow2 79.1%

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

      4. associate-/l* 82.3%

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

   8. Applied egg-rr 82.3%

      \[\leadsto \left(x \cdot \frac{z}{0.5 \cdot \color{blue}{\sqrt{{\left(\frac{a}{\frac{z}{t}}\right)}^{2}}} + -1 \cdot z}\right) \cdot y \]
   9. Step-by-step derivation

      1. unpow2 82.3%

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

      2. rem-sqrt-square 83.7%

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

      3. associate-/r/ 83.7%

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

   10. Simplified 83.7%

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

   11. Taylor expanded in x around 0 62.2%

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

       1. fma-def 62.2%

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

       2. *-commutative 62.2%

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

       3. associate-*r/ 63.6%

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

       4. neg-mul-1 63.6%

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

       5. *-commutative 63.6%

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

       6. associate-*r* 71.2%

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

       7. associate-*r/ 84.7%

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

       8. fma-udef 84.7%

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

       9. associate-*r/ 80.3%

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

       10. *-commutative 80.3%

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

       11. unsub-neg 80.3%

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

   13. Simplified 84.7%

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

   if -4.99999999999999983e-106 < z < 6.3999999999999998e-71

   1. Initial program 80.8%

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

      1. associate-/l* 84.6%

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

   3. Simplified 84.6%

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

   4. Taylor expanded in z around 0 76.2%

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

      1. associate-*r* 74.8%

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

      2. neg-mul-1 74.8%

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

   6. Simplified 76.2%

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

   if 6.3999999999999998e-71 < z

   1. Initial program 65.0%

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

      1. associate-/l* 68.2%

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

   3. Simplified 68.2%

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

   4. Taylor expanded in z around inf 86.3%

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

      1. unpow2 86.3%

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

      2. associate-/l* 91.9%

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

   6. Simplified 91.9%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-106}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{z}{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{-71}:\\ \;\;\;\;\frac{y \cdot x}{\frac{\sqrt{t \cdot \left(-a\right)}}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + \frac{a}{\frac{z \cdot z}{t}} \cdot -0.5}\\ \end{array} \]

Alternative 7: 76.6% accurate, 5.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{-193}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-29}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z + -0.5 \cdot \frac{t \cdot a}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.05e-193)
   (* y (- x))
   (if (<= z 2.4e-29) (* y (/ (* z x) (+ z (* -0.5 (/ (* t a) z))))) (* y x))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.05e-193) {
		tmp = y * -x;
	} else if (z <= 2.4e-29) {
		tmp = y * ((z * x) / (z + (-0.5 * ((t * a) / z))));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.05d-193)) then
        tmp = y * -x
    else if (z <= 2.4d-29) then
        tmp = y * ((z * x) / (z + ((-0.5d0) * ((t * a) / z))))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.05e-193) {
		tmp = y * -x;
	} else if (z <= 2.4e-29) {
		tmp = y * ((z * x) / (z + (-0.5 * ((t * a) / z))));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.05e-193:
		tmp = y * -x
	elif z <= 2.4e-29:
		tmp = y * ((z * x) / (z + (-0.5 * ((t * a) / z))))
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.05e-193)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 2.4e-29)
		tmp = Float64(y * Float64(Float64(z * x) / Float64(z + Float64(-0.5 * Float64(Float64(t * a) / z)))));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.05e-193)
		tmp = y * -x;
	elseif (z <= 2.4e-29)
		tmp = y * ((z * x) / (z + (-0.5 * ((t * a) / 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.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.05e-193], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 2.4e-29], N[(y * N[(N[(z * x), $MachinePrecision] / N[(z + N[(-0.5 * N[(N[(t * a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.05e-193

   1. Initial program 60.3%

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

      1. *-commutative 60.3%

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

      2. associate-*l* 61.7%

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

      3. associate-*r/ 66.1%

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

   3. Simplified 66.1%

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

   4. Taylor expanded in z around -inf 78.5%

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

      1. neg-mul-1 78.5%

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

   6. Simplified 78.5%

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

   if -1.05e-193 < z < 2.39999999999999992e-29

   1. Initial program 81.5%

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

      1. *-commutative 81.5%

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

      2. associate-*l* 75.2%

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

      3. associate-*r/ 78.1%

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

   3. Simplified 78.1%

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

   4. Taylor expanded in z around inf 47.0%

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

   if 2.39999999999999992e-29 < z

   1. Initial program 62.4%

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

      1. *-commutative 62.4%

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

      2. associate-*l* 60.9%

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

      3. associate-*r/ 63.2%

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

   3. Simplified 63.2%

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

   4. Taylor expanded in z around inf 93.3%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{-193}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-29}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z + -0.5 \cdot \frac{t \cdot a}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 8: 77.0% accurate, 5.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-195}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 12.6:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{z + -0.5 \cdot \frac{t \cdot a}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.1e-195)
   (* y (- x))
   (if (<= z 12.6) (* y (* x (/ z (+ z (* -0.5 (/ (* t a) z)))))) (* y x))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.1e-195) {
		tmp = y * -x;
	} else if (z <= 12.6) {
		tmp = y * (x * (z / (z + (-0.5 * ((t * a) / z)))));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-3.1d-195)) then
        tmp = y * -x
    else if (z <= 12.6d0) then
        tmp = y * (x * (z / (z + ((-0.5d0) * ((t * a) / z)))))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.1e-195) {
		tmp = y * -x;
	} else if (z <= 12.6) {
		tmp = y * (x * (z / (z + (-0.5 * ((t * a) / z)))));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.1e-195:
		tmp = y * -x
	elif z <= 12.6:
		tmp = y * (x * (z / (z + (-0.5 * ((t * a) / z)))))
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.1e-195)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 12.6)
		tmp = Float64(y * Float64(x * Float64(z / Float64(z + Float64(-0.5 * Float64(Float64(t * a) / z))))));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -3.1e-195)
		tmp = y * -x;
	elseif (z <= 12.6)
		tmp = y * (x * (z / (z + (-0.5 * ((t * a) / 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.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.1e-195], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 12.6], N[(y * N[(x * N[(z / N[(z + N[(-0.5 * N[(N[(t * a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.10000000000000002e-195

   1. Initial program 60.3%

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

      1. *-commutative 60.3%

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

      2. associate-*l* 61.7%

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

      3. associate-*r/ 66.1%

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

   3. Simplified 66.1%

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

   4. Taylor expanded in z around -inf 78.5%

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

      1. neg-mul-1 78.5%

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

   6. Simplified 78.5%

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

   if -3.10000000000000002e-195 < z < 12.5999999999999996

   1. Initial program 83.0%

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

      1. associate-/l* 85.7%

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

   3. Simplified 85.7%

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

      1. associate-*l/ 84.5%

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

      2. associate-/l* 79.9%

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

      3. div-inv 79.7%

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

      4. associate-*l* 84.4%

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

      5. div-inv 84.4%

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

   5. Applied egg-rr 84.4%

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

   6. Taylor expanded in z around inf 54.1%

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

   if 12.5999999999999996 < z

   1. Initial program 59.5%

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

      1. *-commutative 59.5%

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

      2. associate-*l* 57.9%

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

      3. associate-*r/ 60.4%

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

   3. Simplified 60.4%

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

   4. Taylor expanded in z around inf 94.0%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-195}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 12.6:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{z + -0.5 \cdot \frac{t \cdot a}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 9: 76.0% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{-161}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-177}:\\ \;\;\;\;-2 \cdot \left(\frac{y}{a} \cdot \frac{z \cdot \left(z \cdot x\right)}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.05e-161)
   (* y (- x))
   (if (<= z 2.6e-177) (* -2.0 (* (/ y a) (/ (* z (* z x)) t))) (* y x))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.05e-161) {
		tmp = y * -x;
	} else if (z <= 2.6e-177) {
		tmp = -2.0 * ((y / a) * ((z * (z * x)) / t));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.05d-161)) then
        tmp = y * -x
    else if (z <= 2.6d-177) then
        tmp = (-2.0d0) * ((y / a) * ((z * (z * x)) / t))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.05e-161) {
		tmp = y * -x;
	} else if (z <= 2.6e-177) {
		tmp = -2.0 * ((y / a) * ((z * (z * x)) / t));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.05e-161:
		tmp = y * -x
	elif z <= 2.6e-177:
		tmp = -2.0 * ((y / a) * ((z * (z * x)) / t))
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.05e-161)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 2.6e-177)
		tmp = Float64(-2.0 * Float64(Float64(y / a) * Float64(Float64(z * Float64(z * x)) / t)));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.05e-161)
		tmp = y * -x;
	elseif (z <= 2.6e-177)
		tmp = -2.0 * ((y / a) * ((z * (z * x)) / 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.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.05e-161], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 2.6e-177], N[(-2.0 * N[(N[(y / a), $MachinePrecision] * N[(N[(z * N[(z * x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.05e-161

   1. Initial program 58.9%

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

      1. *-commutative 58.9%

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

      2. associate-*l* 59.4%

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

      3. associate-*r/ 64.1%

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

   3. Simplified 64.1%

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

   4. Taylor expanded in z around -inf 81.9%

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

      1. neg-mul-1 81.9%

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

   6. Simplified 81.9%

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

   if -1.05e-161 < z < 2.6000000000000001e-177

   1. Initial program 77.0%

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

      1. associate-/l* 82.0%

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

   3. Simplified 82.0%

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

      1. *-commutative 82.0%

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

      2. div-inv 81.9%

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

      3. times-frac 74.5%

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

   5. Applied egg-rr 74.5%

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

   6. Taylor expanded in z around inf 36.8%

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

   7. Taylor expanded in z around 0 36.4%

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

      1. times-frac 36.3%

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

      2. unpow2 36.3%

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

      3. associate-*r* 39.0%

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

   9. Simplified 39.0%

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

   if 2.6000000000000001e-177 < z

   1. Initial program 69.5%

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

      1. *-commutative 69.5%

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

      2. associate-*l* 66.5%

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

      3. associate-*r/ 68.2%

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

   3. Simplified 68.2%

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

   4. Taylor expanded in z around inf 81.5%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{-161}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-177}:\\ \;\;\;\;-2 \cdot \left(\frac{y}{a} \cdot \frac{z \cdot \left(z \cdot x\right)}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 10: 76.0% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{-160}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-198}:\\ \;\;\;\;-2 \cdot \frac{\left(z \cdot x\right) \cdot \left(z \cdot \frac{y}{a}\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.3e-160)
   (* y (- x))
   (if (<= z 3.3e-198) (* -2.0 (/ (* (* z x) (* z (/ y a))) t)) (* y x))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.3e-160) {
		tmp = y * -x;
	} else if (z <= 3.3e-198) {
		tmp = -2.0 * (((z * x) * (z * (y / a))) / t);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.3d-160)) then
        tmp = y * -x
    else if (z <= 3.3d-198) then
        tmp = (-2.0d0) * (((z * x) * (z * (y / a))) / t)
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.3e-160) {
		tmp = y * -x;
	} else if (z <= 3.3e-198) {
		tmp = -2.0 * (((z * x) * (z * (y / a))) / t);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.3e-160:
		tmp = y * -x
	elif z <= 3.3e-198:
		tmp = -2.0 * (((z * x) * (z * (y / a))) / t)
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.3e-160)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 3.3e-198)
		tmp = Float64(-2.0 * Float64(Float64(Float64(z * x) * Float64(z * Float64(y / a))) / t));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.3e-160)
		tmp = y * -x;
	elseif (z <= 3.3e-198)
		tmp = -2.0 * (((z * x) * (z * (y / a))) / 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.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.3e-160], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 3.3e-198], N[(-2.0 * N[(N[(N[(z * x), $MachinePrecision] * N[(z * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.30000000000000002e-160

   1. Initial program 58.9%

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

      1. *-commutative 58.9%

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

      2. associate-*l* 59.4%

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

      3. associate-*r/ 64.1%

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

   3. Simplified 64.1%

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

   4. Taylor expanded in z around -inf 81.9%

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

      1. neg-mul-1 81.9%

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

   6. Simplified 81.9%

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

   if -1.30000000000000002e-160 < z < 3.3e-198

   1. Initial program 74.3%

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

      1. associate-/l* 80.0%

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

   3. Simplified 80.0%

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

      1. *-commutative 80.0%

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

      2. div-inv 79.8%

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

      3. times-frac 77.1%

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

   5. Applied egg-rr 77.1%

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

   6. Taylor expanded in z around inf 40.4%

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

   7. Taylor expanded in z around 0 40.1%

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

      1. associate-/r* 40.7%

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

      2. unpow2 40.7%

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

      3. associate-*r* 43.5%

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

      4. associate-*l/ 43.1%

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

      5. associate-*r* 43.3%

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

   9. Simplified 43.3%

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

   if 3.3e-198 < z

   1. Initial program 70.5%

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

      1. *-commutative 70.5%

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

      2. associate-*l* 66.1%

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

      3. associate-*r/ 67.7%

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

   3. Simplified 67.7%

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

   4. Taylor expanded in z around inf 79.0%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{-160}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-198}:\\ \;\;\;\;-2 \cdot \frac{\left(z \cdot x\right) \cdot \left(z \cdot \frac{y}{a}\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 11: 78.0% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 7 \cdot 10^{-198}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{z}{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z 7e-198) (* (* y x) (/ z (- (* 0.5 (* t (/ a z))) z))) (* y x)))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= 7e-198) {
		tmp = (y * x) * (z / ((0.5 * (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.
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 <= 7d-198) then
        tmp = (y * x) * (z / ((0.5d0 * (t * (a / z))) - z))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= 7e-198) {
		tmp = (y * x) * (z / ((0.5 * (t * (a / z))) - z));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= 7e-198:
		tmp = (y * x) * (z / ((0.5 * (t * (a / z))) - z))
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= 7e-198)
		tmp = Float64(Float64(y * x) * Float64(z / Float64(Float64(0.5 * Float64(t * Float64(a / z))) - z)));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= 7e-198)
		tmp = (y * x) * (z / ((0.5 * (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.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, 7e-198], N[(N[(y * x), $MachinePrecision] * N[(z / N[(N[(0.5 * N[(t * N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 7.0000000000000005e-198

   1. Initial program 62.8%

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

      1. associate-/l* 69.1%

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

   3. Simplified 69.1%

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

      1. associate-*l/ 71.1%

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

      2. associate-/l* 68.8%

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

      3. div-inv 68.7%

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

      4. associate-*l* 71.1%

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

      5. div-inv 71.1%

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

   5. Applied egg-rr 71.1%

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

   6. Taylor expanded in z around -inf 69.7%

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

      1. add-sqr-sqrt 46.7%

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

      2. sqrt-unprod 69.6%

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

      3. pow2 69.6%

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

      4. associate-/l* 71.7%

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

   8. Applied egg-rr 71.7%

      \[\leadsto \left(x \cdot \frac{z}{0.5 \cdot \color{blue}{\sqrt{{\left(\frac{a}{\frac{z}{t}}\right)}^{2}}} + -1 \cdot z}\right) \cdot y \]
   9. Step-by-step derivation

      1. unpow2 71.7%

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

      2. rem-sqrt-square 72.9%

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

      3. associate-/r/ 72.9%

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

   10. Simplified 72.9%

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

   11. Taylor expanded in x around 0 57.7%

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

       1. fma-def 57.7%

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

       2. *-commutative 57.7%

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

       3. associate-*r/ 58.6%

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

       4. neg-mul-1 58.6%

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

       5. *-commutative 58.6%

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

       6. associate-*r* 63.8%

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

       7. associate-*r/ 73.6%

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

       8. fma-udef 73.6%

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

       9. associate-*r/ 70.5%

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

       10. *-commutative 70.5%

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

       11. unsub-neg 70.5%

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

   13. Simplified 73.5%

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

   if 7.0000000000000005e-198 < z

   1. Initial program 70.5%

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

      1. *-commutative 70.5%

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

      2. associate-*l* 66.1%

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

      3. associate-*r/ 67.7%

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

   3. Simplified 67.7%

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

   4. Taylor expanded in z around inf 79.0%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7 \cdot 10^{-198}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{z}{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 12: 79.0% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{-278}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{z}{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + \frac{a}{\frac{z \cdot z}{t}} \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.35e-278)
   (* (* y x) (/ z (- (* 0.5 (* t (/ a z))) z)))
   (/ (* y x) (+ 1.0 (* (/ a (/ (* z z) t)) -0.5)))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.35e-278) {
		tmp = (y * x) * (z / ((0.5 * (t * (a / z))) - z));
	} else {
		tmp = (y * x) / (1.0 + ((a / ((z * z) / t)) * -0.5));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.35d-278)) then
        tmp = (y * x) * (z / ((0.5d0 * (t * (a / z))) - z))
    else
        tmp = (y * x) / (1.0d0 + ((a / ((z * z) / t)) * (-0.5d0)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.35e-278) {
		tmp = (y * x) * (z / ((0.5 * (t * (a / z))) - z));
	} else {
		tmp = (y * x) / (1.0 + ((a / ((z * z) / t)) * -0.5));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.35e-278:
		tmp = (y * x) * (z / ((0.5 * (t * (a / z))) - z))
	else:
		tmp = (y * x) / (1.0 + ((a / ((z * z) / t)) * -0.5))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.35e-278)
		tmp = Float64(Float64(y * x) * Float64(z / Float64(Float64(0.5 * Float64(t * Float64(a / z))) - z)));
	else
		tmp = Float64(Float64(y * x) / Float64(1.0 + Float64(Float64(a / Float64(Float64(z * z) / t)) * -0.5)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.35e-278)
		tmp = (y * x) * (z / ((0.5 * (t * (a / z))) - z));
	else
		tmp = (y * x) / (1.0 + ((a / ((z * z) / t)) * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.35e-278], N[(N[(y * x), $MachinePrecision] * N[(z / N[(N[(0.5 * N[(t * N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / N[(1.0 + N[(N[(a / N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3500000000000001e-278

   1. Initial program 60.1%

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

      1. associate-/l* 67.1%

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

   3. Simplified 67.1%

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

      1. associate-*l/ 70.2%

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

      2. associate-/l* 67.6%

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

      3. div-inv 67.5%

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

      4. associate-*l* 70.1%

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

      5. div-inv 70.2%

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

   5. Applied egg-rr 70.2%

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

   6. Taylor expanded in z around -inf 71.0%

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

      1. add-sqr-sqrt 51.1%

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

      2. sqrt-unprod 70.8%

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

      3. pow2 70.8%

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

      4. associate-/l* 73.1%

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

   8. Applied egg-rr 73.1%

      \[\leadsto \left(x \cdot \frac{z}{0.5 \cdot \color{blue}{\sqrt{{\left(\frac{a}{\frac{z}{t}}\right)}^{2}}} + -1 \cdot z}\right) \cdot y \]
   9. Step-by-step derivation

      1. unpow2 73.1%

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

      2. rem-sqrt-square 74.4%

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

      3. associate-/r/ 74.4%

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

   10. Simplified 74.4%

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

   11. Taylor expanded in x around 0 57.5%

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

       1. fma-def 57.5%

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

       2. *-commutative 57.5%

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

       3. associate-*r/ 58.5%

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

       4. neg-mul-1 58.5%

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

       5. *-commutative 58.5%

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

       6. associate-*r* 64.3%

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

       7. associate-*r/ 75.2%

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

       8. fma-udef 75.2%

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

       9. associate-*r/ 71.8%

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

       10. *-commutative 71.8%

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

       11. unsub-neg 71.8%

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

   13. Simplified 75.2%

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

   if -1.3500000000000001e-278 < z

   1. Initial program 72.1%

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

      1. associate-/l* 74.3%

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

   3. Simplified 74.3%

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

   4. Taylor expanded in z around inf 74.6%

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

      1. unpow2 74.6%

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

      2. associate-/l* 78.4%

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

   6. Simplified 78.4%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{-278}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{z}{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + \frac{a}{\frac{z \cdot z}{t}} \cdot -0.5}\\ \end{array} \]

Alternative 13: 74.9% accurate, 9.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-155}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-270}:\\ \;\;\;\;\frac{z \cdot \left(y \cdot x\right)}{-z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.8e-155)
   (* y (- x))
   (if (<= z 1.65e-270) (/ (* z (* y x)) (- z)) (* y x))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.8e-155) {
		tmp = y * -x;
	} else if (z <= 1.65e-270) {
		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.
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 <= (-2.8d-155)) then
        tmp = y * -x
    else if (z <= 1.65d-270) then
        tmp = (z * (y * x)) / -z
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.8e-155) {
		tmp = y * -x;
	} else if (z <= 1.65e-270) {
		tmp = (z * (y * x)) / -z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.8e-155:
		tmp = y * -x
	elif z <= 1.65e-270:
		tmp = (z * (y * x)) / -z
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.8e-155)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.65e-270)
		tmp = Float64(Float64(z * Float64(y * x)) / Float64(-z));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.8e-155)
		tmp = y * -x;
	elseif (z <= 1.65e-270)
		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.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.8e-155], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.65e-270], N[(N[(z * N[(y * x), $MachinePrecision]), $MachinePrecision] / (-z)), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.8e-155

   1. Initial program 58.9%

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

      1. *-commutative 58.9%

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

      2. associate-*l* 59.4%

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

      3. associate-*r/ 64.1%

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

   3. Simplified 64.1%

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

   4. Taylor expanded in z around -inf 81.9%

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

      1. neg-mul-1 81.9%

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

   6. Simplified 81.9%

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

   if -2.8e-155 < z < 1.65000000000000009e-270

   1. Initial program 72.9%

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

   2. Taylor expanded in z around -inf 35.5%

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

      1. neg-mul-1 35.5%

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

   4. Simplified 35.5%

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

   if 1.65000000000000009e-270 < z

   1. Initial program 71.0%

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

      1. *-commutative 71.0%

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

      2. associate-*l* 66.2%

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

      3. associate-*r/ 67.7%

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

   3. Simplified 67.7%

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

   4. Taylor expanded in z around inf 76.1%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-155}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-270}:\\ \;\;\;\;\frac{z \cdot \left(y \cdot x\right)}{-z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 14: 72.9% accurate, 18.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-310}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4e-310) (* y (- x)) (* y x)))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4e-310) {
		tmp = y * -x;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-4d-310)) then
        tmp = y * -x
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4e-310) {
		tmp = y * -x;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -4e-310:
		tmp = y * -x
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4e-310)
		tmp = Float64(y * Float64(-x));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -4e-310)
		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.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4e-310], N[(y * (-x)), $MachinePrecision], N[(y * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.999999999999988e-310

   1. Initial program 61.1%

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

      1. *-commutative 61.1%

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

      2. associate-*l* 63.0%

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

      3. associate-*r/ 68.4%

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

   3. Simplified 68.4%

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

   4. Taylor expanded in z around -inf 70.4%

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

      1. neg-mul-1 70.4%

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

   6. Simplified 70.4%

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

   if -3.999999999999988e-310 < z

   1. Initial program 71.5%

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

      1. *-commutative 71.5%

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

      2. associate-*l* 66.7%

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

      3. associate-*r/ 68.2%

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

   3. Simplified 68.2%

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

   4. Taylor expanded in z around inf 75.0%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-310}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 15: 42.6% accurate, 37.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

x, y = num2cell(sort([x, y])){:}
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.
code[x_, y_, z_, t_, a_] := N[(y * x), $MachinePrecision]

Derivation

1. Initial program 66.4%

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

   1. *-commutative 66.4%

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

   2. associate-*l* 64.9%

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

   3. associate-*r/ 68.3%

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

3. Simplified 68.3%

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

4. Taylor expanded in z around inf 45.9%

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

5. Final simplification 45.9%

   \[\leadsto y \cdot x \]

Developer target: 89.4% 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 2023200 
(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)))))