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

Percentage Accurate: 61.8% → 90.7%

Time: 26.1s

Alternatives: 23

Speedup: 18.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := \frac{a}{\frac{z \cdot z}{t}}\\ \mathbf{if}\;z \leq -1.65 \cdot 10^{+52}:\\ \;\;\;\;\frac{x \cdot y}{\mathsf{fma}\left(0.5, t_1, -1\right)}\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-89}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\sqrt{1 - t_1}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ a (/ (* z z) t))))
   (if (<= z -1.65e+52)
     (/ (* x y) (fma 0.5 t_1 -1.0))
     (if (<= z 9e-89)
       (* y (/ (* z x) (sqrt (- (* z z) (* a t)))))
       (/ (* x y) (sqrt (- 1.0 t_1)))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = a / ((z * z) / t);
	double tmp;
	if (z <= -1.65e+52) {
		tmp = (x * y) / fma(0.5, t_1, -1.0);
	} else if (z <= 9e-89) {
		tmp = y * ((z * x) / sqrt(((z * z) - (a * t))));
	} else {
		tmp = (x * y) / sqrt((1.0 - t_1));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = Float64(a / Float64(Float64(z * z) / t))
	tmp = 0.0
	if (z <= -1.65e+52)
		tmp = Float64(Float64(x * y) / fma(0.5, t_1, -1.0));
	elseif (z <= 9e-89)
		tmp = Float64(y * Float64(Float64(z * x) / sqrt(Float64(Float64(z * z) - Float64(a * t)))));
	else
		tmp = Float64(Float64(x * y) / sqrt(Float64(1.0 - t_1)));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a / N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.65e+52], N[(N[(x * y), $MachinePrecision] / N[(0.5 * t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9e-89], N[(y * N[(N[(z * x), $MachinePrecision] / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.65e52

   1. Initial program 30.0%

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

      1. associate-/l* 36.9%

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

   3. Simplified 36.9%

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

   4. Taylor expanded in z around -inf 83.2%

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

      1. fma-neg 83.2%

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

      2. unpow2 83.2%

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

      3. associate-/l* 95.0%

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

      4. metadata-eval 95.0%

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

   6. Simplified 95.0%

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

   if -1.65e52 < z < 8.9999999999999998e-89

   1. Initial program 88.6%

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

      1. *-commutative 88.6%

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

      2. associate-*l* 83.8%

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

      3. associate-*r/ 86.0%

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

   3. Simplified 86.0%

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

   if 8.9999999999999998e-89 < z

   1. Initial program 62.6%

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

      1. associate-/l* 62.8%

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

   3. Simplified 62.8%

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

      1. add-sqr-sqrt 62.8%

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

      2. sqrt-unprod 62.8%

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

      3. frac-times 57.4%

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

      4. add-sqr-sqrt 57.4%

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

   5. Applied egg-rr 57.4%

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

      1. div-sub 57.4%

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

      2. *-inverses 85.5%

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

      3. *-commutative 85.5%

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

      4. associate-/l* 96.5%

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

   7. Simplified 96.5%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{+52}:\\ \;\;\;\;\frac{x \cdot y}{\mathsf{fma}\left(0.5, \frac{a}{\frac{z \cdot z}{t}}, -1\right)}\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-89}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\sqrt{1 - \frac{a}{\frac{z \cdot z}{t}}}}\\ \end{array} \]

Alternative 2: 90.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{+65}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{+63}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - a \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + \frac{a}{\frac{z \cdot z}{t}} \cdot -0.5}\\ \end{array} \end{array} \]

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 -4.2e+65)
   (* y (- x))
   (if (<= z 8.4e+63)
     (* x (* y (/ z (sqrt (- (* z z) (* a t))))))
     (/ (* x y) (+ 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 <= -4.2e+65) {
		tmp = y * -x;
	} else if (z <= 8.4e+63) {
		tmp = x * (y * (z / sqrt(((z * z) - (a * t)))));
	} else {
		tmp = (x * y) / (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 <= (-4.2d+65)) then
        tmp = y * -x
    else if (z <= 8.4d+63) then
        tmp = x * (y * (z / sqrt(((z * z) - (a * t)))))
    else
        tmp = (x * y) / (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 <= -4.2e+65) {
		tmp = y * -x;
	} else if (z <= 8.4e+63) {
		tmp = x * (y * (z / Math.sqrt(((z * z) - (a * t)))));
	} else {
		tmp = (x * y) / (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 <= -4.2e+65:
		tmp = y * -x
	elif z <= 8.4e+63:
		tmp = x * (y * (z / math.sqrt(((z * z) - (a * t)))))
	else:
		tmp = (x * y) / (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 <= -4.2e+65)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 8.4e+63)
		tmp = Float64(x * Float64(y * Float64(z / sqrt(Float64(Float64(z * z) - Float64(a * t))))));
	else
		tmp = Float64(Float64(x * y) / 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 <= -4.2e+65)
		tmp = y * -x;
	elseif (z <= 8.4e+63)
		tmp = x * (y * (z / sqrt(((z * z) - (a * t)))));
	else
		tmp = (x * y) / (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, -4.2e+65], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 8.4e+63], N[(x * N[(y * N[(z / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $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 < -4.19999999999999983e65

   1. Initial program 30.6%

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

      1. *-commutative 30.6%

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

      2. associate-*l* 27.4%

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

      3. associate-*r/ 30.5%

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

   3. Simplified 30.5%

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

   4. Taylor expanded in z around -inf 94.9%

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

      1. neg-mul-1 94.9%

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

   6. Simplified 94.9%

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

   if -4.19999999999999983e65 < z < 8.4000000000000007e63

   1. Initial program 90.2%

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

      1. associate-/l* 91.1%

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

   3. Simplified 91.1%

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

      1. div-inv 91.0%

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

      2. clear-num 91.4%

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

      3. associate-*l* 90.7%

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

   5. Applied egg-rr 90.7%

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

   if 8.4000000000000007e63 < z

   1. Initial program 41.7%

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

      1. associate-/l* 42.1%

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

   3. Simplified 42.1%

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

   4. Taylor expanded in z around inf 77.4%

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

      1. unpow2 77.4%

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

      2. associate-/l* 88.1%

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

   6. Simplified 88.1%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{+65}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{+63}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - a \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + \frac{a}{\frac{z \cdot z}{t}} \cdot -0.5}\\ \end{array} \]

Alternative 3: 89.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.45 \cdot 10^{+54}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+25}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + \frac{a}{\frac{z \cdot z}{t}} \cdot -0.5}\\ \end{array} \end{array} \]

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 -2.45e+54)
   (* y (- x))
   (if (<= z 5.2e+25)
     (* y (/ (* z x) (sqrt (- (* z z) (* a t)))))
     (/ (* x y) (+ 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 <= -2.45e+54) {
		tmp = y * -x;
	} else if (z <= 5.2e+25) {
		tmp = y * ((z * x) / sqrt(((z * z) - (a * t))));
	} else {
		tmp = (x * y) / (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 <= (-2.45d+54)) then
        tmp = y * -x
    else if (z <= 5.2d+25) then
        tmp = y * ((z * x) / sqrt(((z * z) - (a * t))))
    else
        tmp = (x * y) / (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 <= -2.45e+54) {
		tmp = y * -x;
	} else if (z <= 5.2e+25) {
		tmp = y * ((z * x) / Math.sqrt(((z * z) - (a * t))));
	} else {
		tmp = (x * y) / (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 <= -2.45e+54:
		tmp = y * -x
	elif z <= 5.2e+25:
		tmp = y * ((z * x) / math.sqrt(((z * z) - (a * t))))
	else:
		tmp = (x * y) / (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 <= -2.45e+54)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 5.2e+25)
		tmp = Float64(y * Float64(Float64(z * x) / sqrt(Float64(Float64(z * z) - Float64(a * t)))));
	else
		tmp = Float64(Float64(x * y) / 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 <= -2.45e+54)
		tmp = y * -x;
	elseif (z <= 5.2e+25)
		tmp = y * ((z * x) / sqrt(((z * z) - (a * t))));
	else
		tmp = (x * y) / (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, -2.45e+54], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 5.2e+25], N[(y * N[(N[(z * x), $MachinePrecision] / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $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 < -2.45e54

   1. Initial program 30.0%

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

      1. *-commutative 30.0%

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

      2. associate-*l* 27.0%

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

      3. associate-*r/ 32.4%

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

   3. Simplified 32.4%

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

   4. Taylor expanded in z around -inf 95.0%

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

      1. neg-mul-1 95.0%

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

   6. Simplified 95.0%

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

   if -2.45e54 < z < 5.1999999999999997e25

   1. Initial program 91.0%

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

      1. *-commutative 91.0%

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

      2. associate-*l* 87.1%

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

      3. associate-*r/ 88.8%

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

   3. Simplified 88.8%

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

   if 5.1999999999999997e25 < z

   1. Initial program 48.1%

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

      1. associate-/l* 48.4%

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

   3. Simplified 48.4%

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

   4. Taylor expanded in z around inf 78.4%

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

      1. unpow2 78.4%

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

      2. associate-/l* 88.0%

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

   6. Simplified 88.0%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.45 \cdot 10^{+54}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+25}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + \frac{a}{\frac{z \cdot z}{t}} \cdot -0.5}\\ \end{array} \]

Alternative 4: 90.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{+65}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{+63}:\\ \;\;\;\;\frac{z}{\sqrt{z \cdot z - a \cdot t}} \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + \frac{a}{\frac{z \cdot z}{t}} \cdot -0.5}\\ \end{array} \end{array} \]

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 -4.2e+65)
   (* y (- x))
   (if (<= z 8.4e+63)
     (* (/ z (sqrt (- (* z z) (* a t)))) (* x y))
     (/ (* x y) (+ 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 <= -4.2e+65) {
		tmp = y * -x;
	} else if (z <= 8.4e+63) {
		tmp = (z / sqrt(((z * z) - (a * t)))) * (x * y);
	} else {
		tmp = (x * y) / (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 <= (-4.2d+65)) then
        tmp = y * -x
    else if (z <= 8.4d+63) then
        tmp = (z / sqrt(((z * z) - (a * t)))) * (x * y)
    else
        tmp = (x * y) / (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 <= -4.2e+65) {
		tmp = y * -x;
	} else if (z <= 8.4e+63) {
		tmp = (z / Math.sqrt(((z * z) - (a * t)))) * (x * y);
	} else {
		tmp = (x * y) / (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 <= -4.2e+65:
		tmp = y * -x
	elif z <= 8.4e+63:
		tmp = (z / math.sqrt(((z * z) - (a * t)))) * (x * y)
	else:
		tmp = (x * y) / (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 <= -4.2e+65)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 8.4e+63)
		tmp = Float64(Float64(z / sqrt(Float64(Float64(z * z) - Float64(a * t)))) * Float64(x * y));
	else
		tmp = Float64(Float64(x * y) / 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 <= -4.2e+65)
		tmp = y * -x;
	elseif (z <= 8.4e+63)
		tmp = (z / sqrt(((z * z) - (a * t)))) * (x * y);
	else
		tmp = (x * y) / (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, -4.2e+65], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 8.4e+63], N[(N[(z / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(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 < -4.19999999999999983e65

   1. Initial program 30.6%

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

      1. *-commutative 30.6%

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

      2. associate-*l* 27.4%

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

      3. associate-*r/ 30.5%

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

   3. Simplified 30.5%

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

   4. Taylor expanded in z around -inf 94.9%

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

      1. neg-mul-1 94.9%

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

   6. Simplified 94.9%

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

   if -4.19999999999999983e65 < z < 8.4000000000000007e63

   1. Initial program 90.2%

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

      1. associate-/l* 91.1%

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

   3. Simplified 91.1%

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

      1. clear-num 91.0%

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

      2. associate-/r/ 91.0%

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

      3. clear-num 91.4%

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

   5. Applied egg-rr 91.4%

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

   if 8.4000000000000007e63 < z

   1. Initial program 41.7%

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

      1. associate-/l* 42.1%

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

   3. Simplified 42.1%

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

   4. Taylor expanded in z around inf 77.4%

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

      1. unpow2 77.4%

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

      2. associate-/l* 88.1%

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

   6. Simplified 88.1%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{+65}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{+63}:\\ \;\;\;\;\frac{z}{\sqrt{z \cdot z - a \cdot t}} \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + \frac{a}{\frac{z \cdot z}{t}} \cdot -0.5}\\ \end{array} \]

Alternative 5: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{+65}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+168}:\\ \;\;\;\;\frac{z}{\sqrt{z \cdot z - a \cdot t}} \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{a \cdot \frac{t}{z}}{z}, 1\right) \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4.2e+65)
   (* y (- x))
   (if (<= z 1.15e+168)
     (* (/ z (sqrt (- (* z z) (* a t)))) (* x y))
     (* (fma 0.5 (/ (* a (/ t z)) z) 1.0) (* x y)))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.2e+65) {
		tmp = y * -x;
	} else if (z <= 1.15e+168) {
		tmp = (z / sqrt(((z * z) - (a * t)))) * (x * y);
	} else {
		tmp = fma(0.5, ((a * (t / z)) / z), 1.0) * (x * y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4.2e+65)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.15e+168)
		tmp = Float64(Float64(z / sqrt(Float64(Float64(z * z) - Float64(a * t)))) * Float64(x * y));
	else
		tmp = Float64(fma(0.5, Float64(Float64(a * Float64(t / z)) / z), 1.0) * Float64(x * y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4.2e+65], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.15e+168], N[(N[(z / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(N[(a * N[(t / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.19999999999999983e65

   1. Initial program 30.6%

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

      1. *-commutative 30.6%

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

      2. associate-*l* 27.4%

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

      3. associate-*r/ 30.5%

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

   3. Simplified 30.5%

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

   4. Taylor expanded in z around -inf 94.9%

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

      1. neg-mul-1 94.9%

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

   6. Simplified 94.9%

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

   if -4.19999999999999983e65 < z < 1.15e168

   1. Initial program 87.6%

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

      1. associate-/l* 88.3%

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

   3. Simplified 88.3%

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

      1. clear-num 88.1%

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

      2. associate-/r/ 88.3%

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

      3. clear-num 88.6%

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

   5. Applied egg-rr 88.6%

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

   if 1.15e168 < z

   1. Initial program 11.9%

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

      1. associate-/l* 12.4%

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

   3. Simplified 12.4%

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

      1. clear-num 12.4%

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

      2. associate-/r/ 12.4%

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

      3. clear-num 12.4%

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

   5. Applied egg-rr 12.4%

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

   6. Taylor expanded in z around inf 80.0%

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

      1. fma-def 80.0%

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

      2. *-commutative 80.0%

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

      3. unpow2 80.0%

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

      4. associate-/r* 83.7%

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

      5. *-commutative 83.7%

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

      6. associate-*r/ 99.9%

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

   8. Simplified 99.9%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{+65}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+168}:\\ \;\;\;\;\frac{z}{\sqrt{z \cdot z - a \cdot t}} \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{a \cdot \frac{t}{z}}{z}, 1\right) \cdot \left(x \cdot y\right)\\ \end{array} \]

Alternative 6: 90.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+52}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-89}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\sqrt{1 - \frac{a}{\frac{z \cdot z}{t}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -7e+52)
   (* y (- x))
   (if (<= z 9e-89)
     (* y (/ (* z x) (sqrt (- (* z z) (* a t)))))
     (/ (* x y) (sqrt (- 1.0 (/ a (/ (* z z) t))))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7e+52) {
		tmp = y * -x;
	} else if (z <= 9e-89) {
		tmp = y * ((z * x) / sqrt(((z * z) - (a * t))));
	} else {
		tmp = (x * y) / sqrt((1.0 - (a / ((z * z) / t))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-7d+52)) then
        tmp = y * -x
    else if (z <= 9d-89) then
        tmp = y * ((z * x) / sqrt(((z * z) - (a * t))))
    else
        tmp = (x * y) / sqrt((1.0d0 - (a / ((z * z) / t))))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7e+52) {
		tmp = y * -x;
	} else if (z <= 9e-89) {
		tmp = y * ((z * x) / Math.sqrt(((z * z) - (a * t))));
	} else {
		tmp = (x * y) / Math.sqrt((1.0 - (a / ((z * z) / t))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -7e+52:
		tmp = y * -x
	elif z <= 9e-89:
		tmp = y * ((z * x) / math.sqrt(((z * z) - (a * t))))
	else:
		tmp = (x * y) / math.sqrt((1.0 - (a / ((z * z) / t))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -7e+52)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 9e-89)
		tmp = Float64(y * Float64(Float64(z * x) / sqrt(Float64(Float64(z * z) - Float64(a * t)))));
	else
		tmp = Float64(Float64(x * y) / sqrt(Float64(1.0 - Float64(a / Float64(Float64(z * z) / t)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -7e+52)
		tmp = y * -x;
	elseif (z <= 9e-89)
		tmp = y * ((z * x) / sqrt(((z * z) - (a * t))));
	else
		tmp = (x * y) / sqrt((1.0 - (a / ((z * z) / t))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -7e+52], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 9e-89], N[(y * N[(N[(z * x), $MachinePrecision] / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[Sqrt[N[(1.0 - N[(a / N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7e52

   1. Initial program 30.0%

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

      1. *-commutative 30.0%

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

      2. associate-*l* 27.0%

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

      3. associate-*r/ 32.4%

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

   3. Simplified 32.4%

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

   4. Taylor expanded in z around -inf 95.0%

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

      1. neg-mul-1 95.0%

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

   6. Simplified 95.0%

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

   if -7e52 < z < 8.9999999999999998e-89

   1. Initial program 88.6%

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

      1. *-commutative 88.6%

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

      2. associate-*l* 83.8%

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

      3. associate-*r/ 86.0%

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

   3. Simplified 86.0%

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

   if 8.9999999999999998e-89 < z

   1. Initial program 62.6%

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

      1. associate-/l* 62.8%

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

   3. Simplified 62.8%

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

      1. add-sqr-sqrt 62.8%

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

      2. sqrt-unprod 62.8%

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

      3. frac-times 57.4%

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

      4. add-sqr-sqrt 57.4%

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

   5. Applied egg-rr 57.4%

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

      1. div-sub 57.4%

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

      2. *-inverses 85.5%

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

      3. *-commutative 85.5%

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

      4. associate-/l* 96.5%

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

   7. Simplified 96.5%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+52}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-89}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\sqrt{1 - \frac{a}{\frac{z \cdot z}{t}}}}\\ \end{array} \]

Alternative 7: 82.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-5}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-117}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + \frac{a}{\frac{z \cdot z}{t}} \cdot -0.5}\\ \end{array} \end{array} \]

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 -4.2e-5)
   (* y (- x))
   (if (<= z 1.4e-117)
     (* y (/ (* z x) (sqrt (* a (- t)))))
     (/ (* x y) (+ 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 <= -4.2e-5) {
		tmp = y * -x;
	} else if (z <= 1.4e-117) {
		tmp = y * ((z * x) / sqrt((a * -t)));
	} else {
		tmp = (x * y) / (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 <= (-4.2d-5)) then
        tmp = y * -x
    else if (z <= 1.4d-117) then
        tmp = y * ((z * x) / sqrt((a * -t)))
    else
        tmp = (x * y) / (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 <= -4.2e-5) {
		tmp = y * -x;
	} else if (z <= 1.4e-117) {
		tmp = y * ((z * x) / Math.sqrt((a * -t)));
	} else {
		tmp = (x * y) / (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 <= -4.2e-5:
		tmp = y * -x
	elif z <= 1.4e-117:
		tmp = y * ((z * x) / math.sqrt((a * -t)))
	else:
		tmp = (x * y) / (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 <= -4.2e-5)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.4e-117)
		tmp = Float64(y * Float64(Float64(z * x) / sqrt(Float64(a * Float64(-t)))));
	else
		tmp = Float64(Float64(x * y) / 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 <= -4.2e-5)
		tmp = y * -x;
	elseif (z <= 1.4e-117)
		tmp = y * ((z * x) / sqrt((a * -t)));
	else
		tmp = (x * y) / (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, -4.2e-5], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.4e-117], N[(y * N[(N[(z * x), $MachinePrecision] / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(1.0 + N[(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 < -4.19999999999999977e-5

   1. Initial program 39.0%

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

      1. *-commutative 39.0%

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

      2. associate-*l* 36.4%

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

      3. associate-*r/ 42.2%

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

   3. Simplified 42.2%

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

   4. Taylor expanded in z around -inf 92.4%

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

      1. neg-mul-1 92.4%

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

   6. Simplified 92.4%

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

   if -4.19999999999999977e-5 < z < 1.4e-117

   1. Initial program 87.3%

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

      1. *-commutative 87.3%

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

      2. associate-*l* 81.3%

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

      3. associate-*r/ 82.8%

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

   3. Simplified 82.8%

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

   4. Taylor expanded in z around 0 74.3%

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

      1. associate-*r* 74.3%

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

      2. neg-mul-1 74.3%

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

   6. Simplified 74.3%

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

   if 1.4e-117 < z

   1. Initial program 64.9%

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

      1. associate-/l* 65.2%

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

   3. Simplified 65.2%

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

   4. Taylor expanded in z around inf 79.6%

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

      1. unpow2 79.6%

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

      2. associate-/l* 86.1%

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

   6. Simplified 86.1%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-5}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-117}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + \frac{a}{\frac{z \cdot z}{t}} \cdot -0.5}\\ \end{array} \]

Alternative 8: 82.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -0.00024:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-135}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \frac{y}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + \frac{a}{\frac{z \cdot z}{t}} \cdot -0.5}\\ \end{array} \end{array} \]

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 -0.00024)
   (* y (- x))
   (if (<= z 1.7e-135)
     (* (* z x) (/ y (sqrt (* a (- t)))))
     (/ (* x y) (+ 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 <= -0.00024) {
		tmp = y * -x;
	} else if (z <= 1.7e-135) {
		tmp = (z * x) * (y / sqrt((a * -t)));
	} else {
		tmp = (x * y) / (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 <= (-0.00024d0)) then
        tmp = y * -x
    else if (z <= 1.7d-135) then
        tmp = (z * x) * (y / sqrt((a * -t)))
    else
        tmp = (x * y) / (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 <= -0.00024) {
		tmp = y * -x;
	} else if (z <= 1.7e-135) {
		tmp = (z * x) * (y / Math.sqrt((a * -t)));
	} else {
		tmp = (x * y) / (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 <= -0.00024:
		tmp = y * -x
	elif z <= 1.7e-135:
		tmp = (z * x) * (y / math.sqrt((a * -t)))
	else:
		tmp = (x * y) / (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 <= -0.00024)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.7e-135)
		tmp = Float64(Float64(z * x) * Float64(y / sqrt(Float64(a * Float64(-t)))));
	else
		tmp = Float64(Float64(x * y) / 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 <= -0.00024)
		tmp = y * -x;
	elseif (z <= 1.7e-135)
		tmp = (z * x) * (y / sqrt((a * -t)));
	else
		tmp = (x * y) / (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, -0.00024], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.7e-135], N[(N[(z * x), $MachinePrecision] * N[(y / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $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 < -2.40000000000000006e-4

   1. Initial program 39.0%

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

      1. *-commutative 39.0%

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

      2. associate-*l* 36.4%

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

      3. associate-*r/ 42.2%

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

   3. Simplified 42.2%

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

   4. Taylor expanded in z around -inf 92.4%

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

      1. neg-mul-1 92.4%

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

   6. Simplified 92.4%

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

   if -2.40000000000000006e-4 < z < 1.69999999999999995e-135

   1. Initial program 86.5%

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

      1. *-commutative 86.5%

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

      2. associate-*l* 81.4%

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

      3. associate-*r/ 82.9%

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

   3. Simplified 82.9%

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

      1. clear-num 81.7%

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

      2. un-div-inv 81.7%

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

      3. *-commutative 81.7%

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

   5. Applied egg-rr 81.7%

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

   6. Taylor expanded in z around 0 75.3%

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

      1. associate-*r* 76.4%

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

      2. neg-mul-1 76.4%

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

   8. Simplified 75.3%

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

      1. associate-/r/ 76.5%

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

      2. *-commutative 76.5%

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

   10. Applied egg-rr 76.5%

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

   if 1.69999999999999995e-135 < z

   1. Initial program 66.7%

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

      1. associate-/l* 66.9%

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

   3. Simplified 66.9%

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

   4. Taylor expanded in z around inf 77.9%

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

      1. unpow2 77.9%

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

      2. associate-/l* 84.0%

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

   6. Simplified 84.0%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.00024:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-135}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \frac{y}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + \frac{a}{\frac{z \cdot z}{t}} \cdot -0.5}\\ \end{array} \]

Alternative 9: 82.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-6}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-117}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot x\right)}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + \frac{a}{\frac{z \cdot z}{t}} \cdot -0.5}\\ \end{array} \end{array} \]

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 -6.8e-6)
   (* y (- x))
   (if (<= z 1.6e-117)
     (/ (* y (* z x)) (sqrt (* a (- t))))
     (/ (* x y) (+ 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 <= -6.8e-6) {
		tmp = y * -x;
	} else if (z <= 1.6e-117) {
		tmp = (y * (z * x)) / sqrt((a * -t));
	} else {
		tmp = (x * y) / (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 <= (-6.8d-6)) then
        tmp = y * -x
    else if (z <= 1.6d-117) then
        tmp = (y * (z * x)) / sqrt((a * -t))
    else
        tmp = (x * y) / (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 <= -6.8e-6) {
		tmp = y * -x;
	} else if (z <= 1.6e-117) {
		tmp = (y * (z * x)) / Math.sqrt((a * -t));
	} else {
		tmp = (x * y) / (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 <= -6.8e-6:
		tmp = y * -x
	elif z <= 1.6e-117:
		tmp = (y * (z * x)) / math.sqrt((a * -t))
	else:
		tmp = (x * y) / (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 <= -6.8e-6)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.6e-117)
		tmp = Float64(Float64(y * Float64(z * x)) / sqrt(Float64(a * Float64(-t))));
	else
		tmp = Float64(Float64(x * y) / 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 <= -6.8e-6)
		tmp = y * -x;
	elseif (z <= 1.6e-117)
		tmp = (y * (z * x)) / sqrt((a * -t));
	else
		tmp = (x * y) / (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, -6.8e-6], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.6e-117], N[(N[(y * N[(z * x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $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 < -6.80000000000000012e-6

   1. Initial program 39.0%

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

      1. *-commutative 39.0%

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

      2. associate-*l* 36.4%

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

      3. associate-*r/ 42.2%

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

   3. Simplified 42.2%

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

   4. Taylor expanded in z around -inf 92.4%

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

      1. neg-mul-1 92.4%

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

   6. Simplified 92.4%

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

   if -6.80000000000000012e-6 < z < 1.59999999999999998e-117

   1. Initial program 87.3%

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

      1. *-commutative 87.3%

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

      2. associate-*l* 81.3%

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

      3. associate-*r/ 82.8%

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

   3. Simplified 82.8%

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

      1. clear-num 81.6%

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

      2. un-div-inv 81.6%

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

      3. *-commutative 81.6%

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

   5. Applied egg-rr 81.6%

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

   6. Taylor expanded in z around 0 73.3%

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

      1. associate-*r* 74.3%

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

      2. neg-mul-1 74.3%

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

   8. Simplified 73.3%

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

      1. expm1-log1p-u 59.7%

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

      2. expm1-udef 44.4%

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

      3. associate-/r/ 44.3%

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

      4. *-commutative 44.3%

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

   10. Applied egg-rr 44.3%

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

       1. expm1-def 60.8%

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

       2. expm1-log1p 74.5%

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

       3. associate-*l/ 73.0%

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

       4. distribute-rgt-neg-out 73.0%

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

       5. *-commutative 73.0%

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

       6. distribute-rgt-neg-in 73.0%

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

   12. Simplified 73.0%

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

   if 1.59999999999999998e-117 < z

   1. Initial program 64.9%

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

      1. associate-/l* 65.2%

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

   3. Simplified 65.2%

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

   4. Taylor expanded in z around inf 79.6%

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

      1. unpow2 79.6%

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

      2. associate-/l* 86.1%

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

   6. Simplified 86.1%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-6}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-117}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot x\right)}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + \frac{a}{\frac{z \cdot z}{t}} \cdot -0.5}\\ \end{array} \]

Alternative 10: 76.1% accurate, 5.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -5.6 \cdot 10^{-217}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-61}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z + -0.5 \cdot \frac{a \cdot t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.6e-217)
   (* y (- x))
   (if (<= z 1.2e-61) (* y (/ (* z x) (+ z (* -0.5 (/ (* a t) z))))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.6e-217) {
		tmp = y * -x;
	} else if (z <= 1.2e-61) {
		tmp = y * ((z * x) / (z + (-0.5 * ((a * t) / z))));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-5.6d-217)) then
        tmp = y * -x
    else if (z <= 1.2d-61) then
        tmp = y * ((z * x) / (z + ((-0.5d0) * ((a * t) / z))))
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.6e-217) {
		tmp = y * -x;
	} else if (z <= 1.2e-61) {
		tmp = y * ((z * x) / (z + (-0.5 * ((a * t) / z))));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -5.6e-217:
		tmp = y * -x
	elif z <= 1.2e-61:
		tmp = y * ((z * x) / (z + (-0.5 * ((a * t) / z))))
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.6e-217)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.2e-61)
		tmp = Float64(y * Float64(Float64(z * x) / Float64(z + Float64(-0.5 * Float64(Float64(a * t) / z)))));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -5.6e-217)
		tmp = y * -x;
	elseif (z <= 1.2e-61)
		tmp = y * ((z * x) / (z + (-0.5 * ((a * t) / z))));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.6e-217], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.2e-61], N[(y * N[(N[(z * x), $MachinePrecision] / N[(z + N[(-0.5 * N[(N[(a * t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.6e-217

   1. Initial program 51.3%

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

      1. *-commutative 51.3%

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

      2. associate-*l* 49.3%

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

      3. associate-*r/ 54.6%

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

   3. Simplified 54.6%

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

   4. Taylor expanded in z around -inf 78.8%

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

      1. neg-mul-1 78.8%

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

   6. Simplified 78.8%

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

   if -5.6e-217 < z < 1.2e-61

   1. Initial program 91.3%

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

      1. *-commutative 91.3%

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

      2. associate-*l* 83.3%

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

      3. associate-*r/ 83.2%

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

   3. Simplified 83.2%

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

   4. Taylor expanded in z around inf 50.9%

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

   if 1.2e-61 < z

   1. Initial program 60.4%

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

      1. *-commutative 60.4%

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

      2. associate-*l* 58.9%

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

      3. associate-*r/ 59.2%

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

   3. Simplified 59.2%

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

   4. Taylor expanded in z around inf 85.3%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.6 \cdot 10^{-217}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-61}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z + -0.5 \cdot \frac{a \cdot t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 11: 75.1% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-217}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-140}:\\ \;\;\;\;-2 \cdot \left(\frac{y}{a} \cdot \frac{z}{\frac{\frac{t}{x}}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4.8e-217)
   (* y (- x))
   (if (<= z 3.8e-140) (* -2.0 (* (/ y a) (/ z (/ (/ t x) z)))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.8e-217) {
		tmp = y * -x;
	} else if (z <= 3.8e-140) {
		tmp = -2.0 * ((y / a) * (z / ((t / x) / z)));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-4.8d-217)) then
        tmp = y * -x
    else if (z <= 3.8d-140) then
        tmp = (-2.0d0) * ((y / a) * (z / ((t / x) / z)))
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.8e-217) {
		tmp = y * -x;
	} else if (z <= 3.8e-140) {
		tmp = -2.0 * ((y / a) * (z / ((t / x) / z)));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -4.8e-217:
		tmp = y * -x
	elif z <= 3.8e-140:
		tmp = -2.0 * ((y / a) * (z / ((t / x) / z)))
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4.8e-217)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 3.8e-140)
		tmp = Float64(-2.0 * Float64(Float64(y / a) * Float64(z / Float64(Float64(t / x) / z))));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -4.8e-217)
		tmp = y * -x;
	elseif (z <= 3.8e-140)
		tmp = -2.0 * ((y / a) * (z / ((t / x) / z)));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4.8e-217], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 3.8e-140], N[(-2.0 * N[(N[(y / a), $MachinePrecision] * N[(z / N[(N[(t / x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.7999999999999997e-217

   1. Initial program 51.3%

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

      1. *-commutative 51.3%

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

      2. associate-*l* 49.3%

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

      3. associate-*r/ 54.6%

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

   3. Simplified 54.6%

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

   4. Taylor expanded in z around -inf 78.8%

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

      1. neg-mul-1 78.8%

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

   6. Simplified 78.8%

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

   if -4.7999999999999997e-217 < z < 3.79999999999999998e-140

   1. Initial program 88.2%

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

   2. Taylor expanded in z around inf 45.4%

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

   3. Taylor expanded in z around 0 43.1%

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

      1. times-frac 43.1%

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

      2. associate-/l* 39.9%

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

      3. unpow2 39.9%

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

      4. associate-/l* 40.1%

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

   5. Simplified 40.1%

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

   if 3.79999999999999998e-140 < z

   1. Initial program 66.7%

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

      1. *-commutative 66.7%

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

      2. associate-*l* 64.5%

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

      3. associate-*r/ 64.7%

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

   3. Simplified 64.7%

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

   4. Taylor expanded in z around inf 81.9%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-217}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-140}:\\ \;\;\;\;-2 \cdot \left(\frac{y}{a} \cdot \frac{z}{\frac{\frac{t}{x}}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 12: 74.7% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{-128}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-137}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot x\right)}{0.5 \cdot \frac{a \cdot t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -7.2e-128)
   (* y (- x))
   (if (<= z 6.5e-137) (/ (* y (* z x)) (* 0.5 (/ (* a t) z))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7.2e-128) {
		tmp = y * -x;
	} else if (z <= 6.5e-137) {
		tmp = (y * (z * x)) / (0.5 * ((a * t) / z));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-7.2d-128)) then
        tmp = y * -x
    else if (z <= 6.5d-137) then
        tmp = (y * (z * x)) / (0.5d0 * ((a * t) / z))
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7.2e-128) {
		tmp = y * -x;
	} else if (z <= 6.5e-137) {
		tmp = (y * (z * x)) / (0.5 * ((a * t) / z));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -7.2e-128:
		tmp = y * -x
	elif z <= 6.5e-137:
		tmp = (y * (z * x)) / (0.5 * ((a * t) / z))
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -7.2e-128)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 6.5e-137)
		tmp = Float64(Float64(y * Float64(z * x)) / Float64(0.5 * Float64(Float64(a * t) / z)));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -7.2e-128)
		tmp = y * -x;
	elseif (z <= 6.5e-137)
		tmp = (y * (z * x)) / (0.5 * ((a * t) / z));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -7.2e-128], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 6.5e-137], N[(N[(y * N[(z * x), $MachinePrecision]), $MachinePrecision] / N[(0.5 * N[(N[(a * t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.20000000000000049e-128

   1. Initial program 49.0%

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

      1. *-commutative 49.0%

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

      2. associate-*l* 47.8%

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

      3. associate-*r/ 53.3%

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

   3. Simplified 53.3%

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

   4. Taylor expanded in z around -inf 84.2%

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

      1. neg-mul-1 84.2%

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

   6. Simplified 84.2%

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

   if -7.20000000000000049e-128 < z < 6.49999999999999991e-137

   1. Initial program 86.3%

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

      1. associate-/l* 83.2%

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

   3. Simplified 83.2%

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

   4. Taylor expanded in z around -inf 38.2%

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

   5. Taylor expanded in x around 0 38.4%

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

   6. Taylor expanded in a around inf 38.4%

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

   if 6.49999999999999991e-137 < z

   1. Initial program 66.7%

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

      1. *-commutative 66.7%

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

      2. associate-*l* 64.5%

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

      3. associate-*r/ 64.7%

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

   3. Simplified 64.7%

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

   4. Taylor expanded in z around inf 81.9%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{-128}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-137}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot x\right)}{0.5 \cdot \frac{a \cdot t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 13: 75.3% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-217}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-149}:\\ \;\;\;\;\frac{z \cdot \left(x \cdot y\right)}{-0.5 \cdot \frac{a \cdot t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4.8e-217)
   (* y (- x))
   (if (<= z 2.1e-149) (/ (* z (* x y)) (* -0.5 (/ (* a t) z))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.8e-217) {
		tmp = y * -x;
	} else if (z <= 2.1e-149) {
		tmp = (z * (x * y)) / (-0.5 * ((a * t) / z));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-4.8d-217)) then
        tmp = y * -x
    else if (z <= 2.1d-149) then
        tmp = (z * (x * y)) / ((-0.5d0) * ((a * t) / z))
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.8e-217) {
		tmp = y * -x;
	} else if (z <= 2.1e-149) {
		tmp = (z * (x * y)) / (-0.5 * ((a * t) / z));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -4.8e-217:
		tmp = y * -x
	elif z <= 2.1e-149:
		tmp = (z * (x * y)) / (-0.5 * ((a * t) / z))
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4.8e-217)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 2.1e-149)
		tmp = Float64(Float64(z * Float64(x * y)) / Float64(-0.5 * Float64(Float64(a * t) / z)));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -4.8e-217)
		tmp = y * -x;
	elseif (z <= 2.1e-149)
		tmp = (z * (x * y)) / (-0.5 * ((a * t) / z));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4.8e-217], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 2.1e-149], N[(N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(-0.5 * N[(N[(a * t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.7999999999999997e-217

   1. Initial program 51.3%

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

      1. *-commutative 51.3%

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

      2. associate-*l* 49.3%

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

      3. associate-*r/ 54.6%

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

   3. Simplified 54.6%

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

   4. Taylor expanded in z around -inf 78.8%

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

      1. neg-mul-1 78.8%

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

   6. Simplified 78.8%

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

   if -4.7999999999999997e-217 < z < 2.10000000000000011e-149

   1. Initial program 88.2%

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

   2. Taylor expanded in z around inf 45.4%

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

   3. Taylor expanded in z around 0 45.4%

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

   if 2.10000000000000011e-149 < z

   1. Initial program 66.7%

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

      1. *-commutative 66.7%

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

      2. associate-*l* 64.5%

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

      3. associate-*r/ 64.7%

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

   3. Simplified 64.7%

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

   4. Taylor expanded in z around inf 81.9%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-217}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-149}:\\ \;\;\;\;\frac{z \cdot \left(x \cdot y\right)}{-0.5 \cdot \frac{a \cdot t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 14: 75.4% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -5.6 \cdot 10^{-217}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-139}:\\ \;\;\;\;\frac{z \cdot \left(x \cdot y\right)}{\frac{a \cdot -0.5}{\frac{z}{t}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.6e-217)
   (* y (- x))
   (if (<= z 1.7e-139) (/ (* z (* x y)) (/ (* a -0.5) (/ z t))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.6e-217) {
		tmp = y * -x;
	} else if (z <= 1.7e-139) {
		tmp = (z * (x * y)) / ((a * -0.5) / (z / t));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-5.6d-217)) then
        tmp = y * -x
    else if (z <= 1.7d-139) then
        tmp = (z * (x * y)) / ((a * (-0.5d0)) / (z / t))
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.6e-217) {
		tmp = y * -x;
	} else if (z <= 1.7e-139) {
		tmp = (z * (x * y)) / ((a * -0.5) / (z / t));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -5.6e-217:
		tmp = y * -x
	elif z <= 1.7e-139:
		tmp = (z * (x * y)) / ((a * -0.5) / (z / t))
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.6e-217)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.7e-139)
		tmp = Float64(Float64(z * Float64(x * y)) / Float64(Float64(a * -0.5) / Float64(z / t)));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -5.6e-217)
		tmp = y * -x;
	elseif (z <= 1.7e-139)
		tmp = (z * (x * y)) / ((a * -0.5) / (z / t));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.6e-217], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.7e-139], N[(N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(N[(a * -0.5), $MachinePrecision] / N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.6e-217

   1. Initial program 51.3%

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

      1. *-commutative 51.3%

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

      2. associate-*l* 49.3%

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

      3. associate-*r/ 54.6%

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

   3. Simplified 54.6%

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

   4. Taylor expanded in z around -inf 78.8%

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

      1. neg-mul-1 78.8%

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

   6. Simplified 78.8%

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

   if -5.6e-217 < z < 1.69999999999999999e-139

   1. Initial program 88.2%

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

   2. Taylor expanded in z around inf 45.4%

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

   3. Taylor expanded in z around 0 45.4%

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

      1. associate-/l* 45.4%

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

      2. associate-*r/ 45.4%

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

   5. Simplified 45.4%

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

   if 1.69999999999999999e-139 < z

   1. Initial program 66.7%

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

      1. *-commutative 66.7%

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

      2. associate-*l* 64.5%

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

      3. associate-*r/ 64.7%

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

   3. Simplified 64.7%

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

   4. Taylor expanded in z around inf 81.9%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.6 \cdot 10^{-217}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-139}:\\ \;\;\;\;\frac{z \cdot \left(x \cdot y\right)}{\frac{a \cdot -0.5}{\frac{z}{t}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 15: 76.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -5.6 \cdot 10^{-217}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + \frac{a}{\frac{z \cdot z}{t}} \cdot -0.5}\\ \end{array} \end{array} \]

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 -5.6e-217)
   (* y (- x))
   (/ (* x y) (+ 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 <= -5.6e-217) {
		tmp = y * -x;
	} else {
		tmp = (x * y) / (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 <= (-5.6d-217)) then
        tmp = y * -x
    else
        tmp = (x * y) / (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 <= -5.6e-217) {
		tmp = y * -x;
	} else {
		tmp = (x * y) / (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 <= -5.6e-217:
		tmp = y * -x
	else:
		tmp = (x * y) / (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 <= -5.6e-217)
		tmp = Float64(y * Float64(-x));
	else
		tmp = Float64(Float64(x * y) / 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 <= -5.6e-217)
		tmp = y * -x;
	else
		tmp = (x * y) / (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, -5.6e-217], N[(y * (-x)), $MachinePrecision], N[(N[(x * y), $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 < -5.6e-217

   1. Initial program 51.3%

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

      1. *-commutative 51.3%

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

      2. associate-*l* 49.3%

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

      3. associate-*r/ 54.6%

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

   3. Simplified 54.6%

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

   4. Taylor expanded in z around -inf 78.8%

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

      1. neg-mul-1 78.8%

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

   6. Simplified 78.8%

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

   if -5.6e-217 < z

   1. Initial program 73.0%

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

      1. associate-/l* 71.8%

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

   3. Simplified 71.8%

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

   4. Taylor expanded in z around inf 67.5%

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

      1. unpow2 67.5%

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

      2. associate-/l* 71.8%

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

   6. Simplified 71.8%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.6 \cdot 10^{-217}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + \frac{a}{\frac{z \cdot z}{t}} \cdot -0.5}\\ \end{array} \]

Alternative 16: 78.1% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-264}:\\ \;\;\;\;\frac{x \cdot y}{\frac{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + \frac{a}{\frac{z \cdot z}{t}} \cdot -0.5}\\ \end{array} \end{array} \]

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 -1.9e-264)
   (/ (* x y) (/ (- (* 0.5 (* t (/ a z))) z) z))
   (/ (* x y) (+ 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 <= -1.9e-264) {
		tmp = (x * y) / (((0.5 * (t * (a / z))) - z) / z);
	} else {
		tmp = (x * y) / (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 <= (-1.9d-264)) then
        tmp = (x * y) / (((0.5d0 * (t * (a / z))) - z) / z)
    else
        tmp = (x * y) / (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 <= -1.9e-264) {
		tmp = (x * y) / (((0.5 * (t * (a / z))) - z) / z);
	} else {
		tmp = (x * y) / (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 <= -1.9e-264:
		tmp = (x * y) / (((0.5 * (t * (a / z))) - z) / z)
	else:
		tmp = (x * y) / (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 <= -1.9e-264)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(0.5 * Float64(t * Float64(a / z))) - z) / z));
	else
		tmp = Float64(Float64(x * y) / 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 <= -1.9e-264)
		tmp = (x * y) / (((0.5 * (t * (a / z))) - z) / z);
	else
		tmp = (x * y) / (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, -1.9e-264], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(0.5 * N[(t * N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $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 < -1.90000000000000007e-264

   1. Initial program 54.7%

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

      1. associate-/l* 58.9%

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

   3. Simplified 58.9%

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

   4. Taylor expanded in z around -inf 71.8%

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

      1. *-commutative 71.8%

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

      2. *-un-lft-identity 71.8%

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

      3. times-frac 76.7%

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

   6. Applied egg-rr 76.7%

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

   if -1.90000000000000007e-264 < z

   1. Initial program 71.6%

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

      1. associate-/l* 71.0%

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

   3. Simplified 71.0%

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

   4. Taylor expanded in z around inf 69.2%

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

      1. unpow2 69.2%

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

      2. associate-/l* 73.8%

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

   6. Simplified 73.8%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-264}:\\ \;\;\;\;\frac{x \cdot y}{\frac{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + \frac{a}{\frac{z \cdot z}{t}} \cdot -0.5}\\ \end{array} \]

Alternative 17: 75.0% accurate, 8.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -5.6 \cdot 10^{-217}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-93}:\\ \;\;\;\;\frac{\frac{x \cdot \left(y \cdot \left(-z\right)\right)}{z}}{-1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.6e-217)
   (* y (- x))
   (if (<= z 5e-93) (/ (/ (* x (* y (- z))) z) -1.0) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.6e-217) {
		tmp = y * -x;
	} else if (z <= 5e-93) {
		tmp = ((x * (y * -z)) / z) / -1.0;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-5.6d-217)) then
        tmp = y * -x
    else if (z <= 5d-93) then
        tmp = ((x * (y * -z)) / z) / (-1.0d0)
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.6e-217) {
		tmp = y * -x;
	} else if (z <= 5e-93) {
		tmp = ((x * (y * -z)) / z) / -1.0;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -5.6e-217:
		tmp = y * -x
	elif z <= 5e-93:
		tmp = ((x * (y * -z)) / z) / -1.0
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.6e-217)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 5e-93)
		tmp = Float64(Float64(Float64(x * Float64(y * Float64(-z))) / z) / -1.0);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -5.6e-217)
		tmp = y * -x;
	elseif (z <= 5e-93)
		tmp = ((x * (y * -z)) / z) / -1.0;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.6e-217], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 5e-93], N[(N[(N[(x * N[(y * (-z)), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / -1.0), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.6e-217

   1. Initial program 51.3%

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

      1. *-commutative 51.3%

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

      2. associate-*l* 49.3%

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

      3. associate-*r/ 54.6%

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

   3. Simplified 54.6%

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

   4. Taylor expanded in z around -inf 78.8%

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

      1. neg-mul-1 78.8%

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

   6. Simplified 78.8%

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

   if -5.6e-217 < z < 4.99999999999999994e-93

   1. Initial program 90.5%

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

   2. Taylor expanded in z around -inf 30.0%

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

      1. neg-mul-1 30.0%

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

   4. Simplified 30.0%

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

      1. *-un-lft-identity 30.0%

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

      2. neg-mul-1 30.0%

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

      3. *-commutative 30.0%

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

      4. times-frac 30.0%

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

      5. frac-2neg 30.0%

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

      6. metadata-eval 30.0%

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

      7. add-sqr-sqrt 12.2%

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

      8. sqrt-unprod 16.8%

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

      9. sqr-neg 16.8%

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

      10. sqrt-prod 28.1%

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

      11. add-sqr-sqrt 39.9%

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

      12. *-commutative 39.9%

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

   6. Applied egg-rr 39.9%

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

      1. associate-*r* 41.9%

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

      2. *-commutative 41.9%

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

   8. Simplified 41.9%

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

      1. associate-*r/ 41.9%

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

   10. Applied egg-rr 41.9%

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

       1. associate-*l/ 41.9%

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

       2. mul-1-neg 41.9%

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

       3. associate-*r* 41.8%

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

       4. *-commutative 41.8%

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

       5. distribute-rgt-neg-in 41.8%

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

   12. Simplified 41.8%

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

   if 4.99999999999999994e-93 < z

   1. Initial program 62.6%

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

      1. *-commutative 62.6%

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

      2. associate-*l* 61.2%

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

      3. associate-*r/ 61.4%

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

   3. Simplified 61.4%

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

   4. Taylor expanded in z around inf 84.9%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.6 \cdot 10^{-217}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-93}:\\ \;\;\;\;\frac{\frac{x \cdot \left(y \cdot \left(-z\right)\right)}{z}}{-1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 18: 72.1% accurate, 8.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{-226}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+15}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z}\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+168}:\\ \;\;\;\;z \cdot \frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -7.2e-226)
   (* y (- x))
   (if (<= z 4.2e+15)
     (* y (/ (* z x) z))
     (if (<= z 1.15e+168) (* z (/ y (/ z x))) (* x y)))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7.2e-226) {
		tmp = y * -x;
	} else if (z <= 4.2e+15) {
		tmp = y * ((z * x) / z);
	} else if (z <= 1.15e+168) {
		tmp = z * (y / (z / x));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-7.2d-226)) then
        tmp = y * -x
    else if (z <= 4.2d+15) then
        tmp = y * ((z * x) / z)
    else if (z <= 1.15d+168) then
        tmp = z * (y / (z / x))
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7.2e-226) {
		tmp = y * -x;
	} else if (z <= 4.2e+15) {
		tmp = y * ((z * x) / z);
	} else if (z <= 1.15e+168) {
		tmp = z * (y / (z / x));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -7.2e-226:
		tmp = y * -x
	elif z <= 4.2e+15:
		tmp = y * ((z * x) / z)
	elif z <= 1.15e+168:
		tmp = z * (y / (z / x))
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -7.2e-226)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 4.2e+15)
		tmp = Float64(y * Float64(Float64(z * x) / z));
	elseif (z <= 1.15e+168)
		tmp = Float64(z * Float64(y / Float64(z / x)));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -7.2e-226)
		tmp = y * -x;
	elseif (z <= 4.2e+15)
		tmp = y * ((z * x) / z);
	elseif (z <= 1.15e+168)
		tmp = z * (y / (z / x));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -7.2e-226], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 4.2e+15], N[(y * N[(N[(z * x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.15e+168], N[(z * N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -7.19999999999999988e-226

   1. Initial program 53.3%

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

      1. *-commutative 53.3%

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

      2. associate-*l* 51.4%

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

      3. associate-*r/ 56.5%

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

   3. Simplified 56.5%

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

   4. Taylor expanded in z around -inf 76.5%

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

      1. neg-mul-1 76.5%

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

   6. Simplified 76.5%

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

   if -7.19999999999999988e-226 < z < 4.2e15

   1. Initial program 92.8%

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

      1. *-commutative 92.8%

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

      2. associate-*l* 86.1%

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

      3. associate-*r/ 86.1%

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

   3. Simplified 86.1%

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

   4. Taylor expanded in z around inf 49.3%

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

   if 4.2e15 < z < 1.15e168

   1. Initial program 81.6%

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

      1. associate-*l/ 78.7%

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

   3. Simplified 78.7%

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

   4. Taylor expanded in z around inf 76.8%

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

      1. associate-/l* 71.4%

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

   6. Simplified 71.4%

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

   if 1.15e168 < z

   1. Initial program 11.9%

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

      1. *-commutative 11.9%

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

      2. associate-*l* 11.5%

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

      3. associate-*r/ 11.7%

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

   3. Simplified 11.7%

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

   4. Taylor expanded in z around inf 99.0%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{-226}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+15}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z}\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+168}:\\ \;\;\;\;z \cdot \frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 19: 75.0% accurate, 8.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-217}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-93}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot \left(-x\right)\right)}{-z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4e-217)
   (* y (- x))
   (if (<= z 5e-93) (/ (* y (* z (- x))) (- z)) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4e-217) {
		tmp = y * -x;
	} else if (z <= 5e-93) {
		tmp = (y * (z * -x)) / -z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-4d-217)) then
        tmp = y * -x
    else if (z <= 5d-93) then
        tmp = (y * (z * -x)) / -z
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4e-217) {
		tmp = y * -x;
	} else if (z <= 5e-93) {
		tmp = (y * (z * -x)) / -z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -4e-217:
		tmp = y * -x
	elif z <= 5e-93:
		tmp = (y * (z * -x)) / -z
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4e-217)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 5e-93)
		tmp = Float64(Float64(y * Float64(z * Float64(-x))) / Float64(-z));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -4e-217)
		tmp = y * -x;
	elseif (z <= 5e-93)
		tmp = (y * (z * -x)) / -z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4e-217], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 5e-93], N[(N[(y * N[(z * (-x)), $MachinePrecision]), $MachinePrecision] / (-z)), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.00000000000000033e-217

   1. Initial program 51.3%

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

      1. *-commutative 51.3%

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

      2. associate-*l* 49.3%

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

      3. associate-*r/ 54.6%

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

   3. Simplified 54.6%

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

   4. Taylor expanded in z around -inf 78.8%

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

      1. neg-mul-1 78.8%

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

   6. Simplified 78.8%

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

   if -4.00000000000000033e-217 < z < 4.99999999999999994e-93

   1. Initial program 90.5%

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

   2. Taylor expanded in z around -inf 30.0%

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

      1. neg-mul-1 30.0%

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

   4. Simplified 30.0%

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

      1. *-un-lft-identity 30.0%

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

      2. neg-mul-1 30.0%

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

      3. *-commutative 30.0%

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

      4. times-frac 30.0%

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

      5. frac-2neg 30.0%

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

      6. metadata-eval 30.0%

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

      7. add-sqr-sqrt 12.2%

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

      8. sqrt-unprod 16.8%

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

      9. sqr-neg 16.8%

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

      10. sqrt-prod 28.1%

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

      11. add-sqr-sqrt 39.9%

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

      12. *-commutative 39.9%

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

   6. Applied egg-rr 39.9%

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

      1. associate-*r* 41.9%

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

      2. *-commutative 41.9%

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

   8. Simplified 41.9%

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

      1. expm1-log1p-u 35.7%

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

      2. expm1-udef 41.2%

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

      3. frac-times 41.2%

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

      4. *-commutative 41.2%

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

      5. mul-1-neg 41.2%

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

   10. Applied egg-rr 41.2%

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

       1. expm1-def 35.7%

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

       2. expm1-log1p 41.9%

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

       3. associate-*r* 41.9%

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

       4. mul-1-neg 41.9%

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

   12. Simplified 41.9%

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

   if 4.99999999999999994e-93 < z

   1. Initial program 62.6%

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

      1. *-commutative 62.6%

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

      2. associate-*l* 61.2%

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

      3. associate-*r/ 61.4%

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

   3. Simplified 61.4%

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

   4. Taylor expanded in z around inf 84.9%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-217}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-93}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot \left(-x\right)\right)}{-z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 20: 73.9% accurate, 10.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{-217}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-93}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.6e-217)
   (* y (- x))
   (if (<= z 5e-93) (* x (/ (* z y) z)) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.6e-217) {
		tmp = y * -x;
	} else if (z <= 5e-93) {
		tmp = x * ((z * y) / z);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-2.6d-217)) then
        tmp = y * -x
    else if (z <= 5d-93) then
        tmp = x * ((z * y) / z)
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.6e-217) {
		tmp = y * -x;
	} else if (z <= 5e-93) {
		tmp = x * ((z * y) / z);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.6e-217:
		tmp = y * -x
	elif z <= 5e-93:
		tmp = x * ((z * y) / z)
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.6e-217)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 5e-93)
		tmp = Float64(x * Float64(Float64(z * y) / z));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.6e-217)
		tmp = y * -x;
	elseif (z <= 5e-93)
		tmp = x * ((z * y) / z);
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.6e-217], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 5e-93], N[(x * N[(N[(z * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.59999999999999993e-217

   1. Initial program 51.3%

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

      1. *-commutative 51.3%

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

      2. associate-*l* 49.3%

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

      3. associate-*r/ 54.6%

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

   3. Simplified 54.6%

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

   4. Taylor expanded in z around -inf 78.8%

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

      1. neg-mul-1 78.8%

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

   6. Simplified 78.8%

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

   if -2.59999999999999993e-217 < z < 4.99999999999999994e-93

   1. Initial program 90.5%

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

      1. associate-/l* 87.0%

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

   3. Simplified 87.0%

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

      1. clear-num 87.0%

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

      2. associate-/r/ 86.9%

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

      3. clear-num 87.6%

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

   5. Applied egg-rr 87.6%

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

      1. expm1-log1p-u 77.6%

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

      2. expm1-udef 50.0%

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

      3. cancel-sign-sub-inv 50.0%

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

      4. fma-def 50.0%

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

   7. Applied egg-rr 50.0%

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

      1. expm1-def 77.6%

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

      2. expm1-log1p 87.6%

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

      3. associate-*l/ 90.5%

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

      4. associate-*r* 81.8%

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

      5. *-commutative 81.8%

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

      6. associate-*l/ 80.0%

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

      7. associate-*r* 84.8%

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

      8. associate-*l/ 86.4%

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

      9. distribute-lft-neg-out 86.4%

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

      10. fma-neg 86.4%

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

      11. unpow2 86.4%

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

      12. *-commutative 86.4%

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

      13. unpow2 86.4%

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

   9. Simplified 86.4%

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

   10. Taylor expanded in z around inf 34.7%

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

   if 4.99999999999999994e-93 < z

   1. Initial program 62.6%

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

      1. *-commutative 62.6%

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

      2. associate-*l* 61.2%

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

      3. associate-*r/ 61.4%

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

   3. Simplified 61.4%

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

   4. Taylor expanded in z around inf 84.9%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{-217}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-93}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 21: 74.5% accurate, 10.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{-217}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{z \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.65e-217)
   (* y (- x))
   (if (<= z 1.6e+15) (/ (* z (* x y)) z) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.65e-217) {
		tmp = y * -x;
	} else if (z <= 1.6e+15) {
		tmp = (z * (x * y)) / z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.65d-217)) then
        tmp = y * -x
    else if (z <= 1.6d+15) then
        tmp = (z * (x * y)) / z
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.65e-217) {
		tmp = y * -x;
	} else if (z <= 1.6e+15) {
		tmp = (z * (x * y)) / z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.65e-217:
		tmp = y * -x
	elif z <= 1.6e+15:
		tmp = (z * (x * y)) / z
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.65e-217)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.6e+15)
		tmp = Float64(Float64(z * Float64(x * y)) / z);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.65e-217)
		tmp = y * -x;
	elseif (z <= 1.6e+15)
		tmp = (z * (x * y)) / z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.65e-217], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.6e+15], N[(N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.64999999999999996e-217

   1. Initial program 51.3%

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

      1. *-commutative 51.3%

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

      2. associate-*l* 49.3%

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

      3. associate-*r/ 54.6%

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

   3. Simplified 54.6%

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

   4. Taylor expanded in z around -inf 78.8%

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

      1. neg-mul-1 78.8%

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

   6. Simplified 78.8%

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

   if -1.64999999999999996e-217 < z < 1.6e15

   1. Initial program 93.3%

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

   2. Taylor expanded in z around inf 52.4%

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

   if 1.6e15 < z

   1. Initial program 50.4%

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

      1. *-commutative 50.4%

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

      2. associate-*l* 48.6%

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

      3. associate-*r/ 48.9%

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

   3. Simplified 48.9%

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

   4. Taylor expanded in z around inf 85.7%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{-217}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{z \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 22: 72.5% accurate, 18.6× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2e-310) (* y (- x)) (* x y)))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2e-310) {
		tmp = y * -x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-2d-310)) then
        tmp = y * -x
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2e-310) {
		tmp = y * -x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -2e-310:
		tmp = y * -x
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2e-310)
		tmp = Float64(y * Float64(-x));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2e-310)
		tmp = y * -x;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2e-310], N[(y * (-x)), $MachinePrecision], N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.999999999999994e-310

   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. *-commutative 56.4%

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

      2. associate-*l* 53.3%

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

      3. associate-*r/ 57.9%

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

   3. Simplified 57.9%

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

   4. Taylor expanded in z around -inf 70.1%

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

      1. neg-mul-1 70.1%

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

   6. Simplified 70.1%

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

   if -1.999999999999994e-310 < z

   1. Initial program 70.8%

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

      1. *-commutative 70.8%

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

      2. associate-*l* 67.5%

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

      3. associate-*r/ 67.6%

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

   3. Simplified 67.6%

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

   4. Taylor expanded in z around inf 68.4%

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

4. Final simplification 69.3%

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

Alternative 23: 41.9% accurate, 37.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.3%

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

   1. *-commutative 63.3%

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

   2. associate-*l* 60.2%

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

   3. associate-*r/ 62.6%

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

3. Simplified 62.6%

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

4. Taylor expanded in z around inf 42.1%

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

5. Final simplification 42.1%

   \[\leadsto x \cdot y \]

Developer target: 89.2% 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 2023185 
(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)))))