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

Percentage Accurate: 61.3% → 91.1%

Time: 19.5s

Alternatives: 14

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: 61.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := \frac{a}{\frac{z \cdot z}{t}}\\ \mathbf{if}\;z \leq -1.15 \cdot 10^{+103}:\\ \;\;\;\;\frac{x \cdot y}{\mathsf{fma}\left(0.5, t_1, -1\right)}\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{z \cdot z - a \cdot t}}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\sqrt{1 - t_1}}\\ \end{array} \end{array} \]

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 -1.15e+103)
     (/ (* x y) (fma 0.5 t_1 -1.0))
     (if (<= z 2.4e-14)
       (/ x (/ (sqrt (- (* z z) (* a t))) (* z y)))
       (/ (* x y) (sqrt (- 1.0 t_1)))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = a / ((z * z) / t);
	double tmp;
	if (z <= -1.15e+103) {
		tmp = (x * y) / fma(0.5, t_1, -1.0);
	} else if (z <= 2.4e-14) {
		tmp = x / (sqrt(((z * z) - (a * t))) / (z * y));
	} else {
		tmp = (x * y) / sqrt((1.0 - t_1));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = Float64(a / Float64(Float64(z * z) / t))
	tmp = 0.0
	if (z <= -1.15e+103)
		tmp = Float64(Float64(x * y) / fma(0.5, t_1, -1.0));
	elseif (z <= 2.4e-14)
		tmp = Float64(x / Float64(sqrt(Float64(Float64(z * z) - Float64(a * t))) / Float64(z * y)));
	else
		tmp = Float64(Float64(x * y) / sqrt(Float64(1.0 - t_1)));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a / N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.15e+103], N[(N[(x * y), $MachinePrecision] / N[(0.5 * t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.4e-14], N[(x / N[(N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(z * y), $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 < -1.15000000000000004e103

   1. Initial program 40.9%

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

      1. associate-/l* 41.3%

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

   3. Simplified 41.3%

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

   4. Taylor expanded in z around -inf 87.8%

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

      1. fma-neg 87.8%

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

      2. unpow2 87.8%

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

      3. associate-/l* 96.2%

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

      4. metadata-eval 96.2%

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

   6. Simplified 96.2%

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

   if -1.15000000000000004e103 < z < 2.4e-14

   1. Initial program 75.7%

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

      1. *-commutative 75.7%

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

      2. associate-*l* 78.8%

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

      3. associate-*r/ 82.0%

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

   3. Simplified 82.0%

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

      1. *-commutative 82.0%

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

      2. associate-/l* 82.8%

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

      3. associate-/r/ 83.4%

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

      4. associate-/l/ 82.4%

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

      5. *-commutative 82.4%

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

   5. Applied egg-rr 82.4%

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

   if 2.4e-14 < z

   1. Initial program 45.2%

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

      1. associate-/l* 48.5%

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

   3. Simplified 48.5%

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

      1. add-sqr-sqrt 48.5%

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

      2. sqrt-unprod 48.5%

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

      3. frac-times 38.7%

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

      4. add-sqr-sqrt 38.7%

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

   5. Applied egg-rr 38.7%

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

      1. div-sub 38.7%

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

      2. *-inverses 94.4%

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

      3. *-commutative 94.4%

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

      4. associate-/l* 98.9%

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

   7. Simplified 98.9%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+103}:\\ \;\;\;\;\frac{x \cdot y}{\mathsf{fma}\left(0.5, \frac{a}{\frac{z \cdot z}{t}}, -1\right)}\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{z \cdot z - a \cdot t}}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\sqrt{1 - \frac{a}{\frac{z \cdot z}{t}}}}\\ \end{array} \]

Alternative 2: 91.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+153}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 10^{+96}:\\ \;\;\;\;y \cdot \frac{x}{\frac{\sqrt{z \cdot z - a \cdot t}}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + -0.5 \cdot \frac{a}{z \cdot \frac{z}{t}}}\\ \end{array} \end{array} \]

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 -5e+153)
   (* x (- y))
   (if (<= z 1e+96)
     (* y (/ x (/ (sqrt (- (* z z) (* a t))) z)))
     (/ (* x y) (+ 1.0 (* -0.5 (/ a (* z (/ z t)))))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5e+153) {
		tmp = x * -y;
	} else if (z <= 1e+96) {
		tmp = y * (x / (sqrt(((z * z) - (a * t))) / z));
	} else {
		tmp = (x * y) / (1.0 + (-0.5 * (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 <= (-5d+153)) then
        tmp = x * -y
    else if (z <= 1d+96) then
        tmp = y * (x / (sqrt(((z * z) - (a * t))) / z))
    else
        tmp = (x * y) / (1.0d0 + ((-0.5d0) * (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 <= -5e+153) {
		tmp = x * -y;
	} else if (z <= 1e+96) {
		tmp = y * (x / (Math.sqrt(((z * z) - (a * t))) / z));
	} else {
		tmp = (x * y) / (1.0 + (-0.5 * (a / (z * (z / t)))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -5e+153:
		tmp = x * -y
	elif z <= 1e+96:
		tmp = y * (x / (math.sqrt(((z * z) - (a * t))) / z))
	else:
		tmp = (x * y) / (1.0 + (-0.5 * (a / (z * (z / t)))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5e+153)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 1e+96)
		tmp = Float64(y * Float64(x / Float64(sqrt(Float64(Float64(z * z) - Float64(a * t))) / z)));
	else
		tmp = Float64(Float64(x * y) / Float64(1.0 + Float64(-0.5 * Float64(a / Float64(z * Float64(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 <= -5e+153)
		tmp = x * -y;
	elseif (z <= 1e+96)
		tmp = y * (x / (sqrt(((z * z) - (a * t))) / z));
	else
		tmp = (x * y) / (1.0 + (-0.5 * (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, -5e+153], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 1e+96], N[(y * N[(x / N[(N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(a / N[(z * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.00000000000000018e153

   1. Initial program 14.3%

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

      1. *-commutative 14.3%

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

      2. associate-*l* 14.1%

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

      3. associate-*r/ 14.2%

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

   3. Simplified 14.2%

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

   4. Taylor expanded in z around -inf 97.2%

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

      1. neg-mul-1 97.2%

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

   6. Simplified 97.2%

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

   if -5.00000000000000018e153 < z < 1.00000000000000005e96

   1. Initial program 80.3%

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

      1. *-commutative 80.3%

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

      2. associate-*l* 80.2%

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

      3. associate-*r/ 82.7%

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

   3. Simplified 82.7%

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

      1. associate-/l* 86.3%

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

      2. add-cube-cbrt 85.3%

         \[\leadsto y \cdot \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \]

      3. *-un-lft-identity 85.3%

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

      4. times-frac 85.3%

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

      5. pow2 85.3%

         \[\leadsto y \cdot \left(\frac{\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}}}{1} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\right) \]

   5. Applied egg-rr 85.3%

      \[\leadsto y \cdot \color{blue}{\left(\frac{{\left(\sqrt[3]{x}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\right)} \]
   6. Step-by-step derivation

      1. /-rgt-identity 85.3%

         \[\leadsto y \cdot \left(\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\right) \]

      2. associate-*r/ 85.3%

         \[\leadsto y \cdot \color{blue}{\frac{{\left(\sqrt[3]{x}\right)}^{2} \cdot \sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

      3. unpow2 85.3%

         \[\leadsto y \cdot \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \]

      4. rem-3cbrt-lft 86.3%

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

      5. *-commutative 86.3%

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

   7. Simplified 86.3%

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

   if 1.00000000000000005e96 < z

   1. Initial program 28.7%

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

      1. associate-/l* 31.8%

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

   3. Simplified 31.8%

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

   4. Taylor expanded in z around inf 93.9%

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

      1. unpow2 93.9%

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

      2. associate-/l* 95.7%

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

   6. Simplified 95.7%

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

   7. Taylor expanded in z around 0 95.7%

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

      1. unpow2 95.7%

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

      2. associate-*r/ 95.7%

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

   9. Simplified 95.7%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+153}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 10^{+96}:\\ \;\;\;\;y \cdot \frac{x}{\frac{\sqrt{z \cdot z - a \cdot t}}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + -0.5 \cdot \frac{a}{z \cdot \frac{z}{t}}}\\ \end{array} \]

Alternative 3: 91.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+104}:\\ \;\;\;\;y \cdot \frac{x}{\frac{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}{z}}\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{z \cdot z - a \cdot t}}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\sqrt{1 - \frac{a}{\frac{z \cdot z}{t}}}}\\ \end{array} \end{array} \]

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 -1.5e+104)
   (* y (/ x (/ (- (* 0.5 (* t (/ a z))) z) z)))
   (if (<= z 2.7e-14)
     (/ x (/ (sqrt (- (* z z) (* a t))) (* z y)))
     (/ (* x y) (sqrt (- 1.0 (/ a (/ (* z z) t))))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.5e+104) {
		tmp = y * (x / (((0.5 * (t * (a / z))) - z) / z));
	} else if (z <= 2.7e-14) {
		tmp = x / (sqrt(((z * z) - (a * t))) / (z * y));
	} else {
		tmp = (x * y) / sqrt((1.0 - (a / ((z * z) / t))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.5d+104)) then
        tmp = y * (x / (((0.5d0 * (t * (a / z))) - z) / z))
    else if (z <= 2.7d-14) then
        tmp = x / (sqrt(((z * z) - (a * t))) / (z * y))
    else
        tmp = (x * y) / sqrt((1.0d0 - (a / ((z * z) / t))))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.5e+104) {
		tmp = y * (x / (((0.5 * (t * (a / z))) - z) / z));
	} else if (z <= 2.7e-14) {
		tmp = x / (Math.sqrt(((z * z) - (a * t))) / (z * y));
	} else {
		tmp = (x * y) / Math.sqrt((1.0 - (a / ((z * z) / t))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.5e+104:
		tmp = y * (x / (((0.5 * (t * (a / z))) - z) / z))
	elif z <= 2.7e-14:
		tmp = x / (math.sqrt(((z * z) - (a * t))) / (z * y))
	else:
		tmp = (x * y) / math.sqrt((1.0 - (a / ((z * z) / t))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.5e+104)
		tmp = Float64(y * Float64(x / Float64(Float64(Float64(0.5 * Float64(t * Float64(a / z))) - z) / z)));
	elseif (z <= 2.7e-14)
		tmp = Float64(x / Float64(sqrt(Float64(Float64(z * z) - Float64(a * t))) / Float64(z * y)));
	else
		tmp = Float64(Float64(x * y) / sqrt(Float64(1.0 - Float64(a / Float64(Float64(z * z) / t)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.5e+104)
		tmp = y * (x / (((0.5 * (t * (a / z))) - z) / z));
	elseif (z <= 2.7e-14)
		tmp = x / (sqrt(((z * z) - (a * t))) / (z * y));
	else
		tmp = (x * y) / sqrt((1.0 - (a / ((z * z) / t))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.5e+104], N[(y * N[(x / N[(N[(N[(0.5 * N[(t * N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.7e-14], N[(x / N[(N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(z * y), $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 < -1.49999999999999984e104

   1. Initial program 40.9%

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

      1. *-commutative 40.9%

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

      2. associate-*l* 32.8%

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

      3. associate-*r/ 33.0%

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

   3. Simplified 33.0%

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

      1. associate-/l* 41.4%

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

      2. add-cube-cbrt 41.1%

         \[\leadsto y \cdot \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \]

      3. *-un-lft-identity 41.1%

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

      4. times-frac 41.1%

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

      5. pow2 41.1%

         \[\leadsto y \cdot \left(\frac{\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}}}{1} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\right) \]

   5. Applied egg-rr 41.1%

      \[\leadsto y \cdot \color{blue}{\left(\frac{{\left(\sqrt[3]{x}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\right)} \]
   6. Step-by-step derivation

      1. /-rgt-identity 41.1%

         \[\leadsto y \cdot \left(\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\right) \]

      2. associate-*r/ 41.1%

         \[\leadsto y \cdot \color{blue}{\frac{{\left(\sqrt[3]{x}\right)}^{2} \cdot \sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

      3. unpow2 41.1%

         \[\leadsto y \cdot \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \]

      4. rem-3cbrt-lft 41.4%

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

      5. *-commutative 41.4%

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

   7. Simplified 41.4%

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

   8. Taylor expanded in z around -inf 90.1%

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

      1. associate-/l* 96.2%

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

      2. mul-1-neg 96.2%

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

      3. unsub-neg 96.2%

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

      4. associate-/r/ 96.2%

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

      5. *-commutative 96.2%

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

   10. Simplified 96.2%

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

   if -1.49999999999999984e104 < z < 2.6999999999999999e-14

   1. Initial program 75.7%

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

      1. *-commutative 75.7%

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

      2. associate-*l* 78.8%

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

      3. associate-*r/ 82.0%

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

   3. Simplified 82.0%

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

      1. *-commutative 82.0%

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

      2. associate-/l* 82.8%

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

      3. associate-/r/ 83.4%

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

      4. associate-/l/ 82.4%

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

      5. *-commutative 82.4%

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

   5. Applied egg-rr 82.4%

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

   if 2.6999999999999999e-14 < z

   1. Initial program 45.2%

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

      1. associate-/l* 48.5%

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

   3. Simplified 48.5%

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

      1. add-sqr-sqrt 48.5%

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

      2. sqrt-unprod 48.5%

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

      3. frac-times 38.7%

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

      4. add-sqr-sqrt 38.7%

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

   5. Applied egg-rr 38.7%

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

      1. div-sub 38.7%

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

      2. *-inverses 94.4%

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

      3. *-commutative 94.4%

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

      4. associate-/l* 98.9%

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

   7. Simplified 98.9%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+104}:\\ \;\;\;\;y \cdot \frac{x}{\frac{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}{z}}\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{z \cdot z - a \cdot t}}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\sqrt{1 - \frac{a}{\frac{z \cdot z}{t}}}}\\ \end{array} \]

Alternative 4: 84.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{-97}:\\ \;\;\;\;y \cdot \frac{x}{\frac{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}{z}}\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-134}:\\ \;\;\;\;x \cdot \left(\left(z \cdot y\right) \cdot \frac{1}{\sqrt{a \cdot \left(-t\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + -0.5 \cdot \frac{a}{z \cdot \frac{z}{t}}}\\ \end{array} \end{array} \]

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 -3.6e-97)
   (* y (/ x (/ (- (* 0.5 (* t (/ a z))) z) z)))
   (if (<= z 5.5e-134)
     (* x (* (* z y) (/ 1.0 (sqrt (* a (- t))))))
     (/ (* x y) (+ 1.0 (* -0.5 (/ a (* z (/ z t)))))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.6e-97) {
		tmp = y * (x / (((0.5 * (t * (a / z))) - z) / z));
	} else if (z <= 5.5e-134) {
		tmp = x * ((z * y) * (1.0 / sqrt((a * -t))));
	} else {
		tmp = (x * y) / (1.0 + (-0.5 * (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 <= (-3.6d-97)) then
        tmp = y * (x / (((0.5d0 * (t * (a / z))) - z) / z))
    else if (z <= 5.5d-134) then
        tmp = x * ((z * y) * (1.0d0 / sqrt((a * -t))))
    else
        tmp = (x * y) / (1.0d0 + ((-0.5d0) * (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 <= -3.6e-97) {
		tmp = y * (x / (((0.5 * (t * (a / z))) - z) / z));
	} else if (z <= 5.5e-134) {
		tmp = x * ((z * y) * (1.0 / Math.sqrt((a * -t))));
	} else {
		tmp = (x * y) / (1.0 + (-0.5 * (a / (z * (z / t)))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.6e-97:
		tmp = y * (x / (((0.5 * (t * (a / z))) - z) / z))
	elif z <= 5.5e-134:
		tmp = x * ((z * y) * (1.0 / math.sqrt((a * -t))))
	else:
		tmp = (x * y) / (1.0 + (-0.5 * (a / (z * (z / t)))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.6e-97)
		tmp = Float64(y * Float64(x / Float64(Float64(Float64(0.5 * Float64(t * Float64(a / z))) - z) / z)));
	elseif (z <= 5.5e-134)
		tmp = Float64(x * Float64(Float64(z * y) * Float64(1.0 / sqrt(Float64(a * Float64(-t))))));
	else
		tmp = Float64(Float64(x * y) / Float64(1.0 + Float64(-0.5 * Float64(a / Float64(z * Float64(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 <= -3.6e-97)
		tmp = y * (x / (((0.5 * (t * (a / z))) - z) / z));
	elseif (z <= 5.5e-134)
		tmp = x * ((z * y) * (1.0 / sqrt((a * -t))));
	else
		tmp = (x * y) / (1.0 + (-0.5 * (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, -3.6e-97], N[(y * N[(x / N[(N[(N[(0.5 * N[(t * N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.5e-134], N[(x * N[(N[(z * y), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(a / N[(z * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.59999999999999997e-97

   1. Initial program 64.8%

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

      1. *-commutative 64.8%

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

      2. associate-*l* 60.5%

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

      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. Step-by-step derivation

      1. associate-/l* 67.0%

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

      2. add-cube-cbrt 66.1%

         \[\leadsto y \cdot \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \]

      3. *-un-lft-identity 66.1%

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

      4. times-frac 66.1%

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

      5. pow2 66.1%

         \[\leadsto y \cdot \left(\frac{\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}}}{1} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\right) \]

   5. Applied egg-rr 66.1%

      \[\leadsto y \cdot \color{blue}{\left(\frac{{\left(\sqrt[3]{x}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\right)} \]
   6. Step-by-step derivation

      1. /-rgt-identity 66.1%

         \[\leadsto y \cdot \left(\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\right) \]

      2. associate-*r/ 66.1%

         \[\leadsto y \cdot \color{blue}{\frac{{\left(\sqrt[3]{x}\right)}^{2} \cdot \sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

      3. unpow2 66.1%

         \[\leadsto y \cdot \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \]

      4. rem-3cbrt-lft 67.0%

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

      5. *-commutative 67.0%

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

   7. Simplified 67.0%

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

   8. Taylor expanded in z around -inf 84.6%

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

      1. associate-/l* 87.7%

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

      2. mul-1-neg 87.7%

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

      3. unsub-neg 87.7%

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

      4. associate-/r/ 87.7%

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

      5. *-commutative 87.7%

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

   10. Simplified 87.7%

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

   if -3.59999999999999997e-97 < z < 5.5000000000000002e-134

   1. Initial program 61.9%

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

      1. *-commutative 61.9%

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

      2. associate-*l* 66.9%

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

      3. associate-*r/ 72.0%

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

   3. Simplified 72.0%

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

      1. *-commutative 72.0%

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

      2. associate-/l* 73.5%

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

      3. associate-/r/ 75.0%

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

      4. associate-/l/ 75.9%

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

      5. *-commutative 75.9%

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

   5. Applied egg-rr 75.9%

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

   6. Taylor expanded in z around 0 74.0%

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

      1. associate-*r* 74.0%

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

      2. neg-mul-1 74.0%

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

   8. Simplified 74.0%

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

      1. div-inv 73.9%

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

      2. *-commutative 73.9%

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

   10. Applied egg-rr 73.9%

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

       1. associate-/r/ 74.0%

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

   12. Simplified 74.0%

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

   if 5.5000000000000002e-134 < z

   1. Initial program 51.6%

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

      1. associate-/l* 54.5%

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

   3. Simplified 54.5%

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

   4. Taylor expanded in z around inf 87.1%

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

      1. unpow2 87.1%

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

      2. associate-/l* 88.2%

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

   6. Simplified 88.2%

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

   7. Taylor expanded in z around 0 88.2%

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

      1. unpow2 88.2%

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

      2. associate-*r/ 88.2%

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

   9. Simplified 88.2%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{-97}:\\ \;\;\;\;y \cdot \frac{x}{\frac{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}{z}}\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-134}:\\ \;\;\;\;x \cdot \left(\left(z \cdot y\right) \cdot \frac{1}{\sqrt{a \cdot \left(-t\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + -0.5 \cdot \frac{a}{z \cdot \frac{z}{t}}}\\ \end{array} \]

Alternative 5: 84.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-96}:\\ \;\;\;\;y \cdot \frac{x}{\frac{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}{z}}\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{-100}:\\ \;\;\;\;y \cdot \frac{x}{\frac{\sqrt{a \cdot \left(-t\right)}}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + -0.5 \cdot \frac{a}{z \cdot \frac{z}{t}}}\\ \end{array} \end{array} \]

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 -1e-96)
   (* y (/ x (/ (- (* 0.5 (* t (/ a z))) z) z)))
   (if (<= z 9.2e-100)
     (* y (/ x (/ (sqrt (* a (- t))) z)))
     (/ (* x y) (+ 1.0 (* -0.5 (/ a (* z (/ z t)))))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1e-96) {
		tmp = y * (x / (((0.5 * (t * (a / z))) - z) / z));
	} else if (z <= 9.2e-100) {
		tmp = y * (x / (sqrt((a * -t)) / z));
	} else {
		tmp = (x * y) / (1.0 + (-0.5 * (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 <= (-1d-96)) then
        tmp = y * (x / (((0.5d0 * (t * (a / z))) - z) / z))
    else if (z <= 9.2d-100) then
        tmp = y * (x / (sqrt((a * -t)) / z))
    else
        tmp = (x * y) / (1.0d0 + ((-0.5d0) * (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 <= -1e-96) {
		tmp = y * (x / (((0.5 * (t * (a / z))) - z) / z));
	} else if (z <= 9.2e-100) {
		tmp = y * (x / (Math.sqrt((a * -t)) / z));
	} else {
		tmp = (x * y) / (1.0 + (-0.5 * (a / (z * (z / t)))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1e-96:
		tmp = y * (x / (((0.5 * (t * (a / z))) - z) / z))
	elif z <= 9.2e-100:
		tmp = y * (x / (math.sqrt((a * -t)) / z))
	else:
		tmp = (x * y) / (1.0 + (-0.5 * (a / (z * (z / t)))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1e-96)
		tmp = Float64(y * Float64(x / Float64(Float64(Float64(0.5 * Float64(t * Float64(a / z))) - z) / z)));
	elseif (z <= 9.2e-100)
		tmp = Float64(y * Float64(x / Float64(sqrt(Float64(a * Float64(-t))) / z)));
	else
		tmp = Float64(Float64(x * y) / Float64(1.0 + Float64(-0.5 * Float64(a / Float64(z * Float64(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 <= -1e-96)
		tmp = y * (x / (((0.5 * (t * (a / z))) - z) / z));
	elseif (z <= 9.2e-100)
		tmp = y * (x / (sqrt((a * -t)) / z));
	else
		tmp = (x * y) / (1.0 + (-0.5 * (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, -1e-96], N[(y * N[(x / N[(N[(N[(0.5 * N[(t * N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9.2e-100], N[(y * N[(x / N[(N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(a / N[(z * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.9999999999999991e-97

   1. Initial program 64.8%

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

      1. *-commutative 64.8%

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

      2. associate-*l* 60.5%

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

      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. Step-by-step derivation

      1. associate-/l* 67.0%

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

      2. add-cube-cbrt 66.1%

         \[\leadsto y \cdot \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \]

      3. *-un-lft-identity 66.1%

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

      4. times-frac 66.1%

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

      5. pow2 66.1%

         \[\leadsto y \cdot \left(\frac{\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}}}{1} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\right) \]

   5. Applied egg-rr 66.1%

      \[\leadsto y \cdot \color{blue}{\left(\frac{{\left(\sqrt[3]{x}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\right)} \]
   6. Step-by-step derivation

      1. /-rgt-identity 66.1%

         \[\leadsto y \cdot \left(\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\right) \]

      2. associate-*r/ 66.1%

         \[\leadsto y \cdot \color{blue}{\frac{{\left(\sqrt[3]{x}\right)}^{2} \cdot \sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

      3. unpow2 66.1%

         \[\leadsto y \cdot \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \]

      4. rem-3cbrt-lft 67.0%

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

      5. *-commutative 67.0%

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

   7. Simplified 67.0%

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

   8. Taylor expanded in z around -inf 84.6%

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

      1. associate-/l* 87.7%

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

      2. mul-1-neg 87.7%

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

      3. unsub-neg 87.7%

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

      4. associate-/r/ 87.7%

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

      5. *-commutative 87.7%

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

   10. Simplified 87.7%

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

   if -9.9999999999999991e-97 < z < 9.19999999999999978e-100

   1. Initial program 61.9%

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

      1. *-commutative 61.9%

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

      2. associate-*l* 68.1%

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

      3. associate-*r/ 72.8%

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

   3. Simplified 72.8%

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

      1. associate-/l* 74.1%

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

      2. add-cube-cbrt 73.6%

         \[\leadsto y \cdot \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \]

      3. *-un-lft-identity 73.6%

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

      4. times-frac 73.5%

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

      5. pow2 73.5%

         \[\leadsto y \cdot \left(\frac{\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}}}{1} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\right) \]

   5. Applied egg-rr 73.5%

      \[\leadsto y \cdot \color{blue}{\left(\frac{{\left(\sqrt[3]{x}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\right)} \]
   6. Step-by-step derivation

      1. /-rgt-identity 73.5%

         \[\leadsto y \cdot \left(\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\right) \]

      2. associate-*r/ 73.6%

         \[\leadsto y \cdot \color{blue}{\frac{{\left(\sqrt[3]{x}\right)}^{2} \cdot \sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

      3. unpow2 73.6%

         \[\leadsto y \cdot \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \]

      4. rem-3cbrt-lft 74.1%

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

      5. *-commutative 74.1%

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

   7. Simplified 74.1%

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

   8. Taylor expanded in z around 0 70.7%

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

      1. neg-mul-1 70.7%

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

      2. distribute-lft-neg-in 70.7%

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

      3. *-commutative 70.7%

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

   10. Simplified 70.7%

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

   if 9.19999999999999978e-100 < z

   1. Initial program 51.1%

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

      1. associate-/l* 54.1%

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

   3. Simplified 54.1%

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

   4. Taylor expanded in z around inf 89.2%

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

      1. unpow2 89.2%

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

      2. associate-/l* 90.4%

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

   6. Simplified 90.4%

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

   7. Taylor expanded in z around 0 90.4%

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

      1. unpow2 90.4%

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

      2. associate-*r/ 90.4%

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

   9. Simplified 90.4%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-96}:\\ \;\;\;\;y \cdot \frac{x}{\frac{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}{z}}\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{-100}:\\ \;\;\;\;y \cdot \frac{x}{\frac{\sqrt{a \cdot \left(-t\right)}}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + -0.5 \cdot \frac{a}{z \cdot \frac{z}{t}}}\\ \end{array} \]

Alternative 6: 84.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-97}:\\ \;\;\;\;y \cdot \frac{x}{\frac{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}{z}}\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-133}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{x}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + -0.5 \cdot \frac{a}{z \cdot \frac{z}{t}}}\\ \end{array} \end{array} \]

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 -4.8e-97)
   (* y (/ x (/ (- (* 0.5 (* t (/ a z))) z) z)))
   (if (<= z 1.15e-133)
     (* (* z y) (/ x (sqrt (* a (- t)))))
     (/ (* x y) (+ 1.0 (* -0.5 (/ a (* z (/ z t)))))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.8e-97) {
		tmp = y * (x / (((0.5 * (t * (a / z))) - z) / z));
	} else if (z <= 1.15e-133) {
		tmp = (z * y) * (x / sqrt((a * -t)));
	} else {
		tmp = (x * y) / (1.0 + (-0.5 * (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 <= (-4.8d-97)) then
        tmp = y * (x / (((0.5d0 * (t * (a / z))) - z) / z))
    else if (z <= 1.15d-133) then
        tmp = (z * y) * (x / sqrt((a * -t)))
    else
        tmp = (x * y) / (1.0d0 + ((-0.5d0) * (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 <= -4.8e-97) {
		tmp = y * (x / (((0.5 * (t * (a / z))) - z) / z));
	} else if (z <= 1.15e-133) {
		tmp = (z * y) * (x / Math.sqrt((a * -t)));
	} else {
		tmp = (x * y) / (1.0 + (-0.5 * (a / (z * (z / t)))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -4.8e-97:
		tmp = y * (x / (((0.5 * (t * (a / z))) - z) / z))
	elif z <= 1.15e-133:
		tmp = (z * y) * (x / math.sqrt((a * -t)))
	else:
		tmp = (x * y) / (1.0 + (-0.5 * (a / (z * (z / t)))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4.8e-97)
		tmp = Float64(y * Float64(x / Float64(Float64(Float64(0.5 * Float64(t * Float64(a / z))) - z) / z)));
	elseif (z <= 1.15e-133)
		tmp = Float64(Float64(z * y) * Float64(x / sqrt(Float64(a * Float64(-t)))));
	else
		tmp = Float64(Float64(x * y) / Float64(1.0 + Float64(-0.5 * Float64(a / Float64(z * Float64(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 <= -4.8e-97)
		tmp = y * (x / (((0.5 * (t * (a / z))) - z) / z));
	elseif (z <= 1.15e-133)
		tmp = (z * y) * (x / sqrt((a * -t)));
	else
		tmp = (x * y) / (1.0 + (-0.5 * (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, -4.8e-97], N[(y * N[(x / N[(N[(N[(0.5 * N[(t * N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.15e-133], N[(N[(z * y), $MachinePrecision] * N[(x / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(a / N[(z * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.8e-97

   1. Initial program 64.8%

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

      1. *-commutative 64.8%

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

      2. associate-*l* 60.5%

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

      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. Step-by-step derivation

      1. associate-/l* 67.0%

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

      2. add-cube-cbrt 66.1%

         \[\leadsto y \cdot \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \]

      3. *-un-lft-identity 66.1%

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

      4. times-frac 66.1%

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

      5. pow2 66.1%

         \[\leadsto y \cdot \left(\frac{\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}}}{1} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\right) \]

   5. Applied egg-rr 66.1%

      \[\leadsto y \cdot \color{blue}{\left(\frac{{\left(\sqrt[3]{x}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\right)} \]
   6. Step-by-step derivation

      1. /-rgt-identity 66.1%

         \[\leadsto y \cdot \left(\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\right) \]

      2. associate-*r/ 66.1%

         \[\leadsto y \cdot \color{blue}{\frac{{\left(\sqrt[3]{x}\right)}^{2} \cdot \sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

      3. unpow2 66.1%

         \[\leadsto y \cdot \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \]

      4. rem-3cbrt-lft 67.0%

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

      5. *-commutative 67.0%

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

   7. Simplified 67.0%

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

   8. Taylor expanded in z around -inf 84.6%

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

      1. associate-/l* 87.7%

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

      2. mul-1-neg 87.7%

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

      3. unsub-neg 87.7%

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

      4. associate-/r/ 87.7%

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

      5. *-commutative 87.7%

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

   10. Simplified 87.7%

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

   if -4.8e-97 < z < 1.15e-133

   1. Initial program 61.9%

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

      1. *-commutative 61.9%

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

      2. associate-*l* 66.9%

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

      3. associate-*r/ 72.0%

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

   3. Simplified 72.0%

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

      1. *-commutative 72.0%

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

      2. associate-/l* 73.5%

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

      3. associate-/r/ 75.0%

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

      4. associate-/l/ 75.9%

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

      5. *-commutative 75.9%

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

   5. Applied egg-rr 75.9%

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

   6. Taylor expanded in z around 0 74.0%

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

      1. associate-*r* 74.0%

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

      2. neg-mul-1 74.0%

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

   8. Simplified 74.0%

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

      1. expm1-log1p-u 60.6%

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

      2. expm1-udef 39.9%

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

      3. associate-/r/ 36.8%

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

      4. *-commutative 36.8%

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

   10. Applied egg-rr 36.8%

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

       1. expm1-def 52.7%

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

       2. expm1-log1p 64.1%

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

   12. Simplified 64.1%

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

   if 1.15e-133 < z

   1. Initial program 51.6%

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

      1. associate-/l* 54.5%

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

   3. Simplified 54.5%

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

   4. Taylor expanded in z around inf 87.1%

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

      1. unpow2 87.1%

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

      2. associate-/l* 88.2%

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

   6. Simplified 88.2%

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

   7. Taylor expanded in z around 0 88.2%

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

      1. unpow2 88.2%

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

      2. associate-*r/ 88.2%

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

   9. Simplified 88.2%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-97}:\\ \;\;\;\;y \cdot \frac{x}{\frac{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}{z}}\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-133}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{x}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + -0.5 \cdot \frac{a}{z \cdot \frac{z}{t}}}\\ \end{array} \]

Alternative 7: 84.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -8.8 \cdot 10^{-100}:\\ \;\;\;\;y \cdot \frac{x}{\frac{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}{z}}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-133}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{a \cdot \left(-t\right)}}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + -0.5 \cdot \frac{a}{z \cdot \frac{z}{t}}}\\ \end{array} \end{array} \]

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 -8.8e-100)
   (* y (/ x (/ (- (* 0.5 (* t (/ a z))) z) z)))
   (if (<= z 1.1e-133)
     (/ x (/ (sqrt (* a (- t))) (* z y)))
     (/ (* x y) (+ 1.0 (* -0.5 (/ a (* z (/ z t)))))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.8e-100) {
		tmp = y * (x / (((0.5 * (t * (a / z))) - z) / z));
	} else if (z <= 1.1e-133) {
		tmp = x / (sqrt((a * -t)) / (z * y));
	} else {
		tmp = (x * y) / (1.0 + (-0.5 * (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 <= (-8.8d-100)) then
        tmp = y * (x / (((0.5d0 * (t * (a / z))) - z) / z))
    else if (z <= 1.1d-133) then
        tmp = x / (sqrt((a * -t)) / (z * y))
    else
        tmp = (x * y) / (1.0d0 + ((-0.5d0) * (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 <= -8.8e-100) {
		tmp = y * (x / (((0.5 * (t * (a / z))) - z) / z));
	} else if (z <= 1.1e-133) {
		tmp = x / (Math.sqrt((a * -t)) / (z * y));
	} else {
		tmp = (x * y) / (1.0 + (-0.5 * (a / (z * (z / t)))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -8.8e-100:
		tmp = y * (x / (((0.5 * (t * (a / z))) - z) / z))
	elif z <= 1.1e-133:
		tmp = x / (math.sqrt((a * -t)) / (z * y))
	else:
		tmp = (x * y) / (1.0 + (-0.5 * (a / (z * (z / t)))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -8.8e-100)
		tmp = Float64(y * Float64(x / Float64(Float64(Float64(0.5 * Float64(t * Float64(a / z))) - z) / z)));
	elseif (z <= 1.1e-133)
		tmp = Float64(x / Float64(sqrt(Float64(a * Float64(-t))) / Float64(z * y)));
	else
		tmp = Float64(Float64(x * y) / Float64(1.0 + Float64(-0.5 * Float64(a / Float64(z * Float64(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 <= -8.8e-100)
		tmp = y * (x / (((0.5 * (t * (a / z))) - z) / z));
	elseif (z <= 1.1e-133)
		tmp = x / (sqrt((a * -t)) / (z * y));
	else
		tmp = (x * y) / (1.0 + (-0.5 * (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, -8.8e-100], N[(y * N[(x / N[(N[(N[(0.5 * N[(t * N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.1e-133], N[(x / N[(N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(a / N[(z * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.79999999999999957e-100

   1. Initial program 65.1%

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

      1. *-commutative 65.1%

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

      2. associate-*l* 60.9%

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

      3. associate-*r/ 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. Step-by-step derivation

      1. associate-/l* 66.9%

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

      2. add-cube-cbrt 66.0%

         \[\leadsto y \cdot \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \]

      3. *-un-lft-identity 66.0%

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

      4. times-frac 66.1%

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

      5. pow2 66.1%

         \[\leadsto y \cdot \left(\frac{\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}}}{1} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\right) \]

   5. Applied egg-rr 66.1%

      \[\leadsto y \cdot \color{blue}{\left(\frac{{\left(\sqrt[3]{x}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\right)} \]
   6. Step-by-step derivation

      1. /-rgt-identity 66.1%

         \[\leadsto y \cdot \left(\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\right) \]

      2. associate-*r/ 66.0%

         \[\leadsto y \cdot \color{blue}{\frac{{\left(\sqrt[3]{x}\right)}^{2} \cdot \sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

      3. unpow2 66.0%

         \[\leadsto y \cdot \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \]

      4. rem-3cbrt-lft 66.9%

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

      5. *-commutative 66.9%

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

   7. Simplified 66.9%

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

   8. Taylor expanded in z around -inf 83.8%

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

      1. associate-/l* 86.9%

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

      2. mul-1-neg 86.9%

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

      3. unsub-neg 86.9%

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

      4. associate-/r/ 86.9%

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

      5. *-commutative 86.9%

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

   10. Simplified 86.9%

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

   if -8.79999999999999957e-100 < z < 1.1e-133

   1. Initial program 61.2%

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

      1. *-commutative 61.2%

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

      2. associate-*l* 66.3%

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

      3. associate-*r/ 72.3%

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

   3. Simplified 72.3%

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

      1. *-commutative 72.3%

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

      2. associate-/l* 73.8%

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

      3. associate-/r/ 74.6%

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

      4. associate-/l/ 75.5%

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

      5. *-commutative 75.5%

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

   5. Applied egg-rr 75.5%

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

   6. Taylor expanded in z around 0 73.5%

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

      1. associate-*r* 73.5%

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

      2. neg-mul-1 73.5%

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

   8. Simplified 73.5%

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

   if 1.1e-133 < z

   1. Initial program 51.6%

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

      1. associate-/l* 54.5%

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

   3. Simplified 54.5%

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

   4. Taylor expanded in z around inf 87.1%

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

      1. unpow2 87.1%

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

      2. associate-/l* 88.2%

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

   6. Simplified 88.2%

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

   7. Taylor expanded in z around 0 88.2%

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

      1. unpow2 88.2%

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

      2. associate-*r/ 88.2%

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

   9. Simplified 88.2%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.8 \cdot 10^{-100}:\\ \;\;\;\;y \cdot \frac{x}{\frac{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}{z}}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-133}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{a \cdot \left(-t\right)}}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + -0.5 \cdot \frac{a}{z \cdot \frac{z}{t}}}\\ \end{array} \]

Alternative 8: 75.5% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{-236}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 6.9 \cdot 10^{-217}:\\ \;\;\;\;-2 \cdot \left(\frac{y}{a} \cdot \frac{z \cdot \left(z \cdot x\right)}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.3e-236)
   (* x (- y))
   (if (<= z 6.9e-217) (* -2.0 (* (/ y a) (/ (* z (* z x)) t))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.3e-236) {
		tmp = x * -y;
	} else if (z <= 6.9e-217) {
		tmp = -2.0 * ((y / a) * ((z * (z * x)) / 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 <= (-3.3d-236)) then
        tmp = x * -y
    else if (z <= 6.9d-217) then
        tmp = (-2.0d0) * ((y / a) * ((z * (z * x)) / 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 <= -3.3e-236) {
		tmp = x * -y;
	} else if (z <= 6.9e-217) {
		tmp = -2.0 * ((y / a) * ((z * (z * x)) / t));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.3e-236:
		tmp = x * -y
	elif z <= 6.9e-217:
		tmp = -2.0 * ((y / a) * ((z * (z * x)) / 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 <= -3.3e-236)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 6.9e-217)
		tmp = Float64(-2.0 * Float64(Float64(y / a) * Float64(Float64(z * Float64(z * x)) / 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 <= -3.3e-236)
		tmp = x * -y;
	elseif (z <= 6.9e-217)
		tmp = -2.0 * ((y / a) * ((z * (z * x)) / 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, -3.3e-236], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 6.9e-217], N[(-2.0 * N[(N[(y / a), $MachinePrecision] * N[(N[(z * N[(z * x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.3000000000000001e-236

   1. Initial program 65.0%

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

      1. *-commutative 65.0%

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

      2. associate-*l* 61.4%

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

      3. associate-*r/ 64.4%

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

   3. Simplified 64.4%

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

   4. Taylor expanded in z around -inf 78.3%

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

      1. neg-mul-1 78.3%

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

   6. Simplified 78.3%

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

   if -3.3000000000000001e-236 < z < 6.89999999999999974e-217

   1. Initial program 48.4%

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

      1. associate-/l* 49.9%

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

   3. Simplified 49.9%

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

   4. Taylor expanded in z around inf 34.1%

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

      1. unpow2 34.1%

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

      2. associate-/l* 39.4%

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

   6. Simplified 39.4%

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

   7. Taylor expanded in a around inf 34.7%

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

      1. times-frac 39.4%

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

      2. unpow2 39.4%

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

      3. associate-*l* 39.7%

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

   9. Simplified 39.7%

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

   if 6.89999999999999974e-217 < z

   1. Initial program 54.8%

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

      1. *-commutative 54.8%

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

      2. associate-*l* 55.3%

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

      3. associate-*r/ 57.0%

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

   3. Simplified 57.0%

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

   4. Taylor expanded in z around inf 79.4%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{-236}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 6.9 \cdot 10^{-217}:\\ \;\;\;\;-2 \cdot \left(\frac{y}{a} \cdot \frac{z \cdot \left(z \cdot x\right)}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 9: 77.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{-239}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\frac{z + \left(t \cdot \frac{a}{z}\right) \cdot -0.5}{z}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.4e-239)
   (* x (- y))
   (* y (/ x (/ (+ z (* (* t (/ a z)) -0.5)) z)))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.4e-239) {
		tmp = x * -y;
	} else {
		tmp = y * (x / ((z + ((t * (a / z)) * -0.5)) / z));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.4d-239)) then
        tmp = x * -y
    else
        tmp = y * (x / ((z + ((t * (a / z)) * (-0.5d0))) / z))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.4e-239) {
		tmp = x * -y;
	} else {
		tmp = y * (x / ((z + ((t * (a / z)) * -0.5)) / z));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.4e-239:
		tmp = x * -y
	else:
		tmp = y * (x / ((z + ((t * (a / z)) * -0.5)) / z))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.4e-239)
		tmp = Float64(x * Float64(-y));
	else
		tmp = Float64(y * Float64(x / Float64(Float64(z + Float64(Float64(t * Float64(a / z)) * -0.5)) / z)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.4e-239)
		tmp = x * -y;
	else
		tmp = y * (x / ((z + ((t * (a / z)) * -0.5)) / z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.4e-239], N[(x * (-y)), $MachinePrecision], N[(y * N[(x / N[(N[(z + N[(N[(t * N[(a / z), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.40000000000000006e-239

   1. Initial program 65.0%

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

      1. *-commutative 65.0%

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

      2. associate-*l* 61.4%

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

      3. associate-*r/ 64.4%

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

   3. Simplified 64.4%

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

   4. Taylor expanded in z around -inf 78.3%

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

      1. neg-mul-1 78.3%

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

   6. Simplified 78.3%

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

   if -1.40000000000000006e-239 < z

   1. Initial program 53.9%

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

      1. *-commutative 53.9%

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

      2. associate-*l* 56.2%

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

      3. associate-*r/ 58.1%

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

   3. Simplified 58.1%

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

      1. associate-/l* 59.6%

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

      2. add-cube-cbrt 59.0%

         \[\leadsto y \cdot \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \]

      3. *-un-lft-identity 59.0%

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

      4. times-frac 59.0%

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

      5. pow2 59.0%

         \[\leadsto y \cdot \left(\frac{\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}}}{1} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\right) \]

   5. Applied egg-rr 59.0%

      \[\leadsto y \cdot \color{blue}{\left(\frac{{\left(\sqrt[3]{x}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\right)} \]
   6. Step-by-step derivation

      1. /-rgt-identity 59.0%

         \[\leadsto y \cdot \left(\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\right) \]

      2. associate-*r/ 59.0%

         \[\leadsto y \cdot \color{blue}{\frac{{\left(\sqrt[3]{x}\right)}^{2} \cdot \sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

      3. unpow2 59.0%

         \[\leadsto y \cdot \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \]

      4. rem-3cbrt-lft 59.6%

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

      5. *-commutative 59.6%

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

   7. Simplified 59.6%

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

   8. Taylor expanded in z around inf 74.2%

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

      1. associate-*l/ 74.9%

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

      2. *-commutative 74.9%

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

   10. Simplified 74.9%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{-239}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\frac{z + \left(t \cdot \frac{a}{z}\right) \cdot -0.5}{z}}\\ \end{array} \]

Alternative 10: 79.0% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := t \cdot \frac{a}{z}\\ \mathbf{if}\;z \leq 2 \cdot 10^{-297}:\\ \;\;\;\;y \cdot \frac{x}{\frac{0.5 \cdot t_1 - z}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\frac{z + t_1 \cdot -0.5}{z}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* t (/ a z))))
   (if (<= z 2e-297)
     (* y (/ x (/ (- (* 0.5 t_1) z) z)))
     (* y (/ x (/ (+ z (* t_1 -0.5)) z))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = t * (a / z);
	double tmp;
	if (z <= 2e-297) {
		tmp = y * (x / (((0.5 * t_1) - z) / z));
	} else {
		tmp = y * (x / ((z + (t_1 * -0.5)) / z));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * (a / z)
    if (z <= 2d-297) then
        tmp = y * (x / (((0.5d0 * t_1) - z) / z))
    else
        tmp = y * (x / ((z + (t_1 * (-0.5d0))) / z))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t * (a / z);
	double tmp;
	if (z <= 2e-297) {
		tmp = y * (x / (((0.5 * t_1) - z) / z));
	} else {
		tmp = y * (x / ((z + (t_1 * -0.5)) / z));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	t_1 = t * (a / z)
	tmp = 0
	if z <= 2e-297:
		tmp = y * (x / (((0.5 * t_1) - z) / z))
	else:
		tmp = y * (x / ((z + (t_1 * -0.5)) / z))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = Float64(t * Float64(a / z))
	tmp = 0.0
	if (z <= 2e-297)
		tmp = Float64(y * Float64(x / Float64(Float64(Float64(0.5 * t_1) - z) / z)));
	else
		tmp = Float64(y * Float64(x / Float64(Float64(z + Float64(t_1 * -0.5)) / z)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = t * (a / z);
	tmp = 0.0;
	if (z <= 2e-297)
		tmp = y * (x / (((0.5 * t_1) - z) / z));
	else
		tmp = y * (x / ((z + (t_1 * -0.5)) / z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * N[(a / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 2e-297], N[(y * N[(x / N[(N[(N[(0.5 * t$95$1), $MachinePrecision] - z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / N[(N[(z + N[(t$95$1 * -0.5), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 2.00000000000000008e-297

   1. Initial program 64.2%

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

      1. *-commutative 64.2%

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

      2. associate-*l* 62.4%

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

      3. associate-*r/ 64.9%

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

   3. Simplified 64.9%

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

      1. associate-/l* 68.4%

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

      2. add-cube-cbrt 67.6%

         \[\leadsto y \cdot \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \]

      3. *-un-lft-identity 67.6%

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

      4. times-frac 67.6%

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

      5. pow2 67.6%

         \[\leadsto y \cdot \left(\frac{\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}}}{1} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\right) \]

   5. Applied egg-rr 67.6%

      \[\leadsto y \cdot \color{blue}{\left(\frac{{\left(\sqrt[3]{x}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\right)} \]
   6. Step-by-step derivation

      1. /-rgt-identity 67.6%

         \[\leadsto y \cdot \left(\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\right) \]

      2. associate-*r/ 67.6%

         \[\leadsto y \cdot \color{blue}{\frac{{\left(\sqrt[3]{x}\right)}^{2} \cdot \sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

      3. unpow2 67.6%

         \[\leadsto y \cdot \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \]

      4. rem-3cbrt-lft 68.4%

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

      5. *-commutative 68.4%

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

   7. Simplified 68.4%

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

   8. Taylor expanded in z around -inf 74.0%

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

      1. associate-/l* 76.7%

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

      2. mul-1-neg 76.7%

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

      3. unsub-neg 76.7%

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

      4. associate-/r/ 76.7%

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

      5. *-commutative 76.7%

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

   10. Simplified 76.7%

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

   if 2.00000000000000008e-297 < z

   1. Initial program 53.7%

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

      1. *-commutative 53.7%

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

      2. associate-*l* 54.9%

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

      3. associate-*r/ 57.1%

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

   3. Simplified 57.1%

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

      1. associate-/l* 58.7%

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

      2. add-cube-cbrt 58.1%

         \[\leadsto y \cdot \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \]

      3. *-un-lft-identity 58.1%

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

      4. times-frac 58.1%

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

      5. pow2 58.1%

         \[\leadsto y \cdot \left(\frac{\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}}}{1} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\right) \]

   5. Applied egg-rr 58.1%

      \[\leadsto y \cdot \color{blue}{\left(\frac{{\left(\sqrt[3]{x}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\right)} \]
   6. Step-by-step derivation

      1. /-rgt-identity 58.1%

         \[\leadsto y \cdot \left(\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\right) \]

      2. associate-*r/ 58.1%

         \[\leadsto y \cdot \color{blue}{\frac{{\left(\sqrt[3]{x}\right)}^{2} \cdot \sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

      3. unpow2 58.1%

         \[\leadsto y \cdot \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \]

      4. rem-3cbrt-lft 58.7%

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

      5. *-commutative 58.7%

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

   7. Simplified 58.7%

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

   8. Taylor expanded in z around inf 76.9%

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

      1. associate-*l/ 77.7%

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

      2. *-commutative 77.7%

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

   10. Simplified 77.7%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{-297}:\\ \;\;\;\;y \cdot \frac{x}{\frac{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\frac{z + \left(t \cdot \frac{a}{z}\right) \cdot -0.5}{z}}\\ \end{array} \]

Alternative 11: 79.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 7 \cdot 10^{-284}:\\ \;\;\;\;y \cdot \frac{x}{\frac{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + -0.5 \cdot \frac{a}{z \cdot \frac{z}{t}}}\\ \end{array} \end{array} \]

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-284)
   (* y (/ x (/ (- (* 0.5 (* t (/ a z))) z) z)))
   (/ (* x y) (+ 1.0 (* -0.5 (/ 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-284) {
		tmp = y * (x / (((0.5 * (t * (a / z))) - z) / z));
	} else {
		tmp = (x * y) / (1.0 + (-0.5 * (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-284) then
        tmp = y * (x / (((0.5d0 * (t * (a / z))) - z) / z))
    else
        tmp = (x * y) / (1.0d0 + ((-0.5d0) * (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-284) {
		tmp = y * (x / (((0.5 * (t * (a / z))) - z) / z));
	} else {
		tmp = (x * y) / (1.0 + (-0.5 * (a / (z * (z / t)))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= 7e-284:
		tmp = y * (x / (((0.5 * (t * (a / z))) - z) / z))
	else:
		tmp = (x * y) / (1.0 + (-0.5 * (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-284)
		tmp = Float64(y * Float64(x / Float64(Float64(Float64(0.5 * Float64(t * Float64(a / z))) - z) / z)));
	else
		tmp = Float64(Float64(x * y) / Float64(1.0 + Float64(-0.5 * Float64(a / Float64(z * Float64(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-284)
		tmp = y * (x / (((0.5 * (t * (a / z))) - z) / z));
	else
		tmp = (x * y) / (1.0 + (-0.5 * (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-284], N[(y * N[(x / N[(N[(N[(0.5 * N[(t * N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(a / N[(z * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 6.99999999999999951e-284

   1. Initial program 63.7%

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

      1. *-commutative 63.7%

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

      2. associate-*l* 62.7%

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

      3. associate-*r/ 65.2%

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

   3. Simplified 65.2%

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

      1. associate-/l* 68.6%

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

      2. add-cube-cbrt 67.8%

         \[\leadsto y \cdot \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \]

      3. *-un-lft-identity 67.8%

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

      4. times-frac 67.9%

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

      5. pow2 67.9%

         \[\leadsto y \cdot \left(\frac{\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}}}{1} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\right) \]

   5. Applied egg-rr 67.9%

      \[\leadsto y \cdot \color{blue}{\left(\frac{{\left(\sqrt[3]{x}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\right)} \]
   6. Step-by-step derivation

      1. /-rgt-identity 67.9%

         \[\leadsto y \cdot \left(\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\right) \]

      2. associate-*r/ 67.8%

         \[\leadsto y \cdot \color{blue}{\frac{{\left(\sqrt[3]{x}\right)}^{2} \cdot \sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

      3. unpow2 67.8%

         \[\leadsto y \cdot \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \sqrt[3]{x}}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \]

      4. rem-3cbrt-lft 68.6%

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

      5. *-commutative 68.6%

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

   7. Simplified 68.6%

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

   8. Taylor expanded in z around -inf 73.4%

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

      1. associate-/l* 76.1%

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

      2. mul-1-neg 76.1%

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

      3. unsub-neg 76.1%

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

      4. associate-/r/ 76.1%

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

      5. *-commutative 76.1%

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

   10. Simplified 76.1%

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

   if 6.99999999999999951e-284 < z

   1. Initial program 54.1%

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

      1. associate-/l* 57.6%

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

   3. Simplified 57.6%

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

   4. Taylor expanded in z around inf 74.8%

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

      1. unpow2 74.8%

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

      2. associate-/l* 76.5%

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

   6. Simplified 76.5%

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

   7. Taylor expanded in z around 0 76.5%

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

      1. unpow2 76.5%

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

      2. associate-*r/ 78.8%

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

   9. Simplified 78.8%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7 \cdot 10^{-284}:\\ \;\;\;\;y \cdot \frac{x}{\frac{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + -0.5 \cdot \frac{a}{z \cdot \frac{z}{t}}}\\ \end{array} \]

Alternative 12: 74.4% accurate, 10.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{-241}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-193}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.45e-241)
   (* x (- y))
   (if (<= z 6.6e-193) (* 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 <= -1.45e-241) {
		tmp = x * -y;
	} else if (z <= 6.6e-193) {
		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 <= (-1.45d-241)) then
        tmp = x * -y
    else if (z <= 6.6d-193) 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 <= -1.45e-241) {
		tmp = x * -y;
	} else if (z <= 6.6e-193) {
		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 <= -1.45e-241:
		tmp = x * -y
	elif z <= 6.6e-193:
		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 <= -1.45e-241)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 6.6e-193)
		tmp = Float64(y * Float64(Float64(z * x) / z));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.45e-241)
		tmp = x * -y;
	elseif (z <= 6.6e-193)
		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, -1.45e-241], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 6.6e-193], N[(y * N[(N[(z * x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.45e-241

   1. Initial program 65.0%

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

      1. *-commutative 65.0%

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

      2. associate-*l* 61.4%

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

      3. associate-*r/ 64.4%

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

   3. Simplified 64.4%

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

   4. Taylor expanded in z around -inf 78.3%

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

      1. neg-mul-1 78.3%

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

   6. Simplified 78.3%

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

   if -1.45e-241 < z < 6.5999999999999998e-193

   1. Initial program 49.5%

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

      1. *-commutative 49.5%

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

      2. associate-*l* 59.8%

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

      3. associate-*r/ 65.4%

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

   3. Simplified 65.4%

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

   4. Taylor expanded in z around inf 33.9%

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

   if 6.5999999999999998e-193 < z

   1. Initial program 54.9%

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

      1. *-commutative 54.9%

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

      2. associate-*l* 55.4%

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

      3. associate-*r/ 56.4%

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

   3. Simplified 56.4%

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

   4. Taylor expanded in z around inf 82.3%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{-241}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-193}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 13: 72.9% accurate, 18.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5e-310) (* x (- y)) (* x y)))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5e-310) {
		tmp = x * -y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-5d-310)) then
        tmp = x * -y
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5e-310) {
		tmp = x * -y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -5e-310:
		tmp = x * -y
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5e-310)
		tmp = Float64(x * Float64(-y));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -5e-310)
		tmp = x * -y;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5e-310], N[(x * (-y)), $MachinePrecision], N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.999999999999985e-310

   1. Initial program 63.7%

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

      1. *-commutative 63.7%

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

      2. associate-*l* 61.9%

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

      3. associate-*r/ 64.0%

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

   3. Simplified 64.0%

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

   4. Taylor expanded in z around -inf 75.2%

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

      1. neg-mul-1 75.2%

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

   6. Simplified 75.2%

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

   if -4.999999999999985e-310 < z

   1. Initial program 54.3%

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

      1. *-commutative 54.3%

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

      2. associate-*l* 55.5%

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

      3. associate-*r/ 58.1%

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

   3. Simplified 58.1%

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

   4. Taylor expanded in z around inf 73.2%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 14: 43.0% accurate, 37.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.7%

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

   1. *-commutative 58.7%

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

   2. associate-*l* 58.5%

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

   3. associate-*r/ 60.8%

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

3. Simplified 60.8%

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

4. Taylor expanded in z around inf 45.5%

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

5. Final simplification 45.5%

   \[\leadsto x \cdot y \]

Developer target: 89.9% 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 2023192 
(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)))))