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

Percentage Accurate: 61.2% → 91.9%

Time: 19.7s

Alternatives: 16

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: 61.2% 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: 91.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+129}:\\ \;\;\;\;y \cdot \frac{x}{\mathsf{fma}\left(0.5, \frac{a}{z} \cdot \frac{t}{z}, -1\right)}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-88}:\\ \;\;\;\;y \cdot \frac{x}{\frac{\sqrt{z \cdot z - a \cdot t}}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{\sqrt{1 - \frac{a}{\frac{z \cdot z}{t}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2e+129)
   (* y (/ x (fma 0.5 (* (/ a z) (/ t z)) -1.0)))
   (if (<= z 6e-88)
     (* y (/ x (/ (sqrt (- (* z z) (* a t))) z)))
     (/ (* y x) (sqrt (- 1.0 (/ a (/ (* z z) t))))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2e+129) {
		tmp = y * (x / fma(0.5, ((a / z) * (t / z)), -1.0));
	} else if (z <= 6e-88) {
		tmp = y * (x / (sqrt(((z * z) - (a * t))) / z));
	} else {
		tmp = (y * x) / sqrt((1.0 - (a / ((z * z) / t))));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2e+129)
		tmp = Float64(y * Float64(x / fma(0.5, Float64(Float64(a / z) * Float64(t / z)), -1.0)));
	elseif (z <= 6e-88)
		tmp = Float64(y * Float64(x / Float64(sqrt(Float64(Float64(z * z) - Float64(a * t))) / z)));
	else
		tmp = Float64(Float64(y * x) / sqrt(Float64(1.0 - Float64(a / Float64(Float64(z * z) / t)))));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2e+129], N[(y * N[(x / N[(0.5 * N[(N[(a / z), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6e-88], N[(y * N[(x / N[(N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / N[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 < -2e129

   1. Initial program 23.7%

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

      1. *-commutative 23.7%

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

      2. associate-*l* 23.6%

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

      3. associate-*r/ 26.2%

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

   3. Simplified 26.2%

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

      1. associate-/l* 26.7%

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

      2. add-cube-cbrt 26.6%

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

      3. *-un-lft-identity 26.6%

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

      4. times-frac 26.6%

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

      5. pow2 26.6%

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

   5. Applied egg-rr 26.6%

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

      1. /-rgt-identity 26.6%

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

      2. associate-*r/ 26.6%

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

      3. unpow2 26.6%

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

      4. rem-3cbrt-lft 26.7%

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

      5. *-commutative 26.7%

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

   7. Simplified 26.7%

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

   8. Taylor expanded in z around -inf 81.8%

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

      1. fma-neg 81.8%

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

      2. unpow2 81.8%

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

      3. times-frac 94.9%

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

      4. metadata-eval 94.9%

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

   10. Simplified 94.9%

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

   if -2e129 < z < 5.9999999999999999e-88

   1. Initial program 83.2%

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

      1. *-commutative 83.2%

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

      2. associate-*l* 83.0%

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

      3. associate-*r/ 86.8%

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

   3. Simplified 86.8%

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

      1. associate-/l* 86.4%

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

      2. add-cube-cbrt 85.4%

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

      3. *-un-lft-identity 85.4%

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

      4. times-frac 85.5%

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

      5. pow2 85.5%

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

   5. Applied egg-rr 85.5%

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

      1. /-rgt-identity 85.5%

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

      2. associate-*r/ 85.4%

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

      3. unpow2 85.4%

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

      4. rem-3cbrt-lft 86.4%

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

      5. *-commutative 86.4%

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

   7. Simplified 86.4%

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

   if 5.9999999999999999e-88 < z

   1. Initial program 58.6%

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

      1. associate-/l* 61.9%

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

   3. Simplified 61.9%

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

      1. add-sqr-sqrt 61.9%

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

      2. sqrt-unprod 61.9%

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

      3. frac-times 52.3%

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

      4. add-sqr-sqrt 52.3%

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

   5. Applied egg-rr 52.3%

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

      1. div-sub 52.3%

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

      2. *-inverses 89.5%

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

      3. *-commutative 89.5%

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

      4. associate-/l* 95.8%

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

   7. Simplified 95.8%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+129}:\\ \;\;\;\;y \cdot \frac{x}{\mathsf{fma}\left(0.5, \frac{a}{z} \cdot \frac{t}{z}, -1\right)}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-88}:\\ \;\;\;\;y \cdot \frac{x}{\frac{\sqrt{z \cdot z - a \cdot t}}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{\sqrt{1 - \frac{a}{\frac{z \cdot z}{t}}}}\\ \end{array} \]

Alternative 2: 91.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+153}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+74}:\\ \;\;\;\;y \cdot \frac{x}{\frac{\sqrt{z \cdot z - a \cdot t}}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + \frac{a}{\frac{z \cdot z}{t}} \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5e+153)
   (* y (- x))
   (if (<= z 3e+74)
     (* y (/ x (/ (sqrt (- (* z z) (* a t))) z)))
     (/ (* y x) (+ 1.0 (* (/ a (/ (* z z) t)) -0.5))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5e+153) {
		tmp = y * -x;
	} else if (z <= 3e+74) {
		tmp = y * (x / (sqrt(((z * z) - (a * t))) / z));
	} else {
		tmp = (y * x) / (1.0 + ((a / ((z * z) / t)) * -0.5));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-5d+153)) then
        tmp = y * -x
    else if (z <= 3d+74) then
        tmp = y * (x / (sqrt(((z * z) - (a * t))) / z))
    else
        tmp = (y * x) / (1.0d0 + ((a / ((z * z) / t)) * (-0.5d0)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5e+153) {
		tmp = y * -x;
	} else if (z <= 3e+74) {
		tmp = y * (x / (Math.sqrt(((z * z) - (a * t))) / z));
	} else {
		tmp = (y * x) / (1.0 + ((a / ((z * z) / t)) * -0.5));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -5e+153:
		tmp = y * -x
	elif z <= 3e+74:
		tmp = y * (x / (math.sqrt(((z * z) - (a * t))) / z))
	else:
		tmp = (y * x) / (1.0 + ((a / ((z * z) / t)) * -0.5))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5e+153)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 3e+74)
		tmp = Float64(y * Float64(x / Float64(sqrt(Float64(Float64(z * z) - Float64(a * t))) / z)));
	else
		tmp = Float64(Float64(y * x) / Float64(1.0 + Float64(Float64(a / Float64(Float64(z * z) / t)) * -0.5)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -5e+153)
		tmp = y * -x;
	elseif (z <= 3e+74)
		tmp = y * (x / (sqrt(((z * z) - (a * t))) / z));
	else
		tmp = (y * x) / (1.0 + ((a / ((z * z) / t)) * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5e+153], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 3e+74], N[(y * N[(x / N[(N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / N[(1.0 + N[(N[(a / N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.00000000000000018e153

   1. Initial program 12.5%

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

      1. *-commutative 12.5%

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

      2. associate-*l* 12.4%

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

      3. associate-*r/ 12.5%

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

   3. Simplified 12.5%

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

   4. Taylor expanded in z around -inf 95.6%

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

      1. neg-mul-1 95.6%

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

   6. Simplified 95.6%

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

   if -5.00000000000000018e153 < z < 3e74

   1. Initial program 83.1%

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

      1. *-commutative 83.1%

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

      2. associate-*l* 83.0%

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

      3. associate-*r/ 87.5%

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

   3. Simplified 87.5%

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

      1. associate-/l* 87.3%

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

      2. add-cube-cbrt 86.3%

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

      3. *-un-lft-identity 86.3%

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

      4. times-frac 86.4%

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

      5. pow2 86.4%

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

   5. Applied egg-rr 86.4%

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

      1. /-rgt-identity 86.4%

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

      2. associate-*r/ 86.3%

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

      3. unpow2 86.3%

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

      4. rem-3cbrt-lft 87.3%

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

      5. *-commutative 87.3%

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

   7. Simplified 87.3%

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

   if 3e74 < z

   1. Initial program 45.0%

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

      1. associate-/l* 46.8%

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

   3. Simplified 46.8%

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

   4. Taylor expanded in z around inf 88.7%

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

      1. unpow2 88.7%

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

      2. associate-/l* 97.0%

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

   6. Simplified 97.0%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+153}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+74}:\\ \;\;\;\;y \cdot \frac{x}{\frac{\sqrt{z \cdot z - a \cdot t}}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + \frac{a}{\frac{z \cdot z}{t}} \cdot -0.5}\\ \end{array} \]

Alternative 3: 91.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+154}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.85 \cdot 10^{+76}:\\ \;\;\;\;y \cdot \frac{x}{\frac{\sqrt{z \cdot z - a \cdot t}}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\mathsf{fma}\left(-0.5, \frac{a}{z} \cdot \frac{t}{z}, 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1e+154)
   (* y (- x))
   (if (<= z 2.85e+76)
     (* y (/ x (/ (sqrt (- (* z z) (* a t))) z)))
     (* y (/ x (fma -0.5 (* (/ a z) (/ t z)) 1.0))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1e+154) {
		tmp = y * -x;
	} else if (z <= 2.85e+76) {
		tmp = y * (x / (sqrt(((z * z) - (a * t))) / z));
	} else {
		tmp = y * (x / fma(-0.5, ((a / z) * (t / z)), 1.0));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1e+154)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 2.85e+76)
		tmp = Float64(y * Float64(x / Float64(sqrt(Float64(Float64(z * z) - Float64(a * t))) / z)));
	else
		tmp = Float64(y * Float64(x / fma(-0.5, Float64(Float64(a / z) * Float64(t / z)), 1.0)));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1e+154], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 2.85e+76], N[(y * N[(x / N[(N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / N[(-0.5 * N[(N[(a / z), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.00000000000000004e154

   1. Initial program 12.5%

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

      1. *-commutative 12.5%

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

      2. associate-*l* 12.4%

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

      3. associate-*r/ 12.5%

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

   3. Simplified 12.5%

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

   4. Taylor expanded in z around -inf 95.6%

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

      1. neg-mul-1 95.6%

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

   6. Simplified 95.6%

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

   if -1.00000000000000004e154 < z < 2.85000000000000002e76

   1. Initial program 83.1%

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

      1. *-commutative 83.1%

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

      2. associate-*l* 83.0%

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

      3. associate-*r/ 87.5%

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

   3. Simplified 87.5%

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

      1. associate-/l* 87.3%

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

      2. add-cube-cbrt 86.3%

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

      3. *-un-lft-identity 86.3%

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

      4. times-frac 86.4%

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

      5. pow2 86.4%

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

   5. Applied egg-rr 86.4%

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

      1. /-rgt-identity 86.4%

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

      2. associate-*r/ 86.3%

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

      3. unpow2 86.3%

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

      4. rem-3cbrt-lft 87.3%

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

      5. *-commutative 87.3%

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

   7. Simplified 87.3%

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

   if 2.85000000000000002e76 < z

   1. Initial program 45.0%

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

      1. *-commutative 45.0%

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

      2. associate-*l* 43.3%

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

      3. associate-*r/ 43.6%

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

   3. Simplified 43.6%

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

      1. associate-/l* 47.0%

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

      2. add-cube-cbrt 46.6%

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

      3. *-un-lft-identity 46.6%

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

      4. times-frac 46.6%

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

      5. pow2 46.6%

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

   5. Applied egg-rr 46.6%

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

      1. /-rgt-identity 46.6%

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

      2. associate-*r/ 46.6%

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

      3. unpow2 46.6%

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

      4. rem-3cbrt-lft 47.0%

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

      5. *-commutative 47.0%

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

   7. Simplified 47.0%

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

   8. Taylor expanded in z around inf 88.7%

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

      1. +-commutative 88.7%

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

      2. fma-def 88.7%

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

      3. unpow2 88.7%

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

      4. times-frac 97.0%

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

   10. Simplified 97.0%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+154}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.85 \cdot 10^{+76}:\\ \;\;\;\;y \cdot \frac{x}{\frac{\sqrt{z \cdot z - a \cdot t}}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\mathsf{fma}\left(-0.5, \frac{a}{z} \cdot \frac{t}{z}, 1\right)}\\ \end{array} \]

Alternative 4: 91.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := \frac{a}{z} \cdot \frac{t}{z}\\ \mathbf{if}\;z \leq -2 \cdot 10^{+129}:\\ \;\;\;\;y \cdot \frac{x}{\mathsf{fma}\left(0.5, t_1, -1\right)}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+74}:\\ \;\;\;\;y \cdot \frac{x}{\frac{\sqrt{z \cdot z - a \cdot t}}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\mathsf{fma}\left(-0.5, t_1, 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ a z) (/ t z))))
   (if (<= z -2e+129)
     (* y (/ x (fma 0.5 t_1 -1.0)))
     (if (<= z 4e+74)
       (* y (/ x (/ (sqrt (- (* z z) (* a t))) z)))
       (* y (/ x (fma -0.5 t_1 1.0)))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (a / z) * (t / z);
	double tmp;
	if (z <= -2e+129) {
		tmp = y * (x / fma(0.5, t_1, -1.0));
	} else if (z <= 4e+74) {
		tmp = y * (x / (sqrt(((z * z) - (a * t))) / z));
	} else {
		tmp = y * (x / fma(-0.5, t_1, 1.0));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(a / z) * Float64(t / z))
	tmp = 0.0
	if (z <= -2e+129)
		tmp = Float64(y * Float64(x / fma(0.5, t_1, -1.0)));
	elseif (z <= 4e+74)
		tmp = Float64(y * Float64(x / Float64(sqrt(Float64(Float64(z * z) - Float64(a * t))) / z)));
	else
		tmp = Float64(y * Float64(x / fma(-0.5, t_1, 1.0)));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a / z), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2e+129], N[(y * N[(x / N[(0.5 * t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4e+74], N[(y * N[(x / N[(N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / N[(-0.5 * t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2e129

   1. Initial program 23.7%

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

      1. *-commutative 23.7%

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

      2. associate-*l* 23.6%

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

      3. associate-*r/ 26.2%

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

   3. Simplified 26.2%

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

      1. associate-/l* 26.7%

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

      2. add-cube-cbrt 26.6%

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

      3. *-un-lft-identity 26.6%

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

      4. times-frac 26.6%

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

      5. pow2 26.6%

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

   5. Applied egg-rr 26.6%

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

      1. /-rgt-identity 26.6%

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

      2. associate-*r/ 26.6%

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

      3. unpow2 26.6%

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

      4. rem-3cbrt-lft 26.7%

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

      5. *-commutative 26.7%

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

   7. Simplified 26.7%

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

   8. Taylor expanded in z around -inf 81.8%

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

      1. fma-neg 81.8%

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

      2. unpow2 81.8%

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

      3. times-frac 94.9%

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

      4. metadata-eval 94.9%

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

   10. Simplified 94.9%

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

   if -2e129 < z < 3.99999999999999981e74

   1. Initial program 83.6%

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

      1. *-commutative 83.6%

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

      2. associate-*l* 83.4%

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

      3. associate-*r/ 87.6%

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

   3. Simplified 87.6%

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

      1. associate-/l* 87.4%

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

      2. add-cube-cbrt 86.3%

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

      3. *-un-lft-identity 86.3%

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

      4. times-frac 86.4%

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

      5. pow2 86.4%

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

   5. Applied egg-rr 86.4%

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

      1. /-rgt-identity 86.4%

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

      2. associate-*r/ 86.3%

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

      3. unpow2 86.3%

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

      4. rem-3cbrt-lft 87.4%

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

      5. *-commutative 87.4%

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

   7. Simplified 87.4%

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

   if 3.99999999999999981e74 < z

   1. Initial program 45.0%

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

      1. *-commutative 45.0%

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

      2. associate-*l* 43.3%

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

      3. associate-*r/ 43.6%

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

   3. Simplified 43.6%

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

      1. associate-/l* 47.0%

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

      2. add-cube-cbrt 46.6%

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

      3. *-un-lft-identity 46.6%

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

      4. times-frac 46.6%

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

      5. pow2 46.6%

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

   5. Applied egg-rr 46.6%

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

      1. /-rgt-identity 46.6%

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

      2. associate-*r/ 46.6%

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

      3. unpow2 46.6%

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

      4. rem-3cbrt-lft 47.0%

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

      5. *-commutative 47.0%

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

   7. Simplified 47.0%

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

   8. Taylor expanded in z around inf 88.7%

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

      1. +-commutative 88.7%

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

      2. fma-def 88.7%

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

      3. unpow2 88.7%

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

      4. times-frac 97.0%

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

   10. Simplified 97.0%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+129}:\\ \;\;\;\;y \cdot \frac{x}{\mathsf{fma}\left(0.5, \frac{a}{z} \cdot \frac{t}{z}, -1\right)}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+74}:\\ \;\;\;\;y \cdot \frac{x}{\frac{\sqrt{z \cdot z - a \cdot t}}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\mathsf{fma}\left(-0.5, \frac{a}{z} \cdot \frac{t}{z}, 1\right)}\\ \end{array} \]

Alternative 5: 83.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.65 \cdot 10^{-120}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-76}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + \frac{a}{\frac{z \cdot z}{t}} \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.65e-120)
   (* y (- x))
   (if (<= z 8e-76)
     (* y (/ (* z x) (sqrt (* a (- t)))))
     (/ (* y x) (+ 1.0 (* (/ a (/ (* z z) t)) -0.5))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.65e-120) {
		tmp = y * -x;
	} else if (z <= 8e-76) {
		tmp = y * ((z * x) / sqrt((a * -t)));
	} else {
		tmp = (y * x) / (1.0 + ((a / ((z * z) / t)) * -0.5));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-2.65d-120)) then
        tmp = y * -x
    else if (z <= 8d-76) then
        tmp = y * ((z * x) / sqrt((a * -t)))
    else
        tmp = (y * x) / (1.0d0 + ((a / ((z * z) / t)) * (-0.5d0)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.65e-120) {
		tmp = y * -x;
	} else if (z <= 8e-76) {
		tmp = y * ((z * x) / Math.sqrt((a * -t)));
	} else {
		tmp = (y * x) / (1.0 + ((a / ((z * z) / t)) * -0.5));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.65e-120:
		tmp = y * -x
	elif z <= 8e-76:
		tmp = y * ((z * x) / math.sqrt((a * -t)))
	else:
		tmp = (y * x) / (1.0 + ((a / ((z * z) / t)) * -0.5))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.65e-120)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 8e-76)
		tmp = Float64(y * Float64(Float64(z * x) / sqrt(Float64(a * Float64(-t)))));
	else
		tmp = Float64(Float64(y * x) / Float64(1.0 + Float64(Float64(a / Float64(Float64(z * z) / t)) * -0.5)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.65e-120)
		tmp = y * -x;
	elseif (z <= 8e-76)
		tmp = y * ((z * x) / sqrt((a * -t)));
	else
		tmp = (y * x) / (1.0 + ((a / ((z * z) / t)) * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.65e-120], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 8e-76], N[(y * N[(N[(z * x), $MachinePrecision] / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / N[(1.0 + N[(N[(a / N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.64999999999999999e-120

   1. Initial program 61.7%

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

      1. *-commutative 61.7%

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

      2. associate-*l* 59.6%

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

      3. associate-*r/ 66.3%

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

   3. Simplified 66.3%

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

   4. Taylor expanded in z around -inf 83.7%

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

      1. neg-mul-1 83.7%

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

   6. Simplified 83.7%

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

   if -2.64999999999999999e-120 < z < 7.99999999999999942e-76

   1. Initial program 80.1%

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

      1. *-commutative 80.1%

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

      2. associate-*l* 82.8%

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

      3. associate-*r/ 81.7%

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

   3. Simplified 81.7%

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

   4. Taylor expanded in z around 0 78.9%

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

      1. associate-*r* 78.9%

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

      2. neg-mul-1 78.9%

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

   6. Simplified 78.9%

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

   if 7.99999999999999942e-76 < z

   1. Initial program 58.2%

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

      1. associate-/l* 61.5%

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

   3. Simplified 61.5%

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

   4. Taylor expanded in z around inf 85.5%

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

      1. unpow2 85.5%

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

      2. associate-/l* 91.0%

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

   6. Simplified 91.0%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.65 \cdot 10^{-120}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-76}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + \frac{a}{\frac{z \cdot z}{t}} \cdot -0.5}\\ \end{array} \]

Alternative 6: 83.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-123}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-82}:\\ \;\;\;\;\frac{y}{\frac{\sqrt{a \cdot \left(-t\right)}}{z \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + \frac{a}{\frac{z \cdot z}{t}} \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.5e-123)
   (* y (- x))
   (if (<= z 1.2e-82)
     (/ y (/ (sqrt (* a (- t))) (* z x)))
     (/ (* y x) (+ 1.0 (* (/ a (/ (* z z) t)) -0.5))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.5e-123) {
		tmp = y * -x;
	} else if (z <= 1.2e-82) {
		tmp = y / (sqrt((a * -t)) / (z * x));
	} else {
		tmp = (y * x) / (1.0 + ((a / ((z * z) / t)) * -0.5));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.5d-123)) then
        tmp = y * -x
    else if (z <= 1.2d-82) then
        tmp = y / (sqrt((a * -t)) / (z * x))
    else
        tmp = (y * x) / (1.0d0 + ((a / ((z * z) / t)) * (-0.5d0)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.5e-123) {
		tmp = y * -x;
	} else if (z <= 1.2e-82) {
		tmp = y / (Math.sqrt((a * -t)) / (z * x));
	} else {
		tmp = (y * x) / (1.0 + ((a / ((z * z) / t)) * -0.5));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.5e-123:
		tmp = y * -x
	elif z <= 1.2e-82:
		tmp = y / (math.sqrt((a * -t)) / (z * x))
	else:
		tmp = (y * x) / (1.0 + ((a / ((z * z) / t)) * -0.5))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.5e-123)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.2e-82)
		tmp = Float64(y / Float64(sqrt(Float64(a * Float64(-t))) / Float64(z * x)));
	else
		tmp = Float64(Float64(y * x) / Float64(1.0 + Float64(Float64(a / Float64(Float64(z * z) / t)) * -0.5)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.5e-123)
		tmp = y * -x;
	elseif (z <= 1.2e-82)
		tmp = y / (sqrt((a * -t)) / (z * x));
	else
		tmp = (y * x) / (1.0 + ((a / ((z * z) / t)) * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.5e-123], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.2e-82], N[(y / N[(N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / N[(1.0 + N[(N[(a / N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.49999999999999992e-123

   1. Initial program 61.7%

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

      1. *-commutative 61.7%

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

      2. associate-*l* 59.6%

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

      3. associate-*r/ 66.3%

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

   3. Simplified 66.3%

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

   4. Taylor expanded in z around -inf 83.7%

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

      1. neg-mul-1 83.7%

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

   6. Simplified 83.7%

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

   if -1.49999999999999992e-123 < z < 1.20000000000000004e-82

   1. Initial program 80.1%

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

      1. *-commutative 80.1%

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

      2. associate-*l* 82.8%

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

      3. associate-*r/ 81.7%

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

   3. Simplified 81.7%

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

      1. clear-num 81.3%

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

      2. un-div-inv 81.4%

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

      3. *-commutative 81.4%

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

   5. Applied egg-rr 81.4%

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

   6. Taylor expanded in z around 0 78.5%

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

      1. associate-*r* 78.9%

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

      2. neg-mul-1 78.9%

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

   8. Simplified 78.5%

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

   if 1.20000000000000004e-82 < z

   1. Initial program 58.2%

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

      1. associate-/l* 61.5%

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

   3. Simplified 61.5%

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

   4. Taylor expanded in z around inf 85.5%

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

      1. unpow2 85.5%

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

      2. associate-/l* 91.0%

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

   6. Simplified 91.0%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-123}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-82}:\\ \;\;\;\;\frac{y}{\frac{\sqrt{a \cdot \left(-t\right)}}{z \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + \frac{a}{\frac{z \cdot z}{t}} \cdot -0.5}\\ \end{array} \]

Alternative 7: 76.5% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{-180}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-148}:\\ \;\;\;\;2 \cdot \left(\frac{x}{a} \cdot \frac{y \cdot \left(z \cdot z\right)}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.8e-180)
   (* y (- x))
   (if (<= z 8e-148) (* 2.0 (* (/ x a) (/ (* y (* z z)) t))) (* y x))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.8e-180) {
		tmp = y * -x;
	} else if (z <= 8e-148) {
		tmp = 2.0 * ((x / a) * ((y * (z * z)) / t));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 <= (-1.8d-180)) then
        tmp = y * -x
    else if (z <= 8d-148) then
        tmp = 2.0d0 * ((x / a) * ((y * (z * z)) / t))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.8e-180) {
		tmp = y * -x;
	} else if (z <= 8e-148) {
		tmp = 2.0 * ((x / a) * ((y * (z * z)) / t));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.8e-180:
		tmp = y * -x
	elif z <= 8e-148:
		tmp = 2.0 * ((x / a) * ((y * (z * z)) / t))
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.8e-180)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 8e-148)
		tmp = Float64(2.0 * Float64(Float64(x / a) * Float64(Float64(y * Float64(z * z)) / t)));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.8e-180)
		tmp = y * -x;
	elseif (z <= 8e-148)
		tmp = 2.0 * ((x / a) * ((y * (z * z)) / t));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.8e-180], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 8e-148], N[(2.0 * N[(N[(x / a), $MachinePrecision] * N[(N[(y * N[(z * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.8e-180

   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* 59.9%

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

      3. associate-*r/ 67.0%

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

   3. Simplified 67.0%

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

   4. Taylor expanded in z around -inf 80.4%

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

      1. neg-mul-1 80.4%

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

   6. Simplified 80.4%

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

   if -1.8e-180 < z < 7.99999999999999949e-148

   1. Initial program 80.5%

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

      1. *-commutative 80.5%

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

      2. associate-*l* 84.1%

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

      3. associate-*r/ 80.8%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in z around -inf 38.0%

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

   5. Taylor expanded in z around 0 37.8%

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

      1. associate-*r* 37.8%

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

      2. *-commutative 37.8%

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

      3. times-frac 37.5%

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

      4. unpow2 37.5%

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

   7. Simplified 37.5%

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

   if 7.99999999999999949e-148 < z

   1. Initial program 61.6%

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

      1. *-commutative 61.6%

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

      2. associate-*l* 60.6%

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

      3. associate-*r/ 62.5%

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

   3. Simplified 62.5%

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

   4. Taylor expanded in z around inf 82.1%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{-180}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-148}:\\ \;\;\;\;2 \cdot \left(\frac{x}{a} \cdot \frac{y \cdot \left(z \cdot z\right)}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 8: 76.6% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{-180}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{-148}:\\ \;\;\;\;2 \cdot \left(\frac{z \cdot \left(z \cdot y\right)}{t} \cdot \frac{x}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.85e-180)
   (* y (- x))
   (if (<= z 9.8e-148) (* 2.0 (* (/ (* z (* z y)) t) (/ x a))) (* y x))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.85e-180) {
		tmp = y * -x;
	} else if (z <= 9.8e-148) {
		tmp = 2.0 * (((z * (z * y)) / t) * (x / a));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.85d-180)) then
        tmp = y * -x
    else if (z <= 9.8d-148) then
        tmp = 2.0d0 * (((z * (z * y)) / t) * (x / a))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.85e-180) {
		tmp = y * -x;
	} else if (z <= 9.8e-148) {
		tmp = 2.0 * (((z * (z * y)) / t) * (x / a));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.85e-180:
		tmp = y * -x
	elif z <= 9.8e-148:
		tmp = 2.0 * (((z * (z * y)) / t) * (x / a))
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.85e-180)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 9.8e-148)
		tmp = Float64(2.0 * Float64(Float64(Float64(z * Float64(z * y)) / t) * Float64(x / a)));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.85e-180)
		tmp = y * -x;
	elseif (z <= 9.8e-148)
		tmp = 2.0 * (((z * (z * y)) / t) * (x / a));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.85e-180], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 9.8e-148], N[(2.0 * N[(N[(N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(x / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.85000000000000008e-180

   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* 59.9%

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

      3. associate-*r/ 67.0%

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

   3. Simplified 67.0%

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

   4. Taylor expanded in z around -inf 80.4%

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

      1. neg-mul-1 80.4%

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

   6. Simplified 80.4%

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

   if -1.85000000000000008e-180 < z < 9.8000000000000001e-148

   1. Initial program 80.5%

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

      1. *-commutative 80.5%

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

      2. associate-*l* 84.1%

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

      3. associate-*r/ 80.8%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in z around -inf 38.0%

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

   5. Taylor expanded in z around 0 37.8%

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

      1. associate-*r* 37.8%

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

      2. *-commutative 37.8%

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

      3. times-frac 37.5%

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

      4. unpow2 37.5%

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

   7. Simplified 37.5%

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

   8. Taylor expanded in y around 0 37.5%

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

      1. *-commutative 37.5%

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

      2. unpow2 37.5%

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

      3. associate-*l* 37.7%

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

   10. Simplified 37.7%

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

   if 9.8000000000000001e-148 < z

   1. Initial program 61.6%

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

      1. *-commutative 61.6%

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

      2. associate-*l* 60.6%

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

      3. associate-*r/ 62.5%

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

   3. Simplified 62.5%

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

   4. Taylor expanded in z around inf 82.1%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{-180}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{-148}:\\ \;\;\;\;2 \cdot \left(\frac{z \cdot \left(z \cdot y\right)}{t} \cdot \frac{x}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 9: 76.8% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -6.6 \cdot 10^{-181}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-173}:\\ \;\;\;\;2 \cdot \frac{x \cdot \left(z \cdot \left(z \cdot \frac{y}{t}\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -6.6e-181)
   (* y (- x))
   (if (<= z 1.55e-173) (* 2.0 (/ (* x (* z (* z (/ y t)))) a)) (* y x))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6.6e-181) {
		tmp = y * -x;
	} else if (z <= 1.55e-173) {
		tmp = 2.0 * ((x * (z * (z * (y / t)))) / a);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-6.6d-181)) then
        tmp = y * -x
    else if (z <= 1.55d-173) then
        tmp = 2.0d0 * ((x * (z * (z * (y / t)))) / a)
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6.6e-181) {
		tmp = y * -x;
	} else if (z <= 1.55e-173) {
		tmp = 2.0 * ((x * (z * (z * (y / t)))) / a);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -6.6e-181:
		tmp = y * -x
	elif z <= 1.55e-173:
		tmp = 2.0 * ((x * (z * (z * (y / t)))) / a)
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -6.6e-181)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.55e-173)
		tmp = Float64(2.0 * Float64(Float64(x * Float64(z * Float64(z * Float64(y / t)))) / a));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -6.6e-181)
		tmp = y * -x;
	elseif (z <= 1.55e-173)
		tmp = 2.0 * ((x * (z * (z * (y / t)))) / a);
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -6.6e-181], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.55e-173], N[(2.0 * N[(N[(x * N[(z * N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.60000000000000018e-181

   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* 59.9%

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

      3. associate-*r/ 67.0%

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

   3. Simplified 67.0%

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

   4. Taylor expanded in z around -inf 80.4%

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

      1. neg-mul-1 80.4%

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

   6. Simplified 80.4%

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

   if -6.60000000000000018e-181 < z < 1.55000000000000003e-173

   1. Initial program 82.2%

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

      1. *-commutative 82.2%

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

      2. associate-*l* 88.6%

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

      3. associate-*r/ 84.8%

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

   3. Simplified 84.8%

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

   4. Taylor expanded in z around -inf 40.3%

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

   5. Taylor expanded in z around 0 40.3%

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

      1. associate-*r* 40.3%

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

      2. *-commutative 40.3%

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

      3. times-frac 39.8%

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

      4. unpow2 39.8%

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

   7. Simplified 39.8%

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

      1. associate-*r/ 40.4%

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

      2. associate-/l* 40.4%

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

   9. Applied egg-rr 40.4%

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

   10. Taylor expanded in y around 0 40.4%

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

       1. associate-*l/ 39.9%

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

       2. *-commutative 39.9%

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

       3. unpow2 39.9%

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

       4. associate-*l* 40.3%

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

   12. Simplified 40.3%

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

   if 1.55000000000000003e-173 < z

   1. Initial program 62.0%

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

      1. *-commutative 62.0%

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

      2. associate-*l* 60.2%

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

      3. associate-*r/ 62.0%

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

   3. Simplified 62.0%

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

   4. Taylor expanded in z around inf 78.7%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.6 \cdot 10^{-181}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-173}:\\ \;\;\;\;2 \cdot \frac{x \cdot \left(z \cdot \left(z \cdot \frac{y}{t}\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 10: 76.6% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-180}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-148}:\\ \;\;\;\;2 \cdot \frac{x \cdot \frac{y}{\frac{t}{z \cdot z}}}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.55e-180)
   (* y (- x))
   (if (<= z 9.5e-148) (* 2.0 (/ (* x (/ y (/ t (* z z)))) a)) (* y x))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.55e-180) {
		tmp = y * -x;
	} else if (z <= 9.5e-148) {
		tmp = 2.0 * ((x * (y / (t / (z * z)))) / a);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.55d-180)) then
        tmp = y * -x
    else if (z <= 9.5d-148) then
        tmp = 2.0d0 * ((x * (y / (t / (z * z)))) / a)
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.55e-180) {
		tmp = y * -x;
	} else if (z <= 9.5e-148) {
		tmp = 2.0 * ((x * (y / (t / (z * z)))) / a);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.55e-180:
		tmp = y * -x
	elif z <= 9.5e-148:
		tmp = 2.0 * ((x * (y / (t / (z * z)))) / a)
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.55e-180)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 9.5e-148)
		tmp = Float64(2.0 * Float64(Float64(x * Float64(y / Float64(t / Float64(z * z)))) / a));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.55e-180)
		tmp = y * -x;
	elseif (z <= 9.5e-148)
		tmp = 2.0 * ((x * (y / (t / (z * z)))) / a);
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.55e-180], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 9.5e-148], N[(2.0 * N[(N[(x * N[(y / N[(t / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.5499999999999999e-180

   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* 59.9%

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

      3. associate-*r/ 67.0%

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

   3. Simplified 67.0%

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

   4. Taylor expanded in z around -inf 80.4%

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

      1. neg-mul-1 80.4%

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

   6. Simplified 80.4%

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

   if -1.5499999999999999e-180 < z < 9.50000000000000069e-148

   1. Initial program 80.5%

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

      1. *-commutative 80.5%

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

      2. associate-*l* 84.1%

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

      3. associate-*r/ 80.8%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in z around -inf 38.0%

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

   5. Taylor expanded in z around 0 37.8%

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

      1. associate-*r* 37.8%

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

      2. *-commutative 37.8%

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

      3. times-frac 37.5%

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

      4. unpow2 37.5%

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

   7. Simplified 37.5%

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

      1. associate-*r/ 38.0%

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

      2. associate-/l* 38.0%

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

   9. Applied egg-rr 38.0%

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

   if 9.50000000000000069e-148 < z

   1. Initial program 61.6%

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

      1. *-commutative 61.6%

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

      2. associate-*l* 60.6%

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

      3. associate-*r/ 62.5%

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

   3. Simplified 62.5%

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

   4. Taylor expanded in z around inf 82.1%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-180}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-148}:\\ \;\;\;\;2 \cdot \frac{x \cdot \frac{y}{\frac{t}{z \cdot z}}}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 11: 76.6% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{-180}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.18 \cdot 10^{-147}:\\ \;\;\;\;y \cdot \left(\frac{2}{a} \cdot \frac{x \cdot \left(z \cdot z\right)}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.4e-180)
   (* y (- x))
   (if (<= z 1.18e-147) (* y (* (/ 2.0 a) (/ (* x (* z z)) t))) (* y x))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.4e-180) {
		tmp = y * -x;
	} else if (z <= 1.18e-147) {
		tmp = y * ((2.0 / a) * ((x * (z * z)) / t));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.4d-180)) then
        tmp = y * -x
    else if (z <= 1.18d-147) then
        tmp = y * ((2.0d0 / a) * ((x * (z * z)) / t))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.4e-180) {
		tmp = y * -x;
	} else if (z <= 1.18e-147) {
		tmp = y * ((2.0 / a) * ((x * (z * z)) / t));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.4e-180:
		tmp = y * -x
	elif z <= 1.18e-147:
		tmp = y * ((2.0 / a) * ((x * (z * z)) / t))
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.4e-180)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.18e-147)
		tmp = Float64(y * Float64(Float64(2.0 / a) * Float64(Float64(x * Float64(z * z)) / t)));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.4e-180)
		tmp = y * -x;
	elseif (z <= 1.18e-147)
		tmp = y * ((2.0 / a) * ((x * (z * z)) / t));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.4e-180], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.18e-147], N[(y * N[(N[(2.0 / a), $MachinePrecision] * N[(N[(x * N[(z * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.39999999999999999e-180

   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* 59.9%

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

      3. associate-*r/ 67.0%

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

   3. Simplified 67.0%

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

   4. Taylor expanded in z around -inf 80.4%

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

      1. neg-mul-1 80.4%

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

   6. Simplified 80.4%

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

   if -1.39999999999999999e-180 < z < 1.18000000000000003e-147

   1. Initial program 80.5%

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

      1. *-commutative 80.5%

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

      2. associate-*l* 84.1%

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

      3. associate-*r/ 80.8%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in z around -inf 38.0%

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

      1. expm1-log1p-u 8.5%

         \[\leadsto y \cdot \frac{x \cdot z}{0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a \cdot t}{z}\right)\right)} + -1 \cdot z} \]

      2. expm1-udef 8.5%

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

      3. associate-/l* 8.5%

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

   6. Applied egg-rr 8.5%

      \[\leadsto y \cdot \frac{x \cdot z}{0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{a}{\frac{z}{t}}\right)} - 1\right)} + -1 \cdot z} \]
   7. Step-by-step derivation

      1. expm1-def 8.5%

         \[\leadsto y \cdot \frac{x \cdot z}{0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a}{\frac{z}{t}}\right)\right)} + -1 \cdot z} \]

      2. expm1-log1p 38.0%

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

      3. associate-/r/ 37.9%

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

   8. Simplified 37.9%

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

   9. Taylor expanded in z around 0 37.8%

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

       1. associate-*r/ 37.8%

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

       2. times-frac 38.0%

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

       3. unpow2 38.0%

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

   11. Simplified 38.0%

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

   if 1.18000000000000003e-147 < z

   1. Initial program 61.6%

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

      1. *-commutative 61.6%

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

      2. associate-*l* 60.6%

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

      3. associate-*r/ 62.5%

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

   3. Simplified 62.5%

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

   4. Taylor expanded in z around inf 82.1%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{-180}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.18 \cdot 10^{-147}:\\ \;\;\;\;y \cdot \left(\frac{2}{a} \cdot \frac{x \cdot \left(z \cdot z\right)}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 12: 78.3% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-180}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\frac{z + -0.5 \cdot \left(\frac{a}{z} \cdot t\right)}{z}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.1e-180)
   (* y (- x))
   (* y (/ x (/ (+ z (* -0.5 (* (/ a z) t))) z)))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.1e-180) {
		tmp = y * -x;
	} else {
		tmp = y * (x / ((z + (-0.5 * ((a / z) * t))) / z));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-2.1d-180)) then
        tmp = y * -x
    else
        tmp = y * (x / ((z + ((-0.5d0) * ((a / z) * t))) / z))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.1e-180) {
		tmp = y * -x;
	} else {
		tmp = y * (x / ((z + (-0.5 * ((a / z) * t))) / z));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.1e-180:
		tmp = y * -x
	else:
		tmp = y * (x / ((z + (-0.5 * ((a / z) * t))) / z))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.1e-180)
		tmp = Float64(y * Float64(-x));
	else
		tmp = Float64(y * Float64(x / Float64(Float64(z + Float64(-0.5 * Float64(Float64(a / z) * t))) / z)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.1e-180)
		tmp = y * -x;
	else
		tmp = y * (x / ((z + (-0.5 * ((a / z) * t))) / z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.1e-180], N[(y * (-x)), $MachinePrecision], N[(y * N[(x / N[(N[(z + N[(-0.5 * N[(N[(a / z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.0999999999999999e-180

   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* 59.9%

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

      3. associate-*r/ 67.0%

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

   3. Simplified 67.0%

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

   4. Taylor expanded in z around -inf 80.4%

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

      1. neg-mul-1 80.4%

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

   6. Simplified 80.4%

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

   if -2.0999999999999999e-180 < z

   1. Initial program 67.7%

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

      1. *-commutative 67.7%

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

      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/ 68.4%

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

   3. Simplified 68.4%

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

      1. associate-/l* 68.3%

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

      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 68.3%

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

      5. *-commutative 68.3%

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

   7. Simplified 68.3%

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

   8. Taylor expanded in z around inf 68.5%

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

      1. associate-/l* 71.0%

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

      2. associate-/r/ 71.0%

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

   10. Simplified 71.0%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-180}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\frac{z + -0.5 \cdot \left(\frac{a}{z} \cdot t\right)}{z}}\\ \end{array} \]

Alternative 13: 74.9% accurate, 10.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{-278}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{-195}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4.6e-278)
   (* y (- x))
   (if (<= z 8.2e-195) (* y (/ (* z x) z)) (* y x))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.6e-278) {
		tmp = y * -x;
	} else if (z <= 8.2e-195) {
		tmp = y * ((z * x) / z);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-4.6d-278)) then
        tmp = y * -x
    else if (z <= 8.2d-195) then
        tmp = y * ((z * x) / z)
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.6e-278) {
		tmp = y * -x;
	} else if (z <= 8.2e-195) {
		tmp = y * ((z * x) / z);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -4.6e-278:
		tmp = y * -x
	elif z <= 8.2e-195:
		tmp = y * ((z * x) / z)
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4.6e-278)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 8.2e-195)
		tmp = Float64(y * Float64(Float64(z * x) / z));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -4.6e-278)
		tmp = y * -x;
	elseif (z <= 8.2e-195)
		tmp = y * ((z * x) / z);
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4.6e-278], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 8.2e-195], N[(y * N[(N[(z * x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.60000000000000006e-278

   1. Initial program 62.2%

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

      1. *-commutative 62.2%

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

      2. associate-*l* 62.0%

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

      3. associate-*r/ 68.1%

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

   3. Simplified 68.1%

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

   4. Taylor expanded in z around -inf 71.0%

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

      1. neg-mul-1 71.0%

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

   6. Simplified 71.0%

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

   if -4.60000000000000006e-278 < z < 8.20000000000000024e-195

   1. Initial program 91.8%

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

      1. *-commutative 91.8%

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

      2. associate-*l* 95.9%

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

      3. associate-*r/ 92.0%

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

   3. Simplified 92.0%

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

   4. Taylor expanded in z around inf 38.0%

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

   if 8.20000000000000024e-195 < z

   1. Initial program 63.3%

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

      1. *-commutative 63.3%

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

      2. associate-*l* 61.5%

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

      3. associate-*r/ 62.7%

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

   3. Simplified 62.7%

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

   4. Taylor expanded in z around inf 76.2%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{-278}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{-195}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 14: 75.7% accurate, 10.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{-278}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-120}:\\ \;\;\;\;\frac{z \cdot \left(y \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4.6e-278)
   (* y (- x))
   (if (<= z 1.7e-120) (/ (* z (* y x)) z) (* y x))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.6e-278) {
		tmp = y * -x;
	} else if (z <= 1.7e-120) {
		tmp = (z * (y * x)) / z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-4.6d-278)) then
        tmp = y * -x
    else if (z <= 1.7d-120) then
        tmp = (z * (y * x)) / z
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.6e-278) {
		tmp = y * -x;
	} else if (z <= 1.7e-120) {
		tmp = (z * (y * x)) / z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -4.6e-278:
		tmp = y * -x
	elif z <= 1.7e-120:
		tmp = (z * (y * x)) / z
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4.6e-278)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.7e-120)
		tmp = Float64(Float64(z * Float64(y * x)) / z);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -4.6e-278)
		tmp = y * -x;
	elseif (z <= 1.7e-120)
		tmp = (z * (y * x)) / z;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4.6e-278], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.7e-120], N[(N[(z * N[(y * x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.60000000000000006e-278

   1. Initial program 62.2%

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

      1. *-commutative 62.2%

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

      2. associate-*l* 62.0%

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

      3. associate-*r/ 68.1%

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

   3. Simplified 68.1%

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

   4. Taylor expanded in z around -inf 71.0%

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

      1. neg-mul-1 71.0%

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

   6. Simplified 71.0%

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

   if -4.60000000000000006e-278 < z < 1.70000000000000005e-120

   1. Initial program 89.6%

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

   2. Taylor expanded in z around inf 40.6%

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

   if 1.70000000000000005e-120 < z

   1. Initial program 60.1%

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

      1. *-commutative 60.1%

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

      2. associate-*l* 60.0%

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

      3. associate-*r/ 62.0%

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

   3. Simplified 62.0%

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

   4. Taylor expanded in z around inf 85.2%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{-278}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-120}:\\ \;\;\;\;\frac{z \cdot \left(y \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 15: 73.7% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < -4.999999999999985e-310

   1. Initial program 63.4%

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

      1. *-commutative 63.4%

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

      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/ 68.3%

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

   3. Simplified 68.3%

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

   4. Taylor expanded in z around -inf 68.8%

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

      1. neg-mul-1 68.8%

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

   6. Simplified 68.8%

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

   if -4.999999999999985e-310 < z

   1. Initial program 67.1%

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

      1. *-commutative 67.1%

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

      2. associate-*l* 67.0%

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

      3. associate-*r/ 67.3%

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

   3. Simplified 67.3%

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

   4. Taylor expanded in z around inf 67.4%

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

4. Final simplification 68.1%

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

Alternative 16: 44.4% accurate, 37.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.3%

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

   1. *-commutative 65.3%

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

   2. associate-*l* 64.8%

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

   3. associate-*r/ 67.8%

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

3. Simplified 67.8%

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

4. Taylor expanded in z around inf 40.5%

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

5. Final simplification 40.5%

   \[\leadsto y \cdot x \]

Developer target: 89.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023240 
(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)))))