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

Percentage Accurate: 61.4% → 88.8%

Time: 21.0s

Alternatives: 16

Speedup: 18.6×

• Unsound rule application detected in e-graph. Results may not be sound.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+21}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-272}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{z \cdot z - t \cdot a}}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt{1 - \frac{a}{\frac{z}{\frac{t}{z}}}}}{y \cdot x}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -6.2e+21)
   (* y (- x))
   (if (<= z 3.6e-272)
     (/ x (/ (sqrt (- (* z z) (* t a))) (* z y)))
     (pow (/ (sqrt (- 1.0 (/ a (/ z (/ t z))))) (* y x)) -1.0))))

C

assert(x < y);
assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6.2e+21) {
		tmp = y * -x;
	} else if (z <= 3.6e-272) {
		tmp = x / (sqrt(((z * z) - (t * a))) / (z * y));
	} else {
		tmp = pow((sqrt((1.0 - (a / (z / (t / z))))) / (y * x)), -1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-6.2d+21)) then
        tmp = y * -x
    else if (z <= 3.6d-272) then
        tmp = x / (sqrt(((z * z) - (t * a))) / (z * y))
    else
        tmp = (sqrt((1.0d0 - (a / (z / (t / z))))) / (y * x)) ** (-1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6.2e+21) {
		tmp = y * -x;
	} else if (z <= 3.6e-272) {
		tmp = x / (Math.sqrt(((z * z) - (t * a))) / (z * y));
	} else {
		tmp = Math.pow((Math.sqrt((1.0 - (a / (z / (t / z))))) / (y * x)), -1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -6.2e+21:
		tmp = y * -x
	elif z <= 3.6e-272:
		tmp = x / (math.sqrt(((z * z) - (t * a))) / (z * y))
	else:
		tmp = math.pow((math.sqrt((1.0 - (a / (z / (t / z))))) / (y * x)), -1.0)
	return tmp

Julia

x, y = sort([x, y])
t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -6.2e+21)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 3.6e-272)
		tmp = Float64(x / Float64(sqrt(Float64(Float64(z * z) - Float64(t * a))) / Float64(z * y)));
	else
		tmp = Float64(sqrt(Float64(1.0 - Float64(a / Float64(z / Float64(t / z))))) / Float64(y * x)) ^ -1.0;
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -6.2e+21)
		tmp = y * -x;
	elseif (z <= 3.6e-272)
		tmp = x / (sqrt(((z * z) - (t * a))) / (z * y));
	else
		tmp = (sqrt((1.0 - (a / (z / (t / z))))) / (y * x)) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -6.2e+21], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 3.6e-272], N[(x / N[(N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Sqrt[N[(1.0 - N[(a / N[(z / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(y * x), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.2e21

   1. Initial program 39.4%

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

      1. *-commutative 39.4%

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

      2. associate-*l* 36.4%

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

      3. associate-*r/ 41.7%

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

   3. Simplified 41.7%

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

   4. Taylor expanded in z around -inf 100.0%

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

      1. neg-mul-1 100.0%

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

   6. Simplified 100.0%

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

   if -6.2e21 < z < 3.59999999999999968e-272

   1. Initial program 78.5%

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

      1. *-commutative 78.5%

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

      2. associate-*l* 82.4%

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

      3. associate-*r/ 86.6%

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

   3. Simplified 86.6%

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

      1. *-commutative 86.6%

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

      2. associate-/l* 84.3%

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

      3. associate-/r/ 84.8%

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

      4. associate-/l/ 86.3%

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

      5. *-commutative 86.3%

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

   5. Applied egg-rr 86.3%

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

   if 3.59999999999999968e-272 < z

   1. Initial program 56.3%

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

      1. associate-/l* 60.0%

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

   3. Simplified 60.0%

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

      1. add-sqr-sqrt 60.0%

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

      2. sqrt-unprod 59.1%

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

      3. frac-times 50.2%

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

      4. add-sqr-sqrt 50.2%

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

   5. Applied egg-rr 50.2%

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

      1. div-sub 45.6%

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

      2. *-inverses 79.1%

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

      3. *-commutative 79.1%

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

      4. associate-/l* 85.7%

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

   7. Simplified 85.7%

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

      1. clear-num 85.6%

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

      2. inv-pow 85.6%

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

      3. associate-/l* 90.9%

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

   9. Applied egg-rr 90.9%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+21}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-272}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{z \cdot z - t \cdot a}}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt{1 - \frac{a}{\frac{z}{\frac{t}{z}}}}}{y \cdot x}\right)}^{-1}\\ \end{array} \]

Alternative 2: 81.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} t_1 := \frac{t \cdot a}{z}\\ t_2 := \sqrt{a \cdot \left(-t\right)}\\ t_3 := y \cdot \frac{z \cdot x}{t_2}\\ \mathbf{if}\;z \leq -5 \cdot 10^{-10}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq -5.8 \cdot 10^{-60}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -1.48 \cdot 10^{-92}:\\ \;\;\;\;\frac{y \cdot x}{\frac{0.5 \cdot t_1 - z}{z}}\\ \mathbf{elif}\;z \leq -2.45 \cdot 10^{-226}:\\ \;\;\;\;z \cdot \frac{y \cdot x}{t_2}\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-167}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-80}:\\ \;\;\;\;z \cdot \frac{y \cdot x}{z + -0.5 \cdot t_1}\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{-57}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* t a) z))
        (t_2 (sqrt (* a (- t))))
        (t_3 (* y (/ (* z x) t_2))))
   (if (<= z -5e-10)
     (* y (- x))
     (if (<= z -5.8e-60)
       t_3
       (if (<= z -1.48e-92)
         (/ (* y x) (/ (- (* 0.5 t_1) z) z))
         (if (<= z -2.45e-226)
           (* z (/ (* y x) t_2))
           (if (<= z 1.65e-167)
             t_3
             (if (<= z 5.5e-80)
               (* z (/ (* y x) (+ z (* -0.5 t_1))))
               (if (<= z 2.55e-57)
                 t_3
                 (/ (* y x) (+ 1.0 (* -0.5 (/ a (/ (* z z) t))))))))))))))

C

assert(x < y);
assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (t * a) / z;
	double t_2 = sqrt((a * -t));
	double t_3 = y * ((z * x) / t_2);
	double tmp;
	if (z <= -5e-10) {
		tmp = y * -x;
	} else if (z <= -5.8e-60) {
		tmp = t_3;
	} else if (z <= -1.48e-92) {
		tmp = (y * x) / (((0.5 * t_1) - z) / z);
	} else if (z <= -2.45e-226) {
		tmp = z * ((y * x) / t_2);
	} else if (z <= 1.65e-167) {
		tmp = t_3;
	} else if (z <= 5.5e-80) {
		tmp = z * ((y * x) / (z + (-0.5 * t_1)));
	} else if (z <= 2.55e-57) {
		tmp = t_3;
	} else {
		tmp = (y * x) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (t * a) / z
    t_2 = sqrt((a * -t))
    t_3 = y * ((z * x) / t_2)
    if (z <= (-5d-10)) then
        tmp = y * -x
    else if (z <= (-5.8d-60)) then
        tmp = t_3
    else if (z <= (-1.48d-92)) then
        tmp = (y * x) / (((0.5d0 * t_1) - z) / z)
    else if (z <= (-2.45d-226)) then
        tmp = z * ((y * x) / t_2)
    else if (z <= 1.65d-167) then
        tmp = t_3
    else if (z <= 5.5d-80) then
        tmp = z * ((y * x) / (z + ((-0.5d0) * t_1)))
    else if (z <= 2.55d-57) then
        tmp = t_3
    else
        tmp = (y * x) / (1.0d0 + ((-0.5d0) * (a / ((z * z) / t))))
    end if
    code = tmp
end function

Java

assert x < y;
assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (t * a) / z;
	double t_2 = Math.sqrt((a * -t));
	double t_3 = y * ((z * x) / t_2);
	double tmp;
	if (z <= -5e-10) {
		tmp = y * -x;
	} else if (z <= -5.8e-60) {
		tmp = t_3;
	} else if (z <= -1.48e-92) {
		tmp = (y * x) / (((0.5 * t_1) - z) / z);
	} else if (z <= -2.45e-226) {
		tmp = z * ((y * x) / t_2);
	} else if (z <= 1.65e-167) {
		tmp = t_3;
	} else if (z <= 5.5e-80) {
		tmp = z * ((y * x) / (z + (-0.5 * t_1)));
	} else if (z <= 2.55e-57) {
		tmp = t_3;
	} else {
		tmp = (y * x) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[t, a] = sort([t, a])
def code(x, y, z, t, a):
	t_1 = (t * a) / z
	t_2 = math.sqrt((a * -t))
	t_3 = y * ((z * x) / t_2)
	tmp = 0
	if z <= -5e-10:
		tmp = y * -x
	elif z <= -5.8e-60:
		tmp = t_3
	elif z <= -1.48e-92:
		tmp = (y * x) / (((0.5 * t_1) - z) / z)
	elif z <= -2.45e-226:
		tmp = z * ((y * x) / t_2)
	elif z <= 1.65e-167:
		tmp = t_3
	elif z <= 5.5e-80:
		tmp = z * ((y * x) / (z + (-0.5 * t_1)))
	elif z <= 2.55e-57:
		tmp = t_3
	else:
		tmp = (y * x) / (1.0 + (-0.5 * (a / ((z * z) / t))))
	return tmp

Julia

x, y = sort([x, y])
t, a = sort([t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(t * a) / z)
	t_2 = sqrt(Float64(a * Float64(-t)))
	t_3 = Float64(y * Float64(Float64(z * x) / t_2))
	tmp = 0.0
	if (z <= -5e-10)
		tmp = Float64(y * Float64(-x));
	elseif (z <= -5.8e-60)
		tmp = t_3;
	elseif (z <= -1.48e-92)
		tmp = Float64(Float64(y * x) / Float64(Float64(Float64(0.5 * t_1) - z) / z));
	elseif (z <= -2.45e-226)
		tmp = Float64(z * Float64(Float64(y * x) / t_2));
	elseif (z <= 1.65e-167)
		tmp = t_3;
	elseif (z <= 5.5e-80)
		tmp = Float64(z * Float64(Float64(y * x) / Float64(z + Float64(-0.5 * t_1))));
	elseif (z <= 2.55e-57)
		tmp = t_3;
	else
		tmp = Float64(Float64(y * x) / Float64(1.0 + Float64(-0.5 * Float64(a / Float64(Float64(z * z) / t)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (t * a) / z;
	t_2 = sqrt((a * -t));
	t_3 = y * ((z * x) / t_2);
	tmp = 0.0;
	if (z <= -5e-10)
		tmp = y * -x;
	elseif (z <= -5.8e-60)
		tmp = t_3;
	elseif (z <= -1.48e-92)
		tmp = (y * x) / (((0.5 * t_1) - z) / z);
	elseif (z <= -2.45e-226)
		tmp = z * ((y * x) / t_2);
	elseif (z <= 1.65e-167)
		tmp = t_3;
	elseif (z <= 5.5e-80)
		tmp = z * ((y * x) / (z + (-0.5 * t_1)));
	elseif (z <= 2.55e-57)
		tmp = t_3;
	else
		tmp = (y * x) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t * a), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(y * N[(N[(z * x), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5e-10], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, -5.8e-60], t$95$3, If[LessEqual[z, -1.48e-92], N[(N[(y * x), $MachinePrecision] / N[(N[(N[(0.5 * t$95$1), $MachinePrecision] - z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -2.45e-226], N[(z * N[(N[(y * x), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.65e-167], t$95$3, If[LessEqual[z, 5.5e-80], N[(z * N[(N[(y * x), $MachinePrecision] / N[(z + N[(-0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.55e-57], t$95$3, N[(N[(y * x), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(a / N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if z < -5.00000000000000031e-10

   1. Initial program 45.4%

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

      1. *-commutative 45.4%

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

      2. associate-*l* 42.6%

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

      3. associate-*r/ 47.4%

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

   3. Simplified 47.4%

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

   4. Taylor expanded in z around -inf 98.7%

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

      1. neg-mul-1 98.7%

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

   6. Simplified 98.7%

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

   if -5.00000000000000031e-10 < z < -5.7999999999999999e-60 or -2.44999999999999993e-226 < z < 1.64999999999999998e-167 or 5.4999999999999997e-80 < z < 2.55e-57

   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* 74.3%

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

      3. associate-*r/ 75.3%

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

   3. Simplified 75.3%

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

   4. Taylor expanded in z around 0 75.3%

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

      1. associate-*r* 75.3%

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

      2. neg-mul-1 75.3%

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

   6. Simplified 75.3%

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

   if -5.7999999999999999e-60 < z < -1.48000000000000001e-92

   1. Initial program 89.6%

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

      1. associate-/l* 89.6%

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

   3. Simplified 89.6%

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

   4. Taylor expanded in z around -inf 80.0%

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

   if -1.48000000000000001e-92 < z < -2.44999999999999993e-226

   1. Initial program 80.2%

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

      1. associate-*l/ 94.8%

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

   3. Simplified 94.8%

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

   4. Taylor expanded in z around 0 84.4%

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

      1. associate-*r* 77.9%

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

      2. neg-mul-1 77.9%

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

   6. Simplified 84.4%

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

   if 1.64999999999999998e-167 < z < 5.4999999999999997e-80

   1. Initial program 80.4%

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

      1. associate-*l/ 89.7%

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

   3. Simplified 89.7%

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

   4. Taylor expanded in z around inf 71.9%

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

   if 2.55e-57 < z

   1. Initial program 52.3%

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

      1. associate-/l* 56.1%

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

   3. Simplified 56.1%

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

   4. Taylor expanded in z around inf 86.4%

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

      1. unpow2 86.4%

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

      2. associate-/l* 94.9%

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

   6. Simplified 94.9%

      \[\leadsto \frac{x \cdot y}{\color{blue}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-10}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq -5.8 \cdot 10^{-60}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{elif}\;z \leq -1.48 \cdot 10^{-92}:\\ \;\;\;\;\frac{y \cdot x}{\frac{0.5 \cdot \frac{t \cdot a}{z} - z}{z}}\\ \mathbf{elif}\;z \leq -2.45 \cdot 10^{-226}:\\ \;\;\;\;z \cdot \frac{y \cdot x}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-167}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-80}:\\ \;\;\;\;z \cdot \frac{y \cdot x}{z + -0.5 \cdot \frac{t \cdot a}{z}}\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{-57}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}\\ \end{array} \]

Alternative 3: 81.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} t_1 := \sqrt{a \cdot \left(-t\right)}\\ \mathbf{if}\;z \leq -6.5 \cdot 10^{-10}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq -6 \cdot 10^{-60}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{t_1}\\ \mathbf{elif}\;z \leq -2.3 \cdot 10^{-141}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{-z}\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-167}:\\ \;\;\;\;\frac{x}{\frac{t_1}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (sqrt (* a (- t)))))
   (if (<= z -6.5e-10)
     (* y (- x))
     (if (<= z -6e-60)
       (* y (/ (* z x) t_1))
       (if (<= z -2.3e-141)
         (* y (/ (* z x) (- z)))
         (if (<= z 1.75e-167)
           (/ x (/ t_1 (* z y)))
           (/ (* y x) (+ 1.0 (* -0.5 (/ a (/ (* z z) t)))))))))))

C

assert(x < y);
assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = sqrt((a * -t));
	double tmp;
	if (z <= -6.5e-10) {
		tmp = y * -x;
	} else if (z <= -6e-60) {
		tmp = y * ((z * x) / t_1);
	} else if (z <= -2.3e-141) {
		tmp = y * ((z * x) / -z);
	} else if (z <= 1.75e-167) {
		tmp = x / (t_1 / (z * y));
	} else {
		tmp = (y * x) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((a * -t))
    if (z <= (-6.5d-10)) then
        tmp = y * -x
    else if (z <= (-6d-60)) then
        tmp = y * ((z * x) / t_1)
    else if (z <= (-2.3d-141)) then
        tmp = y * ((z * x) / -z)
    else if (z <= 1.75d-167) then
        tmp = x / (t_1 / (z * y))
    else
        tmp = (y * x) / (1.0d0 + ((-0.5d0) * (a / ((z * z) / t))))
    end if
    code = tmp
end function

Java

assert x < y;
assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.sqrt((a * -t));
	double tmp;
	if (z <= -6.5e-10) {
		tmp = y * -x;
	} else if (z <= -6e-60) {
		tmp = y * ((z * x) / t_1);
	} else if (z <= -2.3e-141) {
		tmp = y * ((z * x) / -z);
	} else if (z <= 1.75e-167) {
		tmp = x / (t_1 / (z * y));
	} else {
		tmp = (y * x) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[t, a] = sort([t, a])
def code(x, y, z, t, a):
	t_1 = math.sqrt((a * -t))
	tmp = 0
	if z <= -6.5e-10:
		tmp = y * -x
	elif z <= -6e-60:
		tmp = y * ((z * x) / t_1)
	elif z <= -2.3e-141:
		tmp = y * ((z * x) / -z)
	elif z <= 1.75e-167:
		tmp = x / (t_1 / (z * y))
	else:
		tmp = (y * x) / (1.0 + (-0.5 * (a / ((z * z) / t))))
	return tmp

Julia

x, y = sort([x, y])
t, a = sort([t, a])
function code(x, y, z, t, a)
	t_1 = sqrt(Float64(a * Float64(-t)))
	tmp = 0.0
	if (z <= -6.5e-10)
		tmp = Float64(y * Float64(-x));
	elseif (z <= -6e-60)
		tmp = Float64(y * Float64(Float64(z * x) / t_1));
	elseif (z <= -2.3e-141)
		tmp = Float64(y * Float64(Float64(z * x) / Float64(-z)));
	elseif (z <= 1.75e-167)
		tmp = Float64(x / Float64(t_1 / Float64(z * y)));
	else
		tmp = Float64(Float64(y * x) / Float64(1.0 + Float64(-0.5 * Float64(a / Float64(Float64(z * z) / t)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = sqrt((a * -t));
	tmp = 0.0;
	if (z <= -6.5e-10)
		tmp = y * -x;
	elseif (z <= -6e-60)
		tmp = y * ((z * x) / t_1);
	elseif (z <= -2.3e-141)
		tmp = y * ((z * x) / -z);
	elseif (z <= 1.75e-167)
		tmp = x / (t_1 / (z * y));
	else
		tmp = (y * x) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -6.5e-10], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, -6e-60], N[(y * N[(N[(z * x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -2.3e-141], N[(y * N[(N[(z * x), $MachinePrecision] / (-z)), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.75e-167], N[(x / N[(t$95$1 / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(a / N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -6.5000000000000003e-10

   1. Initial program 45.4%

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

      1. *-commutative 45.4%

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

      2. associate-*l* 42.6%

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

      3. associate-*r/ 47.4%

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

   3. Simplified 47.4%

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

   4. Taylor expanded in z around -inf 98.7%

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

      1. neg-mul-1 98.7%

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

   6. Simplified 98.7%

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

   if -6.5000000000000003e-10 < z < -6.00000000000000038e-60

   1. Initial program 61.4%

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

      1. *-commutative 61.4%

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

      2. associate-*l* 84.6%

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

      3. associate-*r/ 79.0%

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

   3. Simplified 79.0%

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

   4. Taylor expanded in z around 0 79.0%

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

      1. associate-*r* 79.0%

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

      2. neg-mul-1 79.0%

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

   6. Simplified 79.0%

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

   if -6.00000000000000038e-60 < z < -2.29999999999999995e-141

   1. Initial program 88.8%

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

      1. *-commutative 88.8%

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

      2. associate-*l* 94.2%

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

      3. associate-*r/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around -inf 72.5%

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

      1. neg-mul-1 67.0%

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

   6. Simplified 72.5%

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

   if -2.29999999999999995e-141 < z < 1.75e-167

   1. Initial program 70.8%

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

      1. *-commutative 70.8%

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

      2. associate-*l* 72.7%

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

      3. associate-*r/ 77.5%

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

   3. Simplified 77.5%

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

      1. *-commutative 77.5%

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

      2. associate-/l* 76.1%

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

      3. associate-/r/ 77.8%

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

      4. associate-/l/ 79.7%

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

      5. *-commutative 79.7%

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

   5. Applied egg-rr 79.7%

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

   6. Taylor expanded in z around 0 79.7%

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

      1. associate-*r* 77.5%

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

      2. neg-mul-1 77.5%

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

   8. Simplified 79.7%

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

   if 1.75e-167 < z

   1. Initial program 55.2%

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

      1. associate-/l* 59.3%

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

   3. Simplified 59.3%

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

   4. Taylor expanded in z around inf 80.4%

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

      1. unpow2 80.4%

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

      2. associate-/l* 88.8%

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

   6. Simplified 88.8%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-10}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq -6 \cdot 10^{-60}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{elif}\;z \leq -2.3 \cdot 10^{-141}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{-z}\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-167}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{a \cdot \left(-t\right)}}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}\\ \end{array} \]

Alternative 4: 90.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+160}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+54}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -7e+160)
   (* y (- x))
   (if (<= z 4e+54)
     (* x (* y (/ z (sqrt (- (* z z) (* t a))))))
     (/ (* y x) (+ 1.0 (* -0.5 (/ a (/ (* z z) t))))))))

C

assert(x < y);
assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7e+160) {
		tmp = y * -x;
	} else if (z <= 4e+54) {
		tmp = x * (y * (z / sqrt(((z * z) - (t * a)))));
	} else {
		tmp = (y * x) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-7d+160)) then
        tmp = y * -x
    else if (z <= 4d+54) then
        tmp = x * (y * (z / sqrt(((z * z) - (t * a)))))
    else
        tmp = (y * x) / (1.0d0 + ((-0.5d0) * (a / ((z * z) / t))))
    end if
    code = tmp
end function

Java

assert x < y;
assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7e+160) {
		tmp = y * -x;
	} else if (z <= 4e+54) {
		tmp = x * (y * (z / Math.sqrt(((z * z) - (t * a)))));
	} else {
		tmp = (y * x) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -7e+160:
		tmp = y * -x
	elif z <= 4e+54:
		tmp = x * (y * (z / math.sqrt(((z * z) - (t * a)))))
	else:
		tmp = (y * x) / (1.0 + (-0.5 * (a / ((z * z) / t))))
	return tmp

Julia

x, y = sort([x, y])
t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -7e+160)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 4e+54)
		tmp = Float64(x * Float64(y * Float64(z / sqrt(Float64(Float64(z * z) - Float64(t * a))))));
	else
		tmp = Float64(Float64(y * x) / Float64(1.0 + Float64(-0.5 * Float64(a / Float64(Float64(z * z) / t)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -7e+160)
		tmp = y * -x;
	elseif (z <= 4e+54)
		tmp = x * (y * (z / sqrt(((z * z) - (t * a)))));
	else
		tmp = (y * x) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -7e+160], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 4e+54], N[(x * N[(y * N[(z / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(a / N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.00000000000000051e160

   1. Initial program 5.7%

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

      1. *-commutative 5.7%

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

      2. associate-*l* 5.3%

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

      3. associate-*r/ 5.5%

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

   3. Simplified 5.5%

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

   4. Taylor expanded in z around -inf 100.0%

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

      1. neg-mul-1 100.0%

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

   6. Simplified 100.0%

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

   if -7.00000000000000051e160 < z < 4.0000000000000003e54

   1. Initial program 79.2%

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

      1. associate-/l* 85.9%

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

   3. Simplified 85.9%

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

      1. div-inv 85.9%

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

      2. clear-num 86.5%

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

      3. associate-*l* 87.7%

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

   5. Applied egg-rr 87.7%

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

   if 4.0000000000000003e54 < z

   1. Initial program 40.4%

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

      1. associate-/l* 45.3%

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

   3. Simplified 45.3%

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

   4. Taylor expanded in z around inf 86.0%

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

      1. unpow2 86.0%

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

      2. associate-/l* 97.1%

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

   6. Simplified 97.1%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+160}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+54}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}\\ \end{array} \]

Alternative 5: 89.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+160}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-272}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{1 - \frac{t}{z} \cdot \frac{a}{z}}}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -7e+160)
   (* y (- x))
   (if (<= z 3.6e-272)
     (* x (* y (/ z (sqrt (- (* z z) (* t a))))))
     (/ x (/ (sqrt (- 1.0 (* (/ t z) (/ a z)))) y)))))

C

assert(x < y);
assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7e+160) {
		tmp = y * -x;
	} else if (z <= 3.6e-272) {
		tmp = x * (y * (z / sqrt(((z * z) - (t * a)))));
	} else {
		tmp = x / (sqrt((1.0 - ((t / z) * (a / z)))) / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-7d+160)) then
        tmp = y * -x
    else if (z <= 3.6d-272) then
        tmp = x * (y * (z / sqrt(((z * z) - (t * a)))))
    else
        tmp = x / (sqrt((1.0d0 - ((t / z) * (a / z)))) / y)
    end if
    code = tmp
end function

Java

assert x < y;
assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7e+160) {
		tmp = y * -x;
	} else if (z <= 3.6e-272) {
		tmp = x * (y * (z / Math.sqrt(((z * z) - (t * a)))));
	} else {
		tmp = x / (Math.sqrt((1.0 - ((t / z) * (a / z)))) / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -7e+160:
		tmp = y * -x
	elif z <= 3.6e-272:
		tmp = x * (y * (z / math.sqrt(((z * z) - (t * a)))))
	else:
		tmp = x / (math.sqrt((1.0 - ((t / z) * (a / z)))) / y)
	return tmp

Julia

x, y = sort([x, y])
t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -7e+160)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 3.6e-272)
		tmp = Float64(x * Float64(y * Float64(z / sqrt(Float64(Float64(z * z) - Float64(t * a))))));
	else
		tmp = Float64(x / Float64(sqrt(Float64(1.0 - Float64(Float64(t / z) * Float64(a / z)))) / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -7e+160)
		tmp = y * -x;
	elseif (z <= 3.6e-272)
		tmp = x * (y * (z / sqrt(((z * z) - (t * a)))));
	else
		tmp = x / (sqrt((1.0 - ((t / z) * (a / z)))) / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -7e+160], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 3.6e-272], N[(x * N[(y * N[(z / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[Sqrt[N[(1.0 - N[(N[(t / z), $MachinePrecision] * N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.00000000000000051e160

   1. Initial program 5.7%

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

      1. *-commutative 5.7%

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

      2. associate-*l* 5.3%

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

      3. associate-*r/ 5.5%

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

   3. Simplified 5.5%

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

   4. Taylor expanded in z around -inf 100.0%

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

      1. neg-mul-1 100.0%

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

   6. Simplified 100.0%

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

   if -7.00000000000000051e160 < z < 3.59999999999999968e-272

   1. Initial program 79.7%

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

      1. associate-/l* 88.6%

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

   3. Simplified 88.6%

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

      1. div-inv 88.6%

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

      2. clear-num 89.4%

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

      3. associate-*l* 91.1%

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

   5. Applied egg-rr 91.1%

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

   if 3.59999999999999968e-272 < z

   1. Initial program 56.3%

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

      1. associate-/l* 60.0%

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

   3. Simplified 60.0%

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

      1. add-sqr-sqrt 60.0%

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

      2. sqrt-unprod 59.1%

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

      3. frac-times 50.2%

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

      4. add-sqr-sqrt 50.2%

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

   5. Applied egg-rr 50.2%

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

      1. div-sub 45.6%

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

      2. *-inverses 79.1%

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

      3. *-commutative 79.1%

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

      4. associate-/l* 85.7%

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

   7. Simplified 85.7%

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

      1. expm1-log1p-u 62.0%

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

      2. expm1-udef 39.1%

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

      3. associate-/l* 40.6%

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

   9. Applied egg-rr 40.6%

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

       1. expm1-def 66.4%

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

       2. expm1-log1p 91.0%

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

       3. associate-/l* 90.0%

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

       4. associate-/r/ 93.2%

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

   11. Simplified 93.2%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+160}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-272}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{1 - \frac{t}{z} \cdot \frac{a}{z}}}{y}}\\ \end{array} \]

Alternative 6: 89.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+21}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-273}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{z \cdot z - t \cdot a}}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{1 - \frac{t}{z} \cdot \frac{a}{z}}}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -6.2e+21)
   (* y (- x))
   (if (<= z 6e-273)
     (/ x (/ (sqrt (- (* z z) (* t a))) (* z y)))
     (/ x (/ (sqrt (- 1.0 (* (/ t z) (/ a z)))) y)))))

C

assert(x < y);
assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6.2e+21) {
		tmp = y * -x;
	} else if (z <= 6e-273) {
		tmp = x / (sqrt(((z * z) - (t * a))) / (z * y));
	} else {
		tmp = x / (sqrt((1.0 - ((t / z) * (a / z)))) / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-6.2d+21)) then
        tmp = y * -x
    else if (z <= 6d-273) then
        tmp = x / (sqrt(((z * z) - (t * a))) / (z * y))
    else
        tmp = x / (sqrt((1.0d0 - ((t / z) * (a / z)))) / y)
    end if
    code = tmp
end function

Java

assert x < y;
assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6.2e+21) {
		tmp = y * -x;
	} else if (z <= 6e-273) {
		tmp = x / (Math.sqrt(((z * z) - (t * a))) / (z * y));
	} else {
		tmp = x / (Math.sqrt((1.0 - ((t / z) * (a / z)))) / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -6.2e+21:
		tmp = y * -x
	elif z <= 6e-273:
		tmp = x / (math.sqrt(((z * z) - (t * a))) / (z * y))
	else:
		tmp = x / (math.sqrt((1.0 - ((t / z) * (a / z)))) / y)
	return tmp

Julia

x, y = sort([x, y])
t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -6.2e+21)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 6e-273)
		tmp = Float64(x / Float64(sqrt(Float64(Float64(z * z) - Float64(t * a))) / Float64(z * y)));
	else
		tmp = Float64(x / Float64(sqrt(Float64(1.0 - Float64(Float64(t / z) * Float64(a / z)))) / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -6.2e+21)
		tmp = y * -x;
	elseif (z <= 6e-273)
		tmp = x / (sqrt(((z * z) - (t * a))) / (z * y));
	else
		tmp = x / (sqrt((1.0 - ((t / z) * (a / z)))) / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -6.2e+21], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 6e-273], N[(x / N[(N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[Sqrt[N[(1.0 - N[(N[(t / z), $MachinePrecision] * N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.2e21

   1. Initial program 39.4%

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

      1. *-commutative 39.4%

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

      2. associate-*l* 36.4%

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

      3. associate-*r/ 41.7%

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

   3. Simplified 41.7%

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

   4. Taylor expanded in z around -inf 100.0%

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

      1. neg-mul-1 100.0%

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

   6. Simplified 100.0%

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

   if -6.2e21 < z < 5.99999999999999975e-273

   1. Initial program 78.5%

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

      1. *-commutative 78.5%

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

      2. associate-*l* 82.4%

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

      3. associate-*r/ 86.6%

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

   3. Simplified 86.6%

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

      1. *-commutative 86.6%

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

      2. associate-/l* 84.3%

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

      3. associate-/r/ 84.8%

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

      4. associate-/l/ 86.3%

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

      5. *-commutative 86.3%

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

   5. Applied egg-rr 86.3%

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

   if 5.99999999999999975e-273 < z

   1. Initial program 56.3%

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

      1. associate-/l* 60.0%

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

   3. Simplified 60.0%

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

      1. add-sqr-sqrt 60.0%

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

      2. sqrt-unprod 59.1%

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

      3. frac-times 50.2%

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

      4. add-sqr-sqrt 50.2%

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

   5. Applied egg-rr 50.2%

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

      1. div-sub 45.6%

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

      2. *-inverses 79.1%

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

      3. *-commutative 79.1%

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

      4. associate-/l* 85.7%

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

   7. Simplified 85.7%

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

      1. expm1-log1p-u 62.0%

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

      2. expm1-udef 39.1%

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

      3. associate-/l* 40.6%

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

   9. Applied egg-rr 40.6%

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

       1. expm1-def 66.4%

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

       2. expm1-log1p 91.0%

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

       3. associate-/l* 90.0%

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

       4. associate-/r/ 93.2%

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

   11. Simplified 93.2%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+21}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-273}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{z \cdot z - t \cdot a}}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{1 - \frac{t}{z} \cdot \frac{a}{z}}}{y}}\\ \end{array} \]

Alternative 7: 82.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -5.6 \cdot 10^{-10}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-167}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.6e-10)
   (* y (- x))
   (if (<= z 1.75e-167)
     (* y (/ (* z x) (sqrt (* a (- t)))))
     (/ (* y x) (+ 1.0 (* -0.5 (/ a (/ (* z z) t))))))))

C

assert(x < y);
assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.6e-10) {
		tmp = y * -x;
	} else if (z <= 1.75e-167) {
		tmp = y * ((z * x) / sqrt((a * -t)));
	} else {
		tmp = (y * x) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-5.6d-10)) then
        tmp = y * -x
    else if (z <= 1.75d-167) then
        tmp = y * ((z * x) / sqrt((a * -t)))
    else
        tmp = (y * x) / (1.0d0 + ((-0.5d0) * (a / ((z * z) / t))))
    end if
    code = tmp
end function

Java

assert x < y;
assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.6e-10) {
		tmp = y * -x;
	} else if (z <= 1.75e-167) {
		tmp = y * ((z * x) / Math.sqrt((a * -t)));
	} else {
		tmp = (y * x) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -5.6e-10:
		tmp = y * -x
	elif z <= 1.75e-167:
		tmp = y * ((z * x) / math.sqrt((a * -t)))
	else:
		tmp = (y * x) / (1.0 + (-0.5 * (a / ((z * z) / t))))
	return tmp

Julia

x, y = sort([x, y])
t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.6e-10)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.75e-167)
		tmp = Float64(y * Float64(Float64(z * x) / sqrt(Float64(a * Float64(-t)))));
	else
		tmp = Float64(Float64(y * x) / Float64(1.0 + Float64(-0.5 * Float64(a / Float64(Float64(z * z) / t)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -5.6e-10)
		tmp = y * -x;
	elseif (z <= 1.75e-167)
		tmp = y * ((z * x) / sqrt((a * -t)));
	else
		tmp = (y * x) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.6e-10], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.75e-167], N[(y * N[(N[(z * x), $MachinePrecision] / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(a / N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.60000000000000031e-10

   1. Initial program 45.4%

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

      1. *-commutative 45.4%

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

      2. associate-*l* 42.6%

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

      3. associate-*r/ 47.4%

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

   3. Simplified 47.4%

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

   4. Taylor expanded in z around -inf 98.7%

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

      1. neg-mul-1 98.7%

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

   6. Simplified 98.7%

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

   if -5.60000000000000031e-10 < z < 1.75e-167

   1. Initial program 73.8%

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

      1. *-commutative 73.8%

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

      2. associate-*l* 78.6%

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

      3. associate-*r/ 82.5%

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

   3. Simplified 82.5%

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

   4. Taylor expanded in z around 0 73.0%

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

      1. associate-*r* 73.0%

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

      2. neg-mul-1 73.0%

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

   6. Simplified 73.0%

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

   if 1.75e-167 < z

   1. Initial program 55.2%

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

      1. associate-/l* 59.3%

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

   3. Simplified 59.3%

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

   4. Taylor expanded in z around inf 80.4%

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

      1. unpow2 80.4%

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

      2. associate-/l* 88.8%

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

   6. Simplified 88.8%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.6 \cdot 10^{-10}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-167}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}\\ \end{array} \]

Alternative 8: 76.9% accurate, 5.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.32 \cdot 10^{-198}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+49}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z + -0.5 \cdot \frac{t \cdot a}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.32e-198)
   (* y (- x))
   (if (<= z 1.25e+49)
     (* y (/ (* z x) (+ z (* -0.5 (/ (* t a) z)))))
     (* y x))))

C

assert(x < y);
assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.32e-198) {
		tmp = y * -x;
	} else if (z <= 1.25e+49) {
		tmp = y * ((z * x) / (z + (-0.5 * ((t * a) / z))));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.32d-198)) then
        tmp = y * -x
    else if (z <= 1.25d+49) then
        tmp = y * ((z * x) / (z + ((-0.5d0) * ((t * a) / z))))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.32e-198) {
		tmp = y * -x;
	} else if (z <= 1.25e+49) {
		tmp = y * ((z * x) / (z + (-0.5 * ((t * a) / z))));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.32e-198:
		tmp = y * -x
	elif z <= 1.25e+49:
		tmp = y * ((z * x) / (z + (-0.5 * ((t * a) / z))))
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.32e-198)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.25e+49)
		tmp = Float64(y * Float64(Float64(z * x) / Float64(z + Float64(-0.5 * Float64(Float64(t * a) / z)))));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.32e-198)
		tmp = y * -x;
	elseif (z <= 1.25e+49)
		tmp = y * ((z * x) / (z + (-0.5 * ((t * a) / z))));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.32e-198], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.25e+49], N[(y * N[(N[(z * x), $MachinePrecision] / N[(z + N[(-0.5 * N[(N[(t * a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.3200000000000001e-198

   1. Initial program 55.6%

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

      1. *-commutative 55.6%

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

      2. associate-*l* 56.1%

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

      3. associate-*r/ 60.1%

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

   3. Simplified 60.1%

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

   4. Taylor expanded in z around -inf 83.1%

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

      1. neg-mul-1 83.1%

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

   6. Simplified 83.1%

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

   if -1.3200000000000001e-198 < z < 1.2500000000000001e49

   1. Initial program 74.8%

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

      1. *-commutative 74.8%

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

      2. associate-*l* 74.9%

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

      3. associate-*r/ 79.0%

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

   3. Simplified 79.0%

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

   4. Taylor expanded in z around inf 54.7%

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

   if 1.2500000000000001e49 < z

   1. Initial program 40.4%

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

      1. *-commutative 40.4%

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

      2. associate-*l* 37.3%

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

      3. associate-*r/ 40.5%

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

   3. Simplified 40.5%

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

   4. Taylor expanded in z around inf 95.6%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.32 \cdot 10^{-198}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+49}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z + -0.5 \cdot \frac{t \cdot a}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 9: 76.8% accurate, 5.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{-198}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 6.3 \cdot 10^{+112}:\\ \;\;\;\;z \cdot \frac{y \cdot x}{z + -0.5 \cdot \frac{t \cdot a}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.85e-198)
   (* y (- x))
   (if (<= z 6.3e+112)
     (* z (/ (* y x) (+ z (* -0.5 (/ (* t a) z)))))
     (* y x))))

C

assert(x < y);
assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.85e-198) {
		tmp = y * -x;
	} else if (z <= 6.3e+112) {
		tmp = z * ((y * x) / (z + (-0.5 * ((t * a) / z))));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.85d-198)) then
        tmp = y * -x
    else if (z <= 6.3d+112) then
        tmp = z * ((y * x) / (z + ((-0.5d0) * ((t * a) / z))))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.85e-198) {
		tmp = y * -x;
	} else if (z <= 6.3e+112) {
		tmp = z * ((y * x) / (z + (-0.5 * ((t * a) / z))));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.85e-198:
		tmp = y * -x
	elif z <= 6.3e+112:
		tmp = z * ((y * x) / (z + (-0.5 * ((t * a) / z))))
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.85e-198)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 6.3e+112)
		tmp = Float64(z * Float64(Float64(y * x) / Float64(z + Float64(-0.5 * Float64(Float64(t * a) / z)))));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.85e-198)
		tmp = y * -x;
	elseif (z <= 6.3e+112)
		tmp = z * ((y * x) / (z + (-0.5 * ((t * a) / z))));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.85e-198], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 6.3e+112], N[(z * N[(N[(y * x), $MachinePrecision] / N[(z + N[(-0.5 * N[(N[(t * a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.84999999999999986e-198

   1. Initial program 55.6%

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

      1. *-commutative 55.6%

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

      2. associate-*l* 56.1%

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

      3. associate-*r/ 60.1%

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

   3. Simplified 60.1%

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

   4. Taylor expanded in z around -inf 83.1%

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

      1. neg-mul-1 83.1%

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

   6. Simplified 83.1%

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

   if -1.84999999999999986e-198 < z < 6.2999999999999997e112

   1. Initial program 76.8%

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

      1. associate-*l/ 79.0%

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

   3. Simplified 79.0%

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

   4. Taylor expanded in z around inf 59.3%

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

   if 6.2999999999999997e112 < z

   1. Initial program 30.0%

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

      1. *-commutative 30.0%

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

      2. associate-*l* 28.0%

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

      3. associate-*r/ 31.8%

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

   3. Simplified 31.8%

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

   4. Taylor expanded in z around inf 98.3%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{-198}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 6.3 \cdot 10^{+112}:\\ \;\;\;\;z \cdot \frac{y \cdot x}{z + -0.5 \cdot \frac{t \cdot a}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 10: 76.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 3.5 \cdot 10^{-138}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{0.5 \cdot \frac{t \cdot a}{z} - z}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z 3.5e-138) (* y (* x (/ z (- (* 0.5 (/ (* t a) z)) z)))) (* y x)))

C

assert(x < y);
assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= 3.5e-138) {
		tmp = y * (x * (z / ((0.5 * ((t * a) / z)) - z)));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= 3.5d-138) then
        tmp = y * (x * (z / ((0.5d0 * ((t * a) / z)) - z)))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= 3.5e-138) {
		tmp = y * (x * (z / ((0.5 * ((t * a) / z)) - z)));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= 3.5e-138:
		tmp = y * (x * (z / ((0.5 * ((t * a) / z)) - z)))
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= 3.5e-138)
		tmp = Float64(y * Float64(x * Float64(z / Float64(Float64(0.5 * Float64(Float64(t * a) / z)) - z))));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= 3.5e-138)
		tmp = y * (x * (z / ((0.5 * ((t * a) / z)) - z)));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, 3.5e-138], N[(y * N[(x * N[(z / N[(N[(0.5 * N[(N[(t * a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.4999999999999999e-138

   1. Initial program 59.7%

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

      1. *-commutative 59.7%

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

      2. associate-*l* 60.1%

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

      3. associate-*r/ 64.3%

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

   3. Simplified 64.3%

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

   4. Taylor expanded in z around -inf 62.0%

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

   5. Taylor expanded in x around 0 62.0%

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

      1. fma-def 62.0%

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

      2. *-commutative 62.0%

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

      3. neg-mul-1 62.0%

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

      4. associate-/l* 60.9%

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

      5. associate-/r/ 70.2%

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

      6. *-commutative 70.2%

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

      7. fma-udef 70.2%

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

      8. unsub-neg 70.2%

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

      9. *-commutative 70.2%

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

   7. Simplified 70.2%

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

   if 3.4999999999999999e-138 < z

   1. Initial program 54.3%

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

      1. *-commutative 54.3%

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

      2. associate-*l* 52.1%

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

      3. associate-*r/ 55.4%

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

   3. Simplified 55.4%

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

   4. Taylor expanded in z around inf 85.7%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.5 \cdot 10^{-138}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{0.5 \cdot \frac{t \cdot a}{z} - z}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 11: 76.9% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{-198}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.45e-198)
   (* y (- x))
   (/ (* y x) (+ 1.0 (* -0.5 (/ a (/ (* z z) t)))))))

C

assert(x < y);
assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.45e-198) {
		tmp = y * -x;
	} else {
		tmp = (y * x) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.45d-198)) then
        tmp = y * -x
    else
        tmp = (y * x) / (1.0d0 + ((-0.5d0) * (a / ((z * z) / t))))
    end if
    code = tmp
end function

Java

assert x < y;
assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.45e-198) {
		tmp = y * -x;
	} else {
		tmp = (y * x) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.45e-198:
		tmp = y * -x
	else:
		tmp = (y * x) / (1.0 + (-0.5 * (a / ((z * z) / t))))
	return tmp

Julia

x, y = sort([x, y])
t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.45e-198)
		tmp = Float64(y * Float64(-x));
	else
		tmp = Float64(Float64(y * x) / Float64(1.0 + Float64(-0.5 * Float64(a / Float64(Float64(z * z) / t)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.45e-198)
		tmp = y * -x;
	else
		tmp = (y * x) / (1.0 + (-0.5 * (a / ((z * z) / t))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.45e-198], N[(y * (-x)), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(a / N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.45e-198

   1. Initial program 55.6%

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

      1. *-commutative 55.6%

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

      2. associate-*l* 56.1%

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

      3. associate-*r/ 60.1%

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

   3. Simplified 60.1%

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

   4. Taylor expanded in z around -inf 83.1%

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

      1. neg-mul-1 83.1%

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

   6. Simplified 83.1%

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

   if -1.45e-198 < z

   1. Initial program 59.4%

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

      1. associate-/l* 63.5%

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

   3. Simplified 63.5%

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

   4. Taylor expanded in z around inf 66.4%

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

      1. unpow2 66.4%

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

      2. associate-/l* 72.8%

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

   6. Simplified 72.8%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{-198}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}\\ \end{array} \]

Alternative 12: 75.0% accurate, 10.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{-199}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 10^{-200}:\\ \;\;\;\;y \cdot \left(z \cdot \frac{z}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.85e-199)
   (* y (- x))
   (if (<= z 1e-200) (* y (* z (/ z x))) (* y x))))

C

assert(x < y);
assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.85e-199) {
		tmp = y * -x;
	} else if (z <= 1e-200) {
		tmp = y * (z * (z / x));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.85d-199)) then
        tmp = y * -x
    else if (z <= 1d-200) then
        tmp = y * (z * (z / x))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.85e-199) {
		tmp = y * -x;
	} else if (z <= 1e-200) {
		tmp = y * (z * (z / x));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.85e-199:
		tmp = y * -x
	elif z <= 1e-200:
		tmp = y * (z * (z / x))
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.85e-199)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1e-200)
		tmp = Float64(y * Float64(z * Float64(z / x)));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.85e-199)
		tmp = y * -x;
	elseif (z <= 1e-200)
		tmp = y * (z * (z / x));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.85e-199], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1e-200], N[(y * N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.85e-199

   1. Initial program 55.6%

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

      1. *-commutative 55.6%

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

      2. associate-*l* 56.1%

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

      3. associate-*r/ 60.1%

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

   3. Simplified 60.1%

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

   4. Taylor expanded in z around -inf 83.1%

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

      1. neg-mul-1 83.1%

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

   6. Simplified 83.1%

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

   if -1.85e-199 < z < 9.9999999999999998e-201

   1. Initial program 66.1%

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

      1. *-commutative 66.1%

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

      2. associate-*l* 66.4%

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

      3. associate-*r/ 72.1%

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

   3. Simplified 72.1%

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

   4. Taylor expanded in z around inf 32.8%

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

   6. Applied egg-rr 32.8%

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

   if 9.9999999999999998e-201 < z

   1. Initial program 56.8%

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

      1. *-commutative 56.8%

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

      2. associate-*l* 54.8%

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

      3. associate-*r/ 57.8%

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

   3. Simplified 57.8%

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

   4. Taylor expanded in z around inf 81.6%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{-199}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 10^{-200}:\\ \;\;\;\;y \cdot \left(z \cdot \frac{z}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 13: 75.8% accurate, 10.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{-200}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-143}:\\ \;\;\;\;y \cdot \left(z \cdot \left(z \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -7.2e-200)
   (* y (- x))
   (if (<= z 1.4e-143) (* y (* z (* z x))) (* y x))))

C

assert(x < y);
assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7.2e-200) {
		tmp = y * -x;
	} else if (z <= 1.4e-143) {
		tmp = y * (z * (z * x));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-7.2d-200)) then
        tmp = y * -x
    else if (z <= 1.4d-143) then
        tmp = y * (z * (z * x))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7.2e-200) {
		tmp = y * -x;
	} else if (z <= 1.4e-143) {
		tmp = y * (z * (z * x));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -7.2e-200:
		tmp = y * -x
	elif z <= 1.4e-143:
		tmp = y * (z * (z * x))
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -7.2e-200)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.4e-143)
		tmp = Float64(y * Float64(z * Float64(z * x)));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -7.2e-200)
		tmp = y * -x;
	elseif (z <= 1.4e-143)
		tmp = y * (z * (z * x));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -7.2e-200], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.4e-143], N[(y * N[(z * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.2000000000000003e-200

   1. Initial program 55.6%

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

      1. *-commutative 55.6%

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

      2. associate-*l* 56.1%

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

      3. associate-*r/ 60.1%

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

   3. Simplified 60.1%

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

   4. Taylor expanded in z around -inf 83.1%

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

      1. neg-mul-1 83.1%

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

   6. Simplified 83.1%

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

   if -7.2000000000000003e-200 < z < 1.3999999999999999e-143

   1. Initial program 69.0%

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

      1. *-commutative 69.0%

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

      2. associate-*l* 69.1%

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

      3. associate-*r/ 73.7%

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

   3. Simplified 73.7%

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

   4. Taylor expanded in z around inf 36.7%

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

   6. Applied egg-rr 42.6%

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

   if 1.3999999999999999e-143 < z

   1. Initial program 54.3%

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

      1. *-commutative 54.3%

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

      2. associate-*l* 52.1%

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

      3. associate-*r/ 55.4%

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

   3. Simplified 55.4%

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

   4. Taylor expanded in z around inf 85.7%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{-200}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-143}:\\ \;\;\;\;y \cdot \left(z \cdot \left(z \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 14: 76.2% accurate, 10.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{-198}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-140}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.85e-198)
   (* y (- x))
   (if (<= z 2.6e-140) (* z (* z (* y x))) (* y x))))

C

assert(x < y);
assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.85e-198) {
		tmp = y * -x;
	} else if (z <= 2.6e-140) {
		tmp = z * (z * (y * x));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.85d-198)) then
        tmp = y * -x
    else if (z <= 2.6d-140) then
        tmp = z * (z * (y * x))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.85e-198) {
		tmp = y * -x;
	} else if (z <= 2.6e-140) {
		tmp = z * (z * (y * x));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.85e-198:
		tmp = y * -x
	elif z <= 2.6e-140:
		tmp = z * (z * (y * x))
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.85e-198)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 2.6e-140)
		tmp = Float64(z * Float64(z * Float64(y * x)));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.85e-198)
		tmp = y * -x;
	elseif (z <= 2.6e-140)
		tmp = z * (z * (y * x));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.85e-198], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 2.6e-140], N[(z * N[(z * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.84999999999999986e-198

   1. Initial program 55.6%

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

      1. *-commutative 55.6%

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

      2. associate-*l* 56.1%

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

      3. associate-*r/ 60.1%

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

   3. Simplified 60.1%

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

   4. Taylor expanded in z around -inf 83.1%

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

      1. neg-mul-1 83.1%

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

   6. Simplified 83.1%

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

   if -1.84999999999999986e-198 < z < 2.5999999999999998e-140

   1. Initial program 69.0%

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

   2. Taylor expanded in z around -inf 32.1%

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

      1. neg-mul-1 32.1%

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

   4. Simplified 32.1%

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

   6. Applied egg-rr 44.6%

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

   if 2.5999999999999998e-140 < z

   1. Initial program 54.3%

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

      1. *-commutative 54.3%

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

      2. associate-*l* 52.1%

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

      3. associate-*r/ 55.4%

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

   3. Simplified 55.4%

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

   4. Taylor expanded in z around inf 85.7%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{-198}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-140}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 15: 72.7% accurate, 18.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{-307}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -8.5e-307) (* y (- x)) (* y x)))

C

assert(x < y);
assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.5e-307) {
		tmp = y * -x;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-8.5d-307)) then
        tmp = y * -x
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.5e-307) {
		tmp = y * -x;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -8.5e-307:
		tmp = y * -x
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -8.5e-307)
		tmp = Float64(y * Float64(-x));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -8.5e-307)
		tmp = y * -x;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -8.5e-307], N[(y * (-x)), $MachinePrecision], N[(y * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.4999999999999995e-307

   1. Initial program 57.1%

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

      1. *-commutative 57.1%

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

      2. associate-*l* 56.8%

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

      3. associate-*r/ 61.4%

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

   3. Simplified 61.4%

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

   4. Taylor expanded in z around -inf 76.1%

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

      1. neg-mul-1 76.1%

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

   6. Simplified 76.1%

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

   if -8.4999999999999995e-307 < z

   1. Initial program 58.4%

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

      1. *-commutative 58.4%

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

      2. associate-*l* 57.5%

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

      3. associate-*r/ 60.6%

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

   3. Simplified 60.6%

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

   4. Taylor expanded in z around inf 68.8%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{-307}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 16: 42.8% accurate, 37.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.7%

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

   1. *-commutative 57.7%

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

   2. associate-*l* 57.2%

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

   3. associate-*r/ 61.0%

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

3. Simplified 61.0%

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

4. Taylor expanded in z around inf 41.3%

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

5. Final simplification 41.3%

   \[\leadsto y \cdot x \]

Developer target: 89.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023171 
(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)))))