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

Percentage Accurate: 61.2% → 91.1%

Time: 16.0s

Alternatives: 11

Speedup: 18.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := \frac{a}{z} \cdot \frac{t}{z}\\ \mathbf{if}\;z \leq -4.6 \cdot 10^{+95}:\\ \;\;\;\;\frac{x \cdot y}{\mathsf{fma}\left(0.5, t_1, -1\right)}\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-117}:\\ \;\;\;\;y \cdot \left(z \cdot \left(x \cdot {\left(z \cdot z - a \cdot t\right)}^{-0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\sqrt{1 - t_1}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ a z) (/ t z))))
   (if (<= z -4.6e+95)
     (/ (* x y) (fma 0.5 t_1 -1.0))
     (if (<= z 1.65e-117)
       (* y (* z (* x (pow (- (* z z) (* a t)) -0.5))))
       (* y (/ x (sqrt (- 1.0 t_1))))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (a / z) * (t / z);
	double tmp;
	if (z <= -4.6e+95) {
		tmp = (x * y) / fma(0.5, t_1, -1.0);
	} else if (z <= 1.65e-117) {
		tmp = y * (z * (x * pow(((z * z) - (a * t)), -0.5)));
	} else {
		tmp = y * (x / sqrt((1.0 - t_1)));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(a / z) * Float64(t / z))
	tmp = 0.0
	if (z <= -4.6e+95)
		tmp = Float64(Float64(x * y) / fma(0.5, t_1, -1.0));
	elseif (z <= 1.65e-117)
		tmp = Float64(y * Float64(z * Float64(x * (Float64(Float64(z * z) - Float64(a * t)) ^ -0.5))));
	else
		tmp = Float64(y * Float64(x / sqrt(Float64(1.0 - t_1))));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a / z), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.6e+95], N[(N[(x * y), $MachinePrecision] / N[(0.5 * t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.65e-117], N[(y * N[(z * N[(x * N[Power[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.59999999999999994e95

   1. Initial program 38.6%

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

      1. associate-/l* 40.5%

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

   3. Simplified 40.5%

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

   4. Taylor expanded in z around -inf 92.0%

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

      1. fma-neg 92.0%

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

      2. unpow2 92.0%

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

      3. times-frac 97.0%

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

      4. metadata-eval 97.0%

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

   6. Simplified 97.0%

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

   if -4.59999999999999994e95 < z < 1.65000000000000008e-117

   1. Initial program 80.3%

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

      1. *-commutative 80.3%

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

      2. associate-*l* 79.8%

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

      3. associate-*r/ 84.0%

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

   3. Simplified 84.0%

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

      1. div-inv 83.9%

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

      2. *-commutative 83.9%

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

      3. associate-*l* 86.0%

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

      4. pow1/2 86.0%

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

      5. pow-flip 86.0%

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

      6. metadata-eval 86.0%

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

   5. Applied egg-rr 86.0%

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

   if 1.65000000000000008e-117 < z

   1. Initial program 58.0%

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

      1. associate-/l* 62.0%

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

   3. Simplified 62.0%

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

      1. add-sqr-sqrt 62.0%

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

      2. sqrt-unprod 62.0%

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

      3. frac-times 56.5%

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

      4. add-sqr-sqrt 56.5%

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

   5. Applied egg-rr 56.5%

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

      1. div-sub 56.5%

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

      2. *-inverses 94.9%

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

      3. *-commutative 94.9%

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

      4. times-frac 98.9%

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

   7. Simplified 98.9%

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

      1. expm1-log1p-u 78.2%

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

      2. expm1-udef 38.1%

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

      3. associate-/l* 38.1%

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

      4. associate-*l/ 38.1%

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

   9. Applied egg-rr 38.1%

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

       1. expm1-def 78.1%

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

       2. expm1-log1p 98.7%

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

       3. associate-/r/ 98.9%

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

       4. *-commutative 98.9%

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

       5. associate-*r/ 98.9%

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

   11. Simplified 98.9%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+95}:\\ \;\;\;\;\frac{x \cdot y}{\mathsf{fma}\left(0.5, \frac{a}{z} \cdot \frac{t}{z}, -1\right)}\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-117}:\\ \;\;\;\;y \cdot \left(z \cdot \left(x \cdot {\left(z \cdot z - a \cdot t\right)}^{-0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\sqrt{1 - \frac{a}{z} \cdot \frac{t}{z}}}\\ \end{array} \]

Alternative 2: 90.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+89}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+21}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + -0.5 \cdot \left(\frac{a}{z} \cdot \frac{t}{z}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -6e+89)
   (* x (- y))
   (if (<= z 2.6e+21)
     (* y (/ (* z x) (sqrt (- (* z z) (* a t)))))
     (/ (* x y) (+ 1.0 (* -0.5 (* (/ a z) (/ t z))))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6e+89) {
		tmp = x * -y;
	} else if (z <= 2.6e+21) {
		tmp = y * ((z * x) / sqrt(((z * z) - (a * t))));
	} else {
		tmp = (x * y) / (1.0 + (-0.5 * ((a / z) * (t / z))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-6d+89)) then
        tmp = x * -y
    else if (z <= 2.6d+21) then
        tmp = y * ((z * x) / sqrt(((z * z) - (a * t))))
    else
        tmp = (x * y) / (1.0d0 + ((-0.5d0) * ((a / z) * (t / z))))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6e+89) {
		tmp = x * -y;
	} else if (z <= 2.6e+21) {
		tmp = y * ((z * x) / Math.sqrt(((z * z) - (a * t))));
	} else {
		tmp = (x * y) / (1.0 + (-0.5 * ((a / z) * (t / z))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -6e+89:
		tmp = x * -y
	elif z <= 2.6e+21:
		tmp = y * ((z * x) / math.sqrt(((z * z) - (a * t))))
	else:
		tmp = (x * y) / (1.0 + (-0.5 * ((a / z) * (t / z))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -6e+89)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 2.6e+21)
		tmp = Float64(y * Float64(Float64(z * x) / sqrt(Float64(Float64(z * z) - Float64(a * t)))));
	else
		tmp = Float64(Float64(x * y) / Float64(1.0 + Float64(-0.5 * Float64(Float64(a / z) * Float64(t / z)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -6e+89)
		tmp = x * -y;
	elseif (z <= 2.6e+21)
		tmp = y * ((z * x) / sqrt(((z * z) - (a * t))));
	else
		tmp = (x * y) / (1.0 + (-0.5 * ((a / z) * (t / z))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -6e+89], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 2.6e+21], N[(y * N[(N[(z * x), $MachinePrecision] / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(N[(a / z), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.00000000000000025e89

   1. Initial program 38.6%

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

      1. *-commutative 38.6%

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

      2. associate-*l* 35.2%

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

      3. associate-*r/ 35.5%

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

   3. Simplified 35.5%

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

   4. Taylor expanded in z around -inf 97.0%

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

      1. neg-mul-1 97.0%

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

   6. Simplified 97.0%

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

   if -6.00000000000000025e89 < z < 2.6e21

   1. Initial program 84.2%

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

      1. *-commutative 84.2%

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

      2. associate-*l* 82.8%

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

      3. associate-*r/ 86.0%

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

   3. Simplified 86.0%

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

   if 2.6e21 < z

   1. Initial program 41.2%

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

      1. associate-/l* 47.0%

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

   3. Simplified 47.0%

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

      1. clear-num 47.0%

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

      2. associate-/r/ 47.0%

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

   5. Applied egg-rr 47.0%

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

   6. Taylor expanded in z around inf 92.9%

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

      1. *-commutative 92.9%

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

      2. unpow2 92.9%

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

      3. times-frac 98.7%

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

   8. Simplified 98.7%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+89}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+21}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + -0.5 \cdot \left(\frac{a}{z} \cdot \frac{t}{z}\right)}\\ \end{array} \]

Alternative 3: 91.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -8.6 \cdot 10^{+87}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-117}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\sqrt{1 - \frac{a}{z} \cdot \frac{t}{z}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -8.6e+87)
   (* x (- y))
   (if (<= z 7.8e-117)
     (* y (/ (* z x) (sqrt (- (* z z) (* a t)))))
     (* y (/ x (sqrt (- 1.0 (* (/ a z) (/ t z)))))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.6e+87) {
		tmp = x * -y;
	} else if (z <= 7.8e-117) {
		tmp = y * ((z * x) / sqrt(((z * z) - (a * t))));
	} else {
		tmp = y * (x / sqrt((1.0 - ((a / z) * (t / z)))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-8.6d+87)) then
        tmp = x * -y
    else if (z <= 7.8d-117) then
        tmp = y * ((z * x) / sqrt(((z * z) - (a * t))))
    else
        tmp = y * (x / sqrt((1.0d0 - ((a / z) * (t / z)))))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.6e+87) {
		tmp = x * -y;
	} else if (z <= 7.8e-117) {
		tmp = y * ((z * x) / Math.sqrt(((z * z) - (a * t))));
	} else {
		tmp = y * (x / Math.sqrt((1.0 - ((a / z) * (t / z)))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -8.6e+87:
		tmp = x * -y
	elif z <= 7.8e-117:
		tmp = y * ((z * x) / math.sqrt(((z * z) - (a * t))))
	else:
		tmp = y * (x / math.sqrt((1.0 - ((a / z) * (t / z)))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -8.6e+87)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 7.8e-117)
		tmp = Float64(y * Float64(Float64(z * x) / sqrt(Float64(Float64(z * z) - Float64(a * t)))));
	else
		tmp = Float64(y * Float64(x / sqrt(Float64(1.0 - Float64(Float64(a / z) * Float64(t / z))))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -8.6e+87)
		tmp = x * -y;
	elseif (z <= 7.8e-117)
		tmp = y * ((z * x) / sqrt(((z * z) - (a * t))));
	else
		tmp = y * (x / sqrt((1.0 - ((a / z) * (t / z)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -8.6e+87], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 7.8e-117], N[(y * N[(N[(z * x), $MachinePrecision] / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / N[Sqrt[N[(1.0 - N[(N[(a / z), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.6000000000000002e87

   1. Initial program 38.6%

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

      1. *-commutative 38.6%

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

      2. associate-*l* 35.2%

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

      3. associate-*r/ 35.5%

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

   3. Simplified 35.5%

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

   4. Taylor expanded in z around -inf 97.0%

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

      1. neg-mul-1 97.0%

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

   6. Simplified 97.0%

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

   if -8.6000000000000002e87 < z < 7.79999999999999984e-117

   1. Initial program 80.3%

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

      1. *-commutative 80.3%

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

      2. associate-*l* 79.8%

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

      3. associate-*r/ 84.0%

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

   3. Simplified 84.0%

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

   if 7.79999999999999984e-117 < z

   1. Initial program 58.0%

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

      1. associate-/l* 62.0%

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

   3. Simplified 62.0%

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

      1. add-sqr-sqrt 62.0%

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

      2. sqrt-unprod 62.0%

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

      3. frac-times 56.5%

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

      4. add-sqr-sqrt 56.5%

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

   5. Applied egg-rr 56.5%

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

      1. div-sub 56.5%

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

      2. *-inverses 94.9%

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

      3. *-commutative 94.9%

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

      4. times-frac 98.9%

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

   7. Simplified 98.9%

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

      1. expm1-log1p-u 78.2%

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

      2. expm1-udef 38.1%

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

      3. associate-/l* 38.1%

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

      4. associate-*l/ 38.1%

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

   9. Applied egg-rr 38.1%

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

       1. expm1-def 78.1%

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

       2. expm1-log1p 98.7%

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

       3. associate-/r/ 98.9%

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

       4. *-commutative 98.9%

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

       5. associate-*r/ 98.9%

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

   11. Simplified 98.9%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.6 \cdot 10^{+87}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-117}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\sqrt{1 - \frac{a}{z} \cdot \frac{t}{z}}}\\ \end{array} \]

Alternative 4: 91.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := \frac{a}{z} \cdot \frac{t}{z}\\ \mathbf{if}\;z \leq -2.5 \cdot 10^{+93}:\\ \;\;\;\;\frac{x \cdot y}{\mathsf{fma}\left(0.5, t_1, -1\right)}\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-116}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\sqrt{1 - t_1}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ a z) (/ t z))))
   (if (<= z -2.5e+93)
     (/ (* x y) (fma 0.5 t_1 -1.0))
     (if (<= z 4.4e-116)
       (* y (/ (* z x) (sqrt (- (* z z) (* a t)))))
       (* y (/ x (sqrt (- 1.0 t_1))))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (a / z) * (t / z);
	double tmp;
	if (z <= -2.5e+93) {
		tmp = (x * y) / fma(0.5, t_1, -1.0);
	} else if (z <= 4.4e-116) {
		tmp = y * ((z * x) / sqrt(((z * z) - (a * t))));
	} else {
		tmp = y * (x / sqrt((1.0 - t_1)));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(a / z) * Float64(t / z))
	tmp = 0.0
	if (z <= -2.5e+93)
		tmp = Float64(Float64(x * y) / fma(0.5, t_1, -1.0));
	elseif (z <= 4.4e-116)
		tmp = Float64(y * Float64(Float64(z * x) / sqrt(Float64(Float64(z * z) - Float64(a * t)))));
	else
		tmp = Float64(y * Float64(x / sqrt(Float64(1.0 - t_1))));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a / z), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.5e+93], N[(N[(x * y), $MachinePrecision] / N[(0.5 * t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.4e-116], N[(y * N[(N[(z * x), $MachinePrecision] / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.5000000000000001e93

   1. Initial program 38.6%

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

      1. associate-/l* 40.5%

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

   3. Simplified 40.5%

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

   4. Taylor expanded in z around -inf 92.0%

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

      1. fma-neg 92.0%

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

      2. unpow2 92.0%

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

      3. times-frac 97.0%

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

      4. metadata-eval 97.0%

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

   6. Simplified 97.0%

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

   if -2.5000000000000001e93 < z < 4.4000000000000002e-116

   1. Initial program 80.3%

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

      1. *-commutative 80.3%

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

      2. associate-*l* 79.8%

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

      3. associate-*r/ 84.0%

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

   3. Simplified 84.0%

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

   if 4.4000000000000002e-116 < z

   1. Initial program 58.0%

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

      1. associate-/l* 62.0%

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

   3. Simplified 62.0%

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

      1. add-sqr-sqrt 62.0%

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

      2. sqrt-unprod 62.0%

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

      3. frac-times 56.5%

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

      4. add-sqr-sqrt 56.5%

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

   5. Applied egg-rr 56.5%

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

      1. div-sub 56.5%

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

      2. *-inverses 94.9%

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

      3. *-commutative 94.9%

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

      4. times-frac 98.9%

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

   7. Simplified 98.9%

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

      1. expm1-log1p-u 78.2%

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

      2. expm1-udef 38.1%

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

      3. associate-/l* 38.1%

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

      4. associate-*l/ 38.1%

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

   9. Applied egg-rr 38.1%

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

       1. expm1-def 78.1%

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

       2. expm1-log1p 98.7%

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

       3. associate-/r/ 98.9%

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

       4. *-commutative 98.9%

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

       5. associate-*r/ 98.9%

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

   11. Simplified 98.9%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+93}:\\ \;\;\;\;\frac{x \cdot y}{\mathsf{fma}\left(0.5, \frac{a}{z} \cdot \frac{t}{z}, -1\right)}\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-116}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\sqrt{1 - \frac{a}{z} \cdot \frac{t}{z}}}\\ \end{array} \]

Alternative 5: 82.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -7.4:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{-152}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + -0.5 \cdot \left(\frac{a}{z} \cdot \frac{t}{z}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -7.4)
   (* x (- y))
   (if (<= z 2.35e-152)
     (* y (/ (* z x) (sqrt (* a (- t)))))
     (/ (* x y) (+ 1.0 (* -0.5 (* (/ a z) (/ t z))))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7.4) {
		tmp = x * -y;
	} else if (z <= 2.35e-152) {
		tmp = y * ((z * x) / sqrt((a * -t)));
	} else {
		tmp = (x * y) / (1.0 + (-0.5 * ((a / z) * (t / z))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-7.4d0)) then
        tmp = x * -y
    else if (z <= 2.35d-152) then
        tmp = y * ((z * x) / sqrt((a * -t)))
    else
        tmp = (x * y) / (1.0d0 + ((-0.5d0) * ((a / z) * (t / z))))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7.4) {
		tmp = x * -y;
	} else if (z <= 2.35e-152) {
		tmp = y * ((z * x) / Math.sqrt((a * -t)));
	} else {
		tmp = (x * y) / (1.0 + (-0.5 * ((a / z) * (t / z))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -7.4:
		tmp = x * -y
	elif z <= 2.35e-152:
		tmp = y * ((z * x) / math.sqrt((a * -t)))
	else:
		tmp = (x * y) / (1.0 + (-0.5 * ((a / z) * (t / z))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -7.4)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 2.35e-152)
		tmp = Float64(y * Float64(Float64(z * x) / sqrt(Float64(a * Float64(-t)))));
	else
		tmp = Float64(Float64(x * y) / Float64(1.0 + Float64(-0.5 * Float64(Float64(a / z) * Float64(t / z)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -7.4)
		tmp = x * -y;
	elseif (z <= 2.35e-152)
		tmp = y * ((z * x) / sqrt((a * -t)));
	else
		tmp = (x * y) / (1.0 + (-0.5 * ((a / z) * (t / z))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -7.4], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 2.35e-152], N[(y * N[(N[(z * x), $MachinePrecision] / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(N[(a / z), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.4000000000000004

   1. Initial program 54.1%

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

      1. *-commutative 54.1%

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

      2. associate-*l* 48.1%

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

      3. associate-*r/ 48.3%

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

   3. Simplified 48.3%

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

   4. Taylor expanded in z around -inf 95.5%

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

      1. neg-mul-1 95.5%

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

   6. Simplified 95.5%

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

   if -7.4000000000000004 < z < 2.35000000000000006e-152

   1. Initial program 76.4%

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

      1. *-commutative 76.4%

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

      2. associate-*l* 80.1%

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

      3. associate-*r/ 84.5%

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

   3. Simplified 84.5%

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

   4. Taylor expanded in z around 0 78.6%

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

      1. mul-1-neg 78.6%

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

      2. distribute-rgt-neg-out 78.6%

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

   6. Simplified 78.6%

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

   if 2.35000000000000006e-152 < z

   1. Initial program 58.1%

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

      1. associate-/l* 63.6%

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

   3. Simplified 63.6%

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

      1. clear-num 63.6%

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

      2. associate-/r/ 63.5%

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

   5. Applied egg-rr 63.5%

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

   6. Taylor expanded in z around inf 84.8%

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

      1. *-commutative 84.8%

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

      2. unpow2 84.8%

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

      3. times-frac 88.6%

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

   8. Simplified 88.6%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.4:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{-152}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + -0.5 \cdot \left(\frac{a}{z} \cdot \frac{t}{z}\right)}\\ \end{array} \]

Alternative 6: 77.2% accurate, 5.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-147}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+22}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z + -0.5 \cdot \frac{a \cdot t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.5e-147)
   (* x (- y))
   (if (<= z 1.1e+22) (* y (/ (* z x) (+ z (* -0.5 (/ (* a t) z))))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.5e-147) {
		tmp = x * -y;
	} else if (z <= 1.1e+22) {
		tmp = y * ((z * x) / (z + (-0.5 * ((a * t) / z))));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.5d-147)) then
        tmp = x * -y
    else if (z <= 1.1d+22) then
        tmp = y * ((z * x) / (z + ((-0.5d0) * ((a * t) / z))))
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.5e-147) {
		tmp = x * -y;
	} else if (z <= 1.1e+22) {
		tmp = y * ((z * x) / (z + (-0.5 * ((a * t) / z))));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.5e-147:
		tmp = x * -y
	elif z <= 1.1e+22:
		tmp = y * ((z * x) / (z + (-0.5 * ((a * t) / z))))
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.5e-147)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 1.1e+22)
		tmp = Float64(y * Float64(Float64(z * x) / Float64(z + Float64(-0.5 * Float64(Float64(a * t) / z)))));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.5e-147)
		tmp = x * -y;
	elseif (z <= 1.1e+22)
		tmp = y * ((z * x) / (z + (-0.5 * ((a * t) / z))));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.5e-147], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 1.1e+22], N[(y * N[(N[(z * x), $MachinePrecision] / N[(z + N[(-0.5 * N[(N[(a * t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.5000000000000001e-147

   1. Initial program 60.8%

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

      1. *-commutative 60.8%

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

      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/ 57.3%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in z around -inf 82.7%

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

      1. neg-mul-1 82.7%

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

   6. Simplified 82.7%

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

   if -1.5000000000000001e-147 < z < 1.1e22

   1. Initial program 79.9%

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

      1. *-commutative 79.9%

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

      2. associate-*l* 81.4%

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

      3. associate-*r/ 84.9%

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

   3. Simplified 84.9%

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

   4. Taylor expanded in z around inf 50.1%

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

   if 1.1e22 < z

   1. Initial program 41.2%

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

      1. *-commutative 41.2%

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

      2. associate-*l* 39.7%

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

      3. associate-*r/ 45.3%

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

   3. Simplified 45.3%

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

   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 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-147}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+22}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z + -0.5 \cdot \frac{a \cdot t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 7: 76.0% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{-147}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-152}:\\ \;\;\;\;-2 \cdot \left(\frac{y}{a} \cdot \frac{x \cdot \left(z \cdot z\right)}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -6.2e-147)
   (* x (- y))
   (if (<= z 1.3e-152) (* -2.0 (* (/ y a) (/ (* x (* z z)) t))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6.2e-147) {
		tmp = x * -y;
	} else if (z <= 1.3e-152) {
		tmp = -2.0 * ((y / a) * ((x * (z * z)) / t));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-6.2d-147)) then
        tmp = x * -y
    else if (z <= 1.3d-152) then
        tmp = (-2.0d0) * ((y / a) * ((x * (z * z)) / t))
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6.2e-147) {
		tmp = x * -y;
	} else if (z <= 1.3e-152) {
		tmp = -2.0 * ((y / a) * ((x * (z * z)) / t));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -6.2e-147:
		tmp = x * -y
	elif z <= 1.3e-152:
		tmp = -2.0 * ((y / a) * ((x * (z * z)) / t))
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -6.2e-147)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 1.3e-152)
		tmp = Float64(-2.0 * Float64(Float64(y / a) * Float64(Float64(x * Float64(z * z)) / t)));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -6.2e-147)
		tmp = x * -y;
	elseif (z <= 1.3e-152)
		tmp = -2.0 * ((y / a) * ((x * (z * z)) / t));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -6.2e-147], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 1.3e-152], N[(-2.0 * N[(N[(y / a), $MachinePrecision] * N[(N[(x * N[(z * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.2000000000000005e-147

   1. Initial program 60.8%

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

      1. *-commutative 60.8%

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

      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/ 57.3%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in z around -inf 82.7%

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

      1. neg-mul-1 82.7%

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

   6. Simplified 82.7%

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

   if -6.2000000000000005e-147 < z < 1.30000000000000006e-152

   1. Initial program 71.8%

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

      1. associate-/l* 73.3%

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

   3. Simplified 73.3%

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

      1. clear-num 73.2%

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

      2. associate-/r/ 73.2%

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

   5. Applied egg-rr 73.2%

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

   6. Taylor expanded in z around inf 32.8%

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

      1. *-commutative 32.8%

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

      2. unpow2 32.8%

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

      3. times-frac 35.5%

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

   8. Simplified 35.5%

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

   9. Taylor expanded in t around inf 33.1%

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

       1. times-frac 33.1%

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

       2. unpow2 33.1%

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

   11. Simplified 33.1%

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

   if 1.30000000000000006e-152 < z

   1. Initial program 58.1%

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

      1. *-commutative 58.1%

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

      2. associate-*l* 55.9%

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

      3. associate-*r/ 60.4%

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

   3. Simplified 60.4%

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

   4. Taylor expanded in z around inf 84.3%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{-147}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-152}:\\ \;\;\;\;-2 \cdot \left(\frac{y}{a} \cdot \frac{x \cdot \left(z \cdot z\right)}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 8: 76.1% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-145}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-152}:\\ \;\;\;\;z \cdot \left(-2 \cdot \left(\frac{z}{\frac{t}{y}} \cdot \frac{x}{a}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4.8e-145)
   (* x (- y))
   (if (<= z 2.2e-152) (* z (* -2.0 (* (/ z (/ t y)) (/ x a)))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.8e-145) {
		tmp = x * -y;
	} else if (z <= 2.2e-152) {
		tmp = z * (-2.0 * ((z / (t / y)) * (x / a)));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-4.8d-145)) then
        tmp = x * -y
    else if (z <= 2.2d-152) then
        tmp = z * ((-2.0d0) * ((z / (t / y)) * (x / a)))
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.8e-145) {
		tmp = x * -y;
	} else if (z <= 2.2e-152) {
		tmp = z * (-2.0 * ((z / (t / y)) * (x / a)));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -4.8e-145:
		tmp = x * -y
	elif z <= 2.2e-152:
		tmp = z * (-2.0 * ((z / (t / y)) * (x / a)))
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4.8e-145)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 2.2e-152)
		tmp = Float64(z * Float64(-2.0 * Float64(Float64(z / Float64(t / y)) * Float64(x / a))));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -4.8e-145)
		tmp = x * -y;
	elseif (z <= 2.2e-152)
		tmp = z * (-2.0 * ((z / (t / y)) * (x / a)));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4.8e-145], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 2.2e-152], N[(z * N[(-2.0 * N[(N[(z / N[(t / y), $MachinePrecision]), $MachinePrecision] * N[(x / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.8000000000000003e-145

   1. Initial program 60.8%

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

      1. *-commutative 60.8%

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

      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/ 57.3%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in z around -inf 82.7%

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

      1. neg-mul-1 82.7%

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

   6. Simplified 82.7%

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

   if -4.8000000000000003e-145 < z < 2.19999999999999985e-152

   1. Initial program 71.8%

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

      1. associate-*l/ 75.4%

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

   3. Simplified 75.4%

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

   4. Taylor expanded in z around inf 35.9%

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

   5. Taylor expanded in z around 0 35.5%

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

      1. associate-*r* 35.6%

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

      2. *-commutative 35.6%

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

      3. times-frac 33.6%

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

      4. *-commutative 33.6%

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

      5. associate-/l* 33.5%

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

   7. Simplified 33.5%

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

   if 2.19999999999999985e-152 < z

   1. Initial program 58.1%

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

      1. *-commutative 58.1%

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

      2. associate-*l* 55.9%

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

      3. associate-*r/ 60.4%

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

   3. Simplified 60.4%

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

   4. Taylor expanded in z around inf 84.3%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-145}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-152}:\\ \;\;\;\;z \cdot \left(-2 \cdot \left(\frac{z}{\frac{t}{y}} \cdot \frac{x}{a}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 9: 77.8% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-146}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + -0.5 \cdot \left(\frac{a}{z} \cdot \frac{t}{z}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.55e-146)
   (* x (- y))
   (/ (* x y) (+ 1.0 (* -0.5 (* (/ a z) (/ t z)))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.55e-146) {
		tmp = x * -y;
	} else {
		tmp = (x * y) / (1.0 + (-0.5 * ((a / z) * (t / z))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.55d-146)) then
        tmp = x * -y
    else
        tmp = (x * y) / (1.0d0 + ((-0.5d0) * ((a / z) * (t / z))))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.55e-146) {
		tmp = x * -y;
	} else {
		tmp = (x * y) / (1.0 + (-0.5 * ((a / z) * (t / z))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.55e-146:
		tmp = x * -y
	else:
		tmp = (x * y) / (1.0 + (-0.5 * ((a / z) * (t / z))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.55e-146)
		tmp = Float64(x * Float64(-y));
	else
		tmp = Float64(Float64(x * y) / Float64(1.0 + Float64(-0.5 * Float64(Float64(a / z) * Float64(t / z)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.55e-146)
		tmp = x * -y;
	else
		tmp = (x * y) / (1.0 + (-0.5 * ((a / z) * (t / z))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.55e-146], N[(x * (-y)), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(N[(a / z), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.5499999999999999e-146

   1. Initial program 60.8%

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

      1. *-commutative 60.8%

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

      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/ 57.3%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in z around -inf 82.7%

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

      1. neg-mul-1 82.7%

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

   6. Simplified 82.7%

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

   if -1.5499999999999999e-146 < z

   1. Initial program 62.1%

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

      1. associate-/l* 66.4%

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

   3. Simplified 66.4%

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

      1. clear-num 66.4%

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

      2. associate-/r/ 66.4%

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

   5. Applied egg-rr 66.4%

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

   6. Taylor expanded in z around inf 69.6%

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

      1. *-commutative 69.6%

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

      2. unpow2 69.6%

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

      3. times-frac 73.0%

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

   8. Simplified 73.0%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-146}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + -0.5 \cdot \left(\frac{a}{z} \cdot \frac{t}{z}\right)}\\ \end{array} \]

Alternative 10: 73.3% accurate, 18.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{-304}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.75e-304) (* x (- y)) (* x y)))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.75e-304) {
		tmp = x * -y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.75d-304)) then
        tmp = x * -y
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.75e-304) {
		tmp = x * -y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.75e-304:
		tmp = x * -y
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.75e-304)
		tmp = Float64(x * Float64(-y));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.75e-304)
		tmp = x * -y;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.75e-304], N[(x * (-y)), $MachinePrecision], N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.75e-304

   1. Initial program 63.9%

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

      1. *-commutative 63.9%

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

      2. associate-*l* 59.7%

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

      3. associate-*r/ 60.8%

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

   3. Simplified 60.8%

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

   4. Taylor expanded in z around -inf 70.7%

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

      1. neg-mul-1 70.7%

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

   6. Simplified 70.7%

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

   if -1.75e-304 < z

   1. Initial program 59.3%

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

      1. *-commutative 59.3%

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

      2. associate-*l* 59.7%

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

      3. associate-*r/ 64.9%

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

   3. Simplified 64.9%

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

   4. Taylor expanded in z around inf 73.4%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{-304}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 11: 43.0% accurate, 37.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.6%

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

   1. *-commutative 61.6%

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

   2. associate-*l* 59.7%

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

   3. associate-*r/ 62.8%

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

3. Simplified 62.8%

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

4. Taylor expanded in z around inf 45.9%

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

5. Final simplification 45.9%

   \[\leadsto x \cdot y \]

Developer target: 89.7% 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 2023242 
(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)))))