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

Percentage Accurate: 61.3% → 90.4%

Time: 16.3s

Alternatives: 15

Speedup: 18.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+41}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+45}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.2e+41)
   (* x (- y))
   (if (<= z 1.4e+45) (* x (/ (* z y) (sqrt (- (* z z) (* t a))))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.2e+41) {
		tmp = x * -y;
	} else if (z <= 1.4e+45) {
		tmp = x * ((z * y) / sqrt(((z * z) - (t * 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 <= (-3.2d+41)) then
        tmp = x * -y
    else if (z <= 1.4d+45) then
        tmp = x * ((z * y) / sqrt(((z * z) - (t * 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 <= -3.2e+41) {
		tmp = x * -y;
	} else if (z <= 1.4e+45) {
		tmp = x * ((z * y) / Math.sqrt(((z * z) - (t * a))));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.2e+41:
		tmp = x * -y
	elif z <= 1.4e+45:
		tmp = x * ((z * y) / math.sqrt(((z * z) - (t * 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 <= -3.2e+41)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 1.4e+45)
		tmp = Float64(x * Float64(Float64(z * y) / sqrt(Float64(Float64(z * z) - Float64(t * 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 <= -3.2e+41)
		tmp = x * -y;
	elseif (z <= 1.4e+45)
		tmp = x * ((z * y) / sqrt(((z * z) - (t * 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, -3.2e+41], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 1.4e+45], N[(x * N[(N[(z * y), $MachinePrecision] / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.2000000000000001e41

   1. Initial program 36.5%

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

      1. associate-*l* 31.6%

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

      2. associate-*r/ 31.9%

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

      3. *-commutative 31.9%

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

      4. associate-/l* 31.2%

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

   3. Simplified 31.2%

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

   4. Taylor expanded in z around -inf 93.0%

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

      1. neg-mul-1 93.0%

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

   6. Simplified 93.0%

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

   if -3.2000000000000001e41 < z < 1.4e45

   1. Initial program 86.4%

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

      1. associate-*l* 86.0%

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

      2. associate-*r/ 87.9%

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

   3. Simplified 87.9%

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

   if 1.4e45 < z

   1. Initial program 33.7%

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

      1. associate-*l* 30.4%

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

      2. associate-*r/ 32.2%

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

      3. *-commutative 32.2%

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

      4. associate-/l* 36.0%

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

   3. Simplified 36.0%

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

   4. Taylor expanded in z around inf 94.1%

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

      1. *-commutative 94.1%

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

   6. Simplified 94.1%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+41}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+45}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 2: 90.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+130}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+86}:\\ \;\;\;\;x \cdot \frac{z}{\frac{\sqrt{z \cdot z - t \cdot a}}{y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

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 -6.5e+130)
   (* x (- y))
   (if (<= z 1.5e+86) (* x (/ z (/ (sqrt (- (* z z) (* t a))) y))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6.5e+130) {
		tmp = x * -y;
	} else if (z <= 1.5e+86) {
		tmp = x * (z / (sqrt(((z * z) - (t * a))) / 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 <= (-6.5d+130)) then
        tmp = x * -y
    else if (z <= 1.5d+86) then
        tmp = x * (z / (sqrt(((z * z) - (t * a))) / 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 <= -6.5e+130) {
		tmp = x * -y;
	} else if (z <= 1.5e+86) {
		tmp = x * (z / (Math.sqrt(((z * z) - (t * a))) / y));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -6.5e+130:
		tmp = x * -y
	elif z <= 1.5e+86:
		tmp = x * (z / (math.sqrt(((z * z) - (t * a))) / 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 <= -6.5e+130)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 1.5e+86)
		tmp = Float64(x * Float64(z / Float64(sqrt(Float64(Float64(z * z) - Float64(t * a))) / 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 <= -6.5e+130)
		tmp = x * -y;
	elseif (z <= 1.5e+86)
		tmp = x * (z / (sqrt(((z * z) - (t * a))) / 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, -6.5e+130], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 1.5e+86], N[(x * N[(z / N[(N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.5e130

   1. Initial program 17.1%

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

      1. associate-*l* 14.4%

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

      2. associate-*r/ 14.7%

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

      3. *-commutative 14.7%

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

      4. associate-/l* 15.7%

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

   3. Simplified 15.7%

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

   4. Taylor expanded in z around -inf 98.1%

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

      1. neg-mul-1 98.1%

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

   6. Simplified 98.1%

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

   if -6.5e130 < z < 1.49999999999999988e86

   1. Initial program 86.1%

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

      1. associate-*l* 83.9%

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

      2. associate-*r/ 85.5%

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

      3. *-commutative 85.5%

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

      4. associate-/l* 87.6%

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

   3. Simplified 87.6%

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

   if 1.49999999999999988e86 < z

   1. Initial program 27.1%

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

      1. associate-*l* 25.1%

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

      2. associate-*r/ 27.2%

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

      3. *-commutative 27.2%

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

      4. associate-/l* 29.6%

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

   3. Simplified 29.6%

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

   4. Taylor expanded in z around inf 96.7%

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

      1. *-commutative 96.7%

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

   6. Simplified 96.7%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+130}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+86}:\\ \;\;\;\;x \cdot \frac{z}{\frac{\sqrt{z \cdot z - t \cdot a}}{y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 3: 83.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-119}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-66}:\\ \;\;\;\;x \cdot \frac{z}{\frac{\sqrt{t \cdot \left(-a\right)}}{y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

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-119)
   (* x (- y))
   (if (<= z 3e-66) (* x (/ z (/ (sqrt (* t (- a))) y))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.5e-119) {
		tmp = x * -y;
	} else if (z <= 3e-66) {
		tmp = x * (z / (sqrt((t * -a)) / 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 <= (-1.5d-119)) then
        tmp = x * -y
    else if (z <= 3d-66) then
        tmp = x * (z / (sqrt((t * -a)) / 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 <= -1.5e-119) {
		tmp = x * -y;
	} else if (z <= 3e-66) {
		tmp = x * (z / (Math.sqrt((t * -a)) / y));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.5e-119:
		tmp = x * -y
	elif z <= 3e-66:
		tmp = x * (z / (math.sqrt((t * -a)) / 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 <= -1.5e-119)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 3e-66)
		tmp = Float64(x * Float64(z / Float64(sqrt(Float64(t * Float64(-a))) / 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 <= -1.5e-119)
		tmp = x * -y;
	elseif (z <= 3e-66)
		tmp = x * (z / (sqrt((t * -a)) / 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, -1.5e-119], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 3e-66], N[(x * N[(z / N[(N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.5000000000000001e-119

   1. Initial program 56.0%

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

      1. associate-*l* 52.4%

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

      2. associate-*r/ 53.6%

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

      3. *-commutative 53.6%

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

      4. associate-/l* 52.6%

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

   3. Simplified 52.6%

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

   4. Taylor expanded in z around -inf 85.6%

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

      1. neg-mul-1 85.6%

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

   6. Simplified 85.6%

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

   if -1.5000000000000001e-119 < z < 3.0000000000000002e-66

   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* 81.9%

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

      2. associate-*r/ 82.6%

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

      3. *-commutative 82.6%

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

      4. associate-/l* 87.0%

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

   3. Simplified 87.0%

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

   4. Taylor expanded in z around 0 79.9%

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

      1. mul-1-neg 79.9%

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

      2. distribute-rgt-neg-out 79.9%

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

   6. Simplified 79.9%

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

   if 3.0000000000000002e-66 < z

   1. Initial program 52.2%

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

      1. associate-*l* 48.7%

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

      2. associate-*r/ 51.1%

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

      3. *-commutative 51.1%

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

      4. associate-/l* 54.8%

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

   3. Simplified 54.8%

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

   4. Taylor expanded in z around inf 88.8%

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

      1. *-commutative 88.8%

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

   6. Simplified 88.8%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-119}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-66}:\\ \;\;\;\;x \cdot \frac{z}{\frac{\sqrt{t \cdot \left(-a\right)}}{y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 4: 83.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-154}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-66}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

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-154)
   (* x (- y))
   (if (<= z 7.8e-66) (* y (/ (* z x) (sqrt (* t (- 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-154) {
		tmp = x * -y;
	} else if (z <= 7.8e-66) {
		tmp = y * ((z * x) / sqrt((t * -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-154)) then
        tmp = x * -y
    else if (z <= 7.8d-66) then
        tmp = y * ((z * x) / sqrt((t * -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-154) {
		tmp = x * -y;
	} else if (z <= 7.8e-66) {
		tmp = y * ((z * x) / Math.sqrt((t * -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-154:
		tmp = x * -y
	elif z <= 7.8e-66:
		tmp = y * ((z * x) / math.sqrt((t * -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-154)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 7.8e-66)
		tmp = Float64(y * Float64(Float64(z * x) / sqrt(Float64(t * Float64(-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-154)
		tmp = x * -y;
	elseif (z <= 7.8e-66)
		tmp = y * ((z * x) / sqrt((t * -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-154], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 7.8e-66], N[(y * N[(N[(z * x), $MachinePrecision] / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.79999999999999974e-154

   1. Initial program 57.6%

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

      1. associate-*l* 53.4%

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

      2. associate-*r/ 54.5%

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

      3. *-commutative 54.5%

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

      4. associate-/l* 55.2%

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

   3. Simplified 55.2%

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

   4. Taylor expanded in z around -inf 83.8%

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

      1. neg-mul-1 83.8%

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

   6. Simplified 83.8%

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

   if -4.79999999999999974e-154 < z < 7.79999999999999965e-66

   1. Initial program 80.0%

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

      1. *-commutative 80.0%

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

      2. associate-/l* 79.9%

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

   3. Simplified 79.9%

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

      1. associate-/r/ 79.7%

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

      2. *-commutative 79.7%

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

      3. associate-*r* 80.4%

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

      4. associate-/r/ 85.7%

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

      5. *-commutative 85.7%

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

      6. associate-*r/ 86.6%

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

      7. associate-/r/ 86.7%

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

      8. *-commutative 86.7%

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

   5. Applied egg-rr 86.7%

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

   6. Taylor expanded in z around 0 83.3%

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

      1. mul-1-neg 82.3%

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

      2. distribute-rgt-neg-out 82.3%

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

   8. Simplified 83.3%

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

   if 7.79999999999999965e-66 < z

   1. Initial program 52.2%

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

      1. associate-*l* 48.7%

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

      2. associate-*r/ 51.1%

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

      3. *-commutative 51.1%

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

      4. associate-/l* 54.8%

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

   3. Simplified 54.8%

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

   4. Taylor expanded in z around inf 88.8%

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

      1. *-commutative 88.8%

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

   6. Simplified 88.8%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-154}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-66}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 5: 77.8% accurate, 5.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{-231}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-73}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{z + -0.5 \cdot \frac{t \cdot a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.5e-231)
   (* x (- y))
   (if (<= z 3.2e-73) (* x (/ (* z y) (+ z (* -0.5 (/ (* t a) z))))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.5e-231) {
		tmp = x * -y;
	} else if (z <= 3.2e-73) {
		tmp = x * ((z * y) / (z + (-0.5 * ((t * a) / z))));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-2.5d-231)) then
        tmp = x * -y
    else if (z <= 3.2d-73) then
        tmp = x * ((z * y) / (z + ((-0.5d0) * ((t * a) / z))))
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.5e-231) {
		tmp = x * -y;
	} else if (z <= 3.2e-73) {
		tmp = x * ((z * y) / (z + (-0.5 * ((t * a) / z))));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.5e-231:
		tmp = x * -y
	elif z <= 3.2e-73:
		tmp = x * ((z * y) / (z + (-0.5 * ((t * a) / z))))
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.5e-231)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 3.2e-73)
		tmp = Float64(x * Float64(Float64(z * y) / Float64(z + Float64(-0.5 * Float64(Float64(t * a) / z)))));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.5e-231)
		tmp = x * -y;
	elseif (z <= 3.2e-73)
		tmp = x * ((z * y) / (z + (-0.5 * ((t * a) / z))));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.5e-231], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 3.2e-73], N[(x * N[(N[(z * y), $MachinePrecision] / N[(z + N[(-0.5 * N[(N[(t * a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.50000000000000012e-231

   1. Initial program 60.5%

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

      1. associate-*l* 56.7%

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

      2. associate-*r/ 57.7%

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

      3. *-commutative 57.7%

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

      4. associate-/l* 58.4%

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

   3. Simplified 58.4%

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

   4. Taylor expanded in z around -inf 77.7%

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

      1. neg-mul-1 77.7%

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

   6. Simplified 77.7%

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

   if -2.50000000000000012e-231 < z < 3.19999999999999986e-73

   1. Initial program 79.0%

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

      1. associate-*l* 83.7%

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

      2. associate-*r/ 84.7%

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

   3. Simplified 84.7%

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

   4. Taylor expanded in z around inf 61.5%

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

   if 3.19999999999999986e-73 < z

   1. Initial program 53.2%

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

      1. associate-*l* 49.8%

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

      2. associate-*r/ 52.2%

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

      3. *-commutative 52.2%

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

      4. associate-/l* 55.6%

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

   3. Simplified 55.6%

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

   4. Taylor expanded in z around inf 86.3%

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

      1. *-commutative 86.3%

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

   6. Simplified 86.3%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{-231}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-73}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{z + -0.5 \cdot \frac{t \cdot a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 6: 77.7% accurate, 5.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-25}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{-98}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\frac{a \cdot 0.5}{\frac{z}{t}} - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

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 -6.5e-25)
   (* x (- y))
   (if (<= z 8.2e-98) (* x (/ (* z y) (- (/ (* a 0.5) (/ z t)) z))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6.5e-25) {
		tmp = x * -y;
	} else if (z <= 8.2e-98) {
		tmp = x * ((z * y) / (((a * 0.5) / (z / 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 <= (-6.5d-25)) then
        tmp = x * -y
    else if (z <= 8.2d-98) then
        tmp = x * ((z * y) / (((a * 0.5d0) / (z / 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 <= -6.5e-25) {
		tmp = x * -y;
	} else if (z <= 8.2e-98) {
		tmp = x * ((z * y) / (((a * 0.5) / (z / 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 <= -6.5e-25:
		tmp = x * -y
	elif z <= 8.2e-98:
		tmp = x * ((z * y) / (((a * 0.5) / (z / 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 <= -6.5e-25)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 8.2e-98)
		tmp = Float64(x * Float64(Float64(z * y) / Float64(Float64(Float64(a * 0.5) / Float64(z / 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 <= -6.5e-25)
		tmp = x * -y;
	elseif (z <= 8.2e-98)
		tmp = x * ((z * y) / (((a * 0.5) / (z / 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, -6.5e-25], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 8.2e-98], N[(x * N[(N[(z * y), $MachinePrecision] / N[(N[(N[(a * 0.5), $MachinePrecision] / N[(z / t), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.5e-25

   1. Initial program 47.2%

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

      1. associate-*l* 43.4%

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

      2. associate-*r/ 44.8%

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

      3. *-commutative 44.8%

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

      4. associate-/l* 44.3%

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

   3. Simplified 44.3%

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

   4. Taylor expanded in z around -inf 86.9%

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

      1. neg-mul-1 86.9%

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

   6. Simplified 86.9%

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

   if -6.5e-25 < z < 8.1999999999999996e-98

   1. Initial program 84.8%

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

      1. associate-*l* 85.5%

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

      2. associate-*r/ 86.1%

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

   3. Simplified 86.1%

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

   4. Taylor expanded in z around -inf 59.1%

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

      1. neg-mul-1 59.1%

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

      2. +-commutative 59.1%

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

      3. unsub-neg 59.1%

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

      4. associate-/l* 59.1%

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

      5. associate-*r/ 59.1%

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

   6. Simplified 59.1%

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

   if 8.1999999999999996e-98 < z

   1. Initial program 53.6%

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

      1. associate-*l* 50.4%

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

      2. associate-*r/ 52.7%

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

      3. *-commutative 52.7%

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

      4. associate-/l* 56.0%

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

   3. Simplified 56.0%

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

   4. Taylor expanded in z around inf 85.5%

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

      1. *-commutative 85.5%

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

   6. Simplified 85.5%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-25}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{-98}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\frac{a \cdot 0.5}{\frac{z}{t}} - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 7: 78.3% accurate, 5.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{-229}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 10^{+35}:\\ \;\;\;\;\frac{z}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{x \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.7e-229)
   (* x (- y))
   (if (<= z 1e+35) (/ z (/ (+ z (* -0.5 (/ a (/ z t)))) (* x y))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.7e-229) {
		tmp = x * -y;
	} else if (z <= 1e+35) {
		tmp = z / ((z + (-0.5 * (a / (z / t)))) / (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 <= (-2.7d-229)) then
        tmp = x * -y
    else if (z <= 1d+35) then
        tmp = z / ((z + ((-0.5d0) * (a / (z / t)))) / (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 <= -2.7e-229) {
		tmp = x * -y;
	} else if (z <= 1e+35) {
		tmp = z / ((z + (-0.5 * (a / (z / t)))) / (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 <= -2.7e-229:
		tmp = x * -y
	elif z <= 1e+35:
		tmp = z / ((z + (-0.5 * (a / (z / t)))) / (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 <= -2.7e-229)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 1e+35)
		tmp = Float64(z / Float64(Float64(z + Float64(-0.5 * Float64(a / Float64(z / t)))) / Float64(x * 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 <= -2.7e-229)
		tmp = x * -y;
	elseif (z <= 1e+35)
		tmp = z / ((z + (-0.5 * (a / (z / t)))) / (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, -2.7e-229], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 1e+35], N[(z / N[(N[(z + N[(-0.5 * N[(a / N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.6999999999999998e-229

   1. Initial program 60.5%

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

      1. associate-*l* 56.7%

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

      2. associate-*r/ 57.7%

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

      3. *-commutative 57.7%

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

      4. associate-/l* 58.4%

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

   3. Simplified 58.4%

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

   4. Taylor expanded in z around -inf 77.7%

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

      1. neg-mul-1 77.7%

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

   6. Simplified 77.7%

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

   if -2.6999999999999998e-229 < z < 9.9999999999999997e34

   1. Initial program 83.3%

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

      1. *-commutative 83.3%

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

      2. associate-/l* 83.2%

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

   3. Simplified 83.2%

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

   4. Taylor expanded in z around inf 67.4%

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

      1. associate-/l* 67.6%

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

   6. Simplified 67.6%

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

   if 9.9999999999999997e34 < z

   1. Initial program 37.7%

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

      1. associate-*l* 34.6%

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

      2. associate-*r/ 36.3%

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

      3. *-commutative 36.3%

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

      4. associate-/l* 39.8%

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

   3. Simplified 39.8%

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

   4. Taylor expanded in z around inf 93.1%

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

      1. *-commutative 93.1%

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

   6. Simplified 93.1%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{-229}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 10^{+35}:\\ \;\;\;\;\frac{z}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{x \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 8: 76.1% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{-229}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-77}:\\ \;\;\;\;-2 \cdot \left(\frac{x}{a} \cdot \frac{y \cdot \left(z \cdot z\right)}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

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 -2.2e-229)
   (* x (- y))
   (if (<= z 5e-77) (* -2.0 (* (/ x a) (/ (* y (* 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 <= -2.2e-229) {
		tmp = x * -y;
	} else if (z <= 5e-77) {
		tmp = -2.0 * ((x / a) * ((y * (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 <= (-2.2d-229)) then
        tmp = x * -y
    else if (z <= 5d-77) then
        tmp = (-2.0d0) * ((x / a) * ((y * (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 <= -2.2e-229) {
		tmp = x * -y;
	} else if (z <= 5e-77) {
		tmp = -2.0 * ((x / a) * ((y * (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 <= -2.2e-229:
		tmp = x * -y
	elif z <= 5e-77:
		tmp = -2.0 * ((x / a) * ((y * (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 <= -2.2e-229)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 5e-77)
		tmp = Float64(-2.0 * Float64(Float64(x / a) * Float64(Float64(y * 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 <= -2.2e-229)
		tmp = x * -y;
	elseif (z <= 5e-77)
		tmp = -2.0 * ((x / a) * ((y * (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, -2.2e-229], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 5e-77], N[(-2.0 * N[(N[(x / a), $MachinePrecision] * N[(N[(y * N[(z * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.1999999999999999e-229

   1. Initial program 60.5%

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

      1. associate-*l* 56.7%

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

      2. associate-*r/ 57.7%

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

      3. *-commutative 57.7%

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

      4. associate-/l* 58.4%

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

   3. Simplified 58.4%

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

   4. Taylor expanded in z around -inf 77.7%

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

      1. neg-mul-1 77.7%

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

   6. Simplified 77.7%

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

   if -2.1999999999999999e-229 < z < 4.99999999999999963e-77

   1. Initial program 79.0%

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

      1. *-commutative 79.0%

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

      2. associate-/l* 78.8%

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

   3. Simplified 78.8%

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

   4. Taylor expanded in z around inf 63.4%

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

   5. Taylor expanded in z around 0 55.7%

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

   6. Taylor expanded in z around 0 55.6%

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

      1. times-frac 58.7%

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

      2. unpow2 58.7%

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

   8. Simplified 58.7%

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

   if 4.99999999999999963e-77 < z

   1. Initial program 53.2%

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

      1. associate-*l* 49.8%

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

      2. associate-*r/ 52.2%

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

      3. *-commutative 52.2%

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

      4. associate-/l* 55.6%

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

   3. Simplified 55.6%

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

   4. Taylor expanded in z around inf 86.3%

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

      1. *-commutative 86.3%

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

   6. Simplified 86.3%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{-229}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-77}:\\ \;\;\;\;-2 \cdot \left(\frac{x}{a} \cdot \frac{y \cdot \left(z \cdot z\right)}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 9: 76.6% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-233}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 1.14 \cdot 10^{-97}:\\ \;\;\;\;z \cdot \frac{-2}{\frac{\frac{a}{z} \cdot \frac{t}{y}}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.9e-233)
   (* x (- y))
   (if (<= z 1.14e-97) (* z (/ -2.0 (/ (* (/ a z) (/ t y)) x))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.9e-233) {
		tmp = x * -y;
	} else if (z <= 1.14e-97) {
		tmp = z * (-2.0 / (((a / z) * (t / y)) / x));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.9d-233)) then
        tmp = x * -y
    else if (z <= 1.14d-97) then
        tmp = z * ((-2.0d0) / (((a / z) * (t / y)) / x))
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.9e-233) {
		tmp = x * -y;
	} else if (z <= 1.14e-97) {
		tmp = z * (-2.0 / (((a / z) * (t / y)) / x));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.9e-233:
		tmp = x * -y
	elif z <= 1.14e-97:
		tmp = z * (-2.0 / (((a / z) * (t / y)) / x))
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.9e-233)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 1.14e-97)
		tmp = Float64(z * Float64(-2.0 / Float64(Float64(Float64(a / z) * Float64(t / y)) / x)));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.9e-233)
		tmp = x * -y;
	elseif (z <= 1.14e-97)
		tmp = z * (-2.0 / (((a / z) * (t / y)) / x));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.9e-233], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 1.14e-97], N[(z * N[(-2.0 / N[(N[(N[(a / z), $MachinePrecision] * N[(t / y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.9e-233

   1. Initial program 60.5%

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

      1. associate-*l* 56.7%

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

      2. associate-*r/ 57.7%

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

      3. *-commutative 57.7%

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

      4. associate-/l* 58.4%

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

   3. Simplified 58.4%

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

   4. Taylor expanded in z around -inf 77.7%

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

      1. neg-mul-1 77.7%

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

   6. Simplified 77.7%

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

   if -1.9e-233 < z < 1.14e-97

   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%

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

      2. associate-/l* 80.6%

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

   3. Simplified 80.6%

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

   4. Taylor expanded in z around inf 64.8%

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

   5. Taylor expanded in z around 0 56.9%

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

      1. expm1-log1p-u 56.9%

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

      2. expm1-udef 56.9%

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

      3. *-commutative 56.9%

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

      4. times-frac 60.1%

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

      5. *-commutative 60.1%

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

   7. Applied egg-rr 60.1%

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

      1. expm1-def 60.1%

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

      2. expm1-log1p 60.3%

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

      3. *-rgt-identity 60.3%

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

      4. associate-*r/ 60.3%

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

      5. *-commutative 60.3%

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

      6. associate-/r* 60.3%

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

      7. metadata-eval 60.3%

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

      8. associate-*l/ 60.4%

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

      9. associate-*r/ 56.9%

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

      10. times-frac 60.7%

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

   9. Simplified 60.7%

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

   if 1.14e-97 < z

   1. Initial program 53.6%

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

      1. associate-*l* 50.4%

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

      2. associate-*r/ 52.7%

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

      3. *-commutative 52.7%

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

      4. associate-/l* 56.0%

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

   3. Simplified 56.0%

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

   4. Taylor expanded in z around inf 85.5%

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

      1. *-commutative 85.5%

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

   6. Simplified 85.5%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-233}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 1.14 \cdot 10^{-97}:\\ \;\;\;\;z \cdot \frac{-2}{\frac{\frac{a}{z} \cdot \frac{t}{y}}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 10: 76.8% accurate, 8.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{-238}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-57}:\\ \;\;\;\;\frac{1}{\frac{z}{y \cdot \left(z \cdot x\right)}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -8.2e-238)
   (* x (- y))
   (if (<= z 5e-57) (/ 1.0 (/ z (* y (* z x)))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.2e-238) {
		tmp = x * -y;
	} else if (z <= 5e-57) {
		tmp = 1.0 / (z / (y * (z * x)));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-8.2d-238)) then
        tmp = x * -y
    else if (z <= 5d-57) then
        tmp = 1.0d0 / (z / (y * (z * x)))
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.2e-238) {
		tmp = x * -y;
	} else if (z <= 5e-57) {
		tmp = 1.0 / (z / (y * (z * x)));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -8.2e-238:
		tmp = x * -y
	elif z <= 5e-57:
		tmp = 1.0 / (z / (y * (z * x)))
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -8.2e-238)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 5e-57)
		tmp = Float64(1.0 / Float64(z / Float64(y * Float64(z * x))));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -8.2e-238)
		tmp = x * -y;
	elseif (z <= 5e-57)
		tmp = 1.0 / (z / (y * (z * x)));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -8.2e-238], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 5e-57], N[(1.0 / N[(z / N[(y * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.2000000000000002e-238

   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. associate-*l* 57.0%

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

      2. associate-*r/ 58.1%

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

      3. *-commutative 58.1%

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

      4. associate-/l* 58.7%

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

   3. Simplified 58.7%

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

   4. Taylor expanded in z around -inf 77.1%

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

      1. neg-mul-1 77.1%

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

   6. Simplified 77.1%

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

   if -8.2000000000000002e-238 < z < 5.0000000000000002e-57

   1. Initial program 77.5%

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

      1. *-commutative 77.5%

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

      2. associate-/l* 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in z around -inf 17.9%

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

      1. associate-*r/ 17.9%

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

      2. neg-mul-1 17.9%

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

      3. associate-/r* 17.8%

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

   6. Simplified 17.8%

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

      1. clear-num 17.8%

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

      2. inv-pow 17.8%

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

      3. associate-/l/ 17.9%

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

      4. add-sqr-sqrt 0.7%

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

      5. sqrt-unprod 11.3%

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

      6. sqr-neg 11.3%

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

      7. sqrt-prod 24.8%

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

      8. add-sqr-sqrt 25.2%

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

   8. Applied egg-rr 25.2%

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

      1. unpow-1 25.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{z}{y \cdot x}}{z}}} \]

      2. associate-/l/ 46.2%

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

      3. associate-*r* 50.0%

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

      4. *-commutative 50.0%

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

      5. associate-*l* 48.2%

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

   10. Simplified 48.2%

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

   if 5.0000000000000002e-57 < z

   1. Initial program 51.6%

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

      1. associate-*l* 48.1%

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

      2. associate-*r/ 50.5%

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

      3. *-commutative 50.5%

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

      4. associate-/l* 54.3%

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

   3. Simplified 54.3%

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

   4. Taylor expanded in z around inf 90.5%

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

      1. *-commutative 90.5%

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

   6. Simplified 90.5%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{-238}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-57}:\\ \;\;\;\;\frac{1}{\frac{z}{y \cdot \left(z \cdot x\right)}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 11: 75.8% accurate, 10.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.65 \cdot 10^{-241}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-110}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.65e-241)
   (* x (- y))
   (if (<= z 2.6e-110) (* y (/ (* z x) z)) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.65e-241) {
		tmp = x * -y;
	} else if (z <= 2.6e-110) {
		tmp = y * ((z * x) / z);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-2.65d-241)) then
        tmp = x * -y
    else if (z <= 2.6d-110) then
        tmp = y * ((z * x) / z)
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.65e-241) {
		tmp = x * -y;
	} else if (z <= 2.6e-110) {
		tmp = y * ((z * x) / z);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.65e-241:
		tmp = x * -y
	elif z <= 2.6e-110:
		tmp = y * ((z * x) / z)
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.65e-241)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 2.6e-110)
		tmp = Float64(y * Float64(Float64(z * x) / z));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.65e-241)
		tmp = x * -y;
	elseif (z <= 2.6e-110)
		tmp = y * ((z * x) / z);
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.65e-241], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 2.6e-110], N[(y * N[(N[(z * x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.6499999999999999e-241

   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. associate-*l* 57.0%

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

      2. associate-*r/ 58.1%

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

      3. *-commutative 58.1%

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

      4. associate-/l* 58.7%

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

   3. Simplified 58.7%

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

   4. Taylor expanded in z around -inf 77.1%

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

      1. neg-mul-1 77.1%

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

   6. Simplified 77.1%

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

   if -2.6499999999999999e-241 < z < 2.5999999999999999e-110

   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. *-commutative 76.8%

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

      2. associate-/l* 79.1%

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

   3. Simplified 79.1%

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

      1. associate-/r/ 76.3%

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

      2. *-commutative 76.3%

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

      3. associate-*r* 78.9%

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

      4. associate-/r/ 85.8%

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

      5. *-commutative 85.8%

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

      6. associate-*r/ 84.6%

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

      7. associate-/r/ 84.6%

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

      8. *-commutative 84.6%

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

   5. Applied egg-rr 84.6%

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

   6. Taylor expanded in z around inf 44.5%

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

   if 2.5999999999999999e-110 < z

   1. Initial program 54.6%

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

      1. associate-*l* 51.4%

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

      2. associate-*r/ 53.6%

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

      3. *-commutative 53.6%

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

      4. associate-/l* 56.9%

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

   3. Simplified 56.9%

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

   4. Taylor expanded in z around inf 83.8%

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

      1. *-commutative 83.8%

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

   6. Simplified 83.8%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.65 \cdot 10^{-241}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-110}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 12: 75.3% accurate, 10.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{-286}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-140}:\\ \;\;\;\;\frac{x}{\frac{z}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

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 -4.1e-286)
   (* x (- y))
   (if (<= z 4e-140) (/ x (/ z (* z y))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.1e-286) {
		tmp = x * -y;
	} else if (z <= 4e-140) {
		tmp = x / (z / (z * 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 <= (-4.1d-286)) then
        tmp = x * -y
    else if (z <= 4d-140) then
        tmp = x / (z / (z * 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 <= -4.1e-286) {
		tmp = x * -y;
	} else if (z <= 4e-140) {
		tmp = x / (z / (z * y));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -4.1e-286:
		tmp = x * -y
	elif z <= 4e-140:
		tmp = x / (z / (z * 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 <= -4.1e-286)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 4e-140)
		tmp = Float64(x / Float64(z / Float64(z * 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 <= -4.1e-286)
		tmp = x * -y;
	elseif (z <= 4e-140)
		tmp = x / (z / (z * 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, -4.1e-286], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 4e-140], N[(x / N[(z / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.1e-286

   1. Initial program 59.9%

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

      1. associate-*l* 57.8%

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

      2. associate-*r/ 59.4%

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

      3. *-commutative 59.4%

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

      4. associate-/l* 60.0%

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

   3. Simplified 60.0%

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

   4. Taylor expanded in z around -inf 74.7%

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

      1. neg-mul-1 74.7%

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

   6. Simplified 74.7%

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

   if -4.1e-286 < z < 3.9999999999999999e-140

   1. Initial program 83.6%

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

      1. associate-*l* 83.5%

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

      2. associate-*r/ 82.3%

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

      3. *-commutative 82.3%

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

      4. associate-/l* 85.8%

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

   3. Simplified 85.8%

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

      1. clear-num 83.7%

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

      2. un-div-inv 83.8%

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

      3. associate-/l/ 80.3%

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

   5. Applied egg-rr 80.3%

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

   6. Taylor expanded in z around inf 34.3%

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

   if 3.9999999999999999e-140 < z

   1. Initial program 55.0%

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

      1. associate-*l* 51.9%

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

      2. associate-*r/ 54.1%

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

      3. *-commutative 54.1%

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

      4. associate-/l* 57.3%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in z around inf 82.4%

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

      1. *-commutative 82.4%

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

   6. Simplified 82.4%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{-286}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-140}:\\ \;\;\;\;\frac{x}{\frac{z}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 13: 76.4% accurate, 10.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-268}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-134}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2e-268)
   (* x (- y))
   (if (<= z 5e-134) (/ (* x (* z y)) z) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2e-268) {
		tmp = x * -y;
	} else if (z <= 5e-134) {
		tmp = (x * (z * y)) / z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-2d-268)) then
        tmp = x * -y
    else if (z <= 5d-134) then
        tmp = (x * (z * y)) / z
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2e-268) {
		tmp = x * -y;
	} else if (z <= 5e-134) {
		tmp = (x * (z * y)) / z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -2e-268:
		tmp = x * -y
	elif z <= 5e-134:
		tmp = (x * (z * y)) / z
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2e-268)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 5e-134)
		tmp = Float64(Float64(x * Float64(z * y)) / z);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2e-268)
		tmp = x * -y;
	elseif (z <= 5e-134)
		tmp = (x * (z * y)) / z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2e-268], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 5e-134], N[(N[(x * N[(z * y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.99999999999999992e-268

   1. Initial program 60.7%

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

      1. associate-*l* 57.7%

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

      2. associate-*r/ 58.7%

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

      3. *-commutative 58.7%

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

      4. associate-/l* 59.3%

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

   3. Simplified 59.3%

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

   4. Taylor expanded in z around -inf 75.9%

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

      1. neg-mul-1 75.9%

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

   6. Simplified 75.9%

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

   if -1.99999999999999992e-268 < z < 5.0000000000000003e-134

   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* 82.1%

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

   3. Simplified 82.1%

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

   4. Taylor expanded in z around inf 56.3%

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

   if 5.0000000000000003e-134 < z

   1. Initial program 55.0%

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

      1. associate-*l* 51.9%

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

      2. associate-*r/ 54.1%

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

      3. *-commutative 54.1%

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

      4. associate-/l* 57.3%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in z around inf 82.4%

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

      1. *-commutative 82.4%

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

   6. Simplified 82.4%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-268}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-134}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 14: 74.1% accurate, 18.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-301}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

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 -2.9e-301) (* x (- y)) (* x y)))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.9e-301) {
		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 <= (-2.9d-301)) 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 <= -2.9e-301) {
		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 <= -2.9e-301:
		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 <= -2.9e-301)
		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 <= -2.9e-301)
		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, -2.9e-301], N[(x * (-y)), $MachinePrecision], N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.89999999999999984e-301

   1. Initial program 60.4%

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

      1. associate-*l* 59.1%

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

      2. associate-*r/ 60.6%

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

      3. *-commutative 60.6%

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

      4. associate-/l* 61.2%

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

   3. Simplified 61.2%

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

   4. Taylor expanded in z around -inf 72.5%

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

      1. neg-mul-1 72.5%

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

   6. Simplified 72.5%

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

   if -2.89999999999999984e-301 < z

   1. Initial program 61.1%

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

      1. associate-*l* 57.9%

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

      2. associate-*r/ 59.4%

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

      3. *-commutative 59.4%

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

      4. associate-/l* 62.7%

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

   3. Simplified 62.7%

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

   4. Taylor expanded in z around inf 71.8%

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

      1. *-commutative 71.8%

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

   6. Simplified 71.8%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-301}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 15: 44.1% 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 60.8%

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

   1. associate-*l* 58.5%

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

   2. associate-*r/ 60.0%

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

   3. *-commutative 60.0%

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

   4. associate-/l* 61.9%

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

3. Simplified 61.9%

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

4. Taylor expanded in z around inf 39.8%

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

   1. *-commutative 39.8%

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

6. Simplified 39.8%

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

7. Final simplification 39.8%

   \[\leadsto x \cdot y \]

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 2023293 
(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)))))