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

Percentage Accurate: 61.2% → 90.7%

Time: 17.2s

Alternatives: 18

Speedup: 18.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := \sqrt{z \cdot z - a \cdot t}\\ \mathbf{if}\;z \leq -3 \cdot 10^{+150}:\\ \;\;\;\;\frac{x \cdot y}{\mathsf{fma}\left(0.5, \frac{a}{\frac{z \cdot z}{t}}, -1\right)}\\ \mathbf{elif}\;z \leq -4.2 \cdot 10^{-208}:\\ \;\;\;\;\frac{x \cdot y}{\frac{t_1}{z}}\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+51}:\\ \;\;\;\;\frac{x}{\frac{t_1}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (sqrt (- (* z z) (* a t)))))
   (if (<= z -3e+150)
     (/ (* x y) (fma 0.5 (/ a (/ (* z z) t)) -1.0))
     (if (<= z -4.2e-208)
       (/ (* x y) (/ t_1 z))
       (if (<= z 5.2e+51) (/ x (/ t_1 (* z y))) (* x y))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = sqrt(((z * z) - (a * t)));
	double tmp;
	if (z <= -3e+150) {
		tmp = (x * y) / fma(0.5, (a / ((z * z) / t)), -1.0);
	} else if (z <= -4.2e-208) {
		tmp = (x * y) / (t_1 / z);
	} else if (z <= 5.2e+51) {
		tmp = x / (t_1 / (z * y));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = sqrt(Float64(Float64(z * z) - Float64(a * t)))
	tmp = 0.0
	if (z <= -3e+150)
		tmp = Float64(Float64(x * y) / fma(0.5, Float64(a / Float64(Float64(z * z) / t)), -1.0));
	elseif (z <= -4.2e-208)
		tmp = Float64(Float64(x * y) / Float64(t_1 / z));
	elseif (z <= 5.2e+51)
		tmp = Float64(x / Float64(t_1 / Float64(z * y)));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -3e+150], N[(N[(x * y), $MachinePrecision] / N[(0.5 * N[(a / N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -4.2e-208], N[(N[(x * y), $MachinePrecision] / N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.2e+51], N[(x / N[(t$95$1 / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -3.00000000000000012e150

   1. Initial program 16.7%

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

      1. associate-/l* 17.3%

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

   3. Simplified 17.3%

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

   4. Taylor expanded in z around -inf 85.1%

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

      1. fma-neg 85.1%

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

      2. unpow2 85.1%

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

      3. associate-/l* 97.6%

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

      4. metadata-eval 97.6%

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

   6. Simplified 97.6%

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

   if -3.00000000000000012e150 < z < -4.20000000000000024e-208

   1. Initial program 84.4%

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

      1. associate-/l* 91.2%

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

   3. Simplified 91.2%

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

   if -4.20000000000000024e-208 < z < 5.2000000000000002e51

   1. Initial program 81.9%

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

      1. *-commutative 81.9%

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

      2. associate-*l* 84.3%

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

      3. associate-*r/ 85.4%

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

   3. Simplified 85.4%

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

      1. *-commutative 85.4%

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

      2. associate-/l* 85.3%

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

      3. associate-/r/ 81.4%

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

      4. associate-/l/ 84.5%

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

      5. *-commutative 84.5%

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

   5. Applied egg-rr 84.5%

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

   if 5.2000000000000002e51 < z

   1. Initial program 53.9%

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

      1. *-commutative 53.9%

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

      2. associate-*l* 52.2%

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

      3. associate-*r/ 54.9%

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

   3. Simplified 54.9%

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

   4. Taylor expanded in z around inf 96.3%

      \[\leadsto y \cdot \color{blue}{x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+150}:\\ \;\;\;\;\frac{x \cdot y}{\mathsf{fma}\left(0.5, \frac{a}{\frac{z \cdot z}{t}}, -1\right)}\\ \mathbf{elif}\;z \leq -4.2 \cdot 10^{-208}:\\ \;\;\;\;\frac{x \cdot y}{\frac{\sqrt{z \cdot z - a \cdot t}}{z}}\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+51}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{z \cdot z - a \cdot t}}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 2: 90.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := \sqrt{z \cdot z - a \cdot t}\\ \mathbf{if}\;z \leq -2 \cdot 10^{+151}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq -3.4 \cdot 10^{-206}:\\ \;\;\;\;\frac{x \cdot y}{\frac{t_1}{z}}\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+51}:\\ \;\;\;\;\frac{x}{\frac{t_1}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (sqrt (- (* z z) (* a t)))))
   (if (<= z -2e+151)
     (* x (- y))
     (if (<= z -3.4e-206)
       (/ (* x y) (/ t_1 z))
       (if (<= z 5.2e+51) (/ x (/ t_1 (* z y))) (* x y))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = sqrt(((z * z) - (a * t)));
	double tmp;
	if (z <= -2e+151) {
		tmp = x * -y;
	} else if (z <= -3.4e-206) {
		tmp = (x * y) / (t_1 / z);
	} else if (z <= 5.2e+51) {
		tmp = x / (t_1 / (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) :: t_1
    real(8) :: tmp
    t_1 = sqrt(((z * z) - (a * t)))
    if (z <= (-2d+151)) then
        tmp = x * -y
    else if (z <= (-3.4d-206)) then
        tmp = (x * y) / (t_1 / z)
    else if (z <= 5.2d+51) then
        tmp = x / (t_1 / (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 t_1 = Math.sqrt(((z * z) - (a * t)));
	double tmp;
	if (z <= -2e+151) {
		tmp = x * -y;
	} else if (z <= -3.4e-206) {
		tmp = (x * y) / (t_1 / z);
	} else if (z <= 5.2e+51) {
		tmp = x / (t_1 / (z * y));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	t_1 = math.sqrt(((z * z) - (a * t)))
	tmp = 0
	if z <= -2e+151:
		tmp = x * -y
	elif z <= -3.4e-206:
		tmp = (x * y) / (t_1 / z)
	elif z <= 5.2e+51:
		tmp = x / (t_1 / (z * y))
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = sqrt(Float64(Float64(z * z) - Float64(a * t)))
	tmp = 0.0
	if (z <= -2e+151)
		tmp = Float64(x * Float64(-y));
	elseif (z <= -3.4e-206)
		tmp = Float64(Float64(x * y) / Float64(t_1 / z));
	elseif (z <= 5.2e+51)
		tmp = Float64(x / Float64(t_1 / 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)
	t_1 = sqrt(((z * z) - (a * t)));
	tmp = 0.0;
	if (z <= -2e+151)
		tmp = x * -y;
	elseif (z <= -3.4e-206)
		tmp = (x * y) / (t_1 / z);
	elseif (z <= 5.2e+51)
		tmp = x / (t_1 / (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_] := Block[{t$95$1 = N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -2e+151], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, -3.4e-206], N[(N[(x * y), $MachinePrecision] / N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.2e+51], N[(x / N[(t$95$1 / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -2.00000000000000003e151

   1. Initial program 16.7%

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

      1. *-commutative 16.7%

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

      2. associate-*l* 16.2%

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

      3. associate-*r/ 16.5%

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

   3. Simplified 16.5%

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

   4. Taylor expanded in z around -inf 97.6%

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

      1. neg-mul-1 97.6%

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

   6. Simplified 97.6%

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

   if -2.00000000000000003e151 < z < -3.39999999999999985e-206

   1. Initial program 84.4%

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

      1. associate-/l* 91.2%

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

   3. Simplified 91.2%

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

   if -3.39999999999999985e-206 < z < 5.2000000000000002e51

   1. Initial program 81.9%

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

      1. *-commutative 81.9%

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

      2. associate-*l* 84.3%

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

      3. associate-*r/ 85.4%

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

   3. Simplified 85.4%

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

      1. *-commutative 85.4%

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

      2. associate-/l* 85.3%

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

      3. associate-/r/ 81.4%

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

      4. associate-/l/ 84.5%

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

      5. *-commutative 84.5%

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

   5. Applied egg-rr 84.5%

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

   if 5.2000000000000002e51 < z

   1. Initial program 53.9%

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

      1. *-commutative 53.9%

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

      2. associate-*l* 52.2%

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

      3. associate-*r/ 54.9%

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

   3. Simplified 54.9%

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

   4. Taylor expanded in z around inf 96.3%

      \[\leadsto y \cdot \color{blue}{x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+151}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq -3.4 \cdot 10^{-206}:\\ \;\;\;\;\frac{x \cdot y}{\frac{\sqrt{z \cdot z - a \cdot t}}{z}}\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+51}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{z \cdot z - a \cdot t}}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 3: 89.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+128}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+52}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

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 -1e+128)
   (* x (- y))
   (if (<= z 1.45e+52) (* y (/ (* z x) (sqrt (- (* z z) (* a t))))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1e+128) {
		tmp = x * -y;
	} else if (z <= 1.45e+52) {
		tmp = y * ((z * x) / sqrt(((z * z) - (a * 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 <= (-1d+128)) then
        tmp = x * -y
    else if (z <= 1.45d+52) then
        tmp = y * ((z * x) / sqrt(((z * z) - (a * 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 <= -1e+128) {
		tmp = x * -y;
	} else if (z <= 1.45e+52) {
		tmp = y * ((z * x) / Math.sqrt(((z * z) - (a * t))));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1e+128:
		tmp = x * -y
	elif z <= 1.45e+52:
		tmp = y * ((z * x) / math.sqrt(((z * z) - (a * 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 <= -1e+128)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 1.45e+52)
		tmp = Float64(y * Float64(Float64(z * x) / sqrt(Float64(Float64(z * z) - Float64(a * 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 <= -1e+128)
		tmp = x * -y;
	elseif (z <= 1.45e+52)
		tmp = y * ((z * x) / sqrt(((z * z) - (a * 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, -1e+128], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 1.45e+52], N[(y * N[(N[(z * x), $MachinePrecision] / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.0000000000000001e128

   1. Initial program 20.4%

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

      1. *-commutative 20.4%

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

      2. associate-*l* 19.8%

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

      3. associate-*r/ 20.1%

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

   3. Simplified 20.1%

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

   4. Taylor expanded in z around -inf 97.8%

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

      1. neg-mul-1 97.8%

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

   6. Simplified 97.8%

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

   if -1.0000000000000001e128 < z < 1.45e52

   1. Initial program 83.4%

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

      1. *-commutative 83.4%

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

      2. associate-*l* 82.1%

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

      3. associate-*r/ 83.1%

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

   3. Simplified 83.1%

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

   if 1.45e52 < z

   1. Initial program 53.9%

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

      1. *-commutative 53.9%

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

      2. associate-*l* 52.2%

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

      3. associate-*r/ 54.9%

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

   3. Simplified 54.9%

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

   4. Taylor expanded in z around inf 96.3%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+128}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+52}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 4: 90.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.85 \cdot 10^{+150}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 10^{+44}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{\sqrt{z \cdot z - a \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

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 -2.85e+150)
   (* x (- y))
   (if (<= z 1e+44) (* y (* x (/ z (sqrt (- (* z z) (* a t)))))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.85e+150) {
		tmp = x * -y;
	} else if (z <= 1e+44) {
		tmp = y * (x * (z / sqrt(((z * z) - (a * 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 <= (-2.85d+150)) then
        tmp = x * -y
    else if (z <= 1d+44) then
        tmp = y * (x * (z / sqrt(((z * z) - (a * 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 <= -2.85e+150) {
		tmp = x * -y;
	} else if (z <= 1e+44) {
		tmp = y * (x * (z / Math.sqrt(((z * z) - (a * t)))));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.85e+150:
		tmp = x * -y
	elif z <= 1e+44:
		tmp = y * (x * (z / math.sqrt(((z * z) - (a * 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 <= -2.85e+150)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 1e+44)
		tmp = Float64(y * Float64(x * Float64(z / sqrt(Float64(Float64(z * z) - Float64(a * 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 <= -2.85e+150)
		tmp = x * -y;
	elseif (z <= 1e+44)
		tmp = y * (x * (z / sqrt(((z * z) - (a * 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, -2.85e+150], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 1e+44], N[(y * N[(x * N[(z / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.8500000000000001e150

   1. Initial program 16.7%

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

      1. *-commutative 16.7%

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

      2. associate-*l* 16.2%

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

      3. associate-*r/ 16.5%

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

   3. Simplified 16.5%

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

   4. Taylor expanded in z around -inf 97.6%

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

      1. neg-mul-1 97.6%

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

   6. Simplified 97.6%

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

   if -2.8500000000000001e150 < z < 1.0000000000000001e44

   1. Initial program 83.1%

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

      1. associate-/l* 85.4%

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

   3. Simplified 85.4%

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

      1. associate-*l/ 84.9%

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

      2. associate-/l* 82.8%

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

      3. div-inv 82.8%

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

      4. associate-*l* 85.0%

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

      5. div-inv 85.1%

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

   5. Applied egg-rr 85.1%

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

   if 1.0000000000000001e44 < z

   1. Initial program 53.9%

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

      1. *-commutative 53.9%

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

      2. associate-*l* 52.2%

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

      3. associate-*r/ 54.9%

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

   3. Simplified 54.9%

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

   4. Taylor expanded in z around inf 96.3%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.85 \cdot 10^{+150}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 10^{+44}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{\sqrt{z \cdot z - a \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 5: 89.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{+64}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+51}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{z \cdot z - a \cdot t}}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

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+64)
   (* x (- y))
   (if (<= z 9.5e+51) (/ x (/ (sqrt (- (* z z) (* a t))) (* 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+64) {
		tmp = x * -y;
	} else if (z <= 9.5e+51) {
		tmp = x / (sqrt(((z * z) - (a * t))) / (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+64)) then
        tmp = x * -y
    else if (z <= 9.5d+51) then
        tmp = x / (sqrt(((z * z) - (a * t))) / (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+64) {
		tmp = x * -y;
	} else if (z <= 9.5e+51) {
		tmp = x / (Math.sqrt(((z * z) - (a * t))) / (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+64:
		tmp = x * -y
	elif z <= 9.5e+51:
		tmp = x / (math.sqrt(((z * z) - (a * t))) / (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+64)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 9.5e+51)
		tmp = Float64(x / Float64(sqrt(Float64(Float64(z * z) - Float64(a * t))) / 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+64)
		tmp = x * -y;
	elseif (z <= 9.5e+51)
		tmp = x / (sqrt(((z * z) - (a * t))) / (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+64], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 9.5e+51], N[(x / N[(N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.09999999999999978e64

   1. Initial program 37.8%

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

      1. *-commutative 37.8%

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

      2. associate-*l* 35.6%

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

      3. associate-*r/ 39.1%

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

   3. Simplified 39.1%

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

   4. Taylor expanded in z around -inf 98.4%

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

      1. neg-mul-1 98.4%

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

   6. Simplified 98.4%

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

   if -4.09999999999999978e64 < z < 9.4999999999999999e51

   1. Initial program 82.8%

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

      1. *-commutative 82.8%

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

      2. associate-*l* 82.3%

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

      3. associate-*r/ 81.9%

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

   3. Simplified 81.9%

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

      1. *-commutative 81.9%

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

      2. associate-/l* 82.6%

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

      3. associate-/r/ 82.5%

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

      4. associate-/l/ 84.2%

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

      5. *-commutative 84.2%

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

   5. Applied egg-rr 84.2%

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

   if 9.4999999999999999e51 < z

   1. Initial program 53.9%

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

      1. *-commutative 53.9%

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

      2. associate-*l* 52.2%

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

      3. associate-*r/ 54.9%

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

   3. Simplified 54.9%

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

   4. Taylor expanded in z around inf 96.3%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{+64}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+51}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{z \cdot z - a \cdot t}}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 6: 83.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{-53}:\\ \;\;\;\;\frac{x \cdot y}{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right) - z}{z}}\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{-79}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

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 -1.35e-53)
   (/ (* x y) (/ (- (* 0.5 (* a (/ t z))) z) z))
   (if (<= z 2.05e-79) (* y (/ (* z x) (sqrt (* a (- t))))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.35e-53) {
		tmp = (x * y) / (((0.5 * (a * (t / z))) - z) / z);
	} else if (z <= 2.05e-79) {
		tmp = y * ((z * x) / sqrt((a * -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 <= (-1.35d-53)) then
        tmp = (x * y) / (((0.5d0 * (a * (t / z))) - z) / z)
    else if (z <= 2.05d-79) then
        tmp = y * ((z * x) / sqrt((a * -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 <= -1.35e-53) {
		tmp = (x * y) / (((0.5 * (a * (t / z))) - z) / z);
	} else if (z <= 2.05e-79) {
		tmp = y * ((z * x) / Math.sqrt((a * -t)));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.35e-53:
		tmp = (x * y) / (((0.5 * (a * (t / z))) - z) / z)
	elif z <= 2.05e-79:
		tmp = y * ((z * x) / math.sqrt((a * -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 <= -1.35e-53)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(0.5 * Float64(a * Float64(t / z))) - z) / z));
	elseif (z <= 2.05e-79)
		tmp = Float64(y * Float64(Float64(z * x) / sqrt(Float64(a * Float64(-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 <= -1.35e-53)
		tmp = (x * y) / (((0.5 * (a * (t / z))) - z) / z);
	elseif (z <= 2.05e-79)
		tmp = y * ((z * x) / sqrt((a * -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, -1.35e-53], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(0.5 * N[(a * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.05e-79], N[(y * N[(N[(z * x), $MachinePrecision] / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.35e-53

   1. Initial program 54.7%

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

      1. associate-/l* 58.4%

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

   3. Simplified 58.4%

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

   4. Taylor expanded in z around -inf 88.3%

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

      1. expm1-log1p-u 87.1%

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

      2. expm1-udef 87.1%

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

      3. *-commutative 87.1%

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

   6. Applied egg-rr 87.1%

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

      1. expm1-def 87.1%

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

      2. expm1-log1p 88.3%

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

      3. *-commutative 88.3%

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

      4. associate-*r/ 93.1%

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

   8. Simplified 93.1%

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

   if -1.35e-53 < z < 2.04999999999999997e-79

   1. Initial program 75.5%

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

      1. *-commutative 75.5%

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

      2. associate-*l* 75.8%

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

      3. associate-*r/ 77.5%

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

   3. Simplified 77.5%

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

   4. Taylor expanded in z around 0 74.1%

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

      1. mul-1-neg 74.1%

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

      2. *-commutative 74.1%

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

      3. distribute-rgt-neg-in 74.1%

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

   6. Simplified 74.1%

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

   if 2.04999999999999997e-79 < z

   1. Initial program 62.8%

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

      1. *-commutative 62.8%

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

      2. associate-*l* 60.6%

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

      3. associate-*r/ 62.7%

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

   3. Simplified 62.7%

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

   4. Taylor expanded in z around inf 92.3%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{-53}:\\ \;\;\;\;\frac{x \cdot y}{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right) - z}{z}}\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{-79}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 7: 76.2% accurate, 5.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{-105}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-140}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \frac{y}{\frac{a \cdot \left(t \cdot -0.5\right)}{z} - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

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 -8.2e-105)
   (* x (- y))
   (if (<= z 1.2e-140)
     (* (* z x) (/ y (- (/ (* a (* t -0.5)) z) z)))
     (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.2e-105) {
		tmp = x * -y;
	} else if (z <= 1.2e-140) {
		tmp = (z * x) * (y / (((a * (t * -0.5)) / z) - 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 <= (-8.2d-105)) then
        tmp = x * -y
    else if (z <= 1.2d-140) then
        tmp = (z * x) * (y / (((a * (t * (-0.5d0))) / z) - 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 <= -8.2e-105) {
		tmp = x * -y;
	} else if (z <= 1.2e-140) {
		tmp = (z * x) * (y / (((a * (t * -0.5)) / z) - z));
	} 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-105:
		tmp = x * -y
	elif z <= 1.2e-140:
		tmp = (z * x) * (y / (((a * (t * -0.5)) / z) - 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 <= -8.2e-105)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 1.2e-140)
		tmp = Float64(Float64(z * x) * Float64(y / Float64(Float64(Float64(a * Float64(t * -0.5)) / z) - 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 <= -8.2e-105)
		tmp = x * -y;
	elseif (z <= 1.2e-140)
		tmp = (z * x) * (y / (((a * (t * -0.5)) / z) - 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, -8.2e-105], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 1.2e-140], N[(N[(z * x), $MachinePrecision] * N[(y / N[(N[(N[(a * N[(t * -0.5), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.20000000000000061e-105

   1. Initial program 56.5%

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

      1. *-commutative 56.5%

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

      2. associate-*l* 54.1%

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

      3. associate-*r/ 55.4%

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

   3. Simplified 55.4%

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

   4. Taylor expanded in z around -inf 86.2%

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

      1. neg-mul-1 86.2%

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

   6. Simplified 86.2%

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

   if -8.20000000000000061e-105 < z < 1.19999999999999993e-140

   1. Initial program 72.7%

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

      1. associate-/l* 70.2%

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

   3. Simplified 70.2%

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

   4. Taylor expanded in z around -inf 44.7%

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

      1. div-inv 44.7%

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

      2. *-commutative 44.7%

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

      3. frac-2neg 44.7%

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

      4. metadata-eval 44.7%

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

      5. add-sqr-sqrt 22.2%

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

      6. sqrt-unprod 41.2%

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

      7. sqr-neg 41.2%

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

      8. sqrt-prod 23.0%

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

      9. add-sqr-sqrt 44.8%

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

   6. Applied egg-rr 44.8%

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

      1. associate-*l* 44.8%

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

   8. Simplified 44.8%

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

   9. Taylor expanded in x around 0 43.2%

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

       1. associate-/l* 44.9%

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

       2. associate-/r/ 44.9%

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

       3. neg-mul-1 44.9%

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

       4. +-commutative 44.9%

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

       5. unsub-neg 44.9%

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

       6. associate-*r/ 44.9%

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

       7. *-commutative 44.9%

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

       8. associate-*r* 44.9%

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

   11. Simplified 44.9%

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

   if 1.19999999999999993e-140 < z

   1. Initial program 66.0%

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

      1. *-commutative 66.0%

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

      2. associate-*l* 63.1%

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

      3. associate-*r/ 65.0%

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

   3. Simplified 65.0%

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

   4. Taylor expanded in z around inf 88.0%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{-105}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-140}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \frac{y}{\frac{a \cdot \left(t \cdot -0.5\right)}{z} - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 8: 76.7% accurate, 5.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+49}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-141}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{0.5 \cdot \frac{a \cdot t}{z} - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.9e+49)
   (* x (- y))
   (if (<= z 2.8e-141) (* y (/ (* z x) (- (* 0.5 (/ (* a t) z)) z))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.9e+49) {
		tmp = x * -y;
	} else if (z <= 2.8e-141) {
		tmp = y * ((z * x) / ((0.5 * ((a * t) / z)) - z));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.9d+49)) then
        tmp = x * -y
    else if (z <= 2.8d-141) then
        tmp = y * ((z * x) / ((0.5d0 * ((a * t) / z)) - z))
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.9e+49) {
		tmp = x * -y;
	} else if (z <= 2.8e-141) {
		tmp = y * ((z * x) / ((0.5 * ((a * t) / z)) - z));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.9e+49:
		tmp = x * -y
	elif z <= 2.8e-141:
		tmp = y * ((z * x) / ((0.5 * ((a * t) / z)) - z))
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.9e+49)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 2.8e-141)
		tmp = Float64(y * Float64(Float64(z * x) / Float64(Float64(0.5 * Float64(Float64(a * t) / z)) - z)));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.9e+49)
		tmp = x * -y;
	elseif (z <= 2.8e-141)
		tmp = y * ((z * x) / ((0.5 * ((a * t) / z)) - z));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.9e+49], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 2.8e-141], N[(y * N[(N[(z * x), $MachinePrecision] / N[(N[(0.5 * N[(N[(a * t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.8999999999999999e49

   1. Initial program 39.8%

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

      1. *-commutative 39.8%

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

      2. associate-*l* 37.8%

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

      3. associate-*r/ 41.2%

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

   3. Simplified 41.2%

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

   4. Taylor expanded in z around -inf 96.9%

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

      1. neg-mul-1 96.9%

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

   6. Simplified 96.9%

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

   if -1.8999999999999999e49 < z < 2.80000000000000012e-141

   1. Initial program 78.2%

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

      1. *-commutative 78.2%

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

      2. associate-*l* 79.6%

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

      3. associate-*r/ 79.0%

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

   3. Simplified 79.0%

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

   4. Taylor expanded in z around -inf 53.3%

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

   if 2.80000000000000012e-141 < z

   1. Initial program 66.0%

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

      1. *-commutative 66.0%

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

      2. associate-*l* 63.1%

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

      3. associate-*r/ 65.0%

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

   3. Simplified 65.0%

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

   4. Taylor expanded in z around inf 88.0%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+49}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-141}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{0.5 \cdot \frac{a \cdot t}{z} - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 9: 75.0% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-106}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-140}:\\ \;\;\;\;-2 \cdot \left(\frac{y}{a} \cdot \frac{x \cdot \left(z \cdot z\right)}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -7e-106)
   (* x (- y))
   (if (<= z 2.7e-140) (* -2.0 (* (/ y a) (/ (* x (* z z)) t))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7e-106) {
		tmp = x * -y;
	} else if (z <= 2.7e-140) {
		tmp = -2.0 * ((y / a) * ((x * (z * z)) / t));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-7d-106)) then
        tmp = x * -y
    else if (z <= 2.7d-140) then
        tmp = (-2.0d0) * ((y / a) * ((x * (z * z)) / t))
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7e-106) {
		tmp = x * -y;
	} else if (z <= 2.7e-140) {
		tmp = -2.0 * ((y / a) * ((x * (z * z)) / t));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -7e-106:
		tmp = x * -y
	elif z <= 2.7e-140:
		tmp = -2.0 * ((y / a) * ((x * (z * z)) / t))
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -7e-106)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 2.7e-140)
		tmp = Float64(-2.0 * Float64(Float64(y / a) * Float64(Float64(x * Float64(z * z)) / t)));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -7e-106)
		tmp = x * -y;
	elseif (z <= 2.7e-140)
		tmp = -2.0 * ((y / a) * ((x * (z * z)) / t));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -7e-106], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 2.7e-140], N[(-2.0 * N[(N[(y / a), $MachinePrecision] * N[(N[(x * N[(z * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7e-106

   1. Initial program 56.5%

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

      1. *-commutative 56.5%

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

      2. associate-*l* 54.1%

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

      3. associate-*r/ 55.4%

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

   3. Simplified 55.4%

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

   4. Taylor expanded in z around -inf 86.2%

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

      1. neg-mul-1 86.2%

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

   6. Simplified 86.2%

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

   if -7e-106 < z < 2.7e-140

   1. Initial program 72.7%

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

      1. associate-/l* 70.2%

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

   3. Simplified 70.2%

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

      1. associate-*l/ 73.8%

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

      2. associate-/l* 77.2%

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

      3. div-inv 77.2%

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

      4. associate-*l* 74.2%

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

      5. div-inv 74.3%

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

   5. Applied egg-rr 74.3%

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

   6. Taylor expanded in z around inf 41.3%

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

   7. Taylor expanded in z around 0 41.1%

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

      1. times-frac 41.0%

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

      2. unpow2 41.0%

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

   9. Simplified 41.0%

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

   if 2.7e-140 < z

   1. Initial program 66.0%

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

      1. *-commutative 66.0%

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

      2. associate-*l* 63.1%

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

      3. associate-*r/ 65.0%

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

   3. Simplified 65.0%

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

   4. Taylor expanded in z around inf 88.0%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-106}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-140}:\\ \;\;\;\;-2 \cdot \left(\frac{y}{a} \cdot \frac{x \cdot \left(z \cdot z\right)}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 10: 74.7% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-106}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-142}:\\ \;\;\;\;-2 \cdot \frac{y \cdot \left(z \cdot \left(z \cdot x\right)\right)}{a \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

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 -7e-106)
   (* x (- y))
   (if (<= z 7.2e-142) (* -2.0 (/ (* y (* z (* z x))) (* a t))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7e-106) {
		tmp = x * -y;
	} else if (z <= 7.2e-142) {
		tmp = -2.0 * ((y * (z * (z * x))) / (a * 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 <= (-7d-106)) then
        tmp = x * -y
    else if (z <= 7.2d-142) then
        tmp = (-2.0d0) * ((y * (z * (z * x))) / (a * 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 <= -7e-106) {
		tmp = x * -y;
	} else if (z <= 7.2e-142) {
		tmp = -2.0 * ((y * (z * (z * x))) / (a * t));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -7e-106:
		tmp = x * -y
	elif z <= 7.2e-142:
		tmp = -2.0 * ((y * (z * (z * x))) / (a * 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 <= -7e-106)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 7.2e-142)
		tmp = Float64(-2.0 * Float64(Float64(y * Float64(z * Float64(z * x))) / Float64(a * 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 <= -7e-106)
		tmp = x * -y;
	elseif (z <= 7.2e-142)
		tmp = -2.0 * ((y * (z * (z * x))) / (a * 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, -7e-106], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 7.2e-142], N[(-2.0 * N[(N[(y * N[(z * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7e-106

   1. Initial program 56.5%

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

      1. *-commutative 56.5%

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

      2. associate-*l* 54.1%

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

      3. associate-*r/ 55.4%

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

   3. Simplified 55.4%

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

   4. Taylor expanded in z around -inf 86.2%

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

      1. neg-mul-1 86.2%

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

   6. Simplified 86.2%

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

   if -7e-106 < z < 7.20000000000000001e-142

   1. Initial program 72.7%

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

      1. associate-/l* 70.2%

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

   3. Simplified 70.2%

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

      1. associate-*l/ 73.8%

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

      2. associate-/l* 77.2%

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

      3. div-inv 77.2%

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

      4. associate-*l* 74.2%

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

      5. div-inv 74.3%

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

   5. Applied egg-rr 74.3%

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

   6. Taylor expanded in z around inf 41.3%

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

   7. Taylor expanded in z around 0 41.1%

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

      1. times-frac 41.0%

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

      2. unpow2 41.0%

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

   9. Simplified 41.0%

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

       1. frac-times 41.1%

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

       2. associate-*l* 41.1%

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

       3. *-commutative 41.1%

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

   11. Applied egg-rr 41.1%

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

   if 7.20000000000000001e-142 < z

   1. Initial program 66.0%

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

      1. *-commutative 66.0%

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

      2. associate-*l* 63.1%

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

      3. associate-*r/ 65.0%

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

   3. Simplified 65.0%

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

   4. Taylor expanded in z around inf 88.0%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-106}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-142}:\\ \;\;\;\;-2 \cdot \frac{y \cdot \left(z \cdot \left(z \cdot x\right)\right)}{a \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 11: 75.3% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-100}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-140}:\\ \;\;\;\;2 \cdot \frac{y \cdot \frac{z \cdot \left(z \cdot x\right)}{t}}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

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 -3.1e-100)
   (* x (- y))
   (if (<= z 1.2e-140) (* 2.0 (/ (* y (/ (* z (* z x)) t)) a)) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.1e-100) {
		tmp = x * -y;
	} else if (z <= 1.2e-140) {
		tmp = 2.0 * ((y * ((z * (z * x)) / 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 <= (-3.1d-100)) then
        tmp = x * -y
    else if (z <= 1.2d-140) then
        tmp = 2.0d0 * ((y * ((z * (z * x)) / 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 <= -3.1e-100) {
		tmp = x * -y;
	} else if (z <= 1.2e-140) {
		tmp = 2.0 * ((y * ((z * (z * x)) / 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 <= -3.1e-100:
		tmp = x * -y
	elif z <= 1.2e-140:
		tmp = 2.0 * ((y * ((z * (z * x)) / 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 <= -3.1e-100)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 1.2e-140)
		tmp = Float64(2.0 * Float64(Float64(y * Float64(Float64(z * Float64(z * x)) / 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 <= -3.1e-100)
		tmp = x * -y;
	elseif (z <= 1.2e-140)
		tmp = 2.0 * ((y * ((z * (z * x)) / 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, -3.1e-100], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 1.2e-140], N[(2.0 * N[(N[(y * N[(N[(z * N[(z * x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.0999999999999999e-100

   1. Initial program 54.5%

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

      1. *-commutative 54.5%

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

      2. associate-*l* 52.0%

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

      3. associate-*r/ 53.3%

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

   3. Simplified 53.3%

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

   4. Taylor expanded in z around -inf 86.6%

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

      1. neg-mul-1 86.6%

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

   6. Simplified 86.6%

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

   if -3.0999999999999999e-100 < z < 1.19999999999999993e-140

   1. Initial program 74.5%

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

      1. associate-/l* 72.3%

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

   3. Simplified 72.3%

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

   4. Taylor expanded in z around -inf 46.9%

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

   5. Taylor expanded in a around inf 43.5%

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

      1. times-frac 43.3%

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

      2. associate-*l/ 43.7%

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

      3. unpow2 43.7%

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

      4. associate-*l* 45.4%

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

   7. Simplified 45.4%

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

   if 1.19999999999999993e-140 < z

   1. Initial program 66.0%

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

      1. *-commutative 66.0%

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

      2. associate-*l* 63.1%

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

      3. associate-*r/ 65.0%

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

   3. Simplified 65.0%

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

   4. Taylor expanded in z around inf 88.0%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-100}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-140}:\\ \;\;\;\;2 \cdot \frac{y \cdot \frac{z \cdot \left(z \cdot x\right)}{t}}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 12: 76.0% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 7.5 \cdot 10^{-141}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{0.5 \cdot \frac{a \cdot t}{z} - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

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 7.5e-141) (* y (* x (/ z (- (* 0.5 (/ (* a t) z)) z)))) (* x y)))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= 7.5e-141) {
		tmp = y * (x * (z / ((0.5 * ((a * t) / z)) - 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 <= 7.5d-141) then
        tmp = y * (x * (z / ((0.5d0 * ((a * t) / z)) - 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 <= 7.5e-141) {
		tmp = y * (x * (z / ((0.5 * ((a * t) / z)) - z)));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= 7.5e-141:
		tmp = y * (x * (z / ((0.5 * ((a * t) / z)) - 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 <= 7.5e-141)
		tmp = Float64(y * Float64(x * Float64(z / Float64(Float64(0.5 * Float64(Float64(a * t) / z)) - 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 <= 7.5e-141)
		tmp = y * (x * (z / ((0.5 * ((a * t) / z)) - 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, 7.5e-141], N[(y * N[(x * N[(z / N[(N[(0.5 * N[(N[(a * t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 7.50000000000000046e-141

   1. Initial program 62.6%

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

      1. associate-/l* 65.0%

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

   3. Simplified 65.0%

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

      1. associate-*l/ 64.5%

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

      2. associate-/l* 63.5%

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

      3. div-inv 63.5%

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

      4. associate-*l* 64.6%

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

      5. div-inv 64.6%

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

   5. Applied egg-rr 64.6%

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

   6. Taylor expanded in z around -inf 68.9%

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

   if 7.50000000000000046e-141 < z

   1. Initial program 66.0%

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

      1. *-commutative 66.0%

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

      2. associate-*l* 63.1%

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

      3. associate-*r/ 65.0%

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

   3. Simplified 65.0%

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

   4. Taylor expanded in z around inf 88.0%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7.5 \cdot 10^{-141}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{0.5 \cdot \frac{a \cdot t}{z} - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 13: 77.4% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 7.2 \cdot 10^{-142}:\\ \;\;\;\;\frac{x \cdot y}{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right) - z}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

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 7.2e-142) (/ (* x y) (/ (- (* 0.5 (* a (/ t z))) z) z)) (* x y)))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= 7.2e-142) {
		tmp = (x * y) / (((0.5 * (a * (t / z))) - z) / 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 <= 7.2d-142) then
        tmp = (x * y) / (((0.5d0 * (a * (t / z))) - z) / 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 <= 7.2e-142) {
		tmp = (x * y) / (((0.5 * (a * (t / z))) - z) / z);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= 7.2e-142:
		tmp = (x * y) / (((0.5 * (a * (t / z))) - z) / 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 <= 7.2e-142)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(0.5 * Float64(a * Float64(t / z))) - z) / 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 <= 7.2e-142)
		tmp = (x * y) / (((0.5 * (a * (t / z))) - z) / 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, 7.2e-142], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(0.5 * N[(a * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 7.20000000000000001e-142

   1. Initial program 62.6%

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

      1. associate-/l* 65.0%

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

   3. Simplified 65.0%

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

   4. Taylor expanded in z around -inf 69.4%

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

      1. expm1-log1p-u 61.0%

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

      2. expm1-udef 60.9%

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

      3. *-commutative 60.9%

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

   6. Applied egg-rr 60.9%

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

      1. expm1-def 61.0%

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

      2. expm1-log1p 69.4%

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

      3. *-commutative 69.4%

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

      4. associate-*r/ 72.1%

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

   8. Simplified 72.1%

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

   if 7.20000000000000001e-142 < z

   1. Initial program 66.0%

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

      1. *-commutative 66.0%

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

      2. associate-*l* 63.1%

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

      3. associate-*r/ 65.0%

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

   3. Simplified 65.0%

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

   4. Taylor expanded in z around inf 88.0%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7.2 \cdot 10^{-142}:\\ \;\;\;\;\frac{x \cdot y}{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right) - z}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 14: 73.9% accurate, 9.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+51}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-160}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{-z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

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 -2e+51)
   (* x (- y))
   (if (<= z 9e-160) (* 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 <= -2e+51) {
		tmp = x * -y;
	} else if (z <= 9e-160) {
		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 <= (-2d+51)) then
        tmp = x * -y
    else if (z <= 9d-160) 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 <= -2e+51) {
		tmp = x * -y;
	} else if (z <= 9e-160) {
		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 <= -2e+51:
		tmp = x * -y
	elif z <= 9e-160:
		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 <= -2e+51)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 9e-160)
		tmp = Float64(y * Float64(Float64(z * x) / Float64(-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+51)
		tmp = x * -y;
	elseif (z <= 9e-160)
		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, -2e+51], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 9e-160], N[(y * N[(N[(z * x), $MachinePrecision] / (-z)), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2e51

   1. Initial program 39.8%

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

      1. *-commutative 39.8%

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

      2. associate-*l* 37.8%

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

      3. associate-*r/ 41.2%

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

   3. Simplified 41.2%

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

   4. Taylor expanded in z around -inf 96.9%

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

      1. neg-mul-1 96.9%

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

   6. Simplified 96.9%

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

   if -2e51 < z < 9.00000000000000053e-160

   1. Initial program 78.8%

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

      1. *-commutative 78.8%

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

      2. associate-*l* 79.1%

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

      3. associate-*r/ 78.5%

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

   3. Simplified 78.5%

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

   4. Taylor expanded in z around -inf 44.8%

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

      1. neg-mul-1 44.8%

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

   6. Simplified 44.8%

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

   if 9.00000000000000053e-160 < z

   1. Initial program 65.8%

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

      1. *-commutative 65.8%

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

      2. associate-*l* 63.8%

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

      3. associate-*r/ 65.7%

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

   3. Simplified 65.7%

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

   4. Taylor expanded in z around inf 86.5%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+51}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-160}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{-z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 15: 75.5% accurate, 9.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{-100}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{-159}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot x\right)}{-z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

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 -3.2e-100)
   (* x (- y))
   (if (<= z 2.15e-159) (/ (* 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 <= -3.2e-100) {
		tmp = x * -y;
	} else if (z <= 2.15e-159) {
		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 <= (-3.2d-100)) then
        tmp = x * -y
    else if (z <= 2.15d-159) 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 <= -3.2e-100) {
		tmp = x * -y;
	} else if (z <= 2.15e-159) {
		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 <= -3.2e-100:
		tmp = x * -y
	elif z <= 2.15e-159:
		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 <= -3.2e-100)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 2.15e-159)
		tmp = Float64(Float64(y * Float64(z * x)) / Float64(-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 <= -3.2e-100)
		tmp = x * -y;
	elseif (z <= 2.15e-159)
		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, -3.2e-100], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 2.15e-159], N[(N[(y * N[(z * x), $MachinePrecision]), $MachinePrecision] / (-z)), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.20000000000000017e-100

   1. Initial program 54.5%

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

      1. *-commutative 54.5%

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

      2. associate-*l* 52.0%

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

      3. associate-*r/ 53.3%

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

   3. Simplified 53.3%

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

   4. Taylor expanded in z around -inf 86.6%

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

      1. neg-mul-1 86.6%

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

   6. Simplified 86.6%

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

   if -3.20000000000000017e-100 < z < 2.15e-159

   1. Initial program 75.2%

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

   2. Taylor expanded in z around -inf 30.9%

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

      1. neg-mul-1 35.8%

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

   4. Simplified 30.9%

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

   5. Taylor expanded in x around 0 37.8%

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

   if 2.15e-159 < z

   1. Initial program 65.8%

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

      1. *-commutative 65.8%

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

      2. associate-*l* 63.8%

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

      3. associate-*r/ 65.7%

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

   3. Simplified 65.7%

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

   4. Taylor expanded in z around inf 86.5%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{-100}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{-159}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot x\right)}{-z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 16: 73.0% accurate, 10.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-106}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 10^{-126}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

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 -7e-106)
   (* x (- y))
   (if (<= z 1e-126) (* 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 <= -7e-106) {
		tmp = x * -y;
	} else if (z <= 1e-126) {
		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 <= (-7d-106)) then
        tmp = x * -y
    else if (z <= 1d-126) 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 <= -7e-106) {
		tmp = x * -y;
	} else if (z <= 1e-126) {
		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 <= -7e-106:
		tmp = x * -y
	elif z <= 1e-126:
		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 <= -7e-106)
		tmp = Float64(x * Float64(-y));
	elseif (z <= 1e-126)
		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 <= -7e-106)
		tmp = x * -y;
	elseif (z <= 1e-126)
		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, -7e-106], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, 1e-126], N[(y * N[(N[(z * x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7e-106

   1. Initial program 56.5%

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

      1. *-commutative 56.5%

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

      2. associate-*l* 54.1%

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

      3. associate-*r/ 55.4%

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

   3. Simplified 55.4%

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

   4. Taylor expanded in z around -inf 86.2%

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

      1. neg-mul-1 86.2%

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

   6. Simplified 86.2%

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

   if -7e-106 < z < 9.9999999999999995e-127

   1. Initial program 73.6%

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

      1. *-commutative 73.6%

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

      2. associate-*l* 77.3%

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

      3. associate-*r/ 78.0%

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

   3. Simplified 78.0%

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

   4. Taylor expanded in z around inf 29.0%

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

   if 9.9999999999999995e-127 < z

   1. Initial program 65.4%

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

      1. *-commutative 65.4%

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

      2. associate-*l* 62.4%

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

      3. associate-*r/ 64.4%

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

   3. Simplified 64.4%

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

   4. Taylor expanded in z around inf 88.6%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-106}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 10^{-126}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 17: 72.7% accurate, 18.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4e-310) (* x (- y)) (* x y)))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4e-310) {
		tmp = x * -y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-4d-310)) then
        tmp = x * -y
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4e-310) {
		tmp = x * -y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -4e-310:
		tmp = x * -y
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4e-310)
		tmp = Float64(x * Float64(-y));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -4e-310)
		tmp = x * -y;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4e-310], N[(x * (-y)), $MachinePrecision], N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.999999999999988e-310

   1. Initial program 60.0%

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

      1. *-commutative 60.0%

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

      2. associate-*l* 56.9%

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

      3. associate-*r/ 58.2%

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

   3. Simplified 58.2%

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

   4. Taylor expanded in z around -inf 69.8%

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

      1. neg-mul-1 69.8%

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

   6. Simplified 69.8%

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

   if -3.999999999999988e-310 < z

   1. Initial program 67.7%

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

      1. *-commutative 67.7%

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

      2. associate-*l* 68.0%

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

      3. associate-*r/ 69.5%

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

   3. Simplified 69.5%

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

   4. Taylor expanded in z around inf 73.4%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 18: 42.9% 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 64.0%

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

   1. *-commutative 64.0%

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

   2. associate-*l* 62.8%

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

   3. associate-*r/ 64.2%

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

3. Simplified 64.2%

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

4. Taylor expanded in z around inf 45.0%

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

5. Final simplification 45.0%

   \[\leadsto x \cdot y \]

Developer target: 89.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023195 
(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)))))