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

Percentage Accurate: 61.5% → 91.1%

Time: 14.8s

Alternatives: 13

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: 61.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := \sqrt{z \cdot z - t \cdot a}\\ \mathbf{if}\;z \leq -1.7 \cdot 10^{+84}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-223}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{t_1}\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+106}:\\ \;\;\;\;\frac{y \cdot x}{\frac{t_1}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (sqrt (- (* z z) (* t a)))))
   (if (<= z -1.7e+84)
     (* y (- x))
     (if (<= z 7.5e-223)
       (* y (/ (* z x) t_1))
       (if (<= z 2.8e+106) (/ (* y x) (/ t_1 z)) (* y x))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = sqrt(((z * z) - (t * a)));
	double tmp;
	if (z <= -1.7e+84) {
		tmp = y * -x;
	} else if (z <= 7.5e-223) {
		tmp = y * ((z * x) / t_1);
	} else if (z <= 2.8e+106) {
		tmp = (y * x) / (t_1 / z);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(((z * z) - (t * a)))
    if (z <= (-1.7d+84)) then
        tmp = y * -x
    else if (z <= 7.5d-223) then
        tmp = y * ((z * x) / t_1)
    else if (z <= 2.8d+106) then
        tmp = (y * x) / (t_1 / z)
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.sqrt(((z * z) - (t * a)));
	double tmp;
	if (z <= -1.7e+84) {
		tmp = y * -x;
	} else if (z <= 7.5e-223) {
		tmp = y * ((z * x) / t_1);
	} else if (z <= 2.8e+106) {
		tmp = (y * x) / (t_1 / z);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	t_1 = math.sqrt(((z * z) - (t * a)))
	tmp = 0
	if z <= -1.7e+84:
		tmp = y * -x
	elif z <= 7.5e-223:
		tmp = y * ((z * x) / t_1)
	elif z <= 2.8e+106:
		tmp = (y * x) / (t_1 / z)
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = sqrt(Float64(Float64(z * z) - Float64(t * a)))
	tmp = 0.0
	if (z <= -1.7e+84)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 7.5e-223)
		tmp = Float64(y * Float64(Float64(z * x) / t_1));
	elseif (z <= 2.8e+106)
		tmp = Float64(Float64(y * x) / Float64(t_1 / z));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = sqrt(((z * z) - (t * a)));
	tmp = 0.0;
	if (z <= -1.7e+84)
		tmp = y * -x;
	elseif (z <= 7.5e-223)
		tmp = y * ((z * x) / t_1);
	elseif (z <= 2.8e+106)
		tmp = (y * x) / (t_1 / z);
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -1.7e+84], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 7.5e-223], N[(y * N[(N[(z * x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.8e+106], N[(N[(y * x), $MachinePrecision] / N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.6999999999999999e84

   1. Initial program 38.6%

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

      1. *-commutative 38.6%

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

      2. associate-*l* 34.8%

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

      3. associate-*r/ 38.3%

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

   3. Simplified 38.3%

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

   4. Taylor expanded in z around -inf 98.3%

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

      1. neg-mul-1 98.3%

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

   6. Simplified 98.3%

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

   if -1.6999999999999999e84 < z < 7.50000000000000074e-223

   1. Initial program 87.5%

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

      1. *-commutative 87.5%

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

      2. associate-*l* 88.4%

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

      3. associate-*r/ 90.8%

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

   3. Simplified 90.8%

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

   if 7.50000000000000074e-223 < z < 2.79999999999999993e106

   1. Initial program 96.8%

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

      1. associate-/l* 98.3%

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

   3. Simplified 98.3%

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

   if 2.79999999999999993e106 < z

   1. Initial program 31.6%

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

      1. *-commutative 31.6%

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

      2. associate-*l* 31.4%

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

      3. associate-*r/ 33.3%

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

   3. Simplified 33.3%

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

   4. Taylor expanded in z around inf 98.4%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+84}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-223}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+106}:\\ \;\;\;\;\frac{y \cdot x}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 2: 90.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := \sqrt{z \cdot z - t \cdot a}\\ t_2 := y \cdot \frac{z \cdot x}{t_1}\\ \mathbf{if}\;z \leq -2.3 \cdot 10^{+86}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-192}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{-46}:\\ \;\;\;\;x \cdot \left(z \cdot \frac{y}{t_1}\right)\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+88}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \left(\frac{t}{z} \cdot \frac{a}{z}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (sqrt (- (* z z) (* t a)))) (t_2 (* y (/ (* z x) t_1))))
   (if (<= z -2.3e+86)
     (* y (- x))
     (if (<= z 1.1e-192)
       t_2
       (if (<= z 1.12e-46)
         (* x (* z (/ y t_1)))
         (if (<= z 1.25e+88)
           t_2
           (/ (* y x) (+ 1.0 (* -0.5 (* (/ t z) (/ a z)))))))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = sqrt(((z * z) - (t * a)));
	double t_2 = y * ((z * x) / t_1);
	double tmp;
	if (z <= -2.3e+86) {
		tmp = y * -x;
	} else if (z <= 1.1e-192) {
		tmp = t_2;
	} else if (z <= 1.12e-46) {
		tmp = x * (z * (y / t_1));
	} else if (z <= 1.25e+88) {
		tmp = t_2;
	} else {
		tmp = (y * x) / (1.0 + (-0.5 * ((t / z) * (a / z))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt(((z * z) - (t * a)))
    t_2 = y * ((z * x) / t_1)
    if (z <= (-2.3d+86)) then
        tmp = y * -x
    else if (z <= 1.1d-192) then
        tmp = t_2
    else if (z <= 1.12d-46) then
        tmp = x * (z * (y / t_1))
    else if (z <= 1.25d+88) then
        tmp = t_2
    else
        tmp = (y * x) / (1.0d0 + ((-0.5d0) * ((t / z) * (a / z))))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.sqrt(((z * z) - (t * a)));
	double t_2 = y * ((z * x) / t_1);
	double tmp;
	if (z <= -2.3e+86) {
		tmp = y * -x;
	} else if (z <= 1.1e-192) {
		tmp = t_2;
	} else if (z <= 1.12e-46) {
		tmp = x * (z * (y / t_1));
	} else if (z <= 1.25e+88) {
		tmp = t_2;
	} else {
		tmp = (y * x) / (1.0 + (-0.5 * ((t / z) * (a / z))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	t_1 = math.sqrt(((z * z) - (t * a)))
	t_2 = y * ((z * x) / t_1)
	tmp = 0
	if z <= -2.3e+86:
		tmp = y * -x
	elif z <= 1.1e-192:
		tmp = t_2
	elif z <= 1.12e-46:
		tmp = x * (z * (y / t_1))
	elif z <= 1.25e+88:
		tmp = t_2
	else:
		tmp = (y * x) / (1.0 + (-0.5 * ((t / z) * (a / z))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = sqrt(Float64(Float64(z * z) - Float64(t * a)))
	t_2 = Float64(y * Float64(Float64(z * x) / t_1))
	tmp = 0.0
	if (z <= -2.3e+86)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.1e-192)
		tmp = t_2;
	elseif (z <= 1.12e-46)
		tmp = Float64(x * Float64(z * Float64(y / t_1)));
	elseif (z <= 1.25e+88)
		tmp = t_2;
	else
		tmp = Float64(Float64(y * x) / Float64(1.0 + Float64(-0.5 * Float64(Float64(t / z) * Float64(a / z)))));
	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) - (t * a)));
	t_2 = y * ((z * x) / t_1);
	tmp = 0.0;
	if (z <= -2.3e+86)
		tmp = y * -x;
	elseif (z <= 1.1e-192)
		tmp = t_2;
	elseif (z <= 1.12e-46)
		tmp = x * (z * (y / t_1));
	elseif (z <= 1.25e+88)
		tmp = t_2;
	else
		tmp = (y * x) / (1.0 + (-0.5 * ((t / z) * (a / z))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(N[(z * x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.3e+86], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.1e-192], t$95$2, If[LessEqual[z, 1.12e-46], N[(x * N[(z * N[(y / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.25e+88], t$95$2, N[(N[(y * x), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(N[(t / z), $MachinePrecision] * N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -2.2999999999999999e86

   1. Initial program 38.6%

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

      1. *-commutative 38.6%

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

      2. associate-*l* 34.8%

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

      3. associate-*r/ 38.3%

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

   3. Simplified 38.3%

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

   4. Taylor expanded in z around -inf 98.3%

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

      1. neg-mul-1 98.3%

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

   6. Simplified 98.3%

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

   if -2.2999999999999999e86 < z < 1.10000000000000003e-192 or 1.11999999999999997e-46 < z < 1.24999999999999999e88

   1. Initial program 91.0%

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

      1. *-commutative 91.0%

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

      2. associate-*l* 91.5%

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

      3. associate-*r/ 93.3%

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

   3. Simplified 93.3%

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

   if 1.10000000000000003e-192 < z < 1.11999999999999997e-46

   1. Initial program 96.4%

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

      1. *-commutative 96.4%

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

      2. associate-*l* 96.1%

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

      3. associate-*r/ 96.3%

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

   3. Simplified 96.3%

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

      1. associate-*r/ 96.1%

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

      2. associate-*r* 96.4%

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

      3. *-commutative 96.4%

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

      4. expm1-log1p-u 81.6%

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

      5. expm1-udef 36.0%

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

      6. div-inv 36.0%

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

      7. associate-*l* 36.0%

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

      8. div-inv 36.0%

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

   5. Applied egg-rr 36.0%

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

      1. expm1-def 84.9%

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

      2. expm1-log1p 99.7%

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

      3. associate-*r* 92.4%

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

      4. associate-*r/ 75.8%

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

      5. associate-*l/ 92.6%

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

      6. *-commutative 92.6%

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

   7. Simplified 92.6%

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

   if 1.24999999999999999e88 < z

   1. Initial program 36.4%

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

      1. associate-/l* 38.3%

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

   3. Simplified 38.3%

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

   4. Taylor expanded in z around inf 86.6%

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

      1. unpow2 86.6%

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

      2. associate-*r/ 86.6%

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

   6. Simplified 86.6%

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

   7. Taylor expanded in a around 0 86.6%

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

      1. *-commutative 86.6%

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

      2. unpow2 86.6%

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

      3. times-frac 95.7%

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

   9. Simplified 95.7%

      \[\leadsto \frac{x \cdot y}{1 + \color{blue}{-0.5 \cdot \left(\frac{t}{z} \cdot \frac{a}{z}\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+86}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-192}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{-46}:\\ \;\;\;\;x \cdot \left(z \cdot \frac{y}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+88}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \left(\frac{t}{z} \cdot \frac{a}{z}\right)}\\ \end{array} \]

Alternative 3: 90.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+100}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.65 \cdot 10^{+106}:\\ \;\;\;\;x \cdot \left(z \cdot \frac{y}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.7e+100)
   (* y (- x))
   (if (<= z 2.65e+106) (* x (* z (/ y (sqrt (- (* z z) (* t a)))))) (* y x))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.7e+100) {
		tmp = y * -x;
	} else if (z <= 2.65e+106) {
		tmp = x * (z * (y / sqrt(((z * z) - (t * a)))));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.7d+100)) then
        tmp = y * -x
    else if (z <= 2.65d+106) then
        tmp = x * (z * (y / sqrt(((z * z) - (t * a)))))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.7e+100) {
		tmp = y * -x;
	} else if (z <= 2.65e+106) {
		tmp = x * (z * (y / Math.sqrt(((z * z) - (t * a)))));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.7e+100:
		tmp = y * -x
	elif z <= 2.65e+106:
		tmp = x * (z * (y / math.sqrt(((z * z) - (t * a)))))
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.7e+100)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 2.65e+106)
		tmp = Float64(x * Float64(z * Float64(y / sqrt(Float64(Float64(z * z) - Float64(t * a))))));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.7e+100)
		tmp = y * -x;
	elseif (z <= 2.65e+106)
		tmp = x * (z * (y / sqrt(((z * z) - (t * a)))));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.7e+100], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 2.65e+106], N[(x * N[(z * N[(y / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.69999999999999997e100

   1. Initial program 36.3%

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

      1. *-commutative 36.3%

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

      2. associate-*l* 32.4%

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

      3. associate-*r/ 36.0%

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

   3. Simplified 36.0%

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

   4. Taylor expanded in z around -inf 98.3%

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

      1. neg-mul-1 98.3%

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

   6. Simplified 98.3%

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

   if -1.69999999999999997e100 < z < 2.65e106

   1. Initial program 91.8%

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

      1. *-commutative 91.8%

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

      2. associate-*l* 91.5%

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

      3. associate-*r/ 92.9%

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

   3. Simplified 92.9%

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

      1. associate-*r/ 91.5%

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

      2. associate-*r* 91.8%

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

      3. *-commutative 91.8%

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

      4. expm1-log1p-u 73.5%

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

      5. expm1-udef 39.7%

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

      6. div-inv 39.7%

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

      7. associate-*l* 39.7%

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

      8. div-inv 39.7%

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

   5. Applied egg-rr 39.7%

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

      1. expm1-def 74.7%

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

      2. expm1-log1p 93.1%

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

      3. associate-*r* 90.6%

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

      4. associate-*r/ 86.1%

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

      5. associate-*l/ 89.2%

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

      6. *-commutative 89.2%

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

   7. Simplified 89.2%

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

   if 2.65e106 < z

   1. Initial program 31.6%

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

      1. *-commutative 31.6%

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

      2. associate-*l* 31.4%

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

      3. associate-*r/ 33.3%

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

   3. Simplified 33.3%

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

   4. Taylor expanded in z around inf 98.4%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+100}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.65 \cdot 10^{+106}:\\ \;\;\;\;x \cdot \left(z \cdot \frac{y}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 4: 83.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{-58}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-135}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \left(\frac{t}{z} \cdot \frac{a}{z}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -8.5e-58)
   (* y (- x))
   (if (<= z 3.6e-135)
     (* y (/ (* z x) (sqrt (* t (- a)))))
     (/ (* y x) (+ 1.0 (* -0.5 (* (/ t z) (/ a z))))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.5e-58) {
		tmp = y * -x;
	} else if (z <= 3.6e-135) {
		tmp = y * ((z * x) / sqrt((t * -a)));
	} else {
		tmp = (y * x) / (1.0 + (-0.5 * ((t / z) * (a / z))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-8.5d-58)) then
        tmp = y * -x
    else if (z <= 3.6d-135) then
        tmp = y * ((z * x) / sqrt((t * -a)))
    else
        tmp = (y * x) / (1.0d0 + ((-0.5d0) * ((t / z) * (a / z))))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.5e-58) {
		tmp = y * -x;
	} else if (z <= 3.6e-135) {
		tmp = y * ((z * x) / Math.sqrt((t * -a)));
	} else {
		tmp = (y * x) / (1.0 + (-0.5 * ((t / z) * (a / z))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -8.5e-58:
		tmp = y * -x
	elif z <= 3.6e-135:
		tmp = y * ((z * x) / math.sqrt((t * -a)))
	else:
		tmp = (y * x) / (1.0 + (-0.5 * ((t / z) * (a / z))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -8.5e-58)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 3.6e-135)
		tmp = Float64(y * Float64(Float64(z * x) / sqrt(Float64(t * Float64(-a)))));
	else
		tmp = Float64(Float64(y * x) / Float64(1.0 + Float64(-0.5 * Float64(Float64(t / z) * Float64(a / z)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -8.5e-58)
		tmp = y * -x;
	elseif (z <= 3.6e-135)
		tmp = y * ((z * x) / sqrt((t * -a)));
	else
		tmp = (y * x) / (1.0 + (-0.5 * ((t / z) * (a / z))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -8.5e-58], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 3.6e-135], N[(y * N[(N[(z * x), $MachinePrecision] / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(N[(t / z), $MachinePrecision] * N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.5000000000000004e-58

   1. Initial program 59.0%

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

      1. *-commutative 59.0%

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

      2. associate-*l* 56.4%

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

      3. associate-*r/ 58.8%

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

   3. Simplified 58.8%

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

   4. Taylor expanded in z around -inf 93.7%

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

      1. neg-mul-1 93.7%

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

   6. Simplified 93.7%

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

   if -8.5000000000000004e-58 < z < 3.59999999999999978e-135

   1. Initial program 81.4%

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

      1. *-commutative 81.4%

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

      2. associate-*l* 82.8%

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

      3. associate-*r/ 87.5%

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

   3. Simplified 87.5%

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

   4. Taylor expanded in z around 0 78.9%

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

      1. mul-1-neg 78.9%

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

      2. *-commutative 78.9%

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

      3. distribute-rgt-neg-in 78.9%

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

   6. Simplified 78.9%

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

   if 3.59999999999999978e-135 < z

   1. Initial program 63.6%

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

      1. associate-/l* 64.8%

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

   3. Simplified 64.8%

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

   4. Taylor expanded in z around inf 83.7%

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

      1. unpow2 83.7%

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

      2. associate-*r/ 83.7%

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

   6. Simplified 83.7%

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

   7. Taylor expanded in a around 0 83.7%

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

      1. *-commutative 83.7%

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

      2. unpow2 83.7%

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

      3. times-frac 88.9%

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

   9. Simplified 88.9%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{-58}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-135}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \left(\frac{t}{z} \cdot \frac{a}{z}\right)}\\ \end{array} \]

Alternative 5: 76.9% accurate, 5.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{-141}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+88}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z + -0.5 \cdot \frac{t \cdot a}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.2e-141)
   (* y (- x))
   (if (<= z 1.2e+88) (* y (/ (* z x) (+ z (* -0.5 (/ (* t a) z))))) (* y x))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.2e-141) {
		tmp = y * -x;
	} else if (z <= 1.2e+88) {
		tmp = y * ((z * x) / (z + (-0.5 * ((t * a) / z))));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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-141)) then
        tmp = y * -x
    else if (z <= 1.2d+88) then
        tmp = y * ((z * x) / (z + ((-0.5d0) * ((t * a) / z))))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.2e-141) {
		tmp = y * -x;
	} else if (z <= 1.2e+88) {
		tmp = y * ((z * x) / (z + (-0.5 * ((t * a) / z))));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.2e-141:
		tmp = y * -x
	elif z <= 1.2e+88:
		tmp = y * ((z * x) / (z + (-0.5 * ((t * a) / z))))
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.2e-141)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.2e+88)
		tmp = Float64(y * Float64(Float64(z * x) / Float64(z + Float64(-0.5 * Float64(Float64(t * a) / z)))));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -3.2e-141)
		tmp = y * -x;
	elseif (z <= 1.2e+88)
		tmp = y * ((z * x) / (z + (-0.5 * ((t * a) / z))));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.2e-141], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.2e+88], N[(y * N[(N[(z * x), $MachinePrecision] / N[(z + N[(-0.5 * N[(N[(t * a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.2000000000000001e-141

   1. Initial program 63.3%

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

      1. *-commutative 63.3%

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

      2. associate-*l* 61.1%

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

      3. associate-*r/ 64.1%

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

   3. Simplified 64.1%

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

   4. Taylor expanded in z around -inf 86.5%

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

      1. neg-mul-1 86.5%

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

   6. Simplified 86.5%

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

   if -3.2000000000000001e-141 < z < 1.2e88

   1. Initial program 91.4%

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

      1. *-commutative 91.4%

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

      2. associate-*l* 92.1%

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

      3. associate-*r/ 93.1%

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

   3. Simplified 93.1%

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

   4. Taylor expanded in z around inf 65.9%

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

   if 1.2e88 < z

   1. Initial program 36.4%

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

      1. *-commutative 36.4%

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

      2. associate-*l* 34.6%

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

      3. associate-*r/ 36.4%

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

   3. Simplified 36.4%

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

   4. Taylor expanded in z around inf 95.7%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{-141}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+88}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z + -0.5 \cdot \frac{t \cdot a}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 6: 75.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-136}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-263}:\\ \;\;\;\;-2 \cdot \left(\frac{y}{a} \cdot \frac{x \cdot \left(z \cdot z\right)}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.25e-136)
   (* y (- x))
   (if (<= z 4.2e-263) (* -2.0 (* (/ y a) (/ (* x (* z z)) t))) (* y x))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.25e-136) {
		tmp = y * -x;
	} else if (z <= 4.2e-263) {
		tmp = -2.0 * ((y / a) * ((x * (z * z)) / t));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.25d-136)) then
        tmp = y * -x
    else if (z <= 4.2d-263) then
        tmp = (-2.0d0) * ((y / a) * ((x * (z * z)) / t))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.25e-136) {
		tmp = y * -x;
	} else if (z <= 4.2e-263) {
		tmp = -2.0 * ((y / a) * ((x * (z * z)) / t));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.25e-136:
		tmp = y * -x
	elif z <= 4.2e-263:
		tmp = -2.0 * ((y / a) * ((x * (z * z)) / t))
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.25e-136)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 4.2e-263)
		tmp = Float64(-2.0 * Float64(Float64(y / a) * Float64(Float64(x * Float64(z * z)) / t)));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.25e-136)
		tmp = y * -x;
	elseif (z <= 4.2e-263)
		tmp = -2.0 * ((y / a) * ((x * (z * z)) / t));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.25e-136], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 4.2e-263], N[(-2.0 * N[(N[(y / a), $MachinePrecision] * N[(N[(x * N[(z * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.25e-136

   1. Initial program 63.3%

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

      1. *-commutative 63.3%

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

      2. associate-*l* 61.1%

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

      3. associate-*r/ 64.1%

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

   3. Simplified 64.1%

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

   4. Taylor expanded in z around -inf 86.5%

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

      1. neg-mul-1 86.5%

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

   6. Simplified 86.5%

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

   if -1.25e-136 < z < 4.20000000000000005e-263

   1. Initial program 76.1%

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

   2. Taylor expanded in z around inf 57.7%

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

   3. Taylor expanded in z around 0 57.9%

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

      1. times-frac 57.9%

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

      2. unpow2 57.9%

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

   5. Simplified 57.9%

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

   if 4.20000000000000005e-263 < 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* 64.9%

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

      3. associate-*r/ 66.0%

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

   3. Simplified 66.0%

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

   4. Taylor expanded in z around inf 80.1%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-136}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-263}:\\ \;\;\;\;-2 \cdot \left(\frac{y}{a} \cdot \frac{x \cdot \left(z \cdot z\right)}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 7: 75.9% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-141}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-167}:\\ \;\;\;\;-2 \cdot \frac{y \cdot \left(z \cdot \left(z \cdot x\right)\right)}{t \cdot a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.8e-141)
   (* y (- x))
   (if (<= z 1.25e-167) (* -2.0 (/ (* y (* z (* z x))) (* t a))) (* y x))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.8e-141) {
		tmp = y * -x;
	} else if (z <= 1.25e-167) {
		tmp = -2.0 * ((y * (z * (z * x))) / (t * a));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-2.8d-141)) then
        tmp = y * -x
    else if (z <= 1.25d-167) then
        tmp = (-2.0d0) * ((y * (z * (z * x))) / (t * a))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.8e-141) {
		tmp = y * -x;
	} else if (z <= 1.25e-167) {
		tmp = -2.0 * ((y * (z * (z * x))) / (t * a));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.8e-141:
		tmp = y * -x
	elif z <= 1.25e-167:
		tmp = -2.0 * ((y * (z * (z * x))) / (t * a))
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.8e-141)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.25e-167)
		tmp = Float64(-2.0 * Float64(Float64(y * Float64(z * Float64(z * x))) / Float64(t * a)));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.8e-141)
		tmp = y * -x;
	elseif (z <= 1.25e-167)
		tmp = -2.0 * ((y * (z * (z * x))) / (t * a));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.8e-141], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.25e-167], N[(-2.0 * N[(N[(y * N[(z * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.80000000000000012e-141

   1. Initial program 63.3%

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

      1. *-commutative 63.3%

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

      2. associate-*l* 61.1%

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

      3. associate-*r/ 64.1%

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

   3. Simplified 64.1%

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

   4. Taylor expanded in z around -inf 86.5%

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

      1. neg-mul-1 86.5%

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

   6. Simplified 86.5%

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

   if -2.80000000000000012e-141 < z < 1.25000000000000005e-167

   1. Initial program 80.5%

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

   2. Taylor expanded in z around inf 47.6%

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

   3. Taylor expanded in z around 0 47.5%

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

      1. times-frac 47.5%

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

      2. unpow2 47.5%

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

   5. Simplified 47.5%

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

      1. frac-times 47.5%

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

      2. associate-*l* 47.6%

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

      3. *-commutative 47.6%

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

   7. Applied egg-rr 47.6%

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

   if 1.25000000000000005e-167 < z

   1. Initial program 63.6%

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

      1. *-commutative 63.6%

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

      2. associate-*l* 62.5%

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

      3. associate-*r/ 62.9%

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

   3. Simplified 62.9%

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

   4. Taylor expanded in z around inf 85.7%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-141}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-167}:\\ \;\;\;\;-2 \cdot \frac{y \cdot \left(z \cdot \left(z \cdot x\right)\right)}{t \cdot a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 8: 75.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{-139}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-263}:\\ \;\;\;\;2 \cdot \left(\frac{y}{a} \cdot \left(x \cdot \frac{z \cdot z}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -8.5e-139)
   (* y (- x))
   (if (<= z 4.2e-263) (* 2.0 (* (/ y a) (* x (/ (* z z) t)))) (* y x))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.5e-139) {
		tmp = y * -x;
	} else if (z <= 4.2e-263) {
		tmp = 2.0 * ((y / a) * (x * ((z * z) / t)));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-8.5d-139)) then
        tmp = y * -x
    else if (z <= 4.2d-263) then
        tmp = 2.0d0 * ((y / a) * (x * ((z * z) / t)))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.5e-139) {
		tmp = y * -x;
	} else if (z <= 4.2e-263) {
		tmp = 2.0 * ((y / a) * (x * ((z * z) / t)));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -8.5e-139:
		tmp = y * -x
	elif z <= 4.2e-263:
		tmp = 2.0 * ((y / a) * (x * ((z * z) / t)))
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -8.5e-139)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 4.2e-263)
		tmp = Float64(2.0 * Float64(Float64(y / a) * Float64(x * Float64(Float64(z * z) / t))));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -8.5e-139)
		tmp = y * -x;
	elseif (z <= 4.2e-263)
		tmp = 2.0 * ((y / a) * (x * ((z * z) / t)));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -8.5e-139], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 4.2e-263], N[(2.0 * N[(N[(y / a), $MachinePrecision] * N[(x * N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.5000000000000003e-139

   1. Initial program 63.3%

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

      1. *-commutative 63.3%

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

      2. associate-*l* 61.1%

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

      3. associate-*r/ 64.1%

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

   3. Simplified 64.1%

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

   4. Taylor expanded in z around -inf 86.5%

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

      1. neg-mul-1 86.5%

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

   6. Simplified 86.5%

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

   if -8.5000000000000003e-139 < z < 4.20000000000000005e-263

   1. Initial program 76.1%

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

      1. *-commutative 76.1%

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

      2. associate-*l* 79.7%

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

      3. associate-*r/ 82.2%

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

   3. Simplified 82.2%

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

      1. associate-*r/ 79.7%

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

      2. associate-*r* 76.1%

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

      3. *-commutative 76.1%

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

      4. expm1-log1p-u 72.2%

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

      5. expm1-udef 57.9%

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

      6. div-inv 57.9%

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

      7. associate-*l* 57.9%

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

      8. div-inv 57.9%

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

   5. Applied egg-rr 57.9%

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

      1. expm1-def 74.5%

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

      2. expm1-log1p 78.4%

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

      3. associate-*r* 75.8%

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

      4. associate-*r/ 78.5%

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

      5. associate-*l/ 78.1%

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

      6. *-commutative 78.1%

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

   7. Simplified 78.1%

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

   8. Taylor expanded in z around -inf 58.7%

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

   9. Taylor expanded in a around inf 57.9%

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

       1. times-frac 58.0%

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

       2. associate-/l* 54.0%

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

       3. associate-/r/ 58.0%

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

       4. unpow2 58.0%

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

   11. Simplified 58.0%

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

   if 4.20000000000000005e-263 < 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* 64.9%

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

      3. associate-*r/ 66.0%

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

   3. Simplified 66.0%

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

   4. Taylor expanded in z around inf 80.1%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{-139}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-263}:\\ \;\;\;\;2 \cdot \left(\frac{y}{a} \cdot \left(x \cdot \frac{z \cdot z}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 9: 75.5% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-141}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-263}:\\ \;\;\;\;x \cdot \left(z \cdot \left(2 \cdot \frac{y}{\frac{a}{\frac{z}{t}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4.8e-141)
   (* y (- x))
   (if (<= z 4.2e-263) (* x (* z (* 2.0 (/ y (/ a (/ z t)))))) (* y x))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.8e-141) {
		tmp = y * -x;
	} else if (z <= 4.2e-263) {
		tmp = x * (z * (2.0 * (y / (a / (z / t)))));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-4.8d-141)) then
        tmp = y * -x
    else if (z <= 4.2d-263) then
        tmp = x * (z * (2.0d0 * (y / (a / (z / t)))))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.8e-141) {
		tmp = y * -x;
	} else if (z <= 4.2e-263) {
		tmp = x * (z * (2.0 * (y / (a / (z / t)))));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -4.8e-141:
		tmp = y * -x
	elif z <= 4.2e-263:
		tmp = x * (z * (2.0 * (y / (a / (z / t)))))
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4.8e-141)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 4.2e-263)
		tmp = Float64(x * Float64(z * Float64(2.0 * Float64(y / Float64(a / Float64(z / t))))));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -4.8e-141)
		tmp = y * -x;
	elseif (z <= 4.2e-263)
		tmp = x * (z * (2.0 * (y / (a / (z / t)))));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4.8e-141], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 4.2e-263], N[(x * N[(z * N[(2.0 * N[(y / N[(a / N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.8000000000000002e-141

   1. Initial program 63.3%

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

      1. *-commutative 63.3%

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

      2. associate-*l* 61.1%

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

      3. associate-*r/ 64.1%

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

   3. Simplified 64.1%

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

   4. Taylor expanded in z around -inf 86.5%

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

      1. neg-mul-1 86.5%

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

   6. Simplified 86.5%

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

   if -4.8000000000000002e-141 < z < 4.20000000000000005e-263

   1. Initial program 76.1%

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

      1. *-commutative 76.1%

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

      2. associate-*l* 79.7%

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

      3. associate-*r/ 82.2%

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

   3. Simplified 82.2%

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

      1. associate-*r/ 79.7%

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

      2. associate-*r* 76.1%

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

      3. *-commutative 76.1%

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

      4. expm1-log1p-u 72.2%

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

      5. expm1-udef 57.9%

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

      6. div-inv 57.9%

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

      7. associate-*l* 57.9%

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

      8. div-inv 57.9%

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

   5. Applied egg-rr 57.9%

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

      1. expm1-def 74.5%

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

      2. expm1-log1p 78.4%

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

      3. associate-*r* 75.8%

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

      4. associate-*r/ 78.5%

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

      5. associate-*l/ 78.1%

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

      6. *-commutative 78.1%

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

   7. Simplified 78.1%

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

   8. Taylor expanded in z around -inf 58.7%

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

   9. Taylor expanded in a around inf 58.4%

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

       1. associate-/l* 58.5%

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

       2. associate-/l* 58.7%

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

   11. Simplified 58.7%

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

   if 4.20000000000000005e-263 < 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* 64.9%

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

      3. associate-*r/ 66.0%

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

   3. Simplified 66.0%

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

   4. Taylor expanded in z around inf 80.1%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-141}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-263}:\\ \;\;\;\;x \cdot \left(z \cdot \left(2 \cdot \frac{y}{\frac{a}{\frac{z}{t}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 10: 77.7% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-140}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \left(\frac{t}{z} \cdot \frac{a}{z}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4e-140)
   (* y (- x))
   (/ (* y x) (+ 1.0 (* -0.5 (* (/ t z) (/ a z)))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4e-140) {
		tmp = y * -x;
	} else {
		tmp = (y * x) / (1.0 + (-0.5 * ((t / z) * (a / z))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-4d-140)) then
        tmp = y * -x
    else
        tmp = (y * x) / (1.0d0 + ((-0.5d0) * ((t / z) * (a / z))))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4e-140) {
		tmp = y * -x;
	} else {
		tmp = (y * x) / (1.0 + (-0.5 * ((t / z) * (a / z))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -4e-140:
		tmp = y * -x
	else:
		tmp = (y * x) / (1.0 + (-0.5 * ((t / z) * (a / z))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4e-140)
		tmp = Float64(y * Float64(-x));
	else
		tmp = Float64(Float64(y * x) / Float64(1.0 + Float64(-0.5 * Float64(Float64(t / z) * Float64(a / z)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -4e-140)
		tmp = y * -x;
	else
		tmp = (y * x) / (1.0 + (-0.5 * ((t / z) * (a / z))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4e-140], N[(y * (-x)), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(N[(t / z), $MachinePrecision] * N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.9999999999999999e-140

   1. Initial program 63.3%

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

      1. *-commutative 63.3%

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

      2. associate-*l* 61.1%

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

      3. associate-*r/ 64.1%

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

   3. Simplified 64.1%

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

   4. Taylor expanded in z around -inf 86.5%

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

      1. neg-mul-1 86.5%

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

   6. Simplified 86.5%

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

   if -3.9999999999999999e-140 < z

   1. Initial program 67.8%

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

      1. associate-/l* 69.0%

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

   3. Simplified 69.0%

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

   4. Taylor expanded in z around inf 74.7%

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

      1. unpow2 74.7%

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

      2. associate-*r/ 74.7%

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

   6. Simplified 74.7%

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

   7. Taylor expanded in a around 0 74.7%

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

      1. *-commutative 74.7%

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

      2. unpow2 74.7%

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

      3. times-frac 78.7%

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

   9. Simplified 78.7%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-140}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + -0.5 \cdot \left(\frac{t}{z} \cdot \frac{a}{z}\right)}\\ \end{array} \]

Alternative 11: 73.8% accurate, 10.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-171}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-263}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.9e-171)
   (* y (- x))
   (if (<= z 4.2e-263) (* x (/ (* z y) z)) (* y x))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.9e-171) {
		tmp = y * -x;
	} else if (z <= 4.2e-263) {
		tmp = x * ((z * y) / z);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.9d-171)) then
        tmp = y * -x
    else if (z <= 4.2d-263) then
        tmp = x * ((z * y) / z)
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.9e-171) {
		tmp = y * -x;
	} else if (z <= 4.2e-263) {
		tmp = x * ((z * y) / z);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.9e-171:
		tmp = y * -x
	elif z <= 4.2e-263:
		tmp = x * ((z * y) / z)
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.9e-171)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 4.2e-263)
		tmp = Float64(x * Float64(Float64(z * y) / z));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.9e-171)
		tmp = y * -x;
	elseif (z <= 4.2e-263)
		tmp = x * ((z * y) / z);
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.9e-171], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 4.2e-263], N[(x * N[(N[(z * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.90000000000000011e-171

   1. Initial program 64.7%

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

      1. *-commutative 64.7%

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

      2. associate-*l* 61.8%

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

      3. associate-*r/ 64.6%

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

   3. Simplified 64.6%

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

   4. Taylor expanded in z around -inf 82.2%

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

      1. neg-mul-1 82.2%

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

   6. Simplified 82.2%

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

   if -1.90000000000000011e-171 < z < 4.20000000000000005e-263

   1. Initial program 72.8%

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

      1. *-commutative 72.8%

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

      2. associate-*l* 82.4%

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

      3. associate-*r/ 85.8%

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

   3. Simplified 85.8%

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

      1. associate-*r/ 82.4%

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

      2. associate-*r* 72.8%

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

      3. *-commutative 72.8%

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

      4. expm1-log1p-u 67.5%

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

      5. expm1-udef 62.3%

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

      6. div-inv 62.3%

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

      7. associate-*l* 62.3%

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

      8. div-inv 62.3%

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

   5. Applied egg-rr 62.3%

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

      1. expm1-def 70.5%

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

      2. expm1-log1p 75.7%

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

      3. associate-*r* 77.0%

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

      4. associate-*r/ 84.6%

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

      5. associate-*l/ 84.8%

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

      6. *-commutative 84.8%

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

   7. Simplified 84.8%

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

   8. Taylor expanded in z around inf 17.7%

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

      1. associate-*l/ 48.0%

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

   10. Applied egg-rr 48.0%

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

   if 4.20000000000000005e-263 < 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* 64.9%

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

      3. associate-*r/ 66.0%

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

   3. Simplified 66.0%

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

   4. Taylor expanded in z around inf 80.1%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-171}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-263}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 12: 72.7% accurate, 18.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-311}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4e-311) (* y (- x)) (* y x)))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4e-311) {
		tmp = y * -x;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-4d-311)) then
        tmp = y * -x
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4e-311) {
		tmp = y * -x;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -4e-311:
		tmp = y * -x
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4e-311)
		tmp = Float64(y * Float64(-x));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -4e-311)
		tmp = y * -x;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4e-311], N[(y * (-x)), $MachinePrecision], N[(y * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.99999999999979e-311

   1. Initial program 66.1%

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

      1. *-commutative 66.1%

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

      2. associate-*l* 63.5%

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

      3. associate-*r/ 66.1%

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

   3. Simplified 66.1%

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

   4. Taylor expanded in z around -inf 75.5%

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

      1. neg-mul-1 75.5%

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

   6. Simplified 75.5%

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

   if -3.99999999999979e-311 < z

   1. Initial program 65.9%

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

      1. *-commutative 65.9%

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

      2. associate-*l* 66.2%

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

      3. associate-*r/ 67.7%

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

   3. Simplified 67.7%

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

   4. Taylor expanded in z around inf 76.9%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-311}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 13: 42.7% accurate, 37.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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* 64.9%

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

   3. associate-*r/ 66.9%

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

3. Simplified 66.9%

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

4. Taylor expanded in z around inf 46.4%

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

5. Final simplification 46.4%

   \[\leadsto y \cdot x \]

Developer target: 89.5% 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 2023224 
(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)))))