Data.Colour.Matrix:inverse from colour-2.3.3, B

Percentage Accurate: 91.4% → 95.5%

Time: 9.2s

Alternatives: 11

Speedup: 0.7×

• 28.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot y - z \cdot t}{a} \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return ((x * y) - (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 * t)) / a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot y - z \cdot t}{a} \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return ((x * y) - (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 * t)) / a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+207} \lor \neg \left(t_1 \leq 5 \cdot 10^{+166}\right):\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \frac{y}{\frac{a}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))))
   (if (or (<= t_1 -2e+207) (not (<= t_1 5e+166)))
     (fma -1.0 (/ t (/ a z)) (/ y (/ a x)))
     (/ t_1 a))))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if ((t_1 <= -2e+207) || !(t_1 <= 5e+166)) {
		tmp = fma(-1.0, (t / (a / z)), (y / (a / x)));
	} else {
		tmp = t_1 / a;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if ((t_1 <= -2e+207) || !(t_1 <= 5e+166))
		tmp = fma(-1.0, Float64(t / Float64(a / z)), Float64(y / Float64(a / x)));
	else
		tmp = Float64(t_1 / a);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+207], N[Not[LessEqual[t$95$1, 5e+166]], $MachinePrecision]], N[(-1.0 * N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision] + N[(y / N[(a / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -2.0000000000000001e207 or 5.0000000000000002e166 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 80.5%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 78.0%

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

      1. fma-def 78.0%

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

      2. associate-/l* 83.5%

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

      3. associate-/l* 93.1%

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

   4. Simplified 93.1%

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

   if -2.0000000000000001e207 < (-.f64 (*.f64 x y) (*.f64 z t)) < 5.0000000000000002e166

   1. Initial program 98.3%

      \[\frac{x \cdot y - z \cdot t}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -2 \cdot 10^{+207} \lor \neg \left(x \cdot y - z \cdot t \leq 5 \cdot 10^{+166}\right):\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \frac{y}{\frac{a}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \]

Alternative 2: 96.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 2 \cdot 10^{+248}\right):\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, z \cdot \left(-t\right)\right)}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 2e+248)))
     (- (/ x (/ a y)) (/ z (/ a t)))
     (/ (fma y x (* z (- t))) a))))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 2e+248)) {
		tmp = (x / (a / y)) - (z / (a / t));
	} else {
		tmp = fma(y, x, (z * -t)) / a;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 2e+248))
		tmp = Float64(Float64(x / Float64(a / y)) - Float64(z / Float64(a / t)));
	else
		tmp = Float64(fma(y, x, Float64(z * Float64(-t))) / a);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 2e+248]], $MachinePrecision]], N[(N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision] - N[(z / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * x + N[(z * (-t)), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -inf.0 or 2.00000000000000009e248 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 70.3%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Step-by-step derivation

      1. div-sub 66.4%

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

      2. associate-/l* 78.8%

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

      3. associate-/l* 93.4%

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

   3. Applied egg-rr 93.4%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 2.00000000000000009e248

   1. Initial program 98.6%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 98.6%

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

      1. +-commutative 98.6%

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

      2. mul-1-neg 98.6%

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

      3. sub-neg 98.6%

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

      4. div-sub 98.6%

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

      5. fma-neg 98.6%

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

      6. distribute-rgt-neg-out 98.6%

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

   4. Simplified 98.6%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty \lor \neg \left(x \cdot y - z \cdot t \leq 2 \cdot 10^{+248}\right):\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, z \cdot \left(-t\right)\right)}{a}\\ \end{array} \]

Alternative 3: 96.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 2 \cdot 10^{+248}\right):\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 2e+248)))
     (- (/ x (/ a y)) (/ z (/ a t)))
     (/ t_1 a))))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 2e+248)) {
		tmp = (x / (a / y)) - (z / (a / t));
	} else {
		tmp = t_1 / a;
	}
	return tmp;
}

Java

assert x < y;
assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 2e+248)) {
		tmp = (x / (a / y)) - (z / (a / t));
	} else {
		tmp = t_1 / a;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[z, t] = sort([z, t])
def code(x, y, z, t, a):
	t_1 = (x * y) - (z * t)
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 2e+248):
		tmp = (x / (a / y)) - (z / (a / t))
	else:
		tmp = t_1 / a
	return tmp

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 2e+248))
		tmp = Float64(Float64(x / Float64(a / y)) - Float64(z / Float64(a / t)));
	else
		tmp = Float64(t_1 / a);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (x * y) - (z * t);
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 2e+248)))
		tmp = (x / (a / y)) - (z / (a / t));
	else
		tmp = t_1 / a;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 2e+248]], $MachinePrecision]], N[(N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision] - N[(z / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -inf.0 or 2.00000000000000009e248 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 70.3%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Step-by-step derivation

      1. div-sub 66.4%

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

      2. associate-/l* 78.8%

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

      3. associate-/l* 93.4%

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

   3. Applied egg-rr 93.4%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 2.00000000000000009e248

   1. Initial program 98.6%

      \[\frac{x \cdot y - z \cdot t}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty \lor \neg \left(x \cdot y - z \cdot t \leq 2 \cdot 10^{+248}\right):\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \]

Alternative 4: 68.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.95 \cdot 10^{-56}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{+16}:\\ \;\;\;\;z \cdot \frac{t}{-a}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+62}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;y \leq 10^{+88}:\\ \;\;\;\;\frac{1}{\frac{\frac{a}{-t}}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= y -2.95e-56)
   (/ y (/ a x))
   (if (<= y 4.3e+16)
     (* z (/ t (- a)))
     (if (<= y 3.4e+62)
       (* x (/ y a))
       (if (<= y 1e+88) (/ 1.0 (/ (/ a (- t)) z)) (/ x (/ a y)))))))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -2.95e-56) {
		tmp = y / (a / x);
	} else if (y <= 4.3e+16) {
		tmp = z * (t / -a);
	} else if (y <= 3.4e+62) {
		tmp = x * (y / a);
	} else if (y <= 1e+88) {
		tmp = 1.0 / ((a / -t) / z);
	} else {
		tmp = x / (a / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t 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 (y <= (-2.95d-56)) then
        tmp = y / (a / x)
    else if (y <= 4.3d+16) then
        tmp = z * (t / -a)
    else if (y <= 3.4d+62) then
        tmp = x * (y / a)
    else if (y <= 1d+88) then
        tmp = 1.0d0 / ((a / -t) / z)
    else
        tmp = x / (a / y)
    end if
    code = tmp
end function

Java

assert x < y;
assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -2.95e-56) {
		tmp = y / (a / x);
	} else if (y <= 4.3e+16) {
		tmp = z * (t / -a);
	} else if (y <= 3.4e+62) {
		tmp = x * (y / a);
	} else if (y <= 1e+88) {
		tmp = 1.0 / ((a / -t) / z);
	} else {
		tmp = x / (a / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if y <= -2.95e-56:
		tmp = y / (a / x)
	elif y <= 4.3e+16:
		tmp = z * (t / -a)
	elif y <= 3.4e+62:
		tmp = x * (y / a)
	elif y <= 1e+88:
		tmp = 1.0 / ((a / -t) / z)
	else:
		tmp = x / (a / y)
	return tmp

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= -2.95e-56)
		tmp = Float64(y / Float64(a / x));
	elseif (y <= 4.3e+16)
		tmp = Float64(z * Float64(t / Float64(-a)));
	elseif (y <= 3.4e+62)
		tmp = Float64(x * Float64(y / a));
	elseif (y <= 1e+88)
		tmp = Float64(1.0 / Float64(Float64(a / Float64(-t)) / z));
	else
		tmp = Float64(x / Float64(a / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (y <= -2.95e-56)
		tmp = y / (a / x);
	elseif (y <= 4.3e+16)
		tmp = z * (t / -a);
	elseif (y <= 3.4e+62)
		tmp = x * (y / a);
	elseif (y <= 1e+88)
		tmp = 1.0 / ((a / -t) / z);
	else
		tmp = x / (a / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[y, -2.95e-56], N[(y / N[(a / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.3e+16], N[(z * N[(t / (-a)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.4e+62], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e+88], N[(1.0 / N[(N[(a / (-t)), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -2.9499999999999999e-56

   1. Initial program 85.6%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around inf 59.4%

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

      1. associate-/l* 63.9%

         \[\leadsto \color{blue}{\frac{y}{\frac{a}{x}}} \]

   4. Simplified 63.9%

      \[\leadsto \color{blue}{\frac{y}{\frac{a}{x}}} \]

   if -2.9499999999999999e-56 < y < 4.3e16

   1. Initial program 94.6%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 69.1%

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

      1. associate-*r/ 69.1%

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

      2. mul-1-neg 69.1%

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

      3. distribute-rgt-neg-out 69.1%

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

      4. *-commutative 69.1%

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

      5. associate-/l* 66.8%

         \[\leadsto \color{blue}{\frac{-z}{\frac{a}{t}}} \]

      6. associate-/r/ 70.1%

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

   4. Simplified 70.1%

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

      1. *-commutative 70.1%

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

      2. frac-2neg 70.1%

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

      3. remove-double-neg 70.1%

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

      4. associate-*r/ 69.1%

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

   6. Applied egg-rr 69.1%

      \[\leadsto \color{blue}{\frac{t \cdot z}{-a}} \]
   7. Step-by-step derivation

      1. associate-/l* 70.0%

         \[\leadsto \color{blue}{\frac{t}{\frac{-a}{z}}} \]

      2. associate-/r/ 67.8%

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

   8. Simplified 67.8%

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

   if 4.3e16 < y < 3.40000000000000014e62

   1. Initial program 99.1%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 99.1%

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

      1. +-commutative 99.1%

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

      2. mul-1-neg 99.1%

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

      3. sub-neg 99.1%

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

      4. div-sub 99.1%

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

      5. fma-neg 99.2%

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

      6. distribute-rgt-neg-out 99.2%

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

   4. Simplified 99.2%

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

   5. Taylor expanded in y around inf 72.2%

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

      1. associate-*l/ 72.7%

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

   7. Simplified 72.7%

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

   if 3.40000000000000014e62 < y < 9.99999999999999959e87

   1. Initial program 71.9%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 44.8%

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

      1. associate-*r/ 44.8%

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

      2. mul-1-neg 44.8%

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

      3. distribute-rgt-neg-out 44.8%

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

      4. *-commutative 44.8%

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

      5. associate-/l* 71.6%

         \[\leadsto \color{blue}{\frac{-z}{\frac{a}{t}}} \]

      6. associate-/r/ 71.8%

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

   4. Simplified 71.8%

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

      1. associate-*l/ 44.8%

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

      2. associate-/l* 71.6%

         \[\leadsto \color{blue}{\frac{-z}{\frac{a}{t}}} \]

      3. add-sqr-sqrt 42.6%

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

      4. sqrt-unprod 29.8%

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

      5. sqr-neg 29.8%

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

      6. sqrt-unprod 0.7%

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

      7. add-sqr-sqrt 0.8%

         \[\leadsto \frac{\color{blue}{z}}{\frac{a}{t}} \]

   6. Applied egg-rr 0.8%

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

      1. *-un-lft-identity 0.8%

         \[\leadsto \frac{\color{blue}{1 \cdot z}}{\frac{a}{t}} \]

      2. div-inv 0.8%

         \[\leadsto \frac{1 \cdot z}{\color{blue}{a \cdot \frac{1}{t}}} \]

      3. times-frac 0.7%

         \[\leadsto \color{blue}{\frac{1}{a} \cdot \frac{z}{\frac{1}{t}}} \]

      4. frac-2neg 0.7%

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

      5. metadata-eval 0.7%

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

      6. add-sqr-sqrt 0.4%

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

      7. sqrt-unprod 2.1%

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

      8. sqr-neg 2.1%

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

      9. sqrt-unprod 1.8%

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

      10. add-sqr-sqrt 44.3%

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

   8. Applied egg-rr 44.3%

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

      1. frac-times 71.6%

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

      2. *-un-lft-identity 71.6%

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

      3. clear-num 71.6%

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

      4. frac-2neg 71.6%

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

      5. metadata-eval 71.6%

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

      6. un-div-inv 71.6%

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

   10. Applied egg-rr 71.6%

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

   if 9.99999999999999959e87 < y

   1. Initial program 88.0%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around inf 76.3%

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

      1. associate-*r/ 78.3%

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

   4. Simplified 78.3%

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

      1. associate-*r/ 76.3%

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

      2. *-commutative 76.3%

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

      3. associate-/l* 84.3%

         \[\leadsto \color{blue}{\frac{x}{\frac{a}{y}}} \]

   6. Applied egg-rr 84.3%

      \[\leadsto \color{blue}{\frac{x}{\frac{a}{y}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.95 \cdot 10^{-56}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{+16}:\\ \;\;\;\;z \cdot \frac{t}{-a}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+62}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;y \leq 10^{+88}:\\ \;\;\;\;\frac{1}{\frac{\frac{a}{-t}}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \]

Alternative 5: 68.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := z \cdot \frac{t}{-a}\\ \mathbf{if}\;y \leq -1.9 \cdot 10^{-55}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+61}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;y \leq 1.28 \cdot 10^{+91}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* z (/ t (- a)))))
   (if (<= y -1.9e-55)
     (/ y (/ a x))
     (if (<= y 4.4e+15)
       t_1
       (if (<= y 5e+61)
         (* x (/ y a))
         (if (<= y 1.28e+91) t_1 (/ x (/ a y))))))))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double t_1 = z * (t / -a);
	double tmp;
	if (y <= -1.9e-55) {
		tmp = y / (a / x);
	} else if (y <= 4.4e+15) {
		tmp = t_1;
	} else if (y <= 5e+61) {
		tmp = x * (y / a);
	} else if (y <= 1.28e+91) {
		tmp = t_1;
	} else {
		tmp = x / (a / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t 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 = z * (t / -a)
    if (y <= (-1.9d-55)) then
        tmp = y / (a / x)
    else if (y <= 4.4d+15) then
        tmp = t_1
    else if (y <= 5d+61) then
        tmp = x * (y / a)
    else if (y <= 1.28d+91) then
        tmp = t_1
    else
        tmp = x / (a / y)
    end if
    code = tmp
end function

Java

assert x < y;
assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = z * (t / -a);
	double tmp;
	if (y <= -1.9e-55) {
		tmp = y / (a / x);
	} else if (y <= 4.4e+15) {
		tmp = t_1;
	} else if (y <= 5e+61) {
		tmp = x * (y / a);
	} else if (y <= 1.28e+91) {
		tmp = t_1;
	} else {
		tmp = x / (a / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[z, t] = sort([z, t])
def code(x, y, z, t, a):
	t_1 = z * (t / -a)
	tmp = 0
	if y <= -1.9e-55:
		tmp = y / (a / x)
	elif y <= 4.4e+15:
		tmp = t_1
	elif y <= 5e+61:
		tmp = x * (y / a)
	elif y <= 1.28e+91:
		tmp = t_1
	else:
		tmp = x / (a / y)
	return tmp

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	t_1 = Float64(z * Float64(t / Float64(-a)))
	tmp = 0.0
	if (y <= -1.9e-55)
		tmp = Float64(y / Float64(a / x));
	elseif (y <= 4.4e+15)
		tmp = t_1;
	elseif (y <= 5e+61)
		tmp = Float64(x * Float64(y / a));
	elseif (y <= 1.28e+91)
		tmp = t_1;
	else
		tmp = Float64(x / Float64(a / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = z * (t / -a);
	tmp = 0.0;
	if (y <= -1.9e-55)
		tmp = y / (a / x);
	elseif (y <= 4.4e+15)
		tmp = t_1;
	elseif (y <= 5e+61)
		tmp = x * (y / a);
	elseif (y <= 1.28e+91)
		tmp = t_1;
	else
		tmp = x / (a / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z * N[(t / (-a)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.9e-55], N[(y / N[(a / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.4e+15], t$95$1, If[LessEqual[y, 5e+61], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.28e+91], t$95$1, N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.8999999999999998e-55

   1. Initial program 85.6%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around inf 59.4%

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

      1. associate-/l* 63.9%

         \[\leadsto \color{blue}{\frac{y}{\frac{a}{x}}} \]

   4. Simplified 63.9%

      \[\leadsto \color{blue}{\frac{y}{\frac{a}{x}}} \]

   if -1.8999999999999998e-55 < y < 4.4e15 or 5.00000000000000018e61 < y < 1.27999999999999999e91

   1. Initial program 93.2%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 67.7%

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

      1. associate-*r/ 67.7%

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

      2. mul-1-neg 67.7%

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

      3. distribute-rgt-neg-out 67.7%

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

      4. *-commutative 67.7%

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

      5. associate-/l* 67.1%

         \[\leadsto \color{blue}{\frac{-z}{\frac{a}{t}}} \]

      6. associate-/r/ 70.2%

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

   4. Simplified 70.2%

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

      1. *-commutative 70.2%

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

      2. frac-2neg 70.2%

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

      3. remove-double-neg 70.2%

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

      4. associate-*r/ 67.7%

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

   6. Applied egg-rr 67.7%

      \[\leadsto \color{blue}{\frac{t \cdot z}{-a}} \]
   7. Step-by-step derivation

      1. associate-/l* 70.1%

         \[\leadsto \color{blue}{\frac{t}{\frac{-a}{z}}} \]

      2. associate-/r/ 68.0%

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

   8. Simplified 68.0%

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

   if 4.4e15 < y < 5.00000000000000018e61

   1. Initial program 99.1%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 99.1%

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

      1. +-commutative 99.1%

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

      2. mul-1-neg 99.1%

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

      3. sub-neg 99.1%

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

      4. div-sub 99.1%

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

      5. fma-neg 99.2%

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

      6. distribute-rgt-neg-out 99.2%

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

   4. Simplified 99.2%

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

   5. Taylor expanded in y around inf 72.2%

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

      1. associate-*l/ 72.7%

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

   7. Simplified 72.7%

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

   if 1.27999999999999999e91 < y

   1. Initial program 88.0%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around inf 76.3%

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

      1. associate-*r/ 78.3%

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

   4. Simplified 78.3%

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

      1. associate-*r/ 76.3%

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

      2. *-commutative 76.3%

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

      3. associate-/l* 84.3%

         \[\leadsto \color{blue}{\frac{x}{\frac{a}{y}}} \]

   6. Applied egg-rr 84.3%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-55}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+15}:\\ \;\;\;\;z \cdot \frac{t}{-a}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+61}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;y \leq 1.28 \cdot 10^{+91}:\\ \;\;\;\;z \cdot \frac{t}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \]

Alternative 6: 68.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{-70}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+16}:\\ \;\;\;\;z \cdot \frac{t}{-a}\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{+61}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;y \leq 10^{+88}:\\ \;\;\;\;t \cdot \left(-\frac{z}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= y -1.2e-70)
   (/ y (/ a x))
   (if (<= y 1.05e+16)
     (* z (/ t (- a)))
     (if (<= y 5.1e+61)
       (* x (/ y a))
       (if (<= y 1e+88) (* t (- (/ z a))) (/ x (/ a y)))))))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -1.2e-70) {
		tmp = y / (a / x);
	} else if (y <= 1.05e+16) {
		tmp = z * (t / -a);
	} else if (y <= 5.1e+61) {
		tmp = x * (y / a);
	} else if (y <= 1e+88) {
		tmp = t * -(z / a);
	} else {
		tmp = x / (a / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t 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 (y <= (-1.2d-70)) then
        tmp = y / (a / x)
    else if (y <= 1.05d+16) then
        tmp = z * (t / -a)
    else if (y <= 5.1d+61) then
        tmp = x * (y / a)
    else if (y <= 1d+88) then
        tmp = t * -(z / a)
    else
        tmp = x / (a / y)
    end if
    code = tmp
end function

Java

assert x < y;
assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -1.2e-70) {
		tmp = y / (a / x);
	} else if (y <= 1.05e+16) {
		tmp = z * (t / -a);
	} else if (y <= 5.1e+61) {
		tmp = x * (y / a);
	} else if (y <= 1e+88) {
		tmp = t * -(z / a);
	} else {
		tmp = x / (a / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if y <= -1.2e-70:
		tmp = y / (a / x)
	elif y <= 1.05e+16:
		tmp = z * (t / -a)
	elif y <= 5.1e+61:
		tmp = x * (y / a)
	elif y <= 1e+88:
		tmp = t * -(z / a)
	else:
		tmp = x / (a / y)
	return tmp

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= -1.2e-70)
		tmp = Float64(y / Float64(a / x));
	elseif (y <= 1.05e+16)
		tmp = Float64(z * Float64(t / Float64(-a)));
	elseif (y <= 5.1e+61)
		tmp = Float64(x * Float64(y / a));
	elseif (y <= 1e+88)
		tmp = Float64(t * Float64(-Float64(z / a)));
	else
		tmp = Float64(x / Float64(a / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (y <= -1.2e-70)
		tmp = y / (a / x);
	elseif (y <= 1.05e+16)
		tmp = z * (t / -a);
	elseif (y <= 5.1e+61)
		tmp = x * (y / a);
	elseif (y <= 1e+88)
		tmp = t * -(z / a);
	else
		tmp = x / (a / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[y, -1.2e-70], N[(y / N[(a / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.05e+16], N[(z * N[(t / (-a)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.1e+61], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e+88], N[(t * (-N[(z / a), $MachinePrecision])), $MachinePrecision], N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -1.2000000000000001e-70

   1. Initial program 86.0%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around inf 60.4%

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

      1. associate-/l* 64.8%

         \[\leadsto \color{blue}{\frac{y}{\frac{a}{x}}} \]

   4. Simplified 64.8%

      \[\leadsto \color{blue}{\frac{y}{\frac{a}{x}}} \]

   if -1.2000000000000001e-70 < y < 1.05e16

   1. Initial program 94.5%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 70.3%

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

      1. associate-*r/ 70.3%

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

      2. mul-1-neg 70.3%

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

      3. distribute-rgt-neg-out 70.3%

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

      4. *-commutative 70.3%

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

      5. associate-/l* 67.9%

         \[\leadsto \color{blue}{\frac{-z}{\frac{a}{t}}} \]

      6. associate-/r/ 71.3%

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

   4. Simplified 71.3%

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

      1. *-commutative 71.3%

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

      2. frac-2neg 71.3%

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

      3. remove-double-neg 71.3%

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

      4. associate-*r/ 70.3%

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

   6. Applied egg-rr 70.3%

      \[\leadsto \color{blue}{\frac{t \cdot z}{-a}} \]
   7. Step-by-step derivation

      1. associate-/l* 71.2%

         \[\leadsto \color{blue}{\frac{t}{\frac{-a}{z}}} \]

      2. associate-/r/ 69.0%

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

   8. Simplified 69.0%

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

   if 1.05e16 < y < 5.1000000000000001e61

   1. Initial program 99.1%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 99.1%

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

      1. +-commutative 99.1%

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

      2. mul-1-neg 99.1%

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

      3. sub-neg 99.1%

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

      4. div-sub 99.1%

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

      5. fma-neg 99.2%

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

      6. distribute-rgt-neg-out 99.2%

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

   4. Simplified 99.2%

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

   5. Taylor expanded in y around inf 72.2%

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

      1. associate-*l/ 72.7%

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

   7. Simplified 72.7%

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

   if 5.1000000000000001e61 < y < 9.99999999999999959e87

   1. Initial program 71.9%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 44.8%

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

      1. associate-*r/ 44.8%

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

      2. mul-1-neg 44.8%

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

      3. distribute-rgt-neg-out 44.8%

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

      4. *-commutative 44.8%

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

      5. associate-/l* 71.6%

         \[\leadsto \color{blue}{\frac{-z}{\frac{a}{t}}} \]

      6. associate-/r/ 71.8%

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

   4. Simplified 71.8%

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

   if 9.99999999999999959e87 < y

   1. Initial program 88.0%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around inf 76.3%

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

      1. associate-*r/ 78.3%

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

   4. Simplified 78.3%

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

      1. associate-*r/ 76.3%

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

      2. *-commutative 76.3%

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

      3. associate-/l* 84.3%

         \[\leadsto \color{blue}{\frac{x}{\frac{a}{y}}} \]

   6. Applied egg-rr 84.3%

      \[\leadsto \color{blue}{\frac{x}{\frac{a}{y}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{-70}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+16}:\\ \;\;\;\;z \cdot \frac{t}{-a}\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{+61}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;y \leq 10^{+88}:\\ \;\;\;\;t \cdot \left(-\frac{z}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \]

Alternative 7: 68.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-55}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+16}:\\ \;\;\;\;z \cdot \frac{t}{-a}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+62}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+88}:\\ \;\;\;\;\frac{z}{\frac{-a}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= y -1.8e-55)
   (/ y (/ a x))
   (if (<= y 7.5e+16)
     (* z (/ t (- a)))
     (if (<= y 2.6e+62)
       (* x (/ y a))
       (if (<= y 2.8e+88) (/ z (/ (- a) t)) (/ x (/ a y)))))))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -1.8e-55) {
		tmp = y / (a / x);
	} else if (y <= 7.5e+16) {
		tmp = z * (t / -a);
	} else if (y <= 2.6e+62) {
		tmp = x * (y / a);
	} else if (y <= 2.8e+88) {
		tmp = z / (-a / t);
	} else {
		tmp = x / (a / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t 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 (y <= (-1.8d-55)) then
        tmp = y / (a / x)
    else if (y <= 7.5d+16) then
        tmp = z * (t / -a)
    else if (y <= 2.6d+62) then
        tmp = x * (y / a)
    else if (y <= 2.8d+88) then
        tmp = z / (-a / t)
    else
        tmp = x / (a / y)
    end if
    code = tmp
end function

Java

assert x < y;
assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -1.8e-55) {
		tmp = y / (a / x);
	} else if (y <= 7.5e+16) {
		tmp = z * (t / -a);
	} else if (y <= 2.6e+62) {
		tmp = x * (y / a);
	} else if (y <= 2.8e+88) {
		tmp = z / (-a / t);
	} else {
		tmp = x / (a / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if y <= -1.8e-55:
		tmp = y / (a / x)
	elif y <= 7.5e+16:
		tmp = z * (t / -a)
	elif y <= 2.6e+62:
		tmp = x * (y / a)
	elif y <= 2.8e+88:
		tmp = z / (-a / t)
	else:
		tmp = x / (a / y)
	return tmp

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= -1.8e-55)
		tmp = Float64(y / Float64(a / x));
	elseif (y <= 7.5e+16)
		tmp = Float64(z * Float64(t / Float64(-a)));
	elseif (y <= 2.6e+62)
		tmp = Float64(x * Float64(y / a));
	elseif (y <= 2.8e+88)
		tmp = Float64(z / Float64(Float64(-a) / t));
	else
		tmp = Float64(x / Float64(a / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (y <= -1.8e-55)
		tmp = y / (a / x);
	elseif (y <= 7.5e+16)
		tmp = z * (t / -a);
	elseif (y <= 2.6e+62)
		tmp = x * (y / a);
	elseif (y <= 2.8e+88)
		tmp = z / (-a / t);
	else
		tmp = x / (a / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[y, -1.8e-55], N[(y / N[(a / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.5e+16], N[(z * N[(t / (-a)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.6e+62], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.8e+88], N[(z / N[((-a) / t), $MachinePrecision]), $MachinePrecision], N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -1.8e-55

   1. Initial program 85.6%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around inf 59.4%

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

      1. associate-/l* 63.9%

         \[\leadsto \color{blue}{\frac{y}{\frac{a}{x}}} \]

   4. Simplified 63.9%

      \[\leadsto \color{blue}{\frac{y}{\frac{a}{x}}} \]

   if -1.8e-55 < y < 7.5e16

   1. Initial program 94.6%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 69.1%

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

      1. associate-*r/ 69.1%

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

      2. mul-1-neg 69.1%

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

      3. distribute-rgt-neg-out 69.1%

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

      4. *-commutative 69.1%

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

      5. associate-/l* 66.8%

         \[\leadsto \color{blue}{\frac{-z}{\frac{a}{t}}} \]

      6. associate-/r/ 70.1%

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

   4. Simplified 70.1%

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

      1. *-commutative 70.1%

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

      2. frac-2neg 70.1%

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

      3. remove-double-neg 70.1%

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

      4. associate-*r/ 69.1%

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

   6. Applied egg-rr 69.1%

      \[\leadsto \color{blue}{\frac{t \cdot z}{-a}} \]
   7. Step-by-step derivation

      1. associate-/l* 70.0%

         \[\leadsto \color{blue}{\frac{t}{\frac{-a}{z}}} \]

      2. associate-/r/ 67.8%

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

   8. Simplified 67.8%

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

   if 7.5e16 < y < 2.59999999999999984e62

   1. Initial program 99.1%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 99.1%

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

      1. +-commutative 99.1%

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

      2. mul-1-neg 99.1%

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

      3. sub-neg 99.1%

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

      4. div-sub 99.1%

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

      5. fma-neg 99.2%

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

      6. distribute-rgt-neg-out 99.2%

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

   4. Simplified 99.2%

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

   5. Taylor expanded in y around inf 72.2%

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

      1. associate-*l/ 72.7%

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

   7. Simplified 72.7%

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

   if 2.59999999999999984e62 < y < 2.79999999999999989e88

   1. Initial program 71.9%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 44.8%

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

      1. associate-*r/ 44.8%

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

      2. mul-1-neg 44.8%

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

      3. distribute-rgt-neg-out 44.8%

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

      4. *-commutative 44.8%

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

      5. associate-/l* 71.6%

         \[\leadsto \color{blue}{\frac{-z}{\frac{a}{t}}} \]

      6. associate-/r/ 71.8%

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

   4. Simplified 71.8%

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

      1. associate-*l/ 44.8%

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

      2. associate-/l* 71.6%

         \[\leadsto \color{blue}{\frac{-z}{\frac{a}{t}}} \]

      3. add-sqr-sqrt 42.6%

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

      4. sqrt-unprod 29.8%

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

      5. sqr-neg 29.8%

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

      6. sqrt-unprod 0.7%

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

      7. add-sqr-sqrt 0.8%

         \[\leadsto \frac{\color{blue}{z}}{\frac{a}{t}} \]

   6. Applied egg-rr 0.8%

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

      1. *-un-lft-identity 0.8%

         \[\leadsto \frac{\color{blue}{1 \cdot z}}{\frac{a}{t}} \]

      2. div-inv 0.8%

         \[\leadsto \frac{1 \cdot z}{\color{blue}{a \cdot \frac{1}{t}}} \]

      3. times-frac 0.7%

         \[\leadsto \color{blue}{\frac{1}{a} \cdot \frac{z}{\frac{1}{t}}} \]

      4. frac-2neg 0.7%

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

      5. metadata-eval 0.7%

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

      6. add-sqr-sqrt 0.4%

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

      7. sqrt-unprod 2.1%

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

      8. sqr-neg 2.1%

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

      9. sqrt-unprod 1.8%

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

      10. add-sqr-sqrt 44.3%

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

   8. Applied egg-rr 44.3%

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

      1. clear-num 44.3%

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

      2. frac-times 71.6%

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

      3. *-un-lft-identity 71.6%

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

      4. div-inv 71.6%

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

      5. metadata-eval 71.6%

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

   10. Applied egg-rr 71.6%

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

       1. *-rgt-identity 71.6%

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

       2. *-commutative 71.6%

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

       3. associate-*l/ 71.6%

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

       4. neg-mul-1 71.6%

          \[\leadsto \frac{z}{\frac{\color{blue}{-a}}{t}} \]

   12. Simplified 71.6%

       \[\leadsto \color{blue}{\frac{z}{\frac{-a}{t}}} \]

   if 2.79999999999999989e88 < y

   1. Initial program 88.0%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around inf 76.3%

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

      1. associate-*r/ 78.3%

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

   4. Simplified 78.3%

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

      1. associate-*r/ 76.3%

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

      2. *-commutative 76.3%

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

      3. associate-/l* 84.3%

         \[\leadsto \color{blue}{\frac{x}{\frac{a}{y}}} \]

   6. Applied egg-rr 84.3%

      \[\leadsto \color{blue}{\frac{x}{\frac{a}{y}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-55}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+16}:\\ \;\;\;\;z \cdot \frac{t}{-a}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+62}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+88}:\\ \;\;\;\;\frac{z}{\frac{-a}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \]

Alternative 8: 93.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq 5 \cdot 10^{+248}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) 5e+248) (/ (- (* x y) (* z t)) a) (/ y (/ a x))))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= 5e+248) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = y / (a / x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t 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 ((x * y) <= 5d+248) then
        tmp = ((x * y) - (z * t)) / a
    else
        tmp = y / (a / x)
    end if
    code = tmp
end function

Java

assert x < y;
assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= 5e+248) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = y / (a / x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= 5e+248:
		tmp = ((x * y) - (z * t)) / a
	else:
		tmp = y / (a / x)
	return tmp

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= 5e+248)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a);
	else
		tmp = Float64(y / Float64(a / x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= 5e+248)
		tmp = ((x * y) - (z * t)) / a;
	else
		tmp = y / (a / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], 5e+248], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(y / N[(a / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 4.9999999999999996e248

   1. Initial program 92.7%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   if 4.9999999999999996e248 < (*.f64 x y)

   1. Initial program 69.4%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around inf 69.4%

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

      1. associate-/l* 96.3%

         \[\leadsto \color{blue}{\frac{y}{\frac{a}{x}}} \]

   4. Simplified 96.3%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq 5 \cdot 10^{+248}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \end{array} \]

Alternative 9: 52.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -8.4 \cdot 10^{+218}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= x -8.4e+218) (* y (/ x a)) (* x (/ y a))))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -8.4e+218) {
		tmp = y * (x / a);
	} else {
		tmp = x * (y / a);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t 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 (x <= (-8.4d+218)) then
        tmp = y * (x / a)
    else
        tmp = x * (y / a)
    end if
    code = tmp
end function

Java

assert x < y;
assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -8.4e+218) {
		tmp = y * (x / a);
	} else {
		tmp = x * (y / a);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if x <= -8.4e+218:
		tmp = y * (x / a)
	else:
		tmp = x * (y / a)
	return tmp

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (x <= -8.4e+218)
		tmp = Float64(y * Float64(x / a));
	else
		tmp = Float64(x * Float64(y / a));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (x <= -8.4e+218)
		tmp = y * (x / a);
	else
		tmp = x * (y / a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[x, -8.4e+218], N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.3999999999999995e218

   1. Initial program 89.4%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around inf 76.0%

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

      1. associate-*r/ 89.4%

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

   4. Simplified 89.4%

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

   if -8.3999999999999995e218 < x

   1. Initial program 90.2%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 89.3%

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

      1. +-commutative 89.3%

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

      2. mul-1-neg 89.3%

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

      3. sub-neg 89.3%

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

      4. div-sub 90.2%

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

      5. fma-neg 90.2%

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

      6. distribute-rgt-neg-out 90.2%

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

   4. Simplified 90.2%

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

   5. Taylor expanded in y around inf 47.4%

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

      1. associate-*l/ 49.6%

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

   7. Simplified 49.6%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.4 \cdot 10^{+218}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]

Alternative 10: 52.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+219}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= x -1.35e+219) (/ y (/ a x)) (* x (/ y a))))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -1.35e+219) {
		tmp = y / (a / x);
	} else {
		tmp = x * (y / a);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t 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 (x <= (-1.35d+219)) then
        tmp = y / (a / x)
    else
        tmp = x * (y / a)
    end if
    code = tmp
end function

Java

assert x < y;
assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -1.35e+219) {
		tmp = y / (a / x);
	} else {
		tmp = x * (y / a);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if x <= -1.35e+219:
		tmp = y / (a / x)
	else:
		tmp = x * (y / a)
	return tmp

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (x <= -1.35e+219)
		tmp = Float64(y / Float64(a / x));
	else
		tmp = Float64(x * Float64(y / a));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (x <= -1.35e+219)
		tmp = y / (a / x);
	else
		tmp = x * (y / a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[x, -1.35e+219], N[(y / N[(a / x), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3499999999999999e219

   1. Initial program 89.4%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around inf 76.0%

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

      1. associate-/l* 89.4%

         \[\leadsto \color{blue}{\frac{y}{\frac{a}{x}}} \]

   4. Simplified 89.4%

      \[\leadsto \color{blue}{\frac{y}{\frac{a}{x}}} \]

   if -1.3499999999999999e219 < x

   1. Initial program 90.2%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 89.3%

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

      1. +-commutative 89.3%

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

      2. mul-1-neg 89.3%

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

      3. sub-neg 89.3%

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

      4. div-sub 90.2%

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

      5. fma-neg 90.2%

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

      6. distribute-rgt-neg-out 90.2%

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

   4. Simplified 90.2%

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

   5. Taylor expanded in y around inf 47.4%

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

      1. associate-*l/ 49.6%

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

   7. Simplified 49.6%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+219}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]

Alternative 11: 51.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t 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 / a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.1%

   \[\frac{x \cdot y - z \cdot t}{a} \]

2. Taylor expanded in x around inf 50.6%

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

   1. associate-*r/ 53.1%

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

4. Simplified 53.1%

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

5. Final simplification 53.1%

   \[\leadsto y \cdot \frac{x}{a} \]

Developer target: 91.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \mathbf{if}\;z < -2.468684968699548 \cdot 10^{+170}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z < 6.309831121978371 \cdot 10^{-71}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* (/ y a) x) (* (/ t a) z))))
   (if (< z -2.468684968699548e+170)
     t_1
     (if (< z 6.309831121978371e-71) (/ (- (* x y) (* z t)) a) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((y / a) * x) - ((t / a) * z);
	double tmp;
	if (z < -2.468684968699548e+170) {
		tmp = t_1;
	} else if (z < 6.309831121978371e-71) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = ((y / a) * x) - ((t / a) * z)
    if (z < (-2.468684968699548d+170)) then
        tmp = t_1
    else if (z < 6.309831121978371d-71) then
        tmp = ((x * y) - (z * t)) / a
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = ((y / a) * x) - ((t / a) * z);
	double tmp;
	if (z < -2.468684968699548e+170) {
		tmp = t_1;
	} else if (z < 6.309831121978371e-71) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = ((y / a) * x) - ((t / a) * z)
	tmp = 0
	if z < -2.468684968699548e+170:
		tmp = t_1
	elif z < 6.309831121978371e-71:
		tmp = ((x * y) - (z * t)) / a
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(y / a) * x) - Float64(Float64(t / a) * z))
	tmp = 0.0
	if (z < -2.468684968699548e+170)
		tmp = t_1;
	elseif (z < 6.309831121978371e-71)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((y / a) * x) - ((t / a) * z);
	tmp = 0.0;
	if (z < -2.468684968699548e+170)
		tmp = t_1;
	elseif (z < 6.309831121978371e-71)
		tmp = ((x * y) - (z * t)) / a;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(y / a), $MachinePrecision] * x), $MachinePrecision] - N[(N[(t / a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -2.468684968699548e+170], t$95$1, If[Less[z, 6.309831121978371e-71], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2023229 
(FPCore (x y z t a)
  :name "Data.Colour.Matrix:inverse from colour-2.3.3, B"
  :precision binary64

  :herbie-target
  (if (< z -2.468684968699548e+170) (- (* (/ y a) x) (* (/ t a) z)) (if (< z 6.309831121978371e-71) (/ (- (* x y) (* z t)) a) (- (* (/ y a) x) (* (/ t a) z))))

  (/ (- (* x y) (* z t)) a))