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

Percentage Accurate: 91.6% → 96.9%

Time: 7.9s

Alternatives: 11

Speedup: 0.5×

• 29.0% 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: 91.6% 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: 96.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+228}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a} - t \cdot \frac{z}{a}\\ \end{array} \end{array} \]

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 (<= t_1 -1e+228)
     (- (/ x (/ a y)) (/ z (/ a t)))
     (if (<= t_1 2e+293)
       (/ (fma x y (* z (- t))) a)
       (- (* y (/ x a)) (* t (/ z 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 <= -1e+228) {
		tmp = (x / (a / y)) - (z / (a / t));
	} else if (t_1 <= 2e+293) {
		tmp = fma(x, y, (z * -t)) / a;
	} else {
		tmp = (y * (x / a)) - (t * (z / 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 <= -1e+228)
		tmp = Float64(Float64(x / Float64(a / y)) - Float64(z / Float64(a / t)));
	elseif (t_1 <= 2e+293)
		tmp = Float64(fma(x, y, Float64(z * Float64(-t))) / a);
	else
		tmp = Float64(Float64(y * Float64(x / a)) - Float64(t * Float64(z / 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[LessEqual[t$95$1, -1e+228], N[(N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision] - N[(z / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+293], N[(N[(x * y + N[(z * (-t)), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision] - N[(t * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -9.9999999999999992e227

   1. Initial program 78.1%

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

      1. div-sub 75.7%

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

      2. associate-/l* 91.3%

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

      3. associate-/l* 97.5%

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

   3. Applied egg-rr 97.5%

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

   if -9.9999999999999992e227 < (-.f64 (*.f64 x y) (*.f64 z t)) < 1.9999999999999998e293

   1. Initial program 99.2%

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

      1. fma-neg 99.2%

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

      2. distribute-rgt-neg-out 99.2%

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

   3. Simplified 99.2%

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

   if 1.9999999999999998e293 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 69.9%

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

   2. Taylor expanded in x around 0 64.7%

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

      1. +-commutative 64.7%

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

      2. mul-1-neg 64.7%

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

      3. unsub-neg 64.7%

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

      4. associate-*l/ 79.3%

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

      5. *-commutative 79.3%

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

      6. div-inv 79.3%

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

      7. associate-*l* 86.6%

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

      8. div-inv 86.7%

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

   4. Applied egg-rr 86.7%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -1 \cdot 10^{+228}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a} - t \cdot \frac{z}{a}\\ \end{array} \]

Alternative 2: 97.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+305} \lor \neg \left(t_1 \leq 2 \cdot 10^{+293}\right):\\ \;\;\;\;x \cdot \frac{y}{a} - t \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{a}\\ \end{array} \end{array} \]

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 -5e+305) (not (<= t_1 2e+293)))
     (- (* x (/ y a)) (* t (/ z a)))
     (/ 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 <= -5e+305) || !(t_1 <= 2e+293)) {
		tmp = (x * (y / a)) - (t * (z / a));
	} else {
		tmp = t_1 / 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) :: t_1
    real(8) :: tmp
    t_1 = (x * y) - (z * t)
    if ((t_1 <= (-5d+305)) .or. (.not. (t_1 <= 2d+293))) then
        tmp = (x * (y / a)) - (t * (z / a))
    else
        tmp = t_1 / 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 t_1 = (x * y) - (z * t);
	double tmp;
	if ((t_1 <= -5e+305) || !(t_1 <= 2e+293)) {
		tmp = (x * (y / a)) - (t * (z / a));
	} 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 <= -5e+305) or not (t_1 <= 2e+293):
		tmp = (x * (y / a)) - (t * (z / a))
	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 <= -5e+305) || !(t_1 <= 2e+293))
		tmp = Float64(Float64(x * Float64(y / a)) - Float64(t * Float64(z / a)));
	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 <= -5e+305) || ~((t_1 <= 2e+293)))
		tmp = (x * (y / a)) - (t * (z / a));
	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, -5e+305], N[Not[LessEqual[t$95$1, 2e+293]], $MachinePrecision]], N[(N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision] - N[(t * N[(z / a), $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)) < -5.00000000000000009e305 or 1.9999999999999998e293 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 69.8%

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

   2. Taylor expanded in x around 0 65.5%

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

      1. associate-*l/ 74.9%

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

   4. Applied egg-rr 74.9%

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

      1. +-commutative 74.9%

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

      2. mul-1-neg 74.9%

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

      3. *-commutative 74.9%

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

      4. clear-num 74.9%

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

      5. div-inv 74.9%

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

      6. sub-neg 74.9%

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

      7. associate-*r/ 94.0%

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

      8. associate-/r/ 92.6%

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

      9. *-commutative 92.6%

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

   6. Applied egg-rr 92.6%

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

   if -5.00000000000000009e305 < (-.f64 (*.f64 x y) (*.f64 z t)) < 1.9999999999999998e293

   1. Initial program 99.2%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -5 \cdot 10^{+305} \lor \neg \left(x \cdot y - z \cdot t \leq 2 \cdot 10^{+293}\right):\\ \;\;\;\;x \cdot \frac{y}{a} - t \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \]

Alternative 3: 97.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := t \cdot \frac{z}{a}\\ t_2 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{+305}:\\ \;\;\;\;x \cdot \frac{y}{a} - t_1\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\frac{t_2}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a} - t_1\\ \end{array} \end{array} \]

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 (* t (/ z a))) (t_2 (- (* x y) (* z t))))
   (if (<= t_2 -5e+305)
     (- (* x (/ y a)) t_1)
     (if (<= t_2 2e+293) (/ t_2 a) (- (* y (/ x a)) t_1)))))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double t_1 = t * (z / a);
	double t_2 = (x * y) - (z * t);
	double tmp;
	if (t_2 <= -5e+305) {
		tmp = (x * (y / a)) - t_1;
	} else if (t_2 <= 2e+293) {
		tmp = t_2 / a;
	} else {
		tmp = (y * (x / a)) - t_1;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = t * (z / a)
    t_2 = (x * y) - (z * t)
    if (t_2 <= (-5d+305)) then
        tmp = (x * (y / a)) - t_1
    else if (t_2 <= 2d+293) then
        tmp = t_2 / a
    else
        tmp = (y * (x / a)) - t_1
    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 = t * (z / a);
	double t_2 = (x * y) - (z * t);
	double tmp;
	if (t_2 <= -5e+305) {
		tmp = (x * (y / a)) - t_1;
	} else if (t_2 <= 2e+293) {
		tmp = t_2 / a;
	} else {
		tmp = (y * (x / a)) - t_1;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[z, t] = sort([z, t])
def code(x, y, z, t, a):
	t_1 = t * (z / a)
	t_2 = (x * y) - (z * t)
	tmp = 0
	if t_2 <= -5e+305:
		tmp = (x * (y / a)) - t_1
	elif t_2 <= 2e+293:
		tmp = t_2 / a
	else:
		tmp = (y * (x / a)) - t_1
	return tmp

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	t_1 = Float64(t * Float64(z / a))
	t_2 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if (t_2 <= -5e+305)
		tmp = Float64(Float64(x * Float64(y / a)) - t_1);
	elseif (t_2 <= 2e+293)
		tmp = Float64(t_2 / a);
	else
		tmp = Float64(Float64(y * Float64(x / a)) - t_1);
	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 = t * (z / a);
	t_2 = (x * y) - (z * t);
	tmp = 0.0;
	if (t_2 <= -5e+305)
		tmp = (x * (y / a)) - t_1;
	elseif (t_2 <= 2e+293)
		tmp = t_2 / a;
	else
		tmp = (y * (x / a)) - t_1;
	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[(t * N[(z / a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+305], N[(N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 2e+293], N[(t$95$2 / a), $MachinePrecision], N[(N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -5.00000000000000009e305

   1. Initial program 69.7%

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

   2. Taylor expanded in x around 0 66.5%

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

      1. associate-*l/ 75.3%

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

   4. Applied egg-rr 75.3%

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

      1. +-commutative 75.3%

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

      2. mul-1-neg 75.3%

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

      3. *-commutative 75.3%

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

      4. clear-num 75.3%

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

      5. div-inv 75.2%

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

      6. sub-neg 75.2%

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

      7. associate-*r/ 96.6%

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

      8. associate-/r/ 96.6%

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

      9. *-commutative 96.6%

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

   6. Applied egg-rr 96.6%

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

   if -5.00000000000000009e305 < (-.f64 (*.f64 x y) (*.f64 z t)) < 1.9999999999999998e293

   1. Initial program 99.2%

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

   if 1.9999999999999998e293 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 69.9%

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

   2. Taylor expanded in x around 0 64.7%

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

      1. +-commutative 64.7%

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

      2. mul-1-neg 64.7%

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

      3. unsub-neg 64.7%

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

      4. associate-*l/ 79.3%

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

      5. *-commutative 79.3%

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

      6. div-inv 79.3%

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

      7. associate-*l* 86.6%

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

      8. div-inv 86.7%

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

   4. Applied egg-rr 86.7%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -5 \cdot 10^{+305}:\\ \;\;\;\;x \cdot \frac{y}{a} - t \cdot \frac{z}{a}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a} - t \cdot \frac{z}{a}\\ \end{array} \]

Alternative 4: 96.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+228}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a} - t \cdot \frac{z}{a}\\ \end{array} \end{array} \]

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 (<= t_1 -1e+228)
     (- (/ x (/ a y)) (/ z (/ a t)))
     (if (<= t_1 2e+293) (/ t_1 a) (- (* y (/ x a)) (* t (/ z 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 <= -1e+228) {
		tmp = (x / (a / y)) - (z / (a / t));
	} else if (t_1 <= 2e+293) {
		tmp = t_1 / a;
	} else {
		tmp = (y * (x / a)) - (t * (z / 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) :: t_1
    real(8) :: tmp
    t_1 = (x * y) - (z * t)
    if (t_1 <= (-1d+228)) then
        tmp = (x / (a / y)) - (z / (a / t))
    else if (t_1 <= 2d+293) then
        tmp = t_1 / a
    else
        tmp = (y * (x / a)) - (t * (z / 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 t_1 = (x * y) - (z * t);
	double tmp;
	if (t_1 <= -1e+228) {
		tmp = (x / (a / y)) - (z / (a / t));
	} else if (t_1 <= 2e+293) {
		tmp = t_1 / a;
	} else {
		tmp = (y * (x / a)) - (t * (z / 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 <= -1e+228:
		tmp = (x / (a / y)) - (z / (a / t))
	elif t_1 <= 2e+293:
		tmp = t_1 / a
	else:
		tmp = (y * (x / a)) - (t * (z / 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 <= -1e+228)
		tmp = Float64(Float64(x / Float64(a / y)) - Float64(z / Float64(a / t)));
	elseif (t_1 <= 2e+293)
		tmp = Float64(t_1 / a);
	else
		tmp = Float64(Float64(y * Float64(x / a)) - Float64(t * Float64(z / 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 <= -1e+228)
		tmp = (x / (a / y)) - (z / (a / t));
	elseif (t_1 <= 2e+293)
		tmp = t_1 / a;
	else
		tmp = (y * (x / a)) - (t * (z / 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[LessEqual[t$95$1, -1e+228], N[(N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision] - N[(z / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+293], N[(t$95$1 / a), $MachinePrecision], N[(N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision] - N[(t * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -9.9999999999999992e227

   1. Initial program 78.1%

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

      1. div-sub 75.7%

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

      2. associate-/l* 91.3%

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

      3. associate-/l* 97.5%

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

   3. Applied egg-rr 97.5%

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

   if -9.9999999999999992e227 < (-.f64 (*.f64 x y) (*.f64 z t)) < 1.9999999999999998e293

   1. Initial program 99.2%

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

   if 1.9999999999999998e293 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 69.9%

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

   2. Taylor expanded in x around 0 64.7%

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

      1. +-commutative 64.7%

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

      2. mul-1-neg 64.7%

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

      3. unsub-neg 64.7%

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

      4. associate-*l/ 79.3%

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

      5. *-commutative 79.3%

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

      6. div-inv 79.3%

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

      7. associate-*l* 86.6%

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

      8. div-inv 86.7%

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

   4. Applied egg-rr 86.7%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -1 \cdot 10^{+228}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a} - t \cdot \frac{z}{a}\\ \end{array} \]

Alternative 5: 95.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+217}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \end{array} \]

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 y) (- INFINITY))
   (/ x (/ a y))
   (if (<= (* x y) 5e+217) (/ (- (* x y) (* z t)) 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 * y) <= -((double) INFINITY)) {
		tmp = x / (a / y);
	} else if ((x * y) <= 5e+217) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = x * (y / 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 tmp;
	if ((x * y) <= -Double.POSITIVE_INFINITY) {
		tmp = x / (a / y);
	} else if ((x * y) <= 5e+217) {
		tmp = ((x * y) - (z * t)) / 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 * y) <= -math.inf:
		tmp = x / (a / y)
	elif (x * y) <= 5e+217:
		tmp = ((x * y) - (z * t)) / 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 (Float64(x * y) <= Float64(-Inf))
		tmp = Float64(x / Float64(a / y));
	elseif (Float64(x * y) <= 5e+217)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / 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 * y) <= -Inf)
		tmp = x / (a / y);
	elseif ((x * y) <= 5e+217)
		tmp = ((x * y) - (z * t)) / 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[N[(x * y), $MachinePrecision], (-Infinity)], N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+217], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -inf.0

   1. Initial program 58.5%

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

   2. Taylor expanded in x around inf 58.5%

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

      1. associate-*l/ 93.9%

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

   4. Simplified 93.9%

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

      1. associate-/r/ 94.1%

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

   6. Applied egg-rr 94.1%

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

   if -inf.0 < (*.f64 x y) < 5.00000000000000041e217

   1. Initial program 96.7%

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

   if 5.00000000000000041e217 < (*.f64 x y)

   1. Initial program 73.1%

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

   2. Taylor expanded in x around 0 70.0%

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

   3. Taylor expanded in t around 0 73.2%

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

      1. associate-*r/ 93.7%

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

   5. Simplified 93.7%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+217}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]

Alternative 6: 67.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := \frac{-t}{\frac{a}{z}}\\ \mathbf{if}\;t \leq -1.55 \cdot 10^{-133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7.1 \cdot 10^{-243}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-153}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{+50}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

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 (/ (- t) (/ a z))))
   (if (<= t -1.55e-133)
     t_1
     (if (<= t 7.1e-243)
       (/ y (/ a x))
       (if (<= t 3.3e-153)
         (* x (/ y a))
         (if (<= t 1.3e+50) (* y (/ x a)) t_1))))))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double t_1 = -t / (a / z);
	double tmp;
	if (t <= -1.55e-133) {
		tmp = t_1;
	} else if (t <= 7.1e-243) {
		tmp = y / (a / x);
	} else if (t <= 3.3e-153) {
		tmp = x * (y / a);
	} else if (t <= 1.3e+50) {
		tmp = y * (x / a);
	} else {
		tmp = t_1;
	}
	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 = -t / (a / z)
    if (t <= (-1.55d-133)) then
        tmp = t_1
    else if (t <= 7.1d-243) then
        tmp = y / (a / x)
    else if (t <= 3.3d-153) then
        tmp = x * (y / a)
    else if (t <= 1.3d+50) then
        tmp = y * (x / a)
    else
        tmp = t_1
    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 = -t / (a / z);
	double tmp;
	if (t <= -1.55e-133) {
		tmp = t_1;
	} else if (t <= 7.1e-243) {
		tmp = y / (a / x);
	} else if (t <= 3.3e-153) {
		tmp = x * (y / a);
	} else if (t <= 1.3e+50) {
		tmp = y * (x / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[z, t] = sort([z, t])
def code(x, y, z, t, a):
	t_1 = -t / (a / z)
	tmp = 0
	if t <= -1.55e-133:
		tmp = t_1
	elif t <= 7.1e-243:
		tmp = y / (a / x)
	elif t <= 3.3e-153:
		tmp = x * (y / a)
	elif t <= 1.3e+50:
		tmp = y * (x / a)
	else:
		tmp = t_1
	return tmp

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(-t) / Float64(a / z))
	tmp = 0.0
	if (t <= -1.55e-133)
		tmp = t_1;
	elseif (t <= 7.1e-243)
		tmp = Float64(y / Float64(a / x));
	elseif (t <= 3.3e-153)
		tmp = Float64(x * Float64(y / a));
	elseif (t <= 1.3e+50)
		tmp = Float64(y * Float64(x / a));
	else
		tmp = t_1;
	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 = -t / (a / z);
	tmp = 0.0;
	if (t <= -1.55e-133)
		tmp = t_1;
	elseif (t <= 7.1e-243)
		tmp = y / (a / x);
	elseif (t <= 3.3e-153)
		tmp = x * (y / a);
	elseif (t <= 1.3e+50)
		tmp = y * (x / a);
	else
		tmp = t_1;
	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[((-t) / N[(a / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.55e-133], t$95$1, If[LessEqual[t, 7.1e-243], N[(y / N[(a / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.3e-153], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.3e+50], N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1.55000000000000008e-133 or 1.3000000000000001e50 < t

   1. Initial program 88.2%

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

   2. Taylor expanded in x around 0 66.5%

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

      1. mul-1-neg 66.5%

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

      2. associate-/l* 65.8%

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

   4. Simplified 65.8%

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

   if -1.55000000000000008e-133 < t < 7.1000000000000004e-243

   1. Initial program 96.2%

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

   2. Taylor expanded in x around inf 74.4%

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

      1. associate-*l/ 69.5%

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

   4. Simplified 69.5%

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

      1. *-commutative 69.5%

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

      2. clear-num 69.4%

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

      3. un-div-inv 69.4%

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

   6. Applied egg-rr 69.4%

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

   if 7.1000000000000004e-243 < t < 3.29999999999999988e-153

   1. Initial program 90.7%

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

   2. Taylor expanded in x around 0 86.0%

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

   3. Taylor expanded in t around 0 86.1%

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

      1. associate-*r/ 84.4%

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

   5. Simplified 84.4%

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

   if 3.29999999999999988e-153 < t < 1.3000000000000001e50

   1. Initial program 95.2%

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

   2. Taylor expanded in x around inf 69.4%

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

      1. associate-*l/ 70.6%

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

   4. Simplified 70.6%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.55 \cdot 10^{-133}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{elif}\;t \leq 7.1 \cdot 10^{-243}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-153}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{+50}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \end{array} \]

Alternative 7: 72.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+25} \lor \neg \left(x \cdot y \leq 10^{-31}\right):\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-t}{a}\\ \end{array} \end{array} \]

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 (or (<= (* x y) -5e+25) (not (<= (* x y) 1e-31)))
   (* y (/ x a))
   (* z (/ (- t) a))))

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+25) || !((x * y) <= 1e-31)) {
		tmp = y * (x / a);
	} else {
		tmp = z * (-t / 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 * y) <= (-5d+25)) .or. (.not. ((x * y) <= 1d-31))) then
        tmp = y * (x / a)
    else
        tmp = z * (-t / 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 * y) <= -5e+25) || !((x * y) <= 1e-31)) {
		tmp = y * (x / a);
	} else {
		tmp = z * (-t / 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 * y) <= -5e+25) or not ((x * y) <= 1e-31):
		tmp = y * (x / a)
	else:
		tmp = z * (-t / 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 ((Float64(x * y) <= -5e+25) || !(Float64(x * y) <= 1e-31))
		tmp = Float64(y * Float64(x / a));
	else
		tmp = Float64(z * Float64(Float64(-t) / 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 * y) <= -5e+25) || ~(((x * y) <= 1e-31)))
		tmp = y * (x / a);
	else
		tmp = z * (-t / 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[Or[LessEqual[N[(x * y), $MachinePrecision], -5e+25], N[Not[LessEqual[N[(x * y), $MachinePrecision], 1e-31]], $MachinePrecision]], N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision], N[(z * N[((-t) / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -5.00000000000000024e25 or 1e-31 < (*.f64 x y)

   1. Initial program 87.0%

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

   2. Taylor expanded in x around inf 69.5%

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

      1. associate-*l/ 74.8%

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

   4. Simplified 74.8%

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

   if -5.00000000000000024e25 < (*.f64 x y) < 1e-31

   1. Initial program 96.1%

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

      1. div-sub 96.1%

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

      2. associate-/l* 95.4%

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

      3. associate-/l* 93.0%

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

   3. Applied egg-rr 93.0%

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

   4. Taylor expanded in x around 0 77.3%

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

      1. associate-*r/ 77.3%

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

      2. mul-1-neg 77.3%

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

      3. distribute-lft-neg-out 77.3%

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

      4. associate-*l/ 74.9%

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

      5. *-commutative 74.9%

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

   6. Simplified 74.9%

      \[\leadsto \color{blue}{z \cdot \frac{-t}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+25} \lor \neg \left(x \cdot y \leq 10^{-31}\right):\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-t}{a}\\ \end{array} \]

Alternative 8: 73.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+25} \lor \neg \left(x \cdot y \leq 2 \cdot 10^{-87}\right):\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(-t\right)}{a}\\ \end{array} \end{array} \]

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
 (if (or (<= (* x y) -5e+25) (not (<= (* x y) 2e-87)))
   (* y (/ x a))
   (/ (* z (- t)) a)))

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+25) || !((x * y) <= 2e-87)) {
		tmp = y * (x / a);
	} else {
		tmp = (z * -t) / 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 * y) <= (-5d+25)) .or. (.not. ((x * y) <= 2d-87))) then
        tmp = y * (x / a)
    else
        tmp = (z * -t) / 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 * y) <= -5e+25) || !((x * y) <= 2e-87)) {
		tmp = y * (x / a);
	} else {
		tmp = (z * -t) / 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 * y) <= -5e+25) or not ((x * y) <= 2e-87):
		tmp = y * (x / a)
	else:
		tmp = (z * -t) / 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 ((Float64(x * y) <= -5e+25) || !(Float64(x * y) <= 2e-87))
		tmp = Float64(y * Float64(x / a));
	else
		tmp = Float64(Float64(z * Float64(-t)) / 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 * y) <= -5e+25) || ~(((x * y) <= 2e-87)))
		tmp = y * (x / a);
	else
		tmp = (z * -t) / 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[Or[LessEqual[N[(x * y), $MachinePrecision], -5e+25], N[Not[LessEqual[N[(x * y), $MachinePrecision], 2e-87]], $MachinePrecision]], N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision], N[(N[(z * (-t)), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -5.00000000000000024e25 or 2.00000000000000004e-87 < (*.f64 x y)

   1. Initial program 87.2%

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

   2. Taylor expanded in x around inf 68.2%

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

      1. associate-*l/ 72.5%

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

   4. Simplified 72.5%

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

   if -5.00000000000000024e25 < (*.f64 x y) < 2.00000000000000004e-87

   1. Initial program 96.7%

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

   2. Taylor expanded in x around 0 80.3%

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

      1. mul-1-neg 80.3%

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

      2. distribute-rgt-neg-in 80.3%

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

   4. Simplified 80.3%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+25} \lor \neg \left(x \cdot y \leq 2 \cdot 10^{-87}\right):\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(-t\right)}{a}\\ \end{array} \]

Alternative 9: 53.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-102}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \end{array} \]

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 -2.9e-102) (* 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 <= -2.9e-102) {
		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 <= (-2.9d-102)) 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 <= -2.9e-102) {
		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 <= -2.9e-102:
		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 <= -2.9e-102)
		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 <= -2.9e-102)
		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, -2.9e-102], N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.89999999999999986e-102

   1. Initial program 91.5%

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

   2. Taylor expanded in x around inf 61.7%

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

      1. associate-*l/ 70.0%

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

   4. Simplified 70.0%

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

   if -2.89999999999999986e-102 < x

   1. Initial program 91.2%

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

   2. Taylor expanded in x around 0 88.8%

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

   3. Taylor expanded in t around 0 47.7%

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

      1. associate-*r/ 51.3%

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

   5. Simplified 51.3%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-102}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]

Alternative 10: 53.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-102}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \end{array} \]

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 (<= x -3.6e-102) (* y (/ x 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 (x <= -3.6e-102) {
		tmp = y * (x / 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 (x <= (-3.6d-102)) then
        tmp = y * (x / 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 (x <= -3.6e-102) {
		tmp = y * (x / 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 x <= -3.6e-102:
		tmp = y * (x / 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 (x <= -3.6e-102)
		tmp = Float64(y * Float64(x / 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 (x <= -3.6e-102)
		tmp = y * (x / 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[x, -3.6e-102], N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision], N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.6e-102

   1. Initial program 91.5%

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

   2. Taylor expanded in x around inf 61.7%

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

      1. associate-*l/ 70.0%

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

   4. Simplified 70.0%

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

   if -3.6e-102 < x

   1. Initial program 91.2%

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

   2. Taylor expanded in x around inf 47.7%

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

      1. associate-*l/ 45.0%

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

   4. Simplified 45.0%

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

      1. associate-/r/ 51.3%

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

   6. Applied egg-rr 51.3%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-102}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \]

Alternative 11: 53.0% accurate, 1.8× speedup

Math

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

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 (* x (/ y a)))

C

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 91.3%

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

2. Taylor expanded in x around 0 89.0%

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

3. Taylor expanded in t around 0 52.7%

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

   1. associate-*r/ 55.6%

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

5. Simplified 55.6%

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

6. Final simplification 55.6%

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

Developer target: 91.6% 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 2023336 
(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))