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

Percentage Accurate: 91.2% → 97.4%

Time: 11.1s

Alternatives: 8

Speedup: 0.6×

• 28.0% 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: 91.2% 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: 97.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+282} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+253}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{-z}{a}, t, \frac{y}{a} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a 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+282) (not (<= t_1 2e+253)))
     (fma (/ (- z) a) t (* (/ y a) x))
     (/ t_1 a))))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
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+282) || !(t_1 <= 2e+253)) {
		tmp = fma((-z / a), t, ((y / a) * x));
	} else {
		tmp = t_1 / a;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if ((t_1 <= -2e+282) || !(t_1 <= 2e+253))
		tmp = fma(Float64(Float64(-z) / a), t, Float64(Float64(y / a) * x));
	else
		tmp = Float64(t_1 / a);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a 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+282], N[Not[LessEqual[t$95$1, 2e+253]], $MachinePrecision]], N[(N[((-z) / a), $MachinePrecision] * t + N[(N[(y / a), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -2.00000000000000007e282 or 1.9999999999999999e253 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 74.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. div-add N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 75.7

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 75.7

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

   4. Applied rewrites 75.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 91.3

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

   6. Applied rewrites 91.3%

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

   if -2.00000000000000007e282 < (-.f64 (*.f64 x y) (*.f64 z t)) < 1.9999999999999999e253

   1. Initial program 98.9%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -2 \cdot 10^{+282} \lor \neg \left(x \cdot y - z \cdot t \leq 2 \cdot 10^{+253}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{-z}{a}, t, \frac{y}{a} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 55.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \frac{x \cdot y - z \cdot t}{a}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+274}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a 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)) a)))
   (if (<= t_1 (- INFINITY))
     (* (/ x a) y)
     (if (<= t_1 5e+274) (/ (* x y) a) (* (/ y a) x)))))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = ((x * y) - (z * t)) / a;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (x / a) * y;
	} else if (t_1 <= 5e+274) {
		tmp = (x * y) / a;
	} else {
		tmp = (y / a) * x;
	}
	return tmp;
}

Java

assert x < y && y < z && z < t && t < a;
assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = ((x * y) - (z * t)) / a;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (x / a) * y;
	} else if (t_1 <= 5e+274) {
		tmp = (x * y) / a;
	} else {
		tmp = (y / a) * x;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = ((x * y) - (z * t)) / a
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (x / a) * y
	elif t_1 <= 5e+274:
		tmp = (x * y) / a
	else:
		tmp = (y / a) * x
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(x * y) - Float64(z * t)) / a)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(x / a) * y);
	elseif (t_1 <= 5e+274)
		tmp = Float64(Float64(x * y) / a);
	else
		tmp = Float64(Float64(y / a) * x);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = ((x * y) - (z * t)) / a;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (x / a) * y;
	elseif (t_1 <= 5e+274)
		tmp = (x * y) / a;
	else
		tmp = (y / a) * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(x / a), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$1, 5e+274], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (*.f64 x y) (*.f64 z t)) a) < -inf.0

   1. Initial program 78.0%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 56.4

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

   5. Applied rewrites 56.4%

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

   if -inf.0 < (/.f64 (-.f64 (*.f64 x y) (*.f64 z t)) a) < 4.9999999999999998e274

   1. Initial program 98.8%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 47.5

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

   5. Applied rewrites 47.5%

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

   7. Applied rewrites 47.4%

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

   9. Applied rewrites 55.2%

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

   if 4.9999999999999998e274 < (/.f64 (-.f64 (*.f64 x y) (*.f64 z t)) a)

   1. Initial program 81.7%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 55.3

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

   5. Applied rewrites 55.3%

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

   7. Applied rewrites 50.5%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y - z \cdot t}{a} \leq -\infty:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \mathbf{elif}\;\frac{x \cdot y - z \cdot t}{a} \leq 5 \cdot 10^{+274}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+85}:\\ \;\;\;\;\frac{y}{a} \cdot x\\ \mathbf{elif}\;x \cdot y \leq 10^{-85}:\\ \;\;\;\;\frac{\left(-z\right) \cdot t}{a}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+269}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -1e+85)
   (* (/ y a) x)
   (if (<= (* x y) 1e-85)
     (/ (* (- z) t) a)
     (if (<= (* x y) 2e+269) (/ (* x y) a) (* (/ x a) y)))))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -1e+85) {
		tmp = (y / a) * x;
	} else if ((x * y) <= 1e-85) {
		tmp = (-z * t) / a;
	} else if ((x * y) <= 2e+269) {
		tmp = (x * y) / a;
	} else {
		tmp = (x / a) * y;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a 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) <= (-1d+85)) then
        tmp = (y / a) * x
    else if ((x * y) <= 1d-85) then
        tmp = (-z * t) / a
    else if ((x * y) <= 2d+269) then
        tmp = (x * y) / a
    else
        tmp = (x / a) * y
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -1e+85) {
		tmp = (y / a) * x;
	} else if ((x * y) <= 1e-85) {
		tmp = (-z * t) / a;
	} else if ((x * y) <= 2e+269) {
		tmp = (x * y) / a;
	} else {
		tmp = (x / a) * y;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -1e+85:
		tmp = (y / a) * x
	elif (x * y) <= 1e-85:
		tmp = (-z * t) / a
	elif (x * y) <= 2e+269:
		tmp = (x * y) / a
	else:
		tmp = (x / a) * y
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -1e+85)
		tmp = Float64(Float64(y / a) * x);
	elseif (Float64(x * y) <= 1e-85)
		tmp = Float64(Float64(Float64(-z) * t) / a);
	elseif (Float64(x * y) <= 2e+269)
		tmp = Float64(Float64(x * y) / a);
	else
		tmp = Float64(Float64(x / a) * y);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -1e+85)
		tmp = (y / a) * x;
	elseif ((x * y) <= 1e-85)
		tmp = (-z * t) / a;
	elseif ((x * y) <= 2e+269)
		tmp = (x * y) / a;
	else
		tmp = (x / a) * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -1e+85], N[(N[(y / a), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e-85], N[(N[((-z) * t), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e+269], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision], N[(N[(x / a), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -1e85

   1. Initial program 83.2%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 84.8

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

   5. Applied rewrites 84.8%

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

   7. Applied rewrites 80.8%

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

   if -1e85 < (*.f64 x y) < 9.9999999999999998e-86

   1. Initial program 95.7%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 78.3

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

   5. Applied rewrites 78.3%

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

   if 9.9999999999999998e-86 < (*.f64 x y) < 2.0000000000000001e269

   1. Initial program 98.1%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 64.3

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

   5. Applied rewrites 64.3%

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

   7. Applied rewrites 67.0%

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

   9. Applied rewrites 77.0%

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

   if 2.0000000000000001e269 < (*.f64 x y)

   1. Initial program 61.3%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+85}:\\ \;\;\;\;\frac{y}{a} \cdot x\\ \mathbf{elif}\;x \cdot y \leq 10^{-85}:\\ \;\;\;\;\frac{\left(-z\right) \cdot t}{a}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+269}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+85}:\\ \;\;\;\;\frac{y}{a} \cdot x\\ \mathbf{elif}\;x \cdot y \leq 10^{-85}:\\ \;\;\;\;\left(-z\right) \cdot \frac{t}{a}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+269}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -1e+85)
   (* (/ y a) x)
   (if (<= (* x y) 1e-85)
     (* (- z) (/ t a))
     (if (<= (* x y) 2e+269) (/ (* x y) a) (* (/ x a) y)))))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -1e+85) {
		tmp = (y / a) * x;
	} else if ((x * y) <= 1e-85) {
		tmp = -z * (t / a);
	} else if ((x * y) <= 2e+269) {
		tmp = (x * y) / a;
	} else {
		tmp = (x / a) * y;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a 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) <= (-1d+85)) then
        tmp = (y / a) * x
    else if ((x * y) <= 1d-85) then
        tmp = -z * (t / a)
    else if ((x * y) <= 2d+269) then
        tmp = (x * y) / a
    else
        tmp = (x / a) * y
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -1e+85) {
		tmp = (y / a) * x;
	} else if ((x * y) <= 1e-85) {
		tmp = -z * (t / a);
	} else if ((x * y) <= 2e+269) {
		tmp = (x * y) / a;
	} else {
		tmp = (x / a) * y;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -1e+85:
		tmp = (y / a) * x
	elif (x * y) <= 1e-85:
		tmp = -z * (t / a)
	elif (x * y) <= 2e+269:
		tmp = (x * y) / a
	else:
		tmp = (x / a) * y
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -1e+85)
		tmp = Float64(Float64(y / a) * x);
	elseif (Float64(x * y) <= 1e-85)
		tmp = Float64(Float64(-z) * Float64(t / a));
	elseif (Float64(x * y) <= 2e+269)
		tmp = Float64(Float64(x * y) / a);
	else
		tmp = Float64(Float64(x / a) * y);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -1e+85)
		tmp = (y / a) * x;
	elseif ((x * y) <= 1e-85)
		tmp = -z * (t / a);
	elseif ((x * y) <= 2e+269)
		tmp = (x * y) / a;
	else
		tmp = (x / a) * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -1e+85], N[(N[(y / a), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e-85], N[((-z) * N[(t / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e+269], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision], N[(N[(x / a), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -1e85

   1. Initial program 83.2%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 84.8

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

   5. Applied rewrites 84.8%

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

   7. Applied rewrites 80.8%

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

   if -1e85 < (*.f64 x y) < 9.9999999999999998e-86

   1. Initial program 95.7%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 77.9

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

   5. Applied rewrites 77.9%

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

   if 9.9999999999999998e-86 < (*.f64 x y) < 2.0000000000000001e269

   1. Initial program 98.1%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 64.3

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

   5. Applied rewrites 64.3%

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

   7. Applied rewrites 67.0%

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

   9. Applied rewrites 77.0%

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

   if 2.0000000000000001e269 < (*.f64 x y)

   1. Initial program 61.3%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+85}:\\ \;\;\;\;\frac{y}{a} \cdot x\\ \mathbf{elif}\;x \cdot y \leq 10^{-85}:\\ \;\;\;\;\left(-z\right) \cdot \frac{t}{a}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+269}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+85}:\\ \;\;\;\;\frac{y}{a} \cdot x\\ \mathbf{elif}\;x \cdot y \leq 10^{-85}:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+269}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -1e+85)
   (* (/ y a) x)
   (if (<= (* x y) 1e-85)
     (* t (/ (- z) a))
     (if (<= (* x y) 2e+269) (/ (* x y) a) (* (/ x a) y)))))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -1e+85) {
		tmp = (y / a) * x;
	} else if ((x * y) <= 1e-85) {
		tmp = t * (-z / a);
	} else if ((x * y) <= 2e+269) {
		tmp = (x * y) / a;
	} else {
		tmp = (x / a) * y;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a 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) <= (-1d+85)) then
        tmp = (y / a) * x
    else if ((x * y) <= 1d-85) then
        tmp = t * (-z / a)
    else if ((x * y) <= 2d+269) then
        tmp = (x * y) / a
    else
        tmp = (x / a) * y
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -1e+85) {
		tmp = (y / a) * x;
	} else if ((x * y) <= 1e-85) {
		tmp = t * (-z / a);
	} else if ((x * y) <= 2e+269) {
		tmp = (x * y) / a;
	} else {
		tmp = (x / a) * y;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -1e+85:
		tmp = (y / a) * x
	elif (x * y) <= 1e-85:
		tmp = t * (-z / a)
	elif (x * y) <= 2e+269:
		tmp = (x * y) / a
	else:
		tmp = (x / a) * y
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -1e+85)
		tmp = Float64(Float64(y / a) * x);
	elseif (Float64(x * y) <= 1e-85)
		tmp = Float64(t * Float64(Float64(-z) / a));
	elseif (Float64(x * y) <= 2e+269)
		tmp = Float64(Float64(x * y) / a);
	else
		tmp = Float64(Float64(x / a) * y);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -1e+85)
		tmp = (y / a) * x;
	elseif ((x * y) <= 1e-85)
		tmp = t * (-z / a);
	elseif ((x * y) <= 2e+269)
		tmp = (x * y) / a;
	else
		tmp = (x / a) * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -1e+85], N[(N[(y / a), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e-85], N[(t * N[((-z) / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e+269], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision], N[(N[(x / a), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -1e85

   1. Initial program 83.2%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 84.8

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

   5. Applied rewrites 84.8%

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

   7. Applied rewrites 80.8%

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

   if -1e85 < (*.f64 x y) < 9.9999999999999998e-86

   1. Initial program 95.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. div-add N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 94.6

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 94.6

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

   4. Applied rewrites 94.6%

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

   5. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 78.0

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

   7. Applied rewrites 78.0%

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

   if 9.9999999999999998e-86 < (*.f64 x y) < 2.0000000000000001e269

   1. Initial program 98.1%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 64.3

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

   5. Applied rewrites 64.3%

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

   7. Applied rewrites 67.0%

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

   9. Applied rewrites 77.0%

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

   if 2.0000000000000001e269 < (*.f64 x y)

   1. Initial program 61.3%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+85}:\\ \;\;\;\;\frac{y}{a} \cdot x\\ \mathbf{elif}\;x \cdot y \leq 10^{-85}:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+269}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;\frac{y}{a} \cdot x\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+269}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) (- INFINITY))
   (* (/ y a) x)
   (if (<= (* x y) 2e+269) (/ (- (* x y) (* z t)) a) (* (/ x a) y))))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -((double) INFINITY)) {
		tmp = (y / a) * x;
	} else if ((x * y) <= 2e+269) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = (x / a) * y;
	}
	return tmp;
}

Java

assert x < y && y < z && z < t && t < a;
assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -Double.POSITIVE_INFINITY) {
		tmp = (y / a) * x;
	} else if ((x * y) <= 2e+269) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = (x / a) * y;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -math.inf:
		tmp = (y / a) * x
	elif (x * y) <= 2e+269:
		tmp = ((x * y) - (z * t)) / a
	else:
		tmp = (x / a) * y
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= Float64(-Inf))
		tmp = Float64(Float64(y / a) * x);
	elseif (Float64(x * y) <= 2e+269)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a);
	else
		tmp = Float64(Float64(x / a) * y);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -Inf)
		tmp = (y / a) * x;
	elseif ((x * y) <= 2e+269)
		tmp = ((x * y) - (z * t)) / a;
	else
		tmp = (x / a) * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], (-Infinity)], N[(N[(y / a), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e+269], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(x / a), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 74.6%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 96.2

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

   5. Applied rewrites 96.2%

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

   7. Applied rewrites 96.2%

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

   if -inf.0 < (*.f64 x y) < 2.0000000000000001e269

   1. Initial program 96.4%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   if 2.0000000000000001e269 < (*.f64 x y)

   1. Initial program 61.3%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;\frac{y}{a} \cdot x\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+269}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 5.4 \cdot 10^{-21}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, x, t \cdot \frac{-z}{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= a 5.4e-21)
   (/ (- (* x y) (* z t)) a)
   (fma (/ y a) x (* t (/ (- z) a)))))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= 5.4e-21) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = fma((y / a), x, (t * (-z / a)));
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= 5.4e-21)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a);
	else
		tmp = fma(Float64(y / a), x, Float64(t * Float64(Float64(-z) / a)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[a, 5.4e-21], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * x + N[(t * N[((-z) / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.4000000000000002e-21

   1. Initial program 94.5%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   if 5.4000000000000002e-21 < a

   1. Initial program 82.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-/.f64 94.7

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

   4. Applied rewrites 94.7%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.4 \cdot 10^{-21}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, x, t \cdot \frac{-z}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.2%

   \[\frac{x \cdot y - z \cdot t}{a} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

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

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 50.9

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

5. Applied rewrites 50.9%

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

6. Final simplification 50.9%

   \[\leadsto \frac{x}{a} \cdot y \]
7. Add Preprocessing

Developer Target 1: 91.9% accurate, 0.5× 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 2024342 
(FPCore (x y z t a)
  :name "Data.Colour.Matrix:inverse from colour-2.3.3, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -246868496869954800000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (/ y a) x) (* (/ t a) z)) (if (< z 6309831121978371/100000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (- (* x y) (* z t)) a) (- (* (/ y a) x) (* (/ t a) z)))))

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