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

Percentage Accurate: 91.2% → 94.9%

Time: 7.3s

Alternatives: 9

Speedup: 0.5×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.2% 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: 94.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+267} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+234}\right):\\ \;\;\;\;\frac{-z}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(-z\right) \cdot t\right)}{a}\\ \end{array} \end{array} \]

FPCore

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 (or (<= (* z t) -1e+267) (not (<= (* z t) 2e+234)))
   (* (/ (- z) a) t)
   (/ (fma y x (* (- z) t)) a)))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z * t) <= -1e+267) || !((z * t) <= 2e+234)) {
		tmp = (-z / a) * t;
	} else {
		tmp = fma(y, x, (-z * t)) / a;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if ((Float64(z * t) <= -1e+267) || !(Float64(z * t) <= 2e+234))
		tmp = Float64(Float64(Float64(-z) / a) * t);
	else
		tmp = Float64(fma(y, x, Float64(Float64(-z) * t)) / a);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -1e+267], N[Not[LessEqual[N[(z * t), $MachinePrecision], 2e+234]], $MachinePrecision]], N[(N[((-z) / a), $MachinePrecision] * t), $MachinePrecision], N[(N[(y * x + N[((-z) * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -9.9999999999999997e266 or 2.00000000000000004e234 < (*.f64 z t)

   1. Initial program 72.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 75.3

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

   4. Applied rewrites 75.3%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 96.0

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

   7. Applied rewrites 96.0%

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

   if -9.9999999999999997e266 < (*.f64 z t) < 2.00000000000000004e234

   1. Initial program 97.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 97.5

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

   4. Applied rewrites 97.5%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+267} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+234}\right):\\ \;\;\;\;\frac{-z}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(-z\right) \cdot t\right)}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 74.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \frac{\left(-z\right) \cdot t}{a}\\ \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+267}:\\ \;\;\;\;\frac{-z}{a} \cdot t\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{-66}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 10^{-53}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;z \cdot t \leq 5000000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 10^{+106}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \frac{t}{a}\\ \end{array} \end{array} \]

FPCore

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 (/ (* (- z) t) a)))
   (if (<= (* z t) -1e+267)
     (* (/ (- z) a) t)
     (if (<= (* z t) -1e-66)
       t_1
       (if (<= (* z t) 1e-53)
         (/ (* x y) a)
         (if (<= (* z t) 5000000000000.0)
           t_1
           (if (<= (* z t) 1e+106) (* x (/ y a)) (* (- z) (/ t a)))))))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (-z * t) / a;
	double tmp;
	if ((z * t) <= -1e+267) {
		tmp = (-z / a) * t;
	} else if ((z * t) <= -1e-66) {
		tmp = t_1;
	} else if ((z * t) <= 1e-53) {
		tmp = (x * y) / a;
	} else if ((z * t) <= 5000000000000.0) {
		tmp = t_1;
	} else if ((z * t) <= 1e+106) {
		tmp = x * (y / a);
	} else {
		tmp = -z * (t / a);
	}
	return tmp;
}

Fortran

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) :: t_1
    real(8) :: tmp
    t_1 = (-z * t) / a
    if ((z * t) <= (-1d+267)) then
        tmp = (-z / a) * t
    else if ((z * t) <= (-1d-66)) then
        tmp = t_1
    else if ((z * t) <= 1d-53) then
        tmp = (x * y) / a
    else if ((z * t) <= 5000000000000.0d0) then
        tmp = t_1
    else if ((z * t) <= 1d+106) then
        tmp = x * (y / a)
    else
        tmp = -z * (t / a)
    end if
    code = tmp
end function

Java

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 = (-z * t) / a;
	double tmp;
	if ((z * t) <= -1e+267) {
		tmp = (-z / a) * t;
	} else if ((z * t) <= -1e-66) {
		tmp = t_1;
	} else if ((z * t) <= 1e-53) {
		tmp = (x * y) / a;
	} else if ((z * t) <= 5000000000000.0) {
		tmp = t_1;
	} else if ((z * t) <= 1e+106) {
		tmp = x * (y / a);
	} else {
		tmp = -z * (t / a);
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = (-z * t) / a
	tmp = 0
	if (z * t) <= -1e+267:
		tmp = (-z / a) * t
	elif (z * t) <= -1e-66:
		tmp = t_1
	elif (z * t) <= 1e-53:
		tmp = (x * y) / a
	elif (z * t) <= 5000000000000.0:
		tmp = t_1
	elif (z * t) <= 1e+106:
		tmp = x * (y / a)
	else:
		tmp = -z * (t / a)
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(-z) * t) / a)
	tmp = 0.0
	if (Float64(z * t) <= -1e+267)
		tmp = Float64(Float64(Float64(-z) / a) * t);
	elseif (Float64(z * t) <= -1e-66)
		tmp = t_1;
	elseif (Float64(z * t) <= 1e-53)
		tmp = Float64(Float64(x * y) / a);
	elseif (Float64(z * t) <= 5000000000000.0)
		tmp = t_1;
	elseif (Float64(z * t) <= 1e+106)
		tmp = Float64(x * Float64(y / a));
	else
		tmp = Float64(Float64(-z) * Float64(t / a));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (-z * t) / a;
	tmp = 0.0;
	if ((z * t) <= -1e+267)
		tmp = (-z / a) * t;
	elseif ((z * t) <= -1e-66)
		tmp = t_1;
	elseif ((z * t) <= 1e-53)
		tmp = (x * y) / a;
	elseif ((z * t) <= 5000000000000.0)
		tmp = t_1;
	elseif ((z * t) <= 1e+106)
		tmp = x * (y / a);
	else
		tmp = -z * (t / a);
	end
	tmp_2 = tmp;
end

Wolfram

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[((-z) * t), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -1e+267], N[(N[((-z) / a), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], -1e-66], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 1e-53], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5000000000000.0], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 1e+106], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision], N[((-z) * N[(t / a), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 z t) < -9.9999999999999997e266

   1. Initial program 74.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 78.0

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

   4. Applied rewrites 78.0%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 96.6

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

   7. Applied rewrites 96.6%

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

   if -9.9999999999999997e266 < (*.f64 z t) < -9.9999999999999998e-67 or 1.00000000000000003e-53 < (*.f64 z t) < 5e12

   1. Initial program 98.5%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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 75.6

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

   5. Applied rewrites 75.6%

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

   if -9.9999999999999998e-67 < (*.f64 z t) < 1.00000000000000003e-53

   1. Initial program 97.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. unpow1 N/A

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

      5. metadata-eval N/A

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

      6. sqrt-pow1 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. sqrt-prod N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lower-neg.f64 20.9

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

   4. Applied rewrites 20.9%

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

   5. Taylor expanded in y around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(x \cdot \left(y \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)}}{a} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(x \cdot \left(y \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)}}{a} \]

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

         \[\leadsto \frac{\color{blue}{x \cdot \left(\mathsf{neg}\left(y \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)}}{a} \]

      3. *-commutative N/A

         \[\leadsto \frac{x \cdot \left(\mathsf{neg}\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot y}\right)\right)}{a} \]

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-*.f64 88.4

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

   7. Applied rewrites 88.4%

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

   if 5e12 < (*.f64 z t) < 1.00000000000000009e106

   1. Initial program 90.4%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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 70.6

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

   5. Applied rewrites 70.6%

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

   7. Applied rewrites 65.9%

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

   if 1.00000000000000009e106 < (*.f64 z t)

   1. Initial program 86.1%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 90.6

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

   5. Applied rewrites 90.6%

      \[\leadsto \color{blue}{\left(-z\right) \cdot \frac{t}{a}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+267}:\\ \;\;\;\;\frac{-z}{a} \cdot t\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{-66}:\\ \;\;\;\;\frac{\left(-z\right) \cdot t}{a}\\ \mathbf{elif}\;z \cdot t \leq 10^{-53}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;z \cdot t \leq 5000000000000:\\ \;\;\;\;\frac{\left(-z\right) \cdot t}{a}\\ \mathbf{elif}\;z \cdot t \leq 10^{+106}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \frac{t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq 2 \cdot 10^{+270}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(-z\right) \cdot t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, x, \left(-t\right) \cdot \frac{z}{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (- (* x y) (* z t)) 2e+270)
   (/ (fma y x (* (- z) t)) a)
   (fma (/ y a) x (* (- t) (/ z a)))))

C

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) - (z * t)) <= 2e+270) {
		tmp = fma(y, x, (-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])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(x * y) - Float64(z * t)) <= 2e+270)
		tmp = Float64(fma(y, x, Float64(Float64(-z) * t)) / a);
	else
		tmp = fma(Float64(y / a), x, Float64(Float64(-t) * 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.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision], 2e+270], N[(N[(y * x + 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 (-.f64 (*.f64 x y) (*.f64 z t)) < 2.0000000000000001e270

   1. Initial program 97.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 97.1

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

   4. Applied rewrites 97.1%

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

   if 2.0000000000000001e270 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 68.2%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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 97.1

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

   4. Applied rewrites 97.1%

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

Alternative 4: 94.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+267} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+234}\right):\\ \;\;\;\;\frac{-z}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \end{array} \]

FPCore

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 (or (<= (* z t) -1e+267) (not (<= (* z t) 2e+234)))
   (* (/ (- z) a) t)
   (/ (- (* x y) (* z t)) a)))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z * t) <= -1e+267) || !((z * t) <= 2e+234)) {
		tmp = (-z / a) * t;
	} else {
		tmp = ((x * y) - (z * t)) / a;
	}
	return tmp;
}

Fortran

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 (((z * t) <= (-1d+267)) .or. (.not. ((z * t) <= 2d+234))) then
        tmp = (-z / a) * t
    else
        tmp = ((x * y) - (z * t)) / a
    end if
    code = tmp
end function

Java

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 (((z * t) <= -1e+267) || !((z * t) <= 2e+234)) {
		tmp = (-z / a) * t;
	} else {
		tmp = ((x * y) - (z * t)) / a;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if ((z * t) <= -1e+267) or not ((z * t) <= 2e+234):
		tmp = (-z / a) * t
	else:
		tmp = ((x * y) - (z * t)) / a
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if ((Float64(z * t) <= -1e+267) || !(Float64(z * t) <= 2e+234))
		tmp = Float64(Float64(Float64(-z) / a) * t);
	else
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a);
	end
	return tmp
end

MATLAB

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 (((z * t) <= -1e+267) || ~(((z * t) <= 2e+234)))
		tmp = (-z / a) * t;
	else
		tmp = ((x * y) - (z * t)) / a;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -1e+267], N[Not[LessEqual[N[(z * t), $MachinePrecision], 2e+234]], $MachinePrecision]], N[(N[((-z) / a), $MachinePrecision] * t), $MachinePrecision], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -9.9999999999999997e266 or 2.00000000000000004e234 < (*.f64 z t)

   1. Initial program 72.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 75.3

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

   4. Applied rewrites 75.3%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 96.0

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

   7. Applied rewrites 96.0%

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

   if -9.9999999999999997e266 < (*.f64 z t) < 2.00000000000000004e234

   1. Initial program 97.5%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+267} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+234}\right):\\ \;\;\;\;\frac{-z}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{-66} \lor \neg \left(z \cdot t \leq 10^{+106}\right):\\ \;\;\;\;\frac{-z}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \end{array} \]

FPCore

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 (or (<= (* z t) -1e-66) (not (<= (* z t) 1e+106)))
   (* (/ (- z) a) t)
   (/ (* x y) a)))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z * t) <= -1e-66) || !((z * t) <= 1e+106)) {
		tmp = (-z / a) * t;
	} else {
		tmp = (x * y) / a;
	}
	return tmp;
}

Fortran

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 (((z * t) <= (-1d-66)) .or. (.not. ((z * t) <= 1d+106))) then
        tmp = (-z / a) * t
    else
        tmp = (x * y) / a
    end if
    code = tmp
end function

Java

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 (((z * t) <= -1e-66) || !((z * t) <= 1e+106)) {
		tmp = (-z / a) * t;
	} else {
		tmp = (x * y) / a;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if ((z * t) <= -1e-66) or not ((z * t) <= 1e+106):
		tmp = (-z / a) * t
	else:
		tmp = (x * y) / a
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if ((Float64(z * t) <= -1e-66) || !(Float64(z * t) <= 1e+106))
		tmp = Float64(Float64(Float64(-z) / a) * t);
	else
		tmp = Float64(Float64(x * y) / a);
	end
	return tmp
end

MATLAB

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 (((z * t) <= -1e-66) || ~(((z * t) <= 1e+106)))
		tmp = (-z / a) * t;
	else
		tmp = (x * y) / a;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -1e-66], N[Not[LessEqual[N[(z * t), $MachinePrecision], 1e+106]], $MachinePrecision]], N[(N[((-z) / a), $MachinePrecision] * t), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -9.9999999999999998e-67 or 1.00000000000000009e106 < (*.f64 z t)

   1. Initial program 89.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 90.5

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

   4. Applied rewrites 90.5%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 79.3

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

   7. Applied rewrites 79.3%

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

   if -9.9999999999999998e-67 < (*.f64 z t) < 1.00000000000000009e106

   1. Initial program 96.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. unpow1 N/A

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

      5. metadata-eval N/A

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

      6. sqrt-pow1 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. sqrt-prod N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lower-neg.f64 23.8

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

   4. Applied rewrites 23.8%

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

   5. Taylor expanded in y around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(x \cdot \left(y \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)}}{a} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(x \cdot \left(y \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)}}{a} \]

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

         \[\leadsto \frac{\color{blue}{x \cdot \left(\mathsf{neg}\left(y \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)}}{a} \]

      3. *-commutative N/A

         \[\leadsto \frac{x \cdot \left(\mathsf{neg}\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot y}\right)\right)}{a} \]

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-*.f64 77.9

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

   7. Applied rewrites 77.9%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{-66} \lor \neg \left(z \cdot t \leq 10^{+106}\right):\\ \;\;\;\;\frac{-z}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{-66}:\\ \;\;\;\;\frac{-z}{a} \cdot t\\ \mathbf{elif}\;z \cdot t \leq 10^{-53}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \frac{t}{a}\\ \end{array} \end{array} \]

FPCore

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 (<= (* z t) -1e-66)
   (* (/ (- z) a) t)
   (if (<= (* z t) 1e-53) (/ (* x y) a) (* (- z) (/ t a)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z * t) <= -1e-66) {
		tmp = (-z / a) * t;
	} else if ((z * t) <= 1e-53) {
		tmp = (x * y) / a;
	} else {
		tmp = -z * (t / a);
	}
	return tmp;
}

Fortran

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 ((z * t) <= (-1d-66)) then
        tmp = (-z / a) * t
    else if ((z * t) <= 1d-53) then
        tmp = (x * y) / a
    else
        tmp = -z * (t / a)
    end if
    code = tmp
end function

Java

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 ((z * t) <= -1e-66) {
		tmp = (-z / a) * t;
	} else if ((z * t) <= 1e-53) {
		tmp = (x * y) / a;
	} else {
		tmp = -z * (t / a);
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (z * t) <= -1e-66:
		tmp = (-z / a) * t
	elif (z * t) <= 1e-53:
		tmp = (x * y) / a
	else:
		tmp = -z * (t / a)
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(z * t) <= -1e-66)
		tmp = Float64(Float64(Float64(-z) / a) * t);
	elseif (Float64(z * t) <= 1e-53)
		tmp = Float64(Float64(x * y) / a);
	else
		tmp = Float64(Float64(-z) * Float64(t / a));
	end
	return tmp
end

MATLAB

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 ((z * t) <= -1e-66)
		tmp = (-z / a) * t;
	elseif ((z * t) <= 1e-53)
		tmp = (x * y) / a;
	else
		tmp = -z * (t / a);
	end
	tmp_2 = tmp;
end

Wolfram

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[(z * t), $MachinePrecision], -1e-66], N[(N[((-z) / a), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e-53], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision], N[((-z) * N[(t / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -9.9999999999999998e-67

   1. Initial program 91.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 92.5

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

   4. Applied rewrites 92.5%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 75.1

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

   7. Applied rewrites 75.1%

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

   if -9.9999999999999998e-67 < (*.f64 z t) < 1.00000000000000003e-53

   1. Initial program 97.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. unpow1 N/A

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

      5. metadata-eval N/A

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

      6. sqrt-pow1 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. sqrt-prod N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lower-neg.f64 20.9

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

   4. Applied rewrites 20.9%

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

   5. Taylor expanded in y around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(x \cdot \left(y \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)}}{a} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(x \cdot \left(y \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)}}{a} \]

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

         \[\leadsto \frac{\color{blue}{x \cdot \left(\mathsf{neg}\left(y \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)}}{a} \]

      3. *-commutative N/A

         \[\leadsto \frac{x \cdot \left(\mathsf{neg}\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot y}\right)\right)}{a} \]

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-*.f64 88.4

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

   7. Applied rewrites 88.4%

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

   if 1.00000000000000003e-53 < (*.f64 z t)

   1. Initial program 89.4%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 67.8

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

   5. Applied rewrites 67.8%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{-66}:\\ \;\;\;\;\frac{-z}{a} \cdot t\\ \mathbf{elif}\;z \cdot t \leq 10^{-53}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \frac{t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq 10^{+109}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

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) (* z t)) 1e+109) (/ (* x y) a) (* x (/ y a))))

C

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) - (z * t)) <= 1e+109) {
		tmp = (x * y) / a;
	} else {
		tmp = x * (y / a);
	}
	return tmp;
}

Fortran

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) - (z * t)) <= 1d+109) then
        tmp = (x * y) / a
    else
        tmp = x * (y / a)
    end if
    code = tmp
end function

Java

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) - (z * t)) <= 1e+109) {
		tmp = (x * y) / a;
	} else {
		tmp = x * (y / a);
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if ((x * y) - (z * t)) <= 1e+109:
		tmp = (x * y) / a
	else:
		tmp = x * (y / a)
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(x * y) - Float64(z * t)) <= 1e+109)
		tmp = Float64(Float64(x * y) / a);
	else
		tmp = Float64(x * Float64(y / a));
	end
	return tmp
end

MATLAB

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) - (z * t)) <= 1e+109)
		tmp = (x * y) / a;
	else
		tmp = x * (y / a);
	end
	tmp_2 = tmp;
end

Wolfram

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[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision], 1e+109], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < 9.99999999999999982e108

   1. Initial program 96.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. unpow1 N/A

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

      5. metadata-eval N/A

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

      6. sqrt-pow1 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. sqrt-prod N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lower-neg.f64 28.6

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

   4. Applied rewrites 28.6%

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

   5. Taylor expanded in y around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(x \cdot \left(y \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)}}{a} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(x \cdot \left(y \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)}}{a} \]

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

         \[\leadsto \frac{\color{blue}{x \cdot \left(\mathsf{neg}\left(y \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)}}{a} \]

      3. *-commutative N/A

         \[\leadsto \frac{x \cdot \left(\mathsf{neg}\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot y}\right)\right)}{a} \]

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-*.f64 52.1

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

   7. Applied rewrites 52.1%

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

   if 9.99999999999999982e108 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 84.8%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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 44.5

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

   5. Applied rewrites 44.5%

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

   7. Applied rewrites 41.9%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq 10^{+109}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 8.2 \cdot 10^{-262}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= x 8.2e-262) (* x (/ y a)) (* (/ x a) y)))

C

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 <= 8.2e-262) {
		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.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (x <= 8.2d-262) 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;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= 8.2e-262) {
		tmp = x * (y / a);
	} else {
		tmp = (x / a) * y;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if x <= 8.2e-262:
		tmp = x * (y / a)
	else:
		tmp = (x / a) * y
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (x <= 8.2e-262)
		tmp = Float64(x * Float64(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])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (x <= 8.2e-262)
		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.
code[x_, y_, z_, t_, a_] := If[LessEqual[x, 8.2e-262], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision], N[(N[(x / a), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.20000000000000052e-262

   1. Initial program 90.1%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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 46.8

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

   5. Applied rewrites 46.8%

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

   7. Applied rewrites 48.5%

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

   if 8.20000000000000052e-262 < x

   1. Initial program 96.0%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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 45.1

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

   5. Applied rewrites 45.1%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.2 \cdot 10^{-262}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.3% accurate, 1.5× speedup

Math

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

FPCore

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 (/ y a)))

C

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

Fortran

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 * (y / a)
end function

Java

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 * (y / a);
}

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x, y, z, t, and a 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 93.0%

   \[\frac{x \cdot y - z \cdot t}{a} \]
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 46.0

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

5. Applied rewrites 46.0%

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

7. Applied rewrites 47.3%

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

8. Final simplification 47.3%

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

Developer Target 1: 91.6% 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 2024320 
(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))