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

Percentage Accurate: 91.1% → 95.5%

Time: 3.8s

Alternatives: 8

Speedup: 0.7×

• 29.2% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    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: 91.1% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 8 \cdot 10^{-25}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(-z\right) \cdot t\right)}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a\_m}, x, \left(-z\right) \cdot \frac{t}{a\_m}\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= a_m 8e-25)
    (/ (fma y x (* (- z) t)) a_m)
    (fma (/ y a_m) x (* (- z) (/ t a_m))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (a_m <= 8e-25) {
		tmp = fma(y, x, (-z * t)) / a_m;
	} else {
		tmp = fma((y / a_m), x, (-z * (t / a_m)));
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (a_m <= 8e-25)
		tmp = Float64(fma(y, x, Float64(Float64(-z) * t)) / a_m);
	else
		tmp = fma(Float64(y / a_m), x, Float64(Float64(-z) * Float64(t / a_m)));
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[a$95$m, 8e-25], N[(N[(y * x + N[((-z) * t), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], N[(N[(y / a$95$m), $MachinePrecision] * x + N[((-z) * N[(t / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 8.00000000000000031e-25

   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

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

      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

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

      6. distribute-lft-neg-in N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 89.7

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

   4. Applied rewrites 89.7%

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

   if 8.00000000000000031e-25 < a

   1. Initial program 81.9%

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

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

      4. lift-*.f64 N/A

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

      5. frac-2neg N/A

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

      6. *-commutative N/A

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

      7. frac-2neg N/A

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

      8. *-commutative N/A

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

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

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

      10. div-add N/A

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

      11. *-commutative N/A

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

      12. associate-*l/ N/A

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

      13. distribute-lft-neg-in N/A

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

      14. *-commutative N/A

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

      15. mul-1-neg N/A

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

      16. associate-*r/ N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. associate-*l/ N/A

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

      20. associate-*l* N/A

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

      21. *-commutative N/A

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

      22. associate-*r* N/A

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

      23. *-commutative N/A

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

   4. Applied rewrites 96.7%

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

Alternative 2: 94.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq 3 \cdot 10^{+299}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(-z\right) \cdot t\right)}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{z}, x, -t\right)}{a\_m} \cdot z\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (- (* x y) (* z t)) 3e+299)
    (/ (fma y x (* (- z) t)) a_m)
    (* (/ (fma (/ y z) x (- t)) a_m) z))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (((x * y) - (z * t)) <= 3e+299) {
		tmp = fma(y, x, (-z * t)) / a_m;
	} else {
		tmp = (fma((y / z), x, -t) / a_m) * z;
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(Float64(x * y) - Float64(z * t)) <= 3e+299)
		tmp = Float64(fma(y, x, Float64(Float64(-z) * t)) / a_m);
	else
		tmp = Float64(Float64(fma(Float64(y / z), x, Float64(-t)) / a_m) * z);
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision], 3e+299], N[(N[(y * x + N[((-z) * t), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], N[(N[(N[(N[(y / z), $MachinePrecision] * x + (-t)), $MachinePrecision] / a$95$m), $MachinePrecision] * z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < 3.0000000000000002e299

   1. Initial program 95.5%

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

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

      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

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

      6. distribute-lft-neg-in N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 95.5

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

   4. Applied rewrites 95.5%

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

   if 3.0000000000000002e299 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 44.9%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 87.4%

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

   6. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-neg.f64 90.1

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

   8. Applied rewrites 90.1%

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

Alternative 3: 95.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 2.9 \cdot 10^{-24}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(-z\right) \cdot t\right)}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a\_m}, y, \left(-z\right) \cdot \frac{t}{a\_m}\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= a_m 2.9e-24)
    (/ (fma y x (* (- z) t)) a_m)
    (fma (/ x a_m) y (* (- z) (/ t a_m))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (a_m <= 2.9e-24) {
		tmp = fma(y, x, (-z * t)) / a_m;
	} else {
		tmp = fma((x / a_m), y, (-z * (t / a_m)));
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (a_m <= 2.9e-24)
		tmp = Float64(fma(y, x, Float64(Float64(-z) * t)) / a_m);
	else
		tmp = fma(Float64(x / a_m), y, Float64(Float64(-z) * Float64(t / a_m)));
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[a$95$m, 2.9e-24], N[(N[(y * x + N[((-z) * t), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], N[(N[(x / a$95$m), $MachinePrecision] * y + N[((-z) * N[(t / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 2.8999999999999999e-24

   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

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

      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

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

      6. distribute-lft-neg-in N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 89.7

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

   4. Applied rewrites 89.7%

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

   if 2.8999999999999999e-24 < a

   1. Initial program 81.9%

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

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

      4. lift-*.f64 N/A

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

      5. frac-2neg N/A

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

      6. *-commutative N/A

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

      7. frac-2neg N/A

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

      8. *-commutative N/A

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

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

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

      10. div-add N/A

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

      11. associate-*l/ N/A

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

      12. distribute-lft-neg-in N/A

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

      13. *-commutative N/A

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

      14. mul-1-neg N/A

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

      15. associate-*r/ N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. associate-*l/ N/A

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

      19. associate-*l* N/A

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

      20. *-commutative N/A

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

      21. associate-*r* N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 N/A

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

   4. Applied rewrites 92.9%

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

Alternative 4: 73.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+73}:\\ \;\;\;\;\frac{y}{a\_m} \cdot x\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+24}:\\ \;\;\;\;\left(-z\right) \cdot \frac{t}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a\_m} \cdot y\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* x y) -2e+73)
    (* (/ y a_m) x)
    (if (<= (* x y) 5e+24) (* (- z) (/ t a_m)) (* (/ x a_m) y)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -2e+73) {
		tmp = (y / a_m) * x;
	} else if ((x * y) <= 5e+24) {
		tmp = -z * (t / a_m);
	} else {
		tmp = (x / a_m) * y;
	}
	return a_s * tmp;
}

Fortran

a\_m =     private
a\_s =     private
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a_s, x, y, z, t, a_m)
use fmin_fmax_functions
    real(8), intent (in) :: a_s
    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_m
    real(8) :: tmp
    if ((x * y) <= (-2d+73)) then
        tmp = (y / a_m) * x
    else if ((x * y) <= 5d+24) then
        tmp = -z * (t / a_m)
    else
        tmp = (x / a_m) * y
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -2e+73) {
		tmp = (y / a_m) * x;
	} else if ((x * y) <= 5e+24) {
		tmp = -z * (t / a_m);
	} else {
		tmp = (x / a_m) * y;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if (x * y) <= -2e+73:
		tmp = (y / a_m) * x
	elif (x * y) <= 5e+24:
		tmp = -z * (t / a_m)
	else:
		tmp = (x / a_m) * y
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(x * y) <= -2e+73)
		tmp = Float64(Float64(y / a_m) * x);
	elseif (Float64(x * y) <= 5e+24)
		tmp = Float64(Float64(-z) * Float64(t / a_m));
	else
		tmp = Float64(Float64(x / a_m) * y);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if ((x * y) <= -2e+73)
		tmp = (y / a_m) * x;
	elseif ((x * y) <= 5e+24)
		tmp = -z * (t / a_m);
	else
		tmp = (x / a_m) * y;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], -2e+73], N[(N[(y / a$95$m), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+24], N[((-z) * N[(t / a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x / a$95$m), $MachinePrecision] * y), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1.99999999999999997e73

   1. Initial program 87.6%

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

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

      4. lift-*.f64 N/A

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

      5. frac-2neg N/A

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

      6. *-commutative N/A

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

      7. frac-2neg N/A

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

      8. *-commutative N/A

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

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

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

      10. div-add N/A

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

      11. *-commutative N/A

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

      12. associate-*l/ N/A

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

      13. distribute-lft-neg-in N/A

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

      14. *-commutative N/A

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

      15. mul-1-neg N/A

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

      16. associate-*r/ N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. associate-*l/ N/A

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

      20. associate-*l* N/A

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

      21. *-commutative N/A

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

      22. associate-*r* N/A

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

      23. *-commutative N/A

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

   4. Applied rewrites 85.6%

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

   5. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. div-add N/A

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

      7. lift-*.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-/.f64 78.5

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

   7. Applied rewrites 78.5%

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

   if -1.99999999999999997e73 < (*.f64 x y) < 5.00000000000000045e24

   1. Initial program 90.7%

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

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 72.9

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

   5. Applied rewrites 72.9%

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

   if 5.00000000000000045e24 < (*.f64 x y)

   1. Initial program 80.1%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 71.6%

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

   6. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 78.9

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

   8. Applied rewrites 78.9%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+73}:\\ \;\;\;\;\frac{y}{a} \cdot x\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+24}:\\ \;\;\;\;\left(-z\right) \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+73}:\\ \;\;\;\;\frac{y}{a\_m} \cdot x\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+24}:\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a\_m} \cdot y\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* x y) -2e+73)
    (* (/ y a_m) x)
    (if (<= (* x y) 5e+24) (* (- t) (/ z a_m)) (* (/ x a_m) y)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -2e+73) {
		tmp = (y / a_m) * x;
	} else if ((x * y) <= 5e+24) {
		tmp = -t * (z / a_m);
	} else {
		tmp = (x / a_m) * y;
	}
	return a_s * tmp;
}

Fortran

a\_m =     private
a\_s =     private
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a_s, x, y, z, t, a_m)
use fmin_fmax_functions
    real(8), intent (in) :: a_s
    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_m
    real(8) :: tmp
    if ((x * y) <= (-2d+73)) then
        tmp = (y / a_m) * x
    else if ((x * y) <= 5d+24) then
        tmp = -t * (z / a_m)
    else
        tmp = (x / a_m) * y
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -2e+73) {
		tmp = (y / a_m) * x;
	} else if ((x * y) <= 5e+24) {
		tmp = -t * (z / a_m);
	} else {
		tmp = (x / a_m) * y;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if (x * y) <= -2e+73:
		tmp = (y / a_m) * x
	elif (x * y) <= 5e+24:
		tmp = -t * (z / a_m)
	else:
		tmp = (x / a_m) * y
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(x * y) <= -2e+73)
		tmp = Float64(Float64(y / a_m) * x);
	elseif (Float64(x * y) <= 5e+24)
		tmp = Float64(Float64(-t) * Float64(z / a_m));
	else
		tmp = Float64(Float64(x / a_m) * y);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if ((x * y) <= -2e+73)
		tmp = (y / a_m) * x;
	elseif ((x * y) <= 5e+24)
		tmp = -t * (z / a_m);
	else
		tmp = (x / a_m) * y;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], -2e+73], N[(N[(y / a$95$m), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+24], N[((-t) * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x / a$95$m), $MachinePrecision] * y), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1.99999999999999997e73

   1. Initial program 87.6%

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

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

      4. lift-*.f64 N/A

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

      5. frac-2neg N/A

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

      6. *-commutative N/A

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

      7. frac-2neg N/A

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

      8. *-commutative N/A

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

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

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

      10. div-add N/A

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

      11. *-commutative N/A

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

      12. associate-*l/ N/A

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

      13. distribute-lft-neg-in N/A

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

      14. *-commutative N/A

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

      15. mul-1-neg N/A

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

      16. associate-*r/ N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. associate-*l/ N/A

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

      20. associate-*l* N/A

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

      21. *-commutative N/A

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

      22. associate-*r* N/A

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

      23. *-commutative N/A

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

   4. Applied rewrites 85.6%

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

   5. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. div-add N/A

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

      7. lift-*.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-/.f64 78.5

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

   7. Applied rewrites 78.5%

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

   if -1.99999999999999997e73 < (*.f64 x y) < 5.00000000000000045e24

   1. Initial program 90.7%

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

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

      4. lift-*.f64 N/A

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

      5. frac-2neg N/A

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

      6. *-commutative N/A

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

      7. frac-2neg N/A

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

      8. *-commutative N/A

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

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

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

      10. div-add N/A

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

      11. *-commutative N/A

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

      12. associate-*l/ N/A

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

      13. distribute-lft-neg-in N/A

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

      14. *-commutative N/A

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

      15. mul-1-neg N/A

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

      16. associate-*r/ N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. associate-*l/ N/A

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

      20. associate-*l* N/A

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

      21. *-commutative N/A

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

      22. associate-*r* N/A

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

      23. *-commutative N/A

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

   4. Applied rewrites 87.5%

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

   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

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

      2. lift-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. div-add N/A

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

      7. lift-*.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. associate-/l* N/A

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

      12. distribute-lft-neg-in N/A

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

      13. mul-1-neg N/A

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

      14. lower-*.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-/.f64 75.7

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

   7. Applied rewrites 75.7%

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

   if 5.00000000000000045e24 < (*.f64 x y)

   1. Initial program 80.1%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 71.6%

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

   6. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 78.9

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

   8. Applied rewrites 78.9%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+73}:\\ \;\;\;\;\frac{y}{a} \cdot x\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+24}:\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+293}:\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(-z\right) \cdot t\right)}{a\_m}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* z t) -5e+293) (* (- t) (/ z a_m)) (/ (fma y x (* (- z) t)) a_m))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((z * t) <= -5e+293) {
		tmp = -t * (z / a_m);
	} else {
		tmp = fma(y, x, (-z * t)) / a_m;
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(z * t) <= -5e+293)
		tmp = Float64(Float64(-t) * Float64(z / a_m));
	else
		tmp = Float64(fma(y, x, Float64(Float64(-z) * t)) / a_m);
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(z * t), $MachinePrecision], -5e+293], N[((-t) * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(y * x + N[((-z) * t), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -5.00000000000000033e293

   1. Initial program 48.4%

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

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

      4. lift-*.f64 N/A

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

      5. frac-2neg N/A

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

      6. *-commutative N/A

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

      7. frac-2neg N/A

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

      8. *-commutative N/A

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

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

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

      10. div-add N/A

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

      11. *-commutative N/A

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

      12. associate-*l/ N/A

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

      13. distribute-lft-neg-in N/A

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

      14. *-commutative N/A

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

      15. mul-1-neg N/A

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

      16. associate-*r/ N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. associate-*l/ N/A

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

      20. associate-*l* N/A

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

      21. *-commutative N/A

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

      22. associate-*r* N/A

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

      23. *-commutative N/A

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

   4. Applied rewrites 99.5%

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

   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

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

      2. lift-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. div-add N/A

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

      7. lift-*.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. associate-/l* N/A

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

      12. distribute-lft-neg-in N/A

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

      13. mul-1-neg N/A

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

      14. lower-*.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-/.f64 95.6

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

   7. Applied rewrites 95.6%

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

   if -5.00000000000000033e293 < (*.f64 z t)

   1. Initial program 91.7%

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

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

      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

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

      6. distribute-lft-neg-in N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 91.7

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

   4. Applied rewrites 91.7%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+293}:\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(-z\right) \cdot t\right)}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+293}:\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a\_m}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* z t) -5e+293) (* (- t) (/ z a_m)) (/ (- (* x y) (* z t)) a_m))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((z * t) <= -5e+293) {
		tmp = -t * (z / a_m);
	} else {
		tmp = ((x * y) - (z * t)) / a_m;
	}
	return a_s * tmp;
}

Fortran

a\_m =     private
a\_s =     private
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a_s, x, y, z, t, a_m)
use fmin_fmax_functions
    real(8), intent (in) :: a_s
    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_m
    real(8) :: tmp
    if ((z * t) <= (-5d+293)) then
        tmp = -t * (z / a_m)
    else
        tmp = ((x * y) - (z * t)) / a_m
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((z * t) <= -5e+293) {
		tmp = -t * (z / a_m);
	} else {
		tmp = ((x * y) - (z * t)) / a_m;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if (z * t) <= -5e+293:
		tmp = -t * (z / a_m)
	else:
		tmp = ((x * y) - (z * t)) / a_m
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(z * t) <= -5e+293)
		tmp = Float64(Float64(-t) * Float64(z / a_m));
	else
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a_m);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if ((z * t) <= -5e+293)
		tmp = -t * (z / a_m);
	else
		tmp = ((x * y) - (z * t)) / a_m;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(z * t), $MachinePrecision], -5e+293], N[((-t) * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -5.00000000000000033e293

   1. Initial program 48.4%

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

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

      4. lift-*.f64 N/A

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

      5. frac-2neg N/A

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

      6. *-commutative N/A

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

      7. frac-2neg N/A

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

      8. *-commutative N/A

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

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

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

      10. div-add N/A

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

      11. *-commutative N/A

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

      12. associate-*l/ N/A

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

      13. distribute-lft-neg-in N/A

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

      14. *-commutative N/A

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

      15. mul-1-neg N/A

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

      16. associate-*r/ N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. associate-*l/ N/A

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

      20. associate-*l* N/A

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

      21. *-commutative N/A

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

      22. associate-*r* N/A

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

      23. *-commutative N/A

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

   4. Applied rewrites 99.5%

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

   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

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

      2. lift-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. div-add N/A

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

      7. lift-*.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. associate-/l* N/A

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

      12. distribute-lft-neg-in N/A

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

      13. mul-1-neg N/A

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

      14. lower-*.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-/.f64 95.6

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

   7. Applied rewrites 95.6%

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

   if -5.00000000000000033e293 < (*.f64 z t)

   1. Initial program 91.7%

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

4. Final simplification 92.0%

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

Alternative 8: 51.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \left(\frac{y}{a\_m} \cdot x\right) \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m) :precision binary64 (* a_s (* (/ y a_m) x)))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	return a_s * ((y / a_m) * x);
}

Fortran

a\_m =     private
a\_s =     private
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a_s, x, y, z, t, a_m)
use fmin_fmax_functions
    real(8), intent (in) :: a_s
    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_m
    code = a_s * ((y / a_m) * x)
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	return a_s * ((y / a_m) * x);
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	return a_s * ((y / a_m) * x)

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	return Float64(a_s * Float64(Float64(y / a_m) * x))
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp = code(a_s, x, y, z, t, a_m)
	tmp = a_s * ((y / a_m) * x);
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * N[(N[(y / a$95$m), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.6%

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

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

   4. lift-*.f64 N/A

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

   5. frac-2neg N/A

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

   6. *-commutative N/A

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

   7. frac-2neg N/A

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

   8. *-commutative N/A

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

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

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

   10. div-add N/A

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

   11. *-commutative N/A

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

   12. associate-*l/ N/A

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

   13. distribute-lft-neg-in N/A

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

   14. *-commutative N/A

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

   15. mul-1-neg N/A

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

   16. associate-*r/ N/A

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

   17. lower-fma.f64 N/A

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

   18. lower-/.f64 N/A

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

   19. associate-*l/ N/A

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

   20. associate-*l* N/A

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

   21. *-commutative N/A

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

   22. associate-*r* N/A

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

   23. *-commutative N/A

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

4. Applied rewrites 87.6%

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

5. Taylor expanded in x around inf

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

   1. associate-*l/ N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. div-add N/A

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

   7. lift-*.f64 N/A

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

   8. lift-neg.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/l* N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lift-/.f64 50.8

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

7. Applied rewrites 50.8%

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

8. Final simplification 50.8%

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

Developer Target 1: 91.8% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    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 2025064 
(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))