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

Percentage Accurate: 91.2% → 96.7%

Time: 3.8s

Alternatives: 10

Speedup: 0.5×

• 28.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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: 91.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 96.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{t}, y, -z\right)}{a} \cdot t\\ \mathbf{elif}\;t\_1 \leq 10^{+308}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, t, y \cdot x\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a}, y, z \cdot \left(\frac{-1}{a} \cdot t\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))))
   (if (<= t_1 (- INFINITY))
     (* (/ (fma (/ x t) y (- z)) a) t)
     (if (<= t_1 1e+308)
       (/ (fma (- z) t (* y x)) a)
       (fma (/ x a) y (* z (* (/ -1.0 a) t)))))))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (fma((x / t), y, -z) / a) * t;
	} else if (t_1 <= 1e+308) {
		tmp = fma(-z, t, (y * x)) / a;
	} else {
		tmp = fma((x / a), y, (z * ((-1.0 / a) * t)));
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(fma(Float64(x / t), y, Float64(-z)) / a) * t);
	elseif (t_1 <= 1e+308)
		tmp = Float64(fma(Float64(-z), t, Float64(y * x)) / a);
	else
		tmp = fma(Float64(x / a), y, Float64(z * Float64(Float64(-1.0 / a) * t)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(x / t), $MachinePrecision] * y + (-z)), $MachinePrecision] / a), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 1e+308], N[(N[((-z) * t + N[(y * x), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(x / a), $MachinePrecision] * y + N[(z * N[(N[(-1.0 / a), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 72.9%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 78.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-neg.f64 89.6

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

   7. Applied rewrites 89.6%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 1e308

   1. Initial program 98.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 98.6

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

   4. Applied rewrites 98.6%

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

   if 1e308 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 63.8%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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 99.8%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 99.9

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

   6. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. /-rgt-identity N/A

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

      7. lower-*.f64 N/A

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

      8. lift-neg.f64 99.9

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

   8. Applied rewrites 99.9%

      \[\leadsto \mathsf{fma}\left(\frac{x}{a}, y, \left(-z\right) \cdot \color{blue}{\left(\frac{-1}{a} \cdot \left(-t\right)\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{t}, y, -z\right)}{a} \cdot t\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 10^{+308}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, t, y \cdot x\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a}, y, z \cdot \left(\frac{-1}{a} \cdot t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 5 \cdot 10^{+229}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{t}, y, -z\right)}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, t, y \cdot x\right)}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 5e+229)))
     (* (/ (fma (/ x t) y (- z)) a) t)
     (/ (fma (- z) t (* y x)) a))))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 5e+229)) {
		tmp = (fma((x / t), y, -z) / a) * t;
	} else {
		tmp = fma(-z, t, (y * x)) / a;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 5e+229))
		tmp = Float64(Float64(fma(Float64(x / t), y, Float64(-z)) / a) * t);
	else
		tmp = Float64(fma(Float64(-z), t, Float64(y * x)) / a);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 5e+229]], $MachinePrecision]], N[(N[(N[(N[(x / t), $MachinePrecision] * y + (-z)), $MachinePrecision] / a), $MachinePrecision] * t), $MachinePrecision], N[(N[((-z) * t + N[(y * x), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -inf.0 or 5.0000000000000005e229 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 75.0%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 89.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-neg.f64 91.5

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

   7. Applied rewrites 91.5%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 5.0000000000000005e229

   1. Initial program 98.5%

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

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

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

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 98.5

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

   4. Applied rewrites 98.5%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty \lor \neg \left(x \cdot y - z \cdot t \leq 5 \cdot 10^{+229}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{t}, y, -z\right)}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, t, y \cdot x\right)}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{t}, y, -z\right)}{a} \cdot t\\ \mathbf{elif}\;t\_1 \leq 10^{+308}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, t, y \cdot x\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a}, y, \left(-z\right) \cdot \frac{t}{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))))
   (if (<= t_1 (- INFINITY))
     (* (/ (fma (/ x t) y (- z)) a) t)
     (if (<= t_1 1e+308)
       (/ (fma (- z) t (* y x)) a)
       (fma (/ x a) y (* (- z) (/ t a)))))))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (fma((x / t), y, -z) / a) * t;
	} else if (t_1 <= 1e+308) {
		tmp = fma(-z, t, (y * x)) / a;
	} else {
		tmp = fma((x / a), y, (-z * (t / a)));
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(fma(Float64(x / t), y, Float64(-z)) / a) * t);
	elseif (t_1 <= 1e+308)
		tmp = Float64(fma(Float64(-z), t, Float64(y * x)) / a);
	else
		tmp = fma(Float64(x / a), y, Float64(Float64(-z) * Float64(t / a)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(x / t), $MachinePrecision] * y + (-z)), $MachinePrecision] / a), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 1e+308], N[(N[((-z) * t + N[(y * x), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(x / a), $MachinePrecision] * y + N[((-z) * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 72.9%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 78.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-neg.f64 89.6

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

   7. Applied rewrites 89.6%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 1e308

   1. Initial program 98.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 98.6

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

   4. Applied rewrites 98.6%

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

   if 1e308 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 63.8%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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 99.8%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{t}, y, -z\right)}{a} \cdot t\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 10^{+308}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, t, y \cdot x\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a}, y, \left(-z\right) \cdot \frac{t}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot y - z \cdot t}{a} \leq 10^{+306}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, t, y \cdot x\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-t}{y}, z, x\right)}{a} \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (/ (- (* x y) (* z t)) a) 1e+306)
   (/ (fma (- z) t (* y x)) a)
   (* (/ (fma (/ (- t) y) z x) a) y)))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((((x * y) - (z * t)) / a) <= 1e+306) {
		tmp = fma(-z, t, (y * x)) / a;
	} else {
		tmp = (fma((-t / y), z, x) / a) * y;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(Float64(x * y) - Float64(z * t)) / a) <= 1e+306)
		tmp = Float64(fma(Float64(-z), t, Float64(y * x)) / a);
	else
		tmp = Float64(Float64(fma(Float64(Float64(-t) / y), z, x) / a) * y);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], 1e+306], N[(N[((-z) * t + N[(y * x), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(N[(N[((-t) / y), $MachinePrecision] * z + x), $MachinePrecision] / a), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x y) (*.f64 z t)) a) < 1.00000000000000002e306

   1. Initial program 93.7%

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

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

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

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.7

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

   4. Applied rewrites 93.7%

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

   if 1.00000000000000002e306 < (/.f64 (-.f64 (*.f64 x y) (*.f64 z t)) a)

   1. Initial program 85.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 85.6

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

   4. Applied rewrites 85.6%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 81.0

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

   7. Applied rewrites 81.0%

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

      1. +-commutative 81.0

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

      2. *-commutative 81.0

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

      3. distribute-lft-neg-out 81.0

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

      4. *-commutative 81.0

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

      5. mul-1-neg 81.0

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

      6. div-add-rev 81.0

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

      7. frac-add 81.0

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

      8. *-commutative 81.0

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

      9. *-commutative 81.0

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

      10. mul-1-neg 81.0

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

      11. *-commutative 81.0

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

      12. distribute-lft-neg-out 81.0

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

      13. *-commutative 81.0

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

      14. +-commutative 81.0

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

   9. Applied rewrites 91.9%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y - z \cdot t}{a} \leq 10^{+306}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, t, y \cdot x\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-t}{y}, z, x\right)}{a} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := x \cdot \frac{y}{a}\\ \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+119}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{+93}:\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a}\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{+17}:\\ \;\;\;\;\frac{y \cdot x}{a}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+95}:\\ \;\;\;\;\frac{\left(-z\right) \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* x (/ y a))))
   (if (<= (* x y) -2e+119)
     t_1
     (if (<= (* x y) -5e+93)
       (* (- t) (/ z a))
       (if (<= (* x y) -5e+17)
         (/ (* y x) a)
         (if (<= (* x y) 5e+95) (/ (* (- z) t) a) t_1))))))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = x * (y / a);
	double tmp;
	if ((x * y) <= -2e+119) {
		tmp = t_1;
	} else if ((x * y) <= -5e+93) {
		tmp = -t * (z / a);
	} else if ((x * y) <= -5e+17) {
		tmp = (y * x) / a;
	} else if ((x * y) <= 5e+95) {
		tmp = (-z * t) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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 = x * (y / a)
    if ((x * y) <= (-2d+119)) then
        tmp = t_1
    else if ((x * y) <= (-5d+93)) then
        tmp = -t * (z / a)
    else if ((x * y) <= (-5d+17)) then
        tmp = (y * x) / a
    else if ((x * y) <= 5d+95) then
        tmp = (-z * t) / a
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x * (y / a);
	double tmp;
	if ((x * y) <= -2e+119) {
		tmp = t_1;
	} else if ((x * y) <= -5e+93) {
		tmp = -t * (z / a);
	} else if ((x * y) <= -5e+17) {
		tmp = (y * x) / a;
	} else if ((x * y) <= 5e+95) {
		tmp = (-z * t) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = x * (y / a)
	tmp = 0
	if (x * y) <= -2e+119:
		tmp = t_1
	elif (x * y) <= -5e+93:
		tmp = -t * (z / a)
	elif (x * y) <= -5e+17:
		tmp = (y * x) / a
	elif (x * y) <= 5e+95:
		tmp = (-z * t) / a
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(x * Float64(y / a))
	tmp = 0.0
	if (Float64(x * y) <= -2e+119)
		tmp = t_1;
	elseif (Float64(x * y) <= -5e+93)
		tmp = Float64(Float64(-t) * Float64(z / a));
	elseif (Float64(x * y) <= -5e+17)
		tmp = Float64(Float64(y * x) / a);
	elseif (Float64(x * y) <= 5e+95)
		tmp = Float64(Float64(Float64(-z) * t) / a);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = x * (y / a);
	tmp = 0.0;
	if ((x * y) <= -2e+119)
		tmp = t_1;
	elseif ((x * y) <= -5e+93)
		tmp = -t * (z / a);
	elseif ((x * y) <= -5e+17)
		tmp = (y * x) / a;
	elseif ((x * y) <= 5e+95)
		tmp = (-z * t) / a;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -2e+119], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], -5e+93], N[((-t) * N[(z / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -5e+17], N[(N[(y * x), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+95], N[(N[((-z) * t), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -1.99999999999999989e119 or 5.00000000000000025e95 < (*.f64 x y)

   1. Initial program 88.4%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. 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 89.2%

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

   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. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. div-add N/A

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

   7. Applied rewrites 85.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 86.3

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

   9. Applied rewrites 86.3%

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

   if -1.99999999999999989e119 < (*.f64 x y) < -5.0000000000000001e93

   1. Initial program 90.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 90.3

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 100.0%

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

   if -5.0000000000000001e93 < (*.f64 x y) < -5e17

   1. Initial program 86.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 78.5

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

   5. Applied rewrites 78.5%

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

   if -5e17 < (*.f64 x y) < 5.00000000000000025e95

   1. Initial program 95.1%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 79.6

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

   5. Applied rewrites 79.6%

      \[\leadsto \frac{\color{blue}{\left(-z\right) \cdot t}}{a} \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+119}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{+93}:\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a}\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{+17}:\\ \;\;\;\;\frac{y \cdot x}{a}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+95}:\\ \;\;\;\;\frac{\left(-z\right) \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+301} \lor \neg \left(z \cdot t \leq 10^{+236}\right):\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, t, y \cdot x\right)}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (* z t) -5e+301) (not (<= (* z t) 1e+236)))
   (* (- t) (/ z a))
   (/ (fma (- z) t (* y x)) a)))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z * t) <= -5e+301) || !((z * t) <= 1e+236)) {
		tmp = -t * (z / a);
	} else {
		tmp = fma(-z, t, (y * x)) / a;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if ((Float64(z * t) <= -5e+301) || !(Float64(z * t) <= 1e+236))
		tmp = Float64(Float64(-t) * Float64(z / a));
	else
		tmp = Float64(fma(Float64(-z), t, Float64(y * x)) / a);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -5e+301], N[Not[LessEqual[N[(z * t), $MachinePrecision], 1e+236]], $MachinePrecision]], N[((-t) * N[(z / a), $MachinePrecision]), $MachinePrecision], N[(N[((-z) * t + N[(y * x), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -5.0000000000000004e301 or 1.00000000000000005e236 < (*.f64 z t)

   1. Initial program 74.7%

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

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

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

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 74.7

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

   4. Applied rewrites 74.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 97.6%

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

   if -5.0000000000000004e301 < (*.f64 z t) < 1.00000000000000005e236

   1. Initial program 95.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 95.6

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

   4. Applied rewrites 95.6%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+301} \lor \neg \left(z \cdot t \leq 10^{+236}\right):\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, t, y \cdot x\right)}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+301} \lor \neg \left(z \cdot t \leq 10^{+236}\right):\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (* z t) -5e+301) (not (<= (* z t) 1e+236)))
   (* (- t) (/ z a))
   (/ (- (* x y) (* z t)) a)))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z * t) <= -5e+301) || !((z * t) <= 1e+236)) {
		tmp = -t * (z / a);
	} else {
		tmp = ((x * y) - (z * t)) / a;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a 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(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) :: tmp
    if (((z * t) <= (-5d+301)) .or. (.not. ((z * t) <= 1d+236))) then
        tmp = -t * (z / a)
    else
        tmp = ((x * y) - (z * t)) / a
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z * t) <= -5e+301) || !((z * t) <= 1e+236)) {
		tmp = -t * (z / a);
	} else {
		tmp = ((x * y) - (z * t)) / a;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if ((z * t) <= -5e+301) or not ((z * t) <= 1e+236):
		tmp = -t * (z / a)
	else:
		tmp = ((x * y) - (z * t)) / a
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if ((Float64(z * t) <= -5e+301) || !(Float64(z * t) <= 1e+236))
		tmp = Float64(Float64(-t) * Float64(z / a));
	else
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((z * t) <= -5e+301) || ~(((z * t) <= 1e+236)))
		tmp = -t * (z / a);
	else
		tmp = ((x * y) - (z * t)) / a;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -5e+301], N[Not[LessEqual[N[(z * t), $MachinePrecision], 1e+236]], $MachinePrecision]], N[((-t) * N[(z / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -5.0000000000000004e301 or 1.00000000000000005e236 < (*.f64 z t)

   1. Initial program 74.7%

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

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

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

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 74.7

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

   4. Applied rewrites 74.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 97.6%

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

   if -5.0000000000000004e301 < (*.f64 z t) < 1.00000000000000005e236

   1. Initial program 95.6%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+301} \lor \neg \left(z \cdot t \leq 10^{+236}\right):\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+119} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{+125}\right):\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (* x y) -2e+119) (not (<= (* x y) 5e+125)))
   (* x (/ y a))
   (* (- t) (/ z a))))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((x * y) <= -2e+119) || !((x * y) <= 5e+125)) {
		tmp = x * (y / a);
	} else {
		tmp = -t * (z / a);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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) :: tmp
    if (((x * y) <= (-2d+119)) .or. (.not. ((x * y) <= 5d+125))) then
        tmp = x * (y / a)
    else
        tmp = -t * (z / a)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((x * y) <= -2e+119) || !((x * y) <= 5e+125)) {
		tmp = x * (y / a);
	} else {
		tmp = -t * (z / a);
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if ((x * y) <= -2e+119) or not ((x * y) <= 5e+125):
		tmp = x * (y / a)
	else:
		tmp = -t * (z / a)
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if ((Float64(x * y) <= -2e+119) || !(Float64(x * y) <= 5e+125))
		tmp = Float64(x * Float64(y / a));
	else
		tmp = Float64(Float64(-t) * Float64(z / a));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((x * y) <= -2e+119) || ~(((x * y) <= 5e+125)))
		tmp = x * (y / a);
	else
		tmp = -t * (z / a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -2e+119], N[Not[LessEqual[N[(x * y), $MachinePrecision], 5e+125]], $MachinePrecision]], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision], N[((-t) * N[(z / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.99999999999999989e119 or 4.99999999999999962e125 < (*.f64 x y)

   1. Initial program 88.9%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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 88.6%

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

   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. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. div-add N/A

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

   7. Applied rewrites 87.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 88.6

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

   9. Applied rewrites 88.6%

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

   if -1.99999999999999989e119 < (*.f64 x y) < 4.99999999999999962e125

   1. Initial program 93.7%

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

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

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

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.8

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

   4. Applied rewrites 93.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 73.1%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+119} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{+125}\right):\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-286}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= t 4.2e-286) (* x (/ y a)) (* (/ x a) y)))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 4.2e-286) {
		tmp = x * (y / a);
	} else {
		tmp = (x / a) * y;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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) :: tmp
    if (t <= 4.2d-286) then
        tmp = x * (y / a)
    else
        tmp = (x / a) * y
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 4.2e-286) {
		tmp = x * (y / a);
	} else {
		tmp = (x / a) * y;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if t <= 4.2e-286:
		tmp = x * (y / a)
	else:
		tmp = (x / a) * y
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 4.2e-286)
		tmp = Float64(x * Float64(y / a));
	else
		tmp = Float64(Float64(x / a) * y);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 4.2e-286)
		tmp = x * (y / a);
	else
		tmp = (x / a) * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[t, 4.2e-286], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision], N[(N[(x / a), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.19999999999999977e-286

   1. Initial program 96.1%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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 88.6%

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

   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. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. div-add N/A

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

   7. Applied rewrites 43.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 46.2

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

   9. Applied rewrites 46.2%

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

   if 4.19999999999999977e-286 < t

   1. Initial program 88.4%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. 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 87.0%

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

   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. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. div-add N/A

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

   7. Applied rewrites 47.7%

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

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-286}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.3%

   \[\frac{x \cdot y - z \cdot t}{a} \]
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 87.8%

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

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. lift-*.f64 N/A

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

   4. lift-neg.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-neg.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. associate-*r/ N/A

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

   10. *-commutative N/A

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

   11. div-add N/A

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

7. Applied rewrites 45.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-/.f64 47.7

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

9. Applied rewrites 47.7%

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

10. Final simplification 47.7%

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

Reproduce

herbie shell --seed 2025085 
(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))