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

Percentage Accurate: 91.4% → 97.4%

Time: 4.0s

Alternatives: 10

Speedup: 0.7×

• 28.8% 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: 91.4% 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: 97.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(-z\right) \cdot \frac{t}{a}\\ t_2 := x \cdot y - z \cdot t\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a}, y, t\_1\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+269}:\\ \;\;\;\;\frac{t\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, x, t\_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- z) (/ t a))) (t_2 (- (* x y) (* z t))))
   (if (<= t_2 (- INFINITY))
     (fma (/ x a) y t_1)
     (if (<= t_2 5e+269) (/ t_2 a) (fma (/ y a) x t_1)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = -z * (t / a);
	double t_2 = (x * y) - (z * t);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = fma((x / a), y, t_1);
	} else if (t_2 <= 5e+269) {
		tmp = t_2 / a;
	} else {
		tmp = fma((y / a), x, t_1);
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(-z) * Float64(t / a))
	t_2 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = fma(Float64(x / a), y, t_1);
	elseif (t_2 <= 5e+269)
		tmp = Float64(t_2 / a);
	else
		tmp = fma(Float64(y / a), x, t_1);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[((-z) * N[(t / a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(x / a), $MachinePrecision] * y + t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 5e+269], N[(t$95$2 / a), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * x + t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 64.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. frac-2neg N/A

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

      6. *-commutative N/A

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

      7. frac-2neg N/A

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

      8. *-commutative N/A

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

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

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

      10. div-add N/A

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

      11. 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 95.6%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 5.0000000000000002e269

   1. Initial program 99.0%

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

   if 5.0000000000000002e269 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 64.3%

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

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

      12. associate-*l/ N/A

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

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

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

      14. *-commutative N/A

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

      15. mul-1-neg N/A

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

      16. associate-*r/ N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. associate-*l/ N/A

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

      20. associate-*l* N/A

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

      21. *-commutative N/A

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

      22. associate-*r* N/A

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

      23. *-commutative N/A

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

   4. Applied rewrites 97.0%

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

Alternative 2: 96.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \frac{x \cdot y - z \cdot t}{a}\\ \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 4 \cdot 10^{+282}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{z}, y, -t\right)}{a} \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- (* x y) (* z t)) a)))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 4e+282)))
     (* (/ (fma (/ x z) y (- t)) a) z)
     t_1)))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = ((x * y) - (z * t)) / a;
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 4e+282)) {
		tmp = (fma((x / z), y, -t) / a) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(x * y) - Float64(z * t)) / a)
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 4e+282))
		tmp = Float64(Float64(fma(Float64(x / z), y, Float64(-t)) / a) * z);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 4e+282]], $MachinePrecision]], N[(N[(N[(N[(x / z), $MachinePrecision] * y + (-t)), $MachinePrecision] / a), $MachinePrecision] * z), $MachinePrecision], t$95$1]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x y) (*.f64 z t)) a) < -inf.0 or 4.00000000000000013e282 < (/.f64 (-.f64 (*.f64 x y) (*.f64 z t)) a)

   1. Initial program 77.9%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 89.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-frac-neg N/A

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

      7. mul-1-neg N/A

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

      8. +-commutative N/A

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

      9. associate-*r/ N/A

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

      10. associate-*r/ N/A

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

      11. div-add-rev N/A

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

      12. associate-*l/ N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. associate-*l/ N/A

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

      16. lower-fma.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. mul-1-neg N/A

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

      19. lift-neg.f64 94.0

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

   7. Applied rewrites 94.0%

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

   if -inf.0 < (/.f64 (-.f64 (*.f64 x y) (*.f64 z t)) a) < 4.00000000000000013e282

   1. Initial program 98.8%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y - z \cdot t}{a} \leq -\infty \lor \neg \left(\frac{x \cdot y - z \cdot t}{a} \leq 4 \cdot 10^{+282}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{z}, y, -t\right)}{a} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 10^{+290}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a}, y, \left(-z\right) \cdot \frac{t}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 1e+290)))
     (fma (/ x a) y (* (- z) (/ t a)))
     (/ t_1 a))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 1e+290)) {
		tmp = fma((x / a), y, (-z * (t / a)));
	} else {
		tmp = t_1 / a;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 1e+290))
		tmp = fma(Float64(x / a), y, Float64(Float64(-z) * Float64(t / a)));
	else
		tmp = Float64(t_1 / a);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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, 1e+290]], $MachinePrecision]], N[(N[(x / a), $MachinePrecision] * y + N[((-z) * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -inf.0 or 1.00000000000000006e290 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 61.1%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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 96.2%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 1.00000000000000006e290

   1. Initial program 99.1%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty \lor \neg \left(x \cdot y - z \cdot t \leq 10^{+290}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a}, y, \left(-z\right) \cdot \frac{t}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{z}, y, -t\right)}{a} \cdot z\\ \mathbf{elif}\;t\_1 \leq 10^{+289}:\\ \;\;\;\;\frac{t\_1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-t}{x}, z, y\right)}{a} \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))))
   (if (<= t_1 (- INFINITY))
     (* (/ (fma (/ x z) y (- t)) a) z)
     (if (<= t_1 1e+289) (/ t_1 a) (* (/ (fma (/ (- t) x) z y) a) x)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (fma((x / z), y, -t) / a) * z;
	} else if (t_1 <= 1e+289) {
		tmp = t_1 / a;
	} else {
		tmp = (fma((-t / x), z, y) / a) * x;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(fma(Float64(x / z), y, Float64(-t)) / a) * z);
	elseif (t_1 <= 1e+289)
		tmp = Float64(t_1 / a);
	else
		tmp = Float64(Float64(fma(Float64(Float64(-t) / x), z, y) / a) * x);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := 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 / z), $MachinePrecision] * y + (-t)), $MachinePrecision] / a), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$1, 1e+289], N[(t$95$1 / a), $MachinePrecision], N[(N[(N[(N[((-t) / x), $MachinePrecision] * z + y), $MachinePrecision] / a), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 64.6%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 91.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-frac-neg N/A

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

      7. mul-1-neg N/A

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

      8. +-commutative N/A

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

      9. associate-*r/ N/A

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

      10. associate-*r/ N/A

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

      11. div-add-rev N/A

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

      12. associate-*l/ N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. associate-*l/ N/A

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

      16. lower-fma.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. mul-1-neg N/A

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

      19. lift-neg.f64 91.9

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

   7. Applied rewrites 91.9%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 1.0000000000000001e289

   1. Initial program 99.0%

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

   if 1.0000000000000001e289 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 59.6%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-frac-neg N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r/ N/A

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

      9. div-add-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. distribute-frac-neg N/A

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

      14. lift-/.f64 N/A

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

      15. lift-neg.f64 93.8

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

   7. Applied rewrites 93.8%

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

Alternative 5: 73.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(-z\right) \cdot \frac{t}{a}\\ \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 0:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+40}:\\ \;\;\;\;\frac{y \cdot x}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- z) (/ t a))))
   (if (<= (* z t) -1e+44)
     t_1
     (if (<= (* z t) 0.0)
       (* (/ x a) y)
       (if (<= (* z t) 5e+40) (/ (* y x) a) t_1)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = -z * (t / a);
	double tmp;
	if ((z * t) <= -1e+44) {
		tmp = t_1;
	} else if ((z * t) <= 0.0) {
		tmp = (x / a) * y;
	} else if ((z * t) <= 5e+40) {
		tmp = (y * x) / 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.
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 = -z * (t / a)
    if ((z * t) <= (-1d+44)) then
        tmp = t_1
    else if ((z * t) <= 0.0d0) then
        tmp = (x / a) * y
    else if ((z * t) <= 5d+40) then
        tmp = (y * x) / a
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = -z * (t / a);
	double tmp;
	if ((z * t) <= -1e+44) {
		tmp = t_1;
	} else if ((z * t) <= 0.0) {
		tmp = (x / a) * y;
	} else if ((z * t) <= 5e+40) {
		tmp = (y * x) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = -z * (t / a)
	tmp = 0
	if (z * t) <= -1e+44:
		tmp = t_1
	elif (z * t) <= 0.0:
		tmp = (x / a) * y
	elif (z * t) <= 5e+40:
		tmp = (y * x) / a
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(-z) * Float64(t / a))
	tmp = 0.0
	if (Float64(z * t) <= -1e+44)
		tmp = t_1;
	elseif (Float64(z * t) <= 0.0)
		tmp = Float64(Float64(x / a) * y);
	elseif (Float64(z * t) <= 5e+40)
		tmp = Float64(Float64(y * x) / a);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = -z * (t / a);
	tmp = 0.0;
	if ((z * t) <= -1e+44)
		tmp = t_1;
	elseif ((z * t) <= 0.0)
		tmp = (x / a) * y;
	elseif ((z * t) <= 5e+40)
		tmp = (y * x) / 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.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[((-z) * N[(t / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -1e+44], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 0.0], N[(N[(x / a), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+40], N[(N[(y * x), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -1.0000000000000001e44 or 5.00000000000000003e40 < (*.f64 z t)

   1. Initial program 85.3%

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

   3. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 80.7

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

   5. Applied rewrites 80.7%

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

   if -1.0000000000000001e44 < (*.f64 z t) < 0.0

   1. Initial program 93.1%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 61.1%

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

   6. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 84.9

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

   8. Applied rewrites 84.9%

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

   if 0.0 < (*.f64 z t) < 5.00000000000000003e40

   1. Initial program 99.7%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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 72.2

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

   5. Applied rewrites 72.2%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+44}:\\ \;\;\;\;\left(-z\right) \cdot \frac{t}{a}\\ \mathbf{elif}\;z \cdot t \leq 0:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+40}:\\ \;\;\;\;\frac{y \cdot x}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \frac{t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(-t\right) \cdot \frac{z}{a}\\ \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 0:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+52}:\\ \;\;\;\;\frac{y \cdot x}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- t) (/ z a))))
   (if (<= (* z t) -1e+44)
     t_1
     (if (<= (* z t) 0.0)
       (* (/ x a) y)
       (if (<= (* z t) 2e+52) (/ (* y x) a) t_1)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = -t * (z / a);
	double tmp;
	if ((z * t) <= -1e+44) {
		tmp = t_1;
	} else if ((z * t) <= 0.0) {
		tmp = (x / a) * y;
	} else if ((z * t) <= 2e+52) {
		tmp = (y * x) / 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.
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 = -t * (z / a)
    if ((z * t) <= (-1d+44)) then
        tmp = t_1
    else if ((z * t) <= 0.0d0) then
        tmp = (x / a) * y
    else if ((z * t) <= 2d+52) then
        tmp = (y * x) / a
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = -t * (z / a);
	double tmp;
	if ((z * t) <= -1e+44) {
		tmp = t_1;
	} else if ((z * t) <= 0.0) {
		tmp = (x / a) * y;
	} else if ((z * t) <= 2e+52) {
		tmp = (y * x) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = -t * (z / a)
	tmp = 0
	if (z * t) <= -1e+44:
		tmp = t_1
	elif (z * t) <= 0.0:
		tmp = (x / a) * y
	elif (z * t) <= 2e+52:
		tmp = (y * x) / a
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(-t) * Float64(z / a))
	tmp = 0.0
	if (Float64(z * t) <= -1e+44)
		tmp = t_1;
	elseif (Float64(z * t) <= 0.0)
		tmp = Float64(Float64(x / a) * y);
	elseif (Float64(z * t) <= 2e+52)
		tmp = Float64(Float64(y * x) / a);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = -t * (z / a);
	tmp = 0.0;
	if ((z * t) <= -1e+44)
		tmp = t_1;
	elseif ((z * t) <= 0.0)
		tmp = (x / a) * y;
	elseif ((z * t) <= 2e+52)
		tmp = (y * x) / 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.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[((-t) * N[(z / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -1e+44], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 0.0], N[(N[(x / a), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+52], N[(N[(y * x), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -1.0000000000000001e44 or 2e52 < (*.f64 z t)

   1. Initial program 84.9%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 93.5%

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

   6. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lift-neg.f64 N/A

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

      6. lower-/.f64 80.9

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

   8. Applied rewrites 80.9%

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

   if -1.0000000000000001e44 < (*.f64 z t) < 0.0

   1. Initial program 93.1%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 61.1%

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

   6. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 84.9

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

   8. Applied rewrites 84.9%

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

   if 0.0 < (*.f64 z t) < 2e52

   1. Initial program 99.7%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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 70.1

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

   5. Applied rewrites 70.1%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+44}:\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a}\\ \mathbf{elif}\;z \cdot t \leq 0:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+52}:\\ \;\;\;\;\frac{y \cdot x}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot y - z \cdot t}{a} \leq 2 \cdot 10^{+196}:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (/ (- (* x y) (* z t)) a) 2e+196) (* (/ x a) y) (* (/ y a) x)))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((((x * y) - (z * t)) / a) <= 2e+196) {
		tmp = (x / a) * y;
	} else {
		tmp = (y / a) * x;
	}
	return tmp;
}

Fortran

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) - (z * t)) / a) <= 2d+196) then
        tmp = (x / a) * y
    else
        tmp = (y / a) * x
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((((x * y) - (z * t)) / a) <= 2e+196) {
		tmp = (x / a) * y;
	} else {
		tmp = (y / a) * x;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (((x * y) - (z * t)) / a) <= 2e+196:
		tmp = (x / a) * y
	else:
		tmp = (y / a) * x
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(Float64(x * y) - Float64(z * t)) / a) <= 2e+196)
		tmp = Float64(Float64(x / a) * y);
	else
		tmp = Float64(Float64(y / a) * x);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((((x * y) - (z * t)) / a) <= 2e+196)
		tmp = (x / a) * y;
	else
		tmp = (y / a) * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], 2e+196], N[(N[(x / a), $MachinePrecision] * y), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x y) (*.f64 z t)) a) < 1.9999999999999999e196

   1. Initial program 97.0%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 74.7%

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

   6. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 53.8

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

   8. Applied rewrites 53.8%

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

   if 1.9999999999999999e196 < (/.f64 (-.f64 (*.f64 x y) (*.f64 z t)) a)

   1. Initial program 74.2%

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

      1. lift-/.f64 N/A

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

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

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

      9. mul-1-neg N/A

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

      10. associate-*r* N/A

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

      11. +-commutative N/A

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

      12. div-add-rev N/A

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

      13. associate-*r/ N/A

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

      14. associate-/l* N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

      17. lower-fma.f64 N/A

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

      18. associate-*r/ N/A

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

      19. lower-/.f64 N/A

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

      20. mul-1-neg N/A

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

      21. lower-neg.f64 N/A

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

      22. lower-/.f64 N/A

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

   4. Applied rewrites 79.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 86.7

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

   6. Applied rewrites 86.7%

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

   7. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*l/ N/A

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

      5. div-add-rev N/A

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

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

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 57.5

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

   9. Applied rewrites 57.5%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y - z \cdot t}{a} \leq 2 \cdot 10^{+196}:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) 2e+300) (/ (- (* x y) (* z t)) a) (* (/ y a) x)))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= 2e+300) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = (y / a) * x;
	}
	return tmp;
}

Fortran

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+300) then
        tmp = ((x * y) - (z * t)) / a
    else
        tmp = (y / a) * x
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= 2e+300) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = (y / a) * x;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= 2e+300:
		tmp = ((x * y) - (z * t)) / a
	else:
		tmp = (y / a) * x
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= 2e+300)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a);
	else
		tmp = Float64(Float64(y / a) * x);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= 2e+300)
		tmp = ((x * y) - (z * t)) / a;
	else
		tmp = (y / a) * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], 2e+300], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 2.0000000000000001e300

   1. Initial program 93.6%

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

   if 2.0000000000000001e300 < (*.f64 x y)

   1. Initial program 59.3%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. fp-cancel-sub-sign-inv N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r* N/A

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

      11. +-commutative N/A

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

      12. div-add-rev N/A

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

      13. associate-*r/ N/A

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

      14. associate-/l* N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

      17. lower-fma.f64 N/A

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

      18. associate-*r/ N/A

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

      19. lower-/.f64 N/A

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

      20. mul-1-neg N/A

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

      21. lower-neg.f64 N/A

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

      22. lower-/.f64 N/A

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

   4. Applied rewrites 70.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 94.6

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

   6. Applied rewrites 94.6%

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

   7. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*l/ N/A

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

      5. div-add-rev N/A

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

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

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 94.6

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

   9. Applied rewrites 94.6%

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

Alternative 9: 51.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{-275}:\\ \;\;\;\;\frac{y \cdot x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= t -2.8e-275) (/ (* y x) a) (* (/ x a) y)))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -2.8e-275) {
		tmp = (y * x) / 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.
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 <= (-2.8d-275)) then
        tmp = (y * x) / a
    else
        tmp = (x / a) * y
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -2.8e-275) {
		tmp = (y * x) / a;
	} else {
		tmp = (x / a) * y;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if t <= -2.8e-275:
		tmp = (y * x) / a
	else:
		tmp = (x / a) * y
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -2.8e-275)
		tmp = Float64(Float64(y * x) / a);
	else
		tmp = Float64(Float64(x / a) * y);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -2.8e-275)
		tmp = (y * x) / a;
	else
		tmp = (x / a) * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[t, -2.8e-275], N[(N[(y * x), $MachinePrecision] / a), $MachinePrecision], N[(N[(x / a), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.79999999999999994e-275

   1. Initial program 95.2%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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 57.9

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

   5. Applied rewrites 57.9%

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

   if -2.79999999999999994e-275 < t

   1. Initial program 87.5%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 73.4%

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

   6. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 56.4

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

   8. Applied rewrites 56.4%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{-275}:\\ \;\;\;\;\frac{y \cdot x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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 / a) * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.0%

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

3. Taylor expanded in z around inf

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

5. Applied rewrites 74.5%

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

6. Taylor expanded in x around inf

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

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lift-/.f64 55.1

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

8. Applied rewrites 55.1%

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

9. Final simplification 55.1%

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

Developer Target 1: 91.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \mathbf{if}\;z < -2.468684968699548 \cdot 10^{+170}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z < 6.309831121978371 \cdot 10^{-71}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* (/ y a) x) (* (/ t a) z))))
   (if (< z -2.468684968699548e+170)
     t_1
     (if (< z 6.309831121978371e-71) (/ (- (* x y) (* z t)) a) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((y / a) * x) - ((t / a) * z);
	double tmp;
	if (z < -2.468684968699548e+170) {
		tmp = t_1;
	} else if (z < 6.309831121978371e-71) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((y / a) * x) - ((t / a) * z)
    if (z < (-2.468684968699548d+170)) then
        tmp = t_1
    else if (z < 6.309831121978371d-71) then
        tmp = ((x * y) - (z * t)) / a
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = ((y / a) * x) - ((t / a) * z);
	double tmp;
	if (z < -2.468684968699548e+170) {
		tmp = t_1;
	} else if (z < 6.309831121978371e-71) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = ((y / a) * x) - ((t / a) * z)
	tmp = 0
	if z < -2.468684968699548e+170:
		tmp = t_1
	elif z < 6.309831121978371e-71:
		tmp = ((x * y) - (z * t)) / a
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(y / a) * x) - Float64(Float64(t / a) * z))
	tmp = 0.0
	if (z < -2.468684968699548e+170)
		tmp = t_1;
	elseif (z < 6.309831121978371e-71)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((y / a) * x) - ((t / a) * z);
	tmp = 0.0;
	if (z < -2.468684968699548e+170)
		tmp = t_1;
	elseif (z < 6.309831121978371e-71)
		tmp = ((x * y) - (z * t)) / a;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(y / a), $MachinePrecision] * x), $MachinePrecision] - N[(N[(t / a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -2.468684968699548e+170], t$95$1, If[Less[z, 6.309831121978371e-71], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2025037 
(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))