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

Percentage Accurate: 91.7% → 94.6%

Time: 3.3s

Alternatives: 7

Speedup: 0.7×

• 28.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.7% 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: 94.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+213}:\\ \;\;\;\;\frac{t\_1}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-z}{a}, t, y \cdot \frac{x}{a}\right)\\ \end{array} \end{array} \]

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 5e+213) (/ t_1 a) (fma (/ (- z) a) 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 <= 5e+213) {
		tmp = t_1 / a;
	} else {
		tmp = fma((-z / a), 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 <= 5e+213)
		tmp = Float64(t_1 / a);
	else
		tmp = fma(Float64(Float64(-z) / a), t, Float64(y * Float64(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[LessEqual[t$95$1, 5e+213], N[(t$95$1 / a), $MachinePrecision], N[(N[((-z) / a), $MachinePrecision] * t + N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < 4.9999999999999998e213

   1. Initial program 94.6%

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

   if 4.9999999999999998e213 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 79.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

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

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

      6. div-add N/A

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

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

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. +-commutative N/A

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

      12. associate-*r/ 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. *-commutative N/A

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

      15. associate-*l/ 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

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

      23. *-commutative N/A

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

      24. lower-*.f64 84.3

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

   3. Applied rewrites 84.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 91.6

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

   5. Applied rewrites 91.6%

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

Alternative 2: 94.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+303}:\\ \;\;\;\;\frac{t\_1}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-z}{a}, t, x \cdot \frac{y}{a}\right)\\ \end{array} \end{array} \]

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 5e+303) (/ t_1 a) (fma (/ (- z) a) t (* 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) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if (t_1 <= 5e+303) {
		tmp = t_1 / a;
	} else {
		tmp = fma((-z / a), t, (x * (y / 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 <= 5e+303)
		tmp = Float64(t_1 / a);
	else
		tmp = fma(Float64(Float64(-z) / a), t, Float64(x * Float64(y / 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, 5e+303], N[(t$95$1 / a), $MachinePrecision], N[(N[((-z) / a), $MachinePrecision] * t + N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < 4.9999999999999997e303

   1. Initial program 95.0%

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

   if 4.9999999999999997e303 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 67.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

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

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

      6. div-add N/A

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

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

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. +-commutative N/A

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

      12. associate-*r/ 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. *-commutative N/A

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

      15. associate-*l/ 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

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

      23. *-commutative N/A

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

      24. lower-*.f64 77.3

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

   3. Applied rewrites 77.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 91.3

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

   5. Applied rewrites 91.3%

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

Alternative 3: 94.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t\_1 \leq 10^{+291}:\\ \;\;\;\;\frac{t\_1}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a}, y, \left(-z\right) \cdot \frac{t}{a}\right)\\ \end{array} \end{array} \]

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 1e+291) (/ t_1 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 <= 1e+291) {
		tmp = t_1 / 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 <= 1e+291)
		tmp = Float64(t_1 / 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, 1e+291], N[(t$95$1 / a), $MachinePrecision], N[(N[(x / a), $MachinePrecision] * y + N[((-z) * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < 9.9999999999999996e290

   1. Initial program 95.0%

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

   if 9.9999999999999996e290 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 70.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

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

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

      6. div-add N/A

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

      7. associate-*l/ N/A

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

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

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. associate-*r/ N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. associate-*l/ N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. mul-1-neg N/A

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

      21. lower-neg.f64 N/A

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

      22. lower-/.f64 90.9

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

   3. Applied rewrites 90.9%

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

Alternative 4: 93.6% accurate, 0.6× speedup

Math

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

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* z t) (- INFINITY)) (* (- t) (/ z a)) (/ (fma y x (* (- 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) <= -((double) INFINITY)) {
		tmp = -t * (z / a);
	} else {
		tmp = fma(y, x, (-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) <= Float64(-Inf))
		tmp = Float64(Float64(-t) * Float64(z / a));
	else
		tmp = Float64(fma(y, x, Float64(Float64(-z) * t)) / a);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 68.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

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

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

      6. div-add N/A

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

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

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. +-commutative N/A

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

      12. associate-*r/ 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. *-commutative N/A

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

      15. associate-*l/ 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

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

      23. *-commutative N/A

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

      24. lower-*.f64 88.8

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

   3. Applied rewrites 88.8%

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

   4. Taylor expanded in x around 0

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

      1. lift-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. div-add N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. associate-/l* N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lift-neg.f64 N/A

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

      15. lower-/.f64 97.8

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

   6. Applied rewrites 97.8%

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

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

   1. Initial program 93.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 93.3

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

   3. Applied rewrites 93.3%

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

Alternative 5: 93.5% accurate, 0.7× speedup

Math

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

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 (<= (* z t) (- INFINITY)) (* (- 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) <= -((double) INFINITY)) {
		tmp = -t * (z / a);
	} else {
		tmp = ((x * y) - (z * t)) / a;
	}
	return tmp;
}

Java

assert x < y && y < z && z < t && t < a;
assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z * t) <= -Double.POSITIVE_INFINITY) {
		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) <= -math.inf:
		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) <= Float64(-Inf))
		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) <= -Inf)
		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[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], 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) < -inf.0

   1. Initial program 68.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

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

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

      6. div-add N/A

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

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

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. +-commutative N/A

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

      12. associate-*r/ 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. *-commutative N/A

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

      15. associate-*l/ 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

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

      23. *-commutative N/A

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

      24. lower-*.f64 88.8

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

   3. Applied rewrites 88.8%

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

   4. Taylor expanded in x around 0

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

      1. lift-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. div-add N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. associate-/l* N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lift-neg.f64 N/A

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

      15. lower-/.f64 97.8

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

   6. Applied rewrites 97.8%

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

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

   1. Initial program 93.2%

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

Alternative 6: 70.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \frac{x}{a} \cdot y\\ \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+188}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 a) y)))
   (if (<= (* x y) -1e+188)
     t_1
     (if (<= (* x y) 5e-16) (* (- t) (/ z 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 / a) * y;
	double tmp;
	if ((x * y) <= -1e+188) {
		tmp = t_1;
	} else if ((x * y) <= 5e-16) {
		tmp = -t * (z / 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 / a) * y
    if ((x * y) <= (-1d+188)) then
        tmp = t_1
    else if ((x * y) <= 5d-16) then
        tmp = -t * (z / 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 / a) * y;
	double tmp;
	if ((x * y) <= -1e+188) {
		tmp = t_1;
	} else if ((x * y) <= 5e-16) {
		tmp = -t * (z / 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 / a) * y
	tmp = 0
	if (x * y) <= -1e+188:
		tmp = t_1
	elif (x * y) <= 5e-16:
		tmp = -t * (z / 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(Float64(x / a) * y)
	tmp = 0.0
	if (Float64(x * y) <= -1e+188)
		tmp = t_1;
	elseif (Float64(x * y) <= 5e-16)
		tmp = Float64(Float64(-t) * Float64(z / 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 / a) * y;
	tmp = 0.0;
	if ((x * y) <= -1e+188)
		tmp = t_1;
	elseif ((x * y) <= 5e-16)
		tmp = -t * (z / 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[(N[(x / a), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -1e+188], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 5e-16], N[((-t) * N[(z / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1e188 or 5.0000000000000004e-16 < (*.f64 x y)

   1. Initial program 85.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

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

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

      6. div-add N/A

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

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

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. +-commutative N/A

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

      12. associate-*r/ 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. *-commutative N/A

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

      15. associate-*l/ 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

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

      23. *-commutative N/A

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

      24. lower-*.f64 83.5

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

   3. Applied rewrites 83.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 89.2

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

   5. Applied rewrites 89.2%

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

   6. Taylor expanded in x around inf

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

      1. lift-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. distribute-frac-neg N/A

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

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

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

      10. associate-*l/ N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 77.8

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

   8. Applied rewrites 77.8%

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

   if -1e188 < (*.f64 x y) < 5.0000000000000004e-16

   1. Initial program 95.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

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

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

      6. div-add N/A

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

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

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. +-commutative N/A

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

      12. associate-*r/ 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. *-commutative N/A

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

      15. associate-*l/ 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

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

      23. *-commutative N/A

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

      24. lower-*.f64 91.9

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

   3. Applied rewrites 91.9%

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

   4. Taylor expanded in x around 0

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

      1. lift-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. div-add N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. associate-/l* N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lift-neg.f64 N/A

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

      15. lower-/.f64 66.0

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

   6. Applied rewrites 66.0%

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

Alternative 7: 51.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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;
assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	return (x / a) * y;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

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

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

   6. div-add N/A

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

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

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

   8. *-commutative N/A

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

   9. mul-1-neg N/A

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

   10. associate-*r/ N/A

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

   11. +-commutative N/A

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

   12. associate-*r/ 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. *-commutative N/A

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

   15. associate-*l/ 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

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

   23. *-commutative N/A

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

   24. lower-*.f64 88.7

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

3. Applied rewrites 88.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 87.9

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

5. Applied rewrites 87.9%

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

6. Taylor expanded in x around inf

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

   1. lift-neg.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-*r/ N/A

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

   5. *-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. lift-neg.f64 N/A

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

   8. distribute-frac-neg N/A

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

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

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

   10. associate-*l/ N/A

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

   11. lower-*.f64 N/A

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

   12. lift-/.f64 51.3

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

8. Applied rewrites 51.3%

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

Reproduce

herbie shell --seed 2025120 
(FPCore (x y z t a)
  :name "Data.Colour.Matrix:inverse from colour-2.3.3, B"
  :precision binary64
  (/ (- (* x y) (* z t)) a))