Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, A

Percentage Accurate: 95.5% → 98.9%

Time: 4.3s

Alternatives: 19

Speedup: 0.9×

• 26.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* (* a 27.0) b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((x * 2.0) - (((y * 9.0) * z) * t)) + ((a * 27.0) * b);
}

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, b)
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), intent (in) :: b
    code = ((x * 2.0d0) - (((y * 9.0d0) * z) * t)) + ((a * 27.0d0) * b)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return ((x * 2.0) - (((y * 9.0) * z) * t)) + ((a * 27.0) * b);
}

Python

def code(x, y, z, t, a, b):
	return ((x * 2.0) - (((y * 9.0) * z) * t)) + ((a * 27.0) * b)

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(x * 2.0) - Float64(Float64(Float64(y * 9.0) * z) * t)) + Float64(Float64(a * 27.0) * b))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((x * 2.0) - (((y * 9.0) * z) * t)) + ((a * 27.0) * b);
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(x * 2.0), $MachinePrecision] - N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] + N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]

Initial Program: 95.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* (* a 27.0) b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((x * 2.0) - (((y * 9.0) * z) * t)) + ((a * 27.0) * b);
}

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, b)
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), intent (in) :: b
    code = ((x * 2.0d0) - (((y * 9.0d0) * z) * t)) + ((a * 27.0d0) * b)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return ((x * 2.0) - (((y * 9.0) * z) * t)) + ((a * 27.0) * b);
}

Python

def code(x, y, z, t, a, b):
	return ((x * 2.0) - (((y * 9.0) * z) * t)) + ((a * 27.0) * b)

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(x * 2.0) - Float64(Float64(Float64(y * 9.0) * z) * t)) + Float64(Float64(a * 27.0) * b))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((x * 2.0) - (((y * 9.0) * z) * t)) + ((a * 27.0) * b);
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(x * 2.0), $MachinePrecision] - N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] + N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 21000:\\ \;\;\;\;\mathsf{fma}\left(\left(-9 \cdot t\right) \cdot z, y, \mathsf{fma}\left(b \cdot a, 27, x + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot 27, a, \mathsf{fma}\left(-9 \cdot t, z \cdot y, x + x\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (if (<= z 21000.0)
   (fma (* (* -9.0 t) z) y (fma (* b a) 27.0 (+ x x)))
   (fma (* b 27.0) a (fma (* -9.0 t) (* z y) (+ x x)))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= 21000.0) {
		tmp = fma(((-9.0 * t) * z), y, fma((b * a), 27.0, (x + x)));
	} else {
		tmp = fma((b * 27.0), a, fma((-9.0 * t), (z * y), (x + x)));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= 21000.0)
		tmp = fma(Float64(Float64(-9.0 * t) * z), y, fma(Float64(b * a), 27.0, Float64(x + x)));
	else
		tmp = fma(Float64(b * 27.0), a, fma(Float64(-9.0 * t), Float64(z * y), Float64(x + x)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, 21000.0], N[(N[(N[(-9.0 * t), $MachinePrecision] * z), $MachinePrecision] * y + N[(N[(b * a), $MachinePrecision] * 27.0 + N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * 27.0), $MachinePrecision] * a + N[(N[(-9.0 * t), $MachinePrecision] * N[(z * y), $MachinePrecision] + N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 21000

   1. Initial program 95.2%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot 2} - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

      3. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)} + \left(a \cdot 27\right) \cdot b \]

      4. associate-+l- N/A

         \[\leadsto \color{blue}{x \cdot 2 - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{2 \cdot x} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      6. lift-*.f64 N/A

         \[\leadsto 2 \cdot x - \left(\color{blue}{\left(\left(y \cdot 9\right) \cdot z\right) \cdot t} - \left(a \cdot 27\right) \cdot b\right) \]

      7. lift-*.f64 N/A

         \[\leadsto 2 \cdot x - \left(\color{blue}{\left(\left(y \cdot 9\right) \cdot z\right)} \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      8. lift-*.f64 N/A

         \[\leadsto 2 \cdot x - \left(\left(\color{blue}{\left(y \cdot 9\right)} \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      9. *-commutative N/A

         \[\leadsto 2 \cdot x - \left(\left(\color{blue}{\left(9 \cdot y\right)} \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      10. associate-*r* N/A

          \[\leadsto 2 \cdot x - \left(\color{blue}{\left(9 \cdot \left(y \cdot z\right)\right)} \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      11. associate-*r* N/A

          \[\leadsto 2 \cdot x - \left(\color{blue}{9 \cdot \left(\left(y \cdot z\right) \cdot t\right)} - \left(a \cdot 27\right) \cdot b\right) \]

      12. *-commutative N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \color{blue}{\left(t \cdot \left(y \cdot z\right)\right)} - \left(a \cdot 27\right) \cdot b\right) \]

      13. lift-*.f64 N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \left(t \cdot \left(y \cdot z\right)\right) - \color{blue}{\left(a \cdot 27\right) \cdot b}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \left(t \cdot \left(y \cdot z\right)\right) - \color{blue}{\left(a \cdot 27\right)} \cdot b\right) \]

      15. *-commutative N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \left(t \cdot \left(y \cdot z\right)\right) - \color{blue}{\left(27 \cdot a\right)} \cdot b\right) \]

      16. associate-*r* N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \left(t \cdot \left(y \cdot z\right)\right) - \color{blue}{27 \cdot \left(a \cdot b\right)}\right) \]

      17. associate-+l- N/A

          \[\leadsto \color{blue}{\left(2 \cdot x - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right) + 27 \cdot \left(a \cdot b\right)} \]

   3. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(t \cdot z\right) \cdot -9, y, \mathsf{fma}\left(b \cdot a, 27, x + x\right)\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(t \cdot z\right)} \cdot -9, y, \mathsf{fma}\left(b \cdot a, 27, x + x\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(t \cdot z\right) \cdot -9}, y, \mathsf{fma}\left(b \cdot a, 27, x + x\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-9 \cdot \left(t \cdot z\right)}, y, \mathsf{fma}\left(b \cdot a, 27, x + x\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(-9 \cdot t\right) \cdot z}, y, \mathsf{fma}\left(b \cdot a, 27, x + x\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(-9 \cdot t\right) \cdot z}, y, \mathsf{fma}\left(b \cdot a, 27, x + x\right)\right) \]

      6. lower-*.f64 98.9

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(-9 \cdot t\right)} \cdot z, y, \mathsf{fma}\left(b \cdot a, 27, x + x\right)\right) \]

   5. Applied rewrites 98.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(-9 \cdot t\right) \cdot z}, y, \mathsf{fma}\left(b \cdot a, 27, x + x\right)\right) \]

   if 21000 < z

   1. Initial program 97.0%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot 2} - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

      3. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)} + \left(a \cdot 27\right) \cdot b \]

      4. lift-*.f64 N/A

         \[\leadsto \left(x \cdot 2 - \color{blue}{\left(\left(y \cdot 9\right) \cdot z\right) \cdot t}\right) + \left(a \cdot 27\right) \cdot b \]

      5. lift-*.f64 N/A

         \[\leadsto \left(x \cdot 2 - \color{blue}{\left(\left(y \cdot 9\right) \cdot z\right)} \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

      6. lift-*.f64 N/A

         \[\leadsto \left(x \cdot 2 - \left(\color{blue}{\left(y \cdot 9\right)} \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot 27\right) \cdot b + \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(a \cdot 27\right) \cdot b} + \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(a \cdot 27\right)} \cdot b + \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) \]

      10. associate-*l* N/A

          \[\leadsto \color{blue}{a \cdot \left(27 \cdot b\right)} + \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) \]

      11. *-commutative N/A

          \[\leadsto \color{blue}{\left(27 \cdot b\right) \cdot a} + \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(27 \cdot b, a, x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)} \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot 27}, a, x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot 27}, a, x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot 27, a, x \cdot 2 - \left(\color{blue}{\left(y \cdot 9\right)} \cdot z\right) \cdot t\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot 27, a, x \cdot 2 - \color{blue}{\left(\left(y \cdot 9\right) \cdot z\right)} \cdot t\right) \]

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

          \[\leadsto \mathsf{fma}\left(b \cdot 27, a, \color{blue}{x \cdot 2 + \left(\mathsf{neg}\left(\left(y \cdot 9\right) \cdot z\right)\right) \cdot t}\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b \cdot 27, a, \color{blue}{2 \cdot x} + \left(\mathsf{neg}\left(\left(y \cdot 9\right) \cdot z\right)\right) \cdot t\right) \]

      19. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(b \cdot 27, a, \color{blue}{\left(\mathsf{neg}\left(\left(y \cdot 9\right) \cdot z\right)\right) \cdot t + 2 \cdot x}\right) \]

   3. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot 27, a, \mathsf{fma}\left(-9 \cdot t, z \cdot y, x + x\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 2 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot y, z \cdot -9, \mathsf{fma}\left(b \cdot a, 27, x + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot 27, a, \mathsf{fma}\left(-9 \cdot t, z \cdot y, x + x\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (if (<= t 2e+26)
   (fma (* t y) (* z -9.0) (fma (* b a) 27.0 (+ x x)))
   (fma (* b 27.0) a (fma (* -9.0 t) (* z y) (+ x x)))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 2e+26) {
		tmp = fma((t * y), (z * -9.0), fma((b * a), 27.0, (x + x)));
	} else {
		tmp = fma((b * 27.0), a, fma((-9.0 * t), (z * y), (x + x)));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= 2e+26)
		tmp = fma(Float64(t * y), Float64(z * -9.0), fma(Float64(b * a), 27.0, Float64(x + x)));
	else
		tmp = fma(Float64(b * 27.0), a, fma(Float64(-9.0 * t), Float64(z * y), Float64(x + x)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, 2e+26], N[(N[(t * y), $MachinePrecision] * N[(z * -9.0), $MachinePrecision] + N[(N[(b * a), $MachinePrecision] * 27.0 + N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * 27.0), $MachinePrecision] * a + N[(N[(-9.0 * t), $MachinePrecision] * N[(z * y), $MachinePrecision] + N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.0000000000000001e26

   1. Initial program 92.9%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot 2} - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

      3. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)} + \left(a \cdot 27\right) \cdot b \]

      4. associate-+l- N/A

         \[\leadsto \color{blue}{x \cdot 2 - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{2 \cdot x} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      6. lift-*.f64 N/A

         \[\leadsto 2 \cdot x - \left(\color{blue}{\left(\left(y \cdot 9\right) \cdot z\right) \cdot t} - \left(a \cdot 27\right) \cdot b\right) \]

      7. lift-*.f64 N/A

         \[\leadsto 2 \cdot x - \left(\color{blue}{\left(\left(y \cdot 9\right) \cdot z\right)} \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      8. lift-*.f64 N/A

         \[\leadsto 2 \cdot x - \left(\left(\color{blue}{\left(y \cdot 9\right)} \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      9. *-commutative N/A

         \[\leadsto 2 \cdot x - \left(\left(\color{blue}{\left(9 \cdot y\right)} \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      10. associate-*r* N/A

          \[\leadsto 2 \cdot x - \left(\color{blue}{\left(9 \cdot \left(y \cdot z\right)\right)} \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      11. associate-*r* N/A

          \[\leadsto 2 \cdot x - \left(\color{blue}{9 \cdot \left(\left(y \cdot z\right) \cdot t\right)} - \left(a \cdot 27\right) \cdot b\right) \]

      12. *-commutative N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \color{blue}{\left(t \cdot \left(y \cdot z\right)\right)} - \left(a \cdot 27\right) \cdot b\right) \]

      13. lift-*.f64 N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \left(t \cdot \left(y \cdot z\right)\right) - \color{blue}{\left(a \cdot 27\right) \cdot b}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \left(t \cdot \left(y \cdot z\right)\right) - \color{blue}{\left(a \cdot 27\right)} \cdot b\right) \]

      15. *-commutative N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \left(t \cdot \left(y \cdot z\right)\right) - \color{blue}{\left(27 \cdot a\right)} \cdot b\right) \]

      16. associate-*r* N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \left(t \cdot \left(y \cdot z\right)\right) - \color{blue}{27 \cdot \left(a \cdot b\right)}\right) \]

      17. associate-+l- N/A

          \[\leadsto \color{blue}{\left(2 \cdot x - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right) + 27 \cdot \left(a \cdot b\right)} \]

   3. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot y, z \cdot -9, \mathsf{fma}\left(b \cdot a, 27, x + x\right)\right)} \]

   if 2.0000000000000001e26 < t

   1. Initial program 97.6%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot 2} - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

      3. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)} + \left(a \cdot 27\right) \cdot b \]

      4. lift-*.f64 N/A

         \[\leadsto \left(x \cdot 2 - \color{blue}{\left(\left(y \cdot 9\right) \cdot z\right) \cdot t}\right) + \left(a \cdot 27\right) \cdot b \]

      5. lift-*.f64 N/A

         \[\leadsto \left(x \cdot 2 - \color{blue}{\left(\left(y \cdot 9\right) \cdot z\right)} \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

      6. lift-*.f64 N/A

         \[\leadsto \left(x \cdot 2 - \left(\color{blue}{\left(y \cdot 9\right)} \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot 27\right) \cdot b + \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(a \cdot 27\right) \cdot b} + \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(a \cdot 27\right)} \cdot b + \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) \]

      10. associate-*l* N/A

          \[\leadsto \color{blue}{a \cdot \left(27 \cdot b\right)} + \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) \]

      11. *-commutative N/A

          \[\leadsto \color{blue}{\left(27 \cdot b\right) \cdot a} + \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(27 \cdot b, a, x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)} \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot 27}, a, x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot 27}, a, x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot 27, a, x \cdot 2 - \left(\color{blue}{\left(y \cdot 9\right)} \cdot z\right) \cdot t\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot 27, a, x \cdot 2 - \color{blue}{\left(\left(y \cdot 9\right) \cdot z\right)} \cdot t\right) \]

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

          \[\leadsto \mathsf{fma}\left(b \cdot 27, a, \color{blue}{x \cdot 2 + \left(\mathsf{neg}\left(\left(y \cdot 9\right) \cdot z\right)\right) \cdot t}\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b \cdot 27, a, \color{blue}{2 \cdot x} + \left(\mathsf{neg}\left(\left(y \cdot 9\right) \cdot z\right)\right) \cdot t\right) \]

      19. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(b \cdot 27, a, \color{blue}{\left(\mathsf{neg}\left(\left(y \cdot 9\right) \cdot z\right)\right) \cdot t + 2 \cdot x}\right) \]

   3. Applied rewrites 98.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot 27, a, \mathsf{fma}\left(-9 \cdot t, z \cdot y, x + x\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 97.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;\left(y \cdot 9\right) \cdot z \leq 2 \cdot 10^{+256}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot 27, a, \mathsf{fma}\left(-9 \cdot t, z \cdot y, x + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-9, \left(y \cdot t\right) \cdot z, \left(b \cdot a\right) \cdot 27\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* (* y 9.0) z) 2e+256)
   (fma (* b 27.0) a (fma (* -9.0 t) (* z y) (+ x x)))
   (fma -9.0 (* (* y t) z) (* (* b a) 27.0))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((y * 9.0) * z) <= 2e+256) {
		tmp = fma((b * 27.0), a, fma((-9.0 * t), (z * y), (x + x)));
	} else {
		tmp = fma(-9.0, ((y * t) * z), ((b * a) * 27.0));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(y * 9.0) * z) <= 2e+256)
		tmp = fma(Float64(b * 27.0), a, fma(Float64(-9.0 * t), Float64(z * y), Float64(x + x)));
	else
		tmp = fma(-9.0, Float64(Float64(y * t) * z), Float64(Float64(b * a) * 27.0));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision], 2e+256], N[(N[(b * 27.0), $MachinePrecision] * a + N[(N[(-9.0 * t), $MachinePrecision] * N[(z * y), $MachinePrecision] + N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-9.0 * N[(N[(y * t), $MachinePrecision] * z), $MachinePrecision] + N[(N[(b * a), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y #s(literal 9 binary64)) z) < 2.0000000000000001e256

   1. Initial program 98.6%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot 2} - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

      3. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)} + \left(a \cdot 27\right) \cdot b \]

      4. lift-*.f64 N/A

         \[\leadsto \left(x \cdot 2 - \color{blue}{\left(\left(y \cdot 9\right) \cdot z\right) \cdot t}\right) + \left(a \cdot 27\right) \cdot b \]

      5. lift-*.f64 N/A

         \[\leadsto \left(x \cdot 2 - \color{blue}{\left(\left(y \cdot 9\right) \cdot z\right)} \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

      6. lift-*.f64 N/A

         \[\leadsto \left(x \cdot 2 - \left(\color{blue}{\left(y \cdot 9\right)} \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot 27\right) \cdot b + \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(a \cdot 27\right) \cdot b} + \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(a \cdot 27\right)} \cdot b + \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) \]

      10. associate-*l* N/A

          \[\leadsto \color{blue}{a \cdot \left(27 \cdot b\right)} + \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) \]

      11. *-commutative N/A

          \[\leadsto \color{blue}{\left(27 \cdot b\right) \cdot a} + \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(27 \cdot b, a, x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)} \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot 27}, a, x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot 27}, a, x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot 27, a, x \cdot 2 - \left(\color{blue}{\left(y \cdot 9\right)} \cdot z\right) \cdot t\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot 27, a, x \cdot 2 - \color{blue}{\left(\left(y \cdot 9\right) \cdot z\right)} \cdot t\right) \]

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

          \[\leadsto \mathsf{fma}\left(b \cdot 27, a, \color{blue}{x \cdot 2 + \left(\mathsf{neg}\left(\left(y \cdot 9\right) \cdot z\right)\right) \cdot t}\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b \cdot 27, a, \color{blue}{2 \cdot x} + \left(\mathsf{neg}\left(\left(y \cdot 9\right) \cdot z\right)\right) \cdot t\right) \]

      19. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(b \cdot 27, a, \color{blue}{\left(\mathsf{neg}\left(\left(y \cdot 9\right) \cdot z\right)\right) \cdot t + 2 \cdot x}\right) \]

   3. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot 27, a, \mathsf{fma}\left(-9 \cdot t, z \cdot y, x + x\right)\right)} \]

   if 2.0000000000000001e256 < (*.f64 (*.f64 y #s(literal 9 binary64)) z)

   1. Initial program 71.6%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{27 \cdot \left(a \cdot b\right) - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   3. Step-by-step derivation

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

         \[\leadsto 27 \cdot \left(a \cdot b\right) + \color{blue}{\left(\mathsf{neg}\left(9\right)\right) \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto 27 \cdot \left(a \cdot b\right) + -9 \cdot \left(\color{blue}{t} \cdot \left(y \cdot z\right)\right) \]

      3. +-commutative N/A

         \[\leadsto -9 \cdot \left(t \cdot \left(y \cdot z\right)\right) + \color{blue}{27 \cdot \left(a \cdot b\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-9, \color{blue}{t \cdot \left(y \cdot z\right)}, 27 \cdot \left(a \cdot b\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(y \cdot z\right) \cdot \color{blue}{t}, 27 \cdot \left(a \cdot b\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(y \cdot z\right) \cdot \color{blue}{t}, 27 \cdot \left(a \cdot b\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, 27 \cdot \left(a \cdot b\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, 27 \cdot \left(a \cdot b\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(a \cdot b\right) \cdot 27\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(a \cdot b\right) \cdot 27\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(b \cdot a\right) \cdot 27\right) \]

      12. lower-*.f64 69.3

          \[\leadsto \mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(b \cdot a\right) \cdot 27\right) \]

   4. Applied rewrites 69.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(b \cdot a\right) \cdot 27\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(b \cdot a\right) \cdot 27\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot \color{blue}{t}, \left(b \cdot a\right) \cdot 27\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(y \cdot z\right) \cdot t, \left(b \cdot a\right) \cdot 27\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-9, t \cdot \color{blue}{\left(y \cdot z\right)}, \left(b \cdot a\right) \cdot 27\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(t \cdot y\right) \cdot \color{blue}{z}, \left(b \cdot a\right) \cdot 27\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(t \cdot y\right) \cdot \color{blue}{z}, \left(b \cdot a\right) \cdot 27\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(y \cdot t\right) \cdot z, \left(b \cdot a\right) \cdot 27\right) \]

      8. lower-*.f64 89.0

         \[\leadsto \mathsf{fma}\left(-9, \left(y \cdot t\right) \cdot z, \left(b \cdot a\right) \cdot 27\right) \]

   6. Applied rewrites 89.0%

      \[\leadsto \mathsf{fma}\left(-9, \left(y \cdot t\right) \cdot \color{blue}{z}, \left(b \cdot a\right) \cdot 27\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 86.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(b \cdot a\right) \cdot 27\\ t_2 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+145}:\\ \;\;\;\;\mathsf{fma}\left(-9, \left(y \cdot t\right) \cdot z, t\_1\right)\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+73}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, 27, x + x\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+288}:\\ \;\;\;\;\mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, t \cdot \left(z \cdot -9\right), x + x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* b a) 27.0)) (t_2 (* (* (* y 9.0) z) t)))
   (if (<= t_2 -1e+145)
     (fma -9.0 (* (* y t) z) t_1)
     (if (<= t_2 4e+73)
       (fma (* b a) 27.0 (+ x x))
       (if (<= t_2 1e+288)
         (fma -9.0 (* (* z y) t) t_1)
         (fma y (* t (* z -9.0)) (+ x x)))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (b * a) * 27.0;
	double t_2 = ((y * 9.0) * z) * t;
	double tmp;
	if (t_2 <= -1e+145) {
		tmp = fma(-9.0, ((y * t) * z), t_1);
	} else if (t_2 <= 4e+73) {
		tmp = fma((b * a), 27.0, (x + x));
	} else if (t_2 <= 1e+288) {
		tmp = fma(-9.0, ((z * y) * t), t_1);
	} else {
		tmp = fma(y, (t * (z * -9.0)), (x + x));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(b * a) * 27.0)
	t_2 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if (t_2 <= -1e+145)
		tmp = fma(-9.0, Float64(Float64(y * t) * z), t_1);
	elseif (t_2 <= 4e+73)
		tmp = fma(Float64(b * a), 27.0, Float64(x + x));
	elseif (t_2 <= 1e+288)
		tmp = fma(-9.0, Float64(Float64(z * y) * t), t_1);
	else
		tmp = fma(y, Float64(t * Float64(z * -9.0)), Float64(x + x));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b * a), $MachinePrecision] * 27.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+145], N[(-9.0 * N[(N[(y * t), $MachinePrecision] * z), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 4e+73], N[(N[(b * a), $MachinePrecision] * 27.0 + N[(x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+288], N[(-9.0 * N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] + t$95$1), $MachinePrecision], N[(y * N[(t * N[(z * -9.0), $MachinePrecision]), $MachinePrecision] + N[(x + x), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -9.9999999999999999e144

   1. Initial program 88.6%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{27 \cdot \left(a \cdot b\right) - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   3. Step-by-step derivation

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

         \[\leadsto 27 \cdot \left(a \cdot b\right) + \color{blue}{\left(\mathsf{neg}\left(9\right)\right) \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto 27 \cdot \left(a \cdot b\right) + -9 \cdot \left(\color{blue}{t} \cdot \left(y \cdot z\right)\right) \]

      3. +-commutative N/A

         \[\leadsto -9 \cdot \left(t \cdot \left(y \cdot z\right)\right) + \color{blue}{27 \cdot \left(a \cdot b\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-9, \color{blue}{t \cdot \left(y \cdot z\right)}, 27 \cdot \left(a \cdot b\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(y \cdot z\right) \cdot \color{blue}{t}, 27 \cdot \left(a \cdot b\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(y \cdot z\right) \cdot \color{blue}{t}, 27 \cdot \left(a \cdot b\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, 27 \cdot \left(a \cdot b\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, 27 \cdot \left(a \cdot b\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(a \cdot b\right) \cdot 27\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(a \cdot b\right) \cdot 27\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(b \cdot a\right) \cdot 27\right) \]

      12. lower-*.f64 82.8

          \[\leadsto \mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(b \cdot a\right) \cdot 27\right) \]

   4. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(b \cdot a\right) \cdot 27\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(b \cdot a\right) \cdot 27\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot \color{blue}{t}, \left(b \cdot a\right) \cdot 27\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(y \cdot z\right) \cdot t, \left(b \cdot a\right) \cdot 27\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-9, t \cdot \color{blue}{\left(y \cdot z\right)}, \left(b \cdot a\right) \cdot 27\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(t \cdot y\right) \cdot \color{blue}{z}, \left(b \cdot a\right) \cdot 27\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(t \cdot y\right) \cdot \color{blue}{z}, \left(b \cdot a\right) \cdot 27\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(y \cdot t\right) \cdot z, \left(b \cdot a\right) \cdot 27\right) \]

      8. lower-*.f64 79.3

         \[\leadsto \mathsf{fma}\left(-9, \left(y \cdot t\right) \cdot z, \left(b \cdot a\right) \cdot 27\right) \]

   6. Applied rewrites 79.3%

      \[\leadsto \mathsf{fma}\left(-9, \left(y \cdot t\right) \cdot \color{blue}{z}, \left(b \cdot a\right) \cdot 27\right) \]

   if -9.9999999999999999e144 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 3.99999999999999993e73

   1. Initial program 99.2%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{2 \cdot x + 27 \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 27 \cdot \left(a \cdot b\right) + \color{blue}{2 \cdot x} \]

      2. *-commutative N/A

         \[\leadsto \left(a \cdot b\right) \cdot 27 + \color{blue}{2} \cdot x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot b, \color{blue}{27}, 2 \cdot x\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, 2 \cdot x\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, 2 \cdot x\right) \]

      6. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, x + x\right) \]

      7. lower-+.f64 87.2

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, x + x\right) \]

   4. Applied rewrites 87.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot a, 27, x + x\right)} \]

   if 3.99999999999999993e73 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 1e288

   1. Initial program 99.5%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{27 \cdot \left(a \cdot b\right) - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   3. Step-by-step derivation

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

         \[\leadsto 27 \cdot \left(a \cdot b\right) + \color{blue}{\left(\mathsf{neg}\left(9\right)\right) \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto 27 \cdot \left(a \cdot b\right) + -9 \cdot \left(\color{blue}{t} \cdot \left(y \cdot z\right)\right) \]

      3. +-commutative N/A

         \[\leadsto -9 \cdot \left(t \cdot \left(y \cdot z\right)\right) + \color{blue}{27 \cdot \left(a \cdot b\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-9, \color{blue}{t \cdot \left(y \cdot z\right)}, 27 \cdot \left(a \cdot b\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(y \cdot z\right) \cdot \color{blue}{t}, 27 \cdot \left(a \cdot b\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(y \cdot z\right) \cdot \color{blue}{t}, 27 \cdot \left(a \cdot b\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, 27 \cdot \left(a \cdot b\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, 27 \cdot \left(a \cdot b\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(a \cdot b\right) \cdot 27\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(a \cdot b\right) \cdot 27\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(b \cdot a\right) \cdot 27\right) \]

      12. lower-*.f64 82.8

          \[\leadsto \mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(b \cdot a\right) \cdot 27\right) \]

   4. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(b \cdot a\right) \cdot 27\right)} \]

   if 1e288 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 81.6%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot 2} - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

      3. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)} + \left(a \cdot 27\right) \cdot b \]

      4. associate-+l- N/A

         \[\leadsto \color{blue}{x \cdot 2 - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{2 \cdot x} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      6. lift-*.f64 N/A

         \[\leadsto 2 \cdot x - \left(\color{blue}{\left(\left(y \cdot 9\right) \cdot z\right) \cdot t} - \left(a \cdot 27\right) \cdot b\right) \]

      7. lift-*.f64 N/A

         \[\leadsto 2 \cdot x - \left(\color{blue}{\left(\left(y \cdot 9\right) \cdot z\right)} \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      8. lift-*.f64 N/A

         \[\leadsto 2 \cdot x - \left(\left(\color{blue}{\left(y \cdot 9\right)} \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      9. *-commutative N/A

         \[\leadsto 2 \cdot x - \left(\left(\color{blue}{\left(9 \cdot y\right)} \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      10. associate-*r* N/A

          \[\leadsto 2 \cdot x - \left(\color{blue}{\left(9 \cdot \left(y \cdot z\right)\right)} \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      11. associate-*r* N/A

          \[\leadsto 2 \cdot x - \left(\color{blue}{9 \cdot \left(\left(y \cdot z\right) \cdot t\right)} - \left(a \cdot 27\right) \cdot b\right) \]

      12. *-commutative N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \color{blue}{\left(t \cdot \left(y \cdot z\right)\right)} - \left(a \cdot 27\right) \cdot b\right) \]

      13. lift-*.f64 N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \left(t \cdot \left(y \cdot z\right)\right) - \color{blue}{\left(a \cdot 27\right) \cdot b}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \left(t \cdot \left(y \cdot z\right)\right) - \color{blue}{\left(a \cdot 27\right)} \cdot b\right) \]

      15. *-commutative N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \left(t \cdot \left(y \cdot z\right)\right) - \color{blue}{\left(27 \cdot a\right)} \cdot b\right) \]

      16. associate-*r* N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \left(t \cdot \left(y \cdot z\right)\right) - \color{blue}{27 \cdot \left(a \cdot b\right)}\right) \]

      17. associate-+l- N/A

          \[\leadsto \color{blue}{\left(2 \cdot x - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right) + 27 \cdot \left(a \cdot b\right)} \]

   3. Applied rewrites 93.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(t \cdot y\right) \cdot -9, z, \mathsf{fma}\left(b \cdot a, 27, x + x\right)\right)} \]

   4. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\left(t \cdot y\right) \cdot -9, z, \color{blue}{2 \cdot x}\right) \]
   5. Step-by-step derivation

      1. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(\left(t \cdot y\right) \cdot -9, z, x + \color{blue}{x}\right) \]

      2. lift-+.f64 91.0

         \[\leadsto \mathsf{fma}\left(\left(t \cdot y\right) \cdot -9, z, x + \color{blue}{x}\right) \]

   6. Applied rewrites 91.0%

      \[\leadsto \mathsf{fma}\left(\left(t \cdot y\right) \cdot -9, z, \color{blue}{x + x}\right) \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\left(t \cdot y\right) \cdot -9\right) \cdot z + \left(x + x\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(t \cdot y\right) \cdot -9\right)} \cdot z + \left(x + x\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(t \cdot y\right)} \cdot -9\right) \cdot z + \left(x + x\right) \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(t \cdot y\right) \cdot \left(-9 \cdot z\right)} + \left(x + x\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot t\right)} \cdot \left(-9 \cdot z\right) + \left(x + x\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{y \cdot \left(t \cdot \left(-9 \cdot z\right)\right)} + \left(x + x\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, t \cdot \left(-9 \cdot z\right), x + x\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{t \cdot \left(-9 \cdot z\right)}, x + x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, t \cdot \color{blue}{\left(z \cdot -9\right)}, x + x\right) \]

      10. lower-*.f64 92.9

          \[\leadsto \mathsf{fma}\left(y, t \cdot \color{blue}{\left(z \cdot -9\right)}, x + x\right) \]

      11. *-commutative 92.9

          \[\leadsto \mathsf{fma}\left(y, t \cdot \left(z \cdot -9\right), x + x\right) \]

      12. *-commutative 92.9

          \[\leadsto \mathsf{fma}\left(y, t \cdot \left(z \cdot -9\right), x + x\right) \]

      13. associate-*r* 92.9

          \[\leadsto \mathsf{fma}\left(y, t \cdot \left(z \cdot -9\right), x + x\right) \]

   8. Applied rewrites 92.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, t \cdot \left(z \cdot -9\right), x + x\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 85.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+145}:\\ \;\;\;\;\mathsf{fma}\left(-9, \left(y \cdot t\right) \cdot z, \left(b \cdot a\right) \cdot 27\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+73}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, 27, x + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-9 \cdot t\right) \cdot z, y, \left(a \cdot b\right) \cdot 27\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* y 9.0) z) t)))
   (if (<= t_1 -1e+145)
     (fma -9.0 (* (* y t) z) (* (* b a) 27.0))
     (if (<= t_1 4e+73)
       (fma (* b a) 27.0 (+ x x))
       (fma (* (* -9.0 t) z) y (* (* a b) 27.0))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((y * 9.0) * z) * t;
	double tmp;
	if (t_1 <= -1e+145) {
		tmp = fma(-9.0, ((y * t) * z), ((b * a) * 27.0));
	} else if (t_1 <= 4e+73) {
		tmp = fma((b * a), 27.0, (x + x));
	} else {
		tmp = fma(((-9.0 * t) * z), y, ((a * b) * 27.0));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if (t_1 <= -1e+145)
		tmp = fma(-9.0, Float64(Float64(y * t) * z), Float64(Float64(b * a) * 27.0));
	elseif (t_1 <= 4e+73)
		tmp = fma(Float64(b * a), 27.0, Float64(x + x));
	else
		tmp = fma(Float64(Float64(-9.0 * t) * z), y, Float64(Float64(a * b) * 27.0));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+145], N[(-9.0 * N[(N[(y * t), $MachinePrecision] * z), $MachinePrecision] + N[(N[(b * a), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+73], N[(N[(b * a), $MachinePrecision] * 27.0 + N[(x + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-9.0 * t), $MachinePrecision] * z), $MachinePrecision] * y + N[(N[(a * b), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -9.9999999999999999e144

   1. Initial program 88.6%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{27 \cdot \left(a \cdot b\right) - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   3. Step-by-step derivation

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

         \[\leadsto 27 \cdot \left(a \cdot b\right) + \color{blue}{\left(\mathsf{neg}\left(9\right)\right) \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto 27 \cdot \left(a \cdot b\right) + -9 \cdot \left(\color{blue}{t} \cdot \left(y \cdot z\right)\right) \]

      3. +-commutative N/A

         \[\leadsto -9 \cdot \left(t \cdot \left(y \cdot z\right)\right) + \color{blue}{27 \cdot \left(a \cdot b\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-9, \color{blue}{t \cdot \left(y \cdot z\right)}, 27 \cdot \left(a \cdot b\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(y \cdot z\right) \cdot \color{blue}{t}, 27 \cdot \left(a \cdot b\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(y \cdot z\right) \cdot \color{blue}{t}, 27 \cdot \left(a \cdot b\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, 27 \cdot \left(a \cdot b\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, 27 \cdot \left(a \cdot b\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(a \cdot b\right) \cdot 27\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(a \cdot b\right) \cdot 27\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(b \cdot a\right) \cdot 27\right) \]

      12. lower-*.f64 82.8

          \[\leadsto \mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(b \cdot a\right) \cdot 27\right) \]

   4. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(b \cdot a\right) \cdot 27\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(b \cdot a\right) \cdot 27\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot \color{blue}{t}, \left(b \cdot a\right) \cdot 27\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(y \cdot z\right) \cdot t, \left(b \cdot a\right) \cdot 27\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-9, t \cdot \color{blue}{\left(y \cdot z\right)}, \left(b \cdot a\right) \cdot 27\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(t \cdot y\right) \cdot \color{blue}{z}, \left(b \cdot a\right) \cdot 27\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(t \cdot y\right) \cdot \color{blue}{z}, \left(b \cdot a\right) \cdot 27\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(y \cdot t\right) \cdot z, \left(b \cdot a\right) \cdot 27\right) \]

      8. lower-*.f64 79.3

         \[\leadsto \mathsf{fma}\left(-9, \left(y \cdot t\right) \cdot z, \left(b \cdot a\right) \cdot 27\right) \]

   6. Applied rewrites 79.3%

      \[\leadsto \mathsf{fma}\left(-9, \left(y \cdot t\right) \cdot \color{blue}{z}, \left(b \cdot a\right) \cdot 27\right) \]

   if -9.9999999999999999e144 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 3.99999999999999993e73

   1. Initial program 99.2%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{2 \cdot x + 27 \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 27 \cdot \left(a \cdot b\right) + \color{blue}{2 \cdot x} \]

      2. *-commutative N/A

         \[\leadsto \left(a \cdot b\right) \cdot 27 + \color{blue}{2} \cdot x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot b, \color{blue}{27}, 2 \cdot x\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, 2 \cdot x\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, 2 \cdot x\right) \]

      6. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, x + x\right) \]

      7. lower-+.f64 87.2

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, x + x\right) \]

   4. Applied rewrites 87.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot a, 27, x + x\right)} \]

   if 3.99999999999999993e73 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 90.3%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot 2} - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

      3. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)} + \left(a \cdot 27\right) \cdot b \]

      4. associate-+l- N/A

         \[\leadsto \color{blue}{x \cdot 2 - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{2 \cdot x} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      6. lift-*.f64 N/A

         \[\leadsto 2 \cdot x - \left(\color{blue}{\left(\left(y \cdot 9\right) \cdot z\right) \cdot t} - \left(a \cdot 27\right) \cdot b\right) \]

      7. lift-*.f64 N/A

         \[\leadsto 2 \cdot x - \left(\color{blue}{\left(\left(y \cdot 9\right) \cdot z\right)} \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      8. lift-*.f64 N/A

         \[\leadsto 2 \cdot x - \left(\left(\color{blue}{\left(y \cdot 9\right)} \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      9. *-commutative N/A

         \[\leadsto 2 \cdot x - \left(\left(\color{blue}{\left(9 \cdot y\right)} \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      10. associate-*r* N/A

          \[\leadsto 2 \cdot x - \left(\color{blue}{\left(9 \cdot \left(y \cdot z\right)\right)} \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      11. associate-*r* N/A

          \[\leadsto 2 \cdot x - \left(\color{blue}{9 \cdot \left(\left(y \cdot z\right) \cdot t\right)} - \left(a \cdot 27\right) \cdot b\right) \]

      12. *-commutative N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \color{blue}{\left(t \cdot \left(y \cdot z\right)\right)} - \left(a \cdot 27\right) \cdot b\right) \]

      13. lift-*.f64 N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \left(t \cdot \left(y \cdot z\right)\right) - \color{blue}{\left(a \cdot 27\right) \cdot b}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \left(t \cdot \left(y \cdot z\right)\right) - \color{blue}{\left(a \cdot 27\right)} \cdot b\right) \]

      15. *-commutative N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \left(t \cdot \left(y \cdot z\right)\right) - \color{blue}{\left(27 \cdot a\right)} \cdot b\right) \]

      16. associate-*r* N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \left(t \cdot \left(y \cdot z\right)\right) - \color{blue}{27 \cdot \left(a \cdot b\right)}\right) \]

      17. associate-+l- N/A

          \[\leadsto \color{blue}{\left(2 \cdot x - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right) + 27 \cdot \left(a \cdot b\right)} \]

   3. Applied rewrites 92.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(t \cdot z\right) \cdot -9, y, \mathsf{fma}\left(b \cdot a, 27, x + x\right)\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(t \cdot z\right)} \cdot -9, y, \mathsf{fma}\left(b \cdot a, 27, x + x\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(t \cdot z\right) \cdot -9}, y, \mathsf{fma}\left(b \cdot a, 27, x + x\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-9 \cdot \left(t \cdot z\right)}, y, \mathsf{fma}\left(b \cdot a, 27, x + x\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(-9 \cdot t\right) \cdot z}, y, \mathsf{fma}\left(b \cdot a, 27, x + x\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(-9 \cdot t\right) \cdot z}, y, \mathsf{fma}\left(b \cdot a, 27, x + x\right)\right) \]

      6. lower-*.f64 92.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(-9 \cdot t\right)} \cdot z, y, \mathsf{fma}\left(b \cdot a, 27, x + x\right)\right) \]

   5. Applied rewrites 92.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(-9 \cdot t\right) \cdot z}, y, \mathsf{fma}\left(b \cdot a, 27, x + x\right)\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\left(-9 \cdot t\right) \cdot z, y, \color{blue}{27 \cdot \left(a \cdot b\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(-9 \cdot t\right) \cdot z, y, \left(a \cdot b\right) \cdot \color{blue}{27}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(-9 \cdot t\right) \cdot z, y, \left(a \cdot b\right) \cdot \color{blue}{27}\right) \]

      3. lower-*.f64 83.2

         \[\leadsto \mathsf{fma}\left(\left(-9 \cdot t\right) \cdot z, y, \left(a \cdot b\right) \cdot 27\right) \]

   8. Applied rewrites 83.2%

      \[\leadsto \mathsf{fma}\left(\left(-9 \cdot t\right) \cdot z, y, \color{blue}{\left(a \cdot b\right) \cdot 27}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 85.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-9, \left(y \cdot t\right) \cdot z, \left(b \cdot a\right) \cdot 27\right)\\ t_2 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+145}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+73}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, 27, x + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma -9.0 (* (* y t) z) (* (* b a) 27.0)))
        (t_2 (* (* (* y 9.0) z) t)))
   (if (<= t_2 -1e+145)
     t_1
     (if (<= t_2 4e+73) (fma (* b a) 27.0 (+ x x)) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(-9.0, ((y * t) * z), ((b * a) * 27.0));
	double t_2 = ((y * 9.0) * z) * t;
	double tmp;
	if (t_2 <= -1e+145) {
		tmp = t_1;
	} else if (t_2 <= 4e+73) {
		tmp = fma((b * a), 27.0, (x + x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = fma(-9.0, Float64(Float64(y * t) * z), Float64(Float64(b * a) * 27.0))
	t_2 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if (t_2 <= -1e+145)
		tmp = t_1;
	elseif (t_2 <= 4e+73)
		tmp = fma(Float64(b * a), 27.0, Float64(x + x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(-9.0 * N[(N[(y * t), $MachinePrecision] * z), $MachinePrecision] + N[(N[(b * a), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+145], t$95$1, If[LessEqual[t$95$2, 4e+73], N[(N[(b * a), $MachinePrecision] * 27.0 + N[(x + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -9.9999999999999999e144 or 3.99999999999999993e73 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 89.5%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{27 \cdot \left(a \cdot b\right) - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   3. Step-by-step derivation

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

         \[\leadsto 27 \cdot \left(a \cdot b\right) + \color{blue}{\left(\mathsf{neg}\left(9\right)\right) \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto 27 \cdot \left(a \cdot b\right) + -9 \cdot \left(\color{blue}{t} \cdot \left(y \cdot z\right)\right) \]

      3. +-commutative N/A

         \[\leadsto -9 \cdot \left(t \cdot \left(y \cdot z\right)\right) + \color{blue}{27 \cdot \left(a \cdot b\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-9, \color{blue}{t \cdot \left(y \cdot z\right)}, 27 \cdot \left(a \cdot b\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(y \cdot z\right) \cdot \color{blue}{t}, 27 \cdot \left(a \cdot b\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(y \cdot z\right) \cdot \color{blue}{t}, 27 \cdot \left(a \cdot b\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, 27 \cdot \left(a \cdot b\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, 27 \cdot \left(a \cdot b\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(a \cdot b\right) \cdot 27\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(a \cdot b\right) \cdot 27\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(b \cdot a\right) \cdot 27\right) \]

      12. lower-*.f64 82.6

          \[\leadsto \mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(b \cdot a\right) \cdot 27\right) \]

   4. Applied rewrites 82.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(b \cdot a\right) \cdot 27\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(b \cdot a\right) \cdot 27\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot \color{blue}{t}, \left(b \cdot a\right) \cdot 27\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(y \cdot z\right) \cdot t, \left(b \cdot a\right) \cdot 27\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-9, t \cdot \color{blue}{\left(y \cdot z\right)}, \left(b \cdot a\right) \cdot 27\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(t \cdot y\right) \cdot \color{blue}{z}, \left(b \cdot a\right) \cdot 27\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(t \cdot y\right) \cdot \color{blue}{z}, \left(b \cdot a\right) \cdot 27\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-9, \left(y \cdot t\right) \cdot z, \left(b \cdot a\right) \cdot 27\right) \]

      8. lower-*.f64 78.0

         \[\leadsto \mathsf{fma}\left(-9, \left(y \cdot t\right) \cdot z, \left(b \cdot a\right) \cdot 27\right) \]

   6. Applied rewrites 78.0%

      \[\leadsto \mathsf{fma}\left(-9, \left(y \cdot t\right) \cdot \color{blue}{z}, \left(b \cdot a\right) \cdot 27\right) \]

   if -9.9999999999999999e144 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 3.99999999999999993e73

   1. Initial program 99.2%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{2 \cdot x + 27 \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 27 \cdot \left(a \cdot b\right) + \color{blue}{2 \cdot x} \]

      2. *-commutative N/A

         \[\leadsto \left(a \cdot b\right) \cdot 27 + \color{blue}{2} \cdot x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot b, \color{blue}{27}, 2 \cdot x\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, 2 \cdot x\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, 2 \cdot x\right) \]

      6. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, x + x\right) \]

      7. lower-+.f64 87.2

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, x + x\right) \]

   4. Applied rewrites 87.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot a, 27, x + x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 85.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot -9, t \cdot y, x + x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+93}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, 27, x + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot t, -9 \cdot y, x + x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* y 9.0) z) t)))
   (if (<= t_1 -1e+69)
     (fma (* z -9.0) (* t y) (+ x x))
     (if (<= t_1 1e+93)
       (fma (* b a) 27.0 (+ x x))
       (fma (* z t) (* -9.0 y) (+ x x))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((y * 9.0) * z) * t;
	double tmp;
	if (t_1 <= -1e+69) {
		tmp = fma((z * -9.0), (t * y), (x + x));
	} else if (t_1 <= 1e+93) {
		tmp = fma((b * a), 27.0, (x + x));
	} else {
		tmp = fma((z * t), (-9.0 * y), (x + x));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if (t_1 <= -1e+69)
		tmp = fma(Float64(z * -9.0), Float64(t * y), Float64(x + x));
	elseif (t_1 <= 1e+93)
		tmp = fma(Float64(b * a), 27.0, Float64(x + x));
	else
		tmp = fma(Float64(z * t), Float64(-9.0 * y), Float64(x + x));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+69], N[(N[(z * -9.0), $MachinePrecision] * N[(t * y), $MachinePrecision] + N[(x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+93], N[(N[(b * a), $MachinePrecision] * 27.0 + N[(x + x), $MachinePrecision]), $MachinePrecision], N[(N[(z * t), $MachinePrecision] * N[(-9.0 * y), $MachinePrecision] + N[(x + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -1.0000000000000001e69

   1. Initial program 90.7%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot 2} - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

      3. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)} + \left(a \cdot 27\right) \cdot b \]

      4. associate-+l- N/A

         \[\leadsto \color{blue}{x \cdot 2 - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{2 \cdot x} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      6. lift-*.f64 N/A

         \[\leadsto 2 \cdot x - \left(\color{blue}{\left(\left(y \cdot 9\right) \cdot z\right) \cdot t} - \left(a \cdot 27\right) \cdot b\right) \]

      7. lift-*.f64 N/A

         \[\leadsto 2 \cdot x - \left(\color{blue}{\left(\left(y \cdot 9\right) \cdot z\right)} \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      8. lift-*.f64 N/A

         \[\leadsto 2 \cdot x - \left(\left(\color{blue}{\left(y \cdot 9\right)} \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      9. *-commutative N/A

         \[\leadsto 2 \cdot x - \left(\left(\color{blue}{\left(9 \cdot y\right)} \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      10. associate-*r* N/A

          \[\leadsto 2 \cdot x - \left(\color{blue}{\left(9 \cdot \left(y \cdot z\right)\right)} \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      11. associate-*r* N/A

          \[\leadsto 2 \cdot x - \left(\color{blue}{9 \cdot \left(\left(y \cdot z\right) \cdot t\right)} - \left(a \cdot 27\right) \cdot b\right) \]

      12. *-commutative N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \color{blue}{\left(t \cdot \left(y \cdot z\right)\right)} - \left(a \cdot 27\right) \cdot b\right) \]

      13. lift-*.f64 N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \left(t \cdot \left(y \cdot z\right)\right) - \color{blue}{\left(a \cdot 27\right) \cdot b}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \left(t \cdot \left(y \cdot z\right)\right) - \color{blue}{\left(a \cdot 27\right)} \cdot b\right) \]

      15. *-commutative N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \left(t \cdot \left(y \cdot z\right)\right) - \color{blue}{\left(27 \cdot a\right)} \cdot b\right) \]

      16. associate-*r* N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \left(t \cdot \left(y \cdot z\right)\right) - \color{blue}{27 \cdot \left(a \cdot b\right)}\right) \]

      17. associate-+l- N/A

          \[\leadsto \color{blue}{\left(2 \cdot x - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right) + 27 \cdot \left(a \cdot b\right)} \]

   3. Applied rewrites 85.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(t \cdot y\right) \cdot -9, z, \mathsf{fma}\left(b \cdot a, 27, x + x\right)\right)} \]

   4. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\left(t \cdot y\right) \cdot -9, z, \color{blue}{2 \cdot x}\right) \]
   5. Step-by-step derivation

      1. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(\left(t \cdot y\right) \cdot -9, z, x + \color{blue}{x}\right) \]

      2. lift-+.f64 76.0

         \[\leadsto \mathsf{fma}\left(\left(t \cdot y\right) \cdot -9, z, x + \color{blue}{x}\right) \]

   6. Applied rewrites 76.0%

      \[\leadsto \mathsf{fma}\left(\left(t \cdot y\right) \cdot -9, z, \color{blue}{x + x}\right) \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\left(t \cdot y\right) \cdot -9\right) \cdot z + \left(x + x\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(t \cdot y\right) \cdot -9\right)} \cdot z + \left(x + x\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(t \cdot y\right)} \cdot -9\right) \cdot z + \left(x + x\right) \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(t \cdot y\right) \cdot \left(-9 \cdot z\right)} + \left(x + x\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(-9 \cdot z\right) \cdot \left(t \cdot y\right)} + \left(x + x\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-9 \cdot z, t \cdot y, x + x\right)} \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot -9}, t \cdot y, x + x\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot -9}, t \cdot y, x + x\right) \]

      9. lift-*.f64 76.1

         \[\leadsto \mathsf{fma}\left(z \cdot -9, \color{blue}{t \cdot y}, x + x\right) \]

      10. *-commutative 76.1

          \[\leadsto \mathsf{fma}\left(z \cdot -9, t \cdot y, x + x\right) \]

      11. *-commutative 76.1

          \[\leadsto \mathsf{fma}\left(z \cdot -9, t \cdot y, x + x\right) \]

      12. associate-*r* 76.1

          \[\leadsto \mathsf{fma}\left(z \cdot -9, t \cdot y, x + x\right) \]

   8. Applied rewrites 76.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot -9, t \cdot y, x + x\right)} \]

   if -1.0000000000000001e69 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 1.00000000000000004e93

   1. Initial program 99.1%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{2 \cdot x + 27 \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 27 \cdot \left(a \cdot b\right) + \color{blue}{2 \cdot x} \]

      2. *-commutative N/A

         \[\leadsto \left(a \cdot b\right) \cdot 27 + \color{blue}{2} \cdot x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot b, \color{blue}{27}, 2 \cdot x\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, 2 \cdot x\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, 2 \cdot x\right) \]

      6. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, x + x\right) \]

      7. lower-+.f64 88.6

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, x + x\right) \]

   4. Applied rewrites 88.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot a, 27, x + x\right)} \]

   if 1.00000000000000004e93 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 89.8%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot 2} - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

      3. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)} + \left(a \cdot 27\right) \cdot b \]

      4. associate-+l- N/A

         \[\leadsto \color{blue}{x \cdot 2 - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{2 \cdot x} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      6. lift-*.f64 N/A

         \[\leadsto 2 \cdot x - \left(\color{blue}{\left(\left(y \cdot 9\right) \cdot z\right) \cdot t} - \left(a \cdot 27\right) \cdot b\right) \]

      7. lift-*.f64 N/A

         \[\leadsto 2 \cdot x - \left(\color{blue}{\left(\left(y \cdot 9\right) \cdot z\right)} \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      8. lift-*.f64 N/A

         \[\leadsto 2 \cdot x - \left(\left(\color{blue}{\left(y \cdot 9\right)} \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      9. *-commutative N/A

         \[\leadsto 2 \cdot x - \left(\left(\color{blue}{\left(9 \cdot y\right)} \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      10. associate-*r* N/A

          \[\leadsto 2 \cdot x - \left(\color{blue}{\left(9 \cdot \left(y \cdot z\right)\right)} \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      11. associate-*r* N/A

          \[\leadsto 2 \cdot x - \left(\color{blue}{9 \cdot \left(\left(y \cdot z\right) \cdot t\right)} - \left(a \cdot 27\right) \cdot b\right) \]

      12. *-commutative N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \color{blue}{\left(t \cdot \left(y \cdot z\right)\right)} - \left(a \cdot 27\right) \cdot b\right) \]

      13. lift-*.f64 N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \left(t \cdot \left(y \cdot z\right)\right) - \color{blue}{\left(a \cdot 27\right) \cdot b}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \left(t \cdot \left(y \cdot z\right)\right) - \color{blue}{\left(a \cdot 27\right)} \cdot b\right) \]

      15. *-commutative N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \left(t \cdot \left(y \cdot z\right)\right) - \color{blue}{\left(27 \cdot a\right)} \cdot b\right) \]

      16. associate-*r* N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \left(t \cdot \left(y \cdot z\right)\right) - \color{blue}{27 \cdot \left(a \cdot b\right)}\right) \]

      17. associate-+l- N/A

          \[\leadsto \color{blue}{\left(2 \cdot x - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right) + 27 \cdot \left(a \cdot b\right)} \]

   3. Applied rewrites 91.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(t \cdot z\right) \cdot -9, y, \mathsf{fma}\left(b \cdot a, 27, x + x\right)\right)} \]

   4. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\left(t \cdot z\right) \cdot -9, y, \color{blue}{2 \cdot x}\right) \]
   5. Step-by-step derivation

      1. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(\left(t \cdot z\right) \cdot -9, y, x + \color{blue}{x}\right) \]

      2. lift-+.f64 80.9

         \[\leadsto \mathsf{fma}\left(\left(t \cdot z\right) \cdot -9, y, x + \color{blue}{x}\right) \]

   6. Applied rewrites 80.9%

      \[\leadsto \mathsf{fma}\left(\left(t \cdot z\right) \cdot -9, y, \color{blue}{x + x}\right) \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\left(t \cdot z\right) \cdot -9\right) \cdot y + \left(x + x\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(t \cdot z\right) \cdot -9\right)} \cdot y + \left(x + x\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(t \cdot z\right)} \cdot -9\right) \cdot y + \left(x + x\right) \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(t \cdot z\right) \cdot \left(-9 \cdot y\right)} + \left(x + x\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot z, -9 \cdot y, x + x\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot t}, -9 \cdot y, x + x\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot t}, -9 \cdot y, x + x\right) \]

      8. lower-*.f64 80.7

         \[\leadsto \mathsf{fma}\left(z \cdot t, \color{blue}{-9 \cdot y}, x + x\right) \]

   8. Applied rewrites 80.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot t, -9 \cdot y, x + x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 85.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot -9, t \cdot y, x + x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+93}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, 27, x + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, t \cdot \left(z \cdot -9\right), x + x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* y 9.0) z) t)))
   (if (<= t_1 -1e+69)
     (fma (* z -9.0) (* t y) (+ x x))
     (if (<= t_1 1e+93)
       (fma (* b a) 27.0 (+ x x))
       (fma y (* t (* z -9.0)) (+ x x))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((y * 9.0) * z) * t;
	double tmp;
	if (t_1 <= -1e+69) {
		tmp = fma((z * -9.0), (t * y), (x + x));
	} else if (t_1 <= 1e+93) {
		tmp = fma((b * a), 27.0, (x + x));
	} else {
		tmp = fma(y, (t * (z * -9.0)), (x + x));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if (t_1 <= -1e+69)
		tmp = fma(Float64(z * -9.0), Float64(t * y), Float64(x + x));
	elseif (t_1 <= 1e+93)
		tmp = fma(Float64(b * a), 27.0, Float64(x + x));
	else
		tmp = fma(y, Float64(t * Float64(z * -9.0)), Float64(x + x));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+69], N[(N[(z * -9.0), $MachinePrecision] * N[(t * y), $MachinePrecision] + N[(x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+93], N[(N[(b * a), $MachinePrecision] * 27.0 + N[(x + x), $MachinePrecision]), $MachinePrecision], N[(y * N[(t * N[(z * -9.0), $MachinePrecision]), $MachinePrecision] + N[(x + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -1.0000000000000001e69

   1. Initial program 90.7%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot 2} - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

      3. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)} + \left(a \cdot 27\right) \cdot b \]

      4. associate-+l- N/A

         \[\leadsto \color{blue}{x \cdot 2 - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{2 \cdot x} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      6. lift-*.f64 N/A

         \[\leadsto 2 \cdot x - \left(\color{blue}{\left(\left(y \cdot 9\right) \cdot z\right) \cdot t} - \left(a \cdot 27\right) \cdot b\right) \]

      7. lift-*.f64 N/A

         \[\leadsto 2 \cdot x - \left(\color{blue}{\left(\left(y \cdot 9\right) \cdot z\right)} \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      8. lift-*.f64 N/A

         \[\leadsto 2 \cdot x - \left(\left(\color{blue}{\left(y \cdot 9\right)} \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      9. *-commutative N/A

         \[\leadsto 2 \cdot x - \left(\left(\color{blue}{\left(9 \cdot y\right)} \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      10. associate-*r* N/A

          \[\leadsto 2 \cdot x - \left(\color{blue}{\left(9 \cdot \left(y \cdot z\right)\right)} \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      11. associate-*r* N/A

          \[\leadsto 2 \cdot x - \left(\color{blue}{9 \cdot \left(\left(y \cdot z\right) \cdot t\right)} - \left(a \cdot 27\right) \cdot b\right) \]

      12. *-commutative N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \color{blue}{\left(t \cdot \left(y \cdot z\right)\right)} - \left(a \cdot 27\right) \cdot b\right) \]

      13. lift-*.f64 N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \left(t \cdot \left(y \cdot z\right)\right) - \color{blue}{\left(a \cdot 27\right) \cdot b}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \left(t \cdot \left(y \cdot z\right)\right) - \color{blue}{\left(a \cdot 27\right)} \cdot b\right) \]

      15. *-commutative N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \left(t \cdot \left(y \cdot z\right)\right) - \color{blue}{\left(27 \cdot a\right)} \cdot b\right) \]

      16. associate-*r* N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \left(t \cdot \left(y \cdot z\right)\right) - \color{blue}{27 \cdot \left(a \cdot b\right)}\right) \]

      17. associate-+l- N/A

          \[\leadsto \color{blue}{\left(2 \cdot x - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right) + 27 \cdot \left(a \cdot b\right)} \]

   3. Applied rewrites 85.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(t \cdot y\right) \cdot -9, z, \mathsf{fma}\left(b \cdot a, 27, x + x\right)\right)} \]

   4. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\left(t \cdot y\right) \cdot -9, z, \color{blue}{2 \cdot x}\right) \]
   5. Step-by-step derivation

      1. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(\left(t \cdot y\right) \cdot -9, z, x + \color{blue}{x}\right) \]

      2. lift-+.f64 76.0

         \[\leadsto \mathsf{fma}\left(\left(t \cdot y\right) \cdot -9, z, x + \color{blue}{x}\right) \]

   6. Applied rewrites 76.0%

      \[\leadsto \mathsf{fma}\left(\left(t \cdot y\right) \cdot -9, z, \color{blue}{x + x}\right) \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\left(t \cdot y\right) \cdot -9\right) \cdot z + \left(x + x\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(t \cdot y\right) \cdot -9\right)} \cdot z + \left(x + x\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(t \cdot y\right)} \cdot -9\right) \cdot z + \left(x + x\right) \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(t \cdot y\right) \cdot \left(-9 \cdot z\right)} + \left(x + x\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(-9 \cdot z\right) \cdot \left(t \cdot y\right)} + \left(x + x\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-9 \cdot z, t \cdot y, x + x\right)} \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot -9}, t \cdot y, x + x\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot -9}, t \cdot y, x + x\right) \]

      9. lift-*.f64 76.1

         \[\leadsto \mathsf{fma}\left(z \cdot -9, \color{blue}{t \cdot y}, x + x\right) \]

      10. *-commutative 76.1

          \[\leadsto \mathsf{fma}\left(z \cdot -9, t \cdot y, x + x\right) \]

      11. *-commutative 76.1

          \[\leadsto \mathsf{fma}\left(z \cdot -9, t \cdot y, x + x\right) \]

      12. associate-*r* 76.1

          \[\leadsto \mathsf{fma}\left(z \cdot -9, t \cdot y, x + x\right) \]

   8. Applied rewrites 76.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot -9, t \cdot y, x + x\right)} \]

   if -1.0000000000000001e69 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 1.00000000000000004e93

   1. Initial program 99.1%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{2 \cdot x + 27 \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 27 \cdot \left(a \cdot b\right) + \color{blue}{2 \cdot x} \]

      2. *-commutative N/A

         \[\leadsto \left(a \cdot b\right) \cdot 27 + \color{blue}{2} \cdot x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot b, \color{blue}{27}, 2 \cdot x\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, 2 \cdot x\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, 2 \cdot x\right) \]

      6. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, x + x\right) \]

      7. lower-+.f64 88.6

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, x + x\right) \]

   4. Applied rewrites 88.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot a, 27, x + x\right)} \]

   if 1.00000000000000004e93 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 89.8%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot 2} - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

      3. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)} + \left(a \cdot 27\right) \cdot b \]

      4. associate-+l- N/A

         \[\leadsto \color{blue}{x \cdot 2 - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{2 \cdot x} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      6. lift-*.f64 N/A

         \[\leadsto 2 \cdot x - \left(\color{blue}{\left(\left(y \cdot 9\right) \cdot z\right) \cdot t} - \left(a \cdot 27\right) \cdot b\right) \]

      7. lift-*.f64 N/A

         \[\leadsto 2 \cdot x - \left(\color{blue}{\left(\left(y \cdot 9\right) \cdot z\right)} \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      8. lift-*.f64 N/A

         \[\leadsto 2 \cdot x - \left(\left(\color{blue}{\left(y \cdot 9\right)} \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      9. *-commutative N/A

         \[\leadsto 2 \cdot x - \left(\left(\color{blue}{\left(9 \cdot y\right)} \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      10. associate-*r* N/A

          \[\leadsto 2 \cdot x - \left(\color{blue}{\left(9 \cdot \left(y \cdot z\right)\right)} \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      11. associate-*r* N/A

          \[\leadsto 2 \cdot x - \left(\color{blue}{9 \cdot \left(\left(y \cdot z\right) \cdot t\right)} - \left(a \cdot 27\right) \cdot b\right) \]

      12. *-commutative N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \color{blue}{\left(t \cdot \left(y \cdot z\right)\right)} - \left(a \cdot 27\right) \cdot b\right) \]

      13. lift-*.f64 N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \left(t \cdot \left(y \cdot z\right)\right) - \color{blue}{\left(a \cdot 27\right) \cdot b}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \left(t \cdot \left(y \cdot z\right)\right) - \color{blue}{\left(a \cdot 27\right)} \cdot b\right) \]

      15. *-commutative N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \left(t \cdot \left(y \cdot z\right)\right) - \color{blue}{\left(27 \cdot a\right)} \cdot b\right) \]

      16. associate-*r* N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \left(t \cdot \left(y \cdot z\right)\right) - \color{blue}{27 \cdot \left(a \cdot b\right)}\right) \]

      17. associate-+l- N/A

          \[\leadsto \color{blue}{\left(2 \cdot x - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right) + 27 \cdot \left(a \cdot b\right)} \]

   3. Applied rewrites 84.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(t \cdot y\right) \cdot -9, z, \mathsf{fma}\left(b \cdot a, 27, x + x\right)\right)} \]

   4. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\left(t \cdot y\right) \cdot -9, z, \color{blue}{2 \cdot x}\right) \]
   5. Step-by-step derivation

      1. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(\left(t \cdot y\right) \cdot -9, z, x + \color{blue}{x}\right) \]

      2. lift-+.f64 75.8

         \[\leadsto \mathsf{fma}\left(\left(t \cdot y\right) \cdot -9, z, x + \color{blue}{x}\right) \]

   6. Applied rewrites 75.8%

      \[\leadsto \mathsf{fma}\left(\left(t \cdot y\right) \cdot -9, z, \color{blue}{x + x}\right) \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\left(t \cdot y\right) \cdot -9\right) \cdot z + \left(x + x\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(t \cdot y\right) \cdot -9\right)} \cdot z + \left(x + x\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(t \cdot y\right)} \cdot -9\right) \cdot z + \left(x + x\right) \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(t \cdot y\right) \cdot \left(-9 \cdot z\right)} + \left(x + x\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot t\right)} \cdot \left(-9 \cdot z\right) + \left(x + x\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{y \cdot \left(t \cdot \left(-9 \cdot z\right)\right)} + \left(x + x\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, t \cdot \left(-9 \cdot z\right), x + x\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{t \cdot \left(-9 \cdot z\right)}, x + x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, t \cdot \color{blue}{\left(z \cdot -9\right)}, x + x\right) \]

      10. lower-*.f64 80.9

          \[\leadsto \mathsf{fma}\left(y, t \cdot \color{blue}{\left(z \cdot -9\right)}, x + x\right) \]

      11. *-commutative 80.9

          \[\leadsto \mathsf{fma}\left(y, t \cdot \left(z \cdot -9\right), x + x\right) \]

      12. *-commutative 80.9

          \[\leadsto \mathsf{fma}\left(y, t \cdot \left(z \cdot -9\right), x + x\right) \]

      13. associate-*r* 80.9

          \[\leadsto \mathsf{fma}\left(y, t \cdot \left(z \cdot -9\right), x + x\right) \]

   8. Applied rewrites 80.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, t \cdot \left(z \cdot -9\right), x + x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 84.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot z\right) \cdot -9, t, x + x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+93}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, 27, x + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, t \cdot \left(z \cdot -9\right), x + x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* y 9.0) z) t)))
   (if (<= t_1 -1e+69)
     (fma (* (* y z) -9.0) t (+ x x))
     (if (<= t_1 1e+93)
       (fma (* b a) 27.0 (+ x x))
       (fma y (* t (* z -9.0)) (+ x x))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((y * 9.0) * z) * t;
	double tmp;
	if (t_1 <= -1e+69) {
		tmp = fma(((y * z) * -9.0), t, (x + x));
	} else if (t_1 <= 1e+93) {
		tmp = fma((b * a), 27.0, (x + x));
	} else {
		tmp = fma(y, (t * (z * -9.0)), (x + x));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if (t_1 <= -1e+69)
		tmp = fma(Float64(Float64(y * z) * -9.0), t, Float64(x + x));
	elseif (t_1 <= 1e+93)
		tmp = fma(Float64(b * a), 27.0, Float64(x + x));
	else
		tmp = fma(y, Float64(t * Float64(z * -9.0)), Float64(x + x));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+69], N[(N[(N[(y * z), $MachinePrecision] * -9.0), $MachinePrecision] * t + N[(x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+93], N[(N[(b * a), $MachinePrecision] * 27.0 + N[(x + x), $MachinePrecision]), $MachinePrecision], N[(y * N[(t * N[(z * -9.0), $MachinePrecision]), $MachinePrecision] + N[(x + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -1.0000000000000001e69

   1. Initial program 90.7%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot x} \]
   3. Step-by-step derivation

      1. count-2-rev N/A

         \[\leadsto x + \color{blue}{x} \]

      2. lower-+.f64 13.1

         \[\leadsto x + \color{blue}{x} \]

   4. Applied rewrites 13.1%

      \[\leadsto \color{blue}{x + x} \]

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{2 \cdot x - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   6. Step-by-step derivation

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

         \[\leadsto 2 \cdot x + \color{blue}{\left(\mathsf{neg}\left(9\right)\right) \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto 2 \cdot x + -9 \cdot \left(\color{blue}{t} \cdot \left(y \cdot z\right)\right) \]

      3. +-commutative N/A

         \[\leadsto -9 \cdot \left(t \cdot \left(y \cdot z\right)\right) + \color{blue}{2 \cdot x} \]

      4. *-commutative N/A

         \[\leadsto -9 \cdot \left(\left(y \cdot z\right) \cdot t\right) + 2 \cdot x \]

      5. *-commutative N/A

         \[\leadsto -9 \cdot \left(\left(z \cdot y\right) \cdot t\right) + 2 \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \left(-9 \cdot \left(z \cdot y\right)\right) \cdot t + \color{blue}{2} \cdot x \]

      7. *-commutative N/A

         \[\leadsto \left(-9 \cdot \left(y \cdot z\right)\right) \cdot t + 2 \cdot x \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-9 \cdot \left(y \cdot z\right), \color{blue}{t}, 2 \cdot x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(y \cdot z\right) \cdot -9, t, 2 \cdot x\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(y \cdot z\right) \cdot -9, t, 2 \cdot x\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(y \cdot z\right) \cdot -9, t, 2 \cdot x\right) \]

      12. count-2-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(y \cdot z\right) \cdot -9, t, x + x\right) \]

      13. lift-+.f64 80.9

          \[\leadsto \mathsf{fma}\left(\left(y \cdot z\right) \cdot -9, t, x + x\right) \]

   7. Applied rewrites 80.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y \cdot z\right) \cdot -9, t, x + x\right)} \]

   if -1.0000000000000001e69 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 1.00000000000000004e93

   1. Initial program 99.1%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{2 \cdot x + 27 \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 27 \cdot \left(a \cdot b\right) + \color{blue}{2 \cdot x} \]

      2. *-commutative N/A

         \[\leadsto \left(a \cdot b\right) \cdot 27 + \color{blue}{2} \cdot x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot b, \color{blue}{27}, 2 \cdot x\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, 2 \cdot x\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, 2 \cdot x\right) \]

      6. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, x + x\right) \]

      7. lower-+.f64 88.6

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, x + x\right) \]

   4. Applied rewrites 88.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot a, 27, x + x\right)} \]

   if 1.00000000000000004e93 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 89.8%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot 2} - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

      3. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)} + \left(a \cdot 27\right) \cdot b \]

      4. associate-+l- N/A

         \[\leadsto \color{blue}{x \cdot 2 - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{2 \cdot x} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      6. lift-*.f64 N/A

         \[\leadsto 2 \cdot x - \left(\color{blue}{\left(\left(y \cdot 9\right) \cdot z\right) \cdot t} - \left(a \cdot 27\right) \cdot b\right) \]

      7. lift-*.f64 N/A

         \[\leadsto 2 \cdot x - \left(\color{blue}{\left(\left(y \cdot 9\right) \cdot z\right)} \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      8. lift-*.f64 N/A

         \[\leadsto 2 \cdot x - \left(\left(\color{blue}{\left(y \cdot 9\right)} \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      9. *-commutative N/A

         \[\leadsto 2 \cdot x - \left(\left(\color{blue}{\left(9 \cdot y\right)} \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      10. associate-*r* N/A

          \[\leadsto 2 \cdot x - \left(\color{blue}{\left(9 \cdot \left(y \cdot z\right)\right)} \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      11. associate-*r* N/A

          \[\leadsto 2 \cdot x - \left(\color{blue}{9 \cdot \left(\left(y \cdot z\right) \cdot t\right)} - \left(a \cdot 27\right) \cdot b\right) \]

      12. *-commutative N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \color{blue}{\left(t \cdot \left(y \cdot z\right)\right)} - \left(a \cdot 27\right) \cdot b\right) \]

      13. lift-*.f64 N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \left(t \cdot \left(y \cdot z\right)\right) - \color{blue}{\left(a \cdot 27\right) \cdot b}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \left(t \cdot \left(y \cdot z\right)\right) - \color{blue}{\left(a \cdot 27\right)} \cdot b\right) \]

      15. *-commutative N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \left(t \cdot \left(y \cdot z\right)\right) - \color{blue}{\left(27 \cdot a\right)} \cdot b\right) \]

      16. associate-*r* N/A

          \[\leadsto 2 \cdot x - \left(9 \cdot \left(t \cdot \left(y \cdot z\right)\right) - \color{blue}{27 \cdot \left(a \cdot b\right)}\right) \]

      17. associate-+l- N/A

          \[\leadsto \color{blue}{\left(2 \cdot x - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right) + 27 \cdot \left(a \cdot b\right)} \]

   3. Applied rewrites 84.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(t \cdot y\right) \cdot -9, z, \mathsf{fma}\left(b \cdot a, 27, x + x\right)\right)} \]

   4. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\left(t \cdot y\right) \cdot -9, z, \color{blue}{2 \cdot x}\right) \]
   5. Step-by-step derivation

      1. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(\left(t \cdot y\right) \cdot -9, z, x + \color{blue}{x}\right) \]

      2. lift-+.f64 75.8

         \[\leadsto \mathsf{fma}\left(\left(t \cdot y\right) \cdot -9, z, x + \color{blue}{x}\right) \]

   6. Applied rewrites 75.8%

      \[\leadsto \mathsf{fma}\left(\left(t \cdot y\right) \cdot -9, z, \color{blue}{x + x}\right) \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\left(t \cdot y\right) \cdot -9\right) \cdot z + \left(x + x\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(t \cdot y\right) \cdot -9\right)} \cdot z + \left(x + x\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(t \cdot y\right)} \cdot -9\right) \cdot z + \left(x + x\right) \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(t \cdot y\right) \cdot \left(-9 \cdot z\right)} + \left(x + x\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot t\right)} \cdot \left(-9 \cdot z\right) + \left(x + x\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{y \cdot \left(t \cdot \left(-9 \cdot z\right)\right)} + \left(x + x\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, t \cdot \left(-9 \cdot z\right), x + x\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{t \cdot \left(-9 \cdot z\right)}, x + x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, t \cdot \color{blue}{\left(z \cdot -9\right)}, x + x\right) \]

      10. lower-*.f64 80.9

          \[\leadsto \mathsf{fma}\left(y, t \cdot \color{blue}{\left(z \cdot -9\right)}, x + x\right) \]

      11. *-commutative 80.9

          \[\leadsto \mathsf{fma}\left(y, t \cdot \left(z \cdot -9\right), x + x\right) \]

      12. *-commutative 80.9

          \[\leadsto \mathsf{fma}\left(y, t \cdot \left(z \cdot -9\right), x + x\right) \]

      13. associate-*r* 80.9

          \[\leadsto \mathsf{fma}\left(y, t \cdot \left(z \cdot -9\right), x + x\right) \]

   8. Applied rewrites 80.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, t \cdot \left(z \cdot -9\right), x + x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 84.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot z\right) \cdot -9, t, x + x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+93}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, 27, x + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-9 \cdot t, z \cdot y, x + x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* y 9.0) z) t)))
   (if (<= t_1 -1e+69)
     (fma (* (* y z) -9.0) t (+ x x))
     (if (<= t_1 1e+93)
       (fma (* b a) 27.0 (+ x x))
       (fma (* -9.0 t) (* z y) (+ x x))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((y * 9.0) * z) * t;
	double tmp;
	if (t_1 <= -1e+69) {
		tmp = fma(((y * z) * -9.0), t, (x + x));
	} else if (t_1 <= 1e+93) {
		tmp = fma((b * a), 27.0, (x + x));
	} else {
		tmp = fma((-9.0 * t), (z * y), (x + x));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if (t_1 <= -1e+69)
		tmp = fma(Float64(Float64(y * z) * -9.0), t, Float64(x + x));
	elseif (t_1 <= 1e+93)
		tmp = fma(Float64(b * a), 27.0, Float64(x + x));
	else
		tmp = fma(Float64(-9.0 * t), Float64(z * y), Float64(x + x));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+69], N[(N[(N[(y * z), $MachinePrecision] * -9.0), $MachinePrecision] * t + N[(x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+93], N[(N[(b * a), $MachinePrecision] * 27.0 + N[(x + x), $MachinePrecision]), $MachinePrecision], N[(N[(-9.0 * t), $MachinePrecision] * N[(z * y), $MachinePrecision] + N[(x + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -1.0000000000000001e69

   1. Initial program 90.7%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot x} \]
   3. Step-by-step derivation

      1. count-2-rev N/A

         \[\leadsto x + \color{blue}{x} \]

      2. lower-+.f64 13.1

         \[\leadsto x + \color{blue}{x} \]

   4. Applied rewrites 13.1%

      \[\leadsto \color{blue}{x + x} \]

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{2 \cdot x - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   6. Step-by-step derivation

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

         \[\leadsto 2 \cdot x + \color{blue}{\left(\mathsf{neg}\left(9\right)\right) \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto 2 \cdot x + -9 \cdot \left(\color{blue}{t} \cdot \left(y \cdot z\right)\right) \]

      3. +-commutative N/A

         \[\leadsto -9 \cdot \left(t \cdot \left(y \cdot z\right)\right) + \color{blue}{2 \cdot x} \]

      4. *-commutative N/A

         \[\leadsto -9 \cdot \left(\left(y \cdot z\right) \cdot t\right) + 2 \cdot x \]

      5. *-commutative N/A

         \[\leadsto -9 \cdot \left(\left(z \cdot y\right) \cdot t\right) + 2 \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \left(-9 \cdot \left(z \cdot y\right)\right) \cdot t + \color{blue}{2} \cdot x \]

      7. *-commutative N/A

         \[\leadsto \left(-9 \cdot \left(y \cdot z\right)\right) \cdot t + 2 \cdot x \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-9 \cdot \left(y \cdot z\right), \color{blue}{t}, 2 \cdot x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(y \cdot z\right) \cdot -9, t, 2 \cdot x\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(y \cdot z\right) \cdot -9, t, 2 \cdot x\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(y \cdot z\right) \cdot -9, t, 2 \cdot x\right) \]

      12. count-2-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(y \cdot z\right) \cdot -9, t, x + x\right) \]

      13. lift-+.f64 80.9

          \[\leadsto \mathsf{fma}\left(\left(y \cdot z\right) \cdot -9, t, x + x\right) \]

   7. Applied rewrites 80.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y \cdot z\right) \cdot -9, t, x + x\right)} \]

   if -1.0000000000000001e69 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 1.00000000000000004e93

   1. Initial program 99.1%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{2 \cdot x + 27 \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 27 \cdot \left(a \cdot b\right) + \color{blue}{2 \cdot x} \]

      2. *-commutative N/A

         \[\leadsto \left(a \cdot b\right) \cdot 27 + \color{blue}{2} \cdot x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot b, \color{blue}{27}, 2 \cdot x\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, 2 \cdot x\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, 2 \cdot x\right) \]

      6. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, x + x\right) \]

      7. lower-+.f64 88.6

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, x + x\right) \]

   4. Applied rewrites 88.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot a, 27, x + x\right)} \]

   if 1.00000000000000004e93 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 89.8%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{2 \cdot x - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   3. Step-by-step derivation

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

         \[\leadsto 2 \cdot x + \color{blue}{\left(\mathsf{neg}\left(9\right)\right) \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto 2 \cdot x + -9 \cdot \left(\color{blue}{t} \cdot \left(y \cdot z\right)\right) \]

      3. +-commutative N/A

         \[\leadsto -9 \cdot \left(t \cdot \left(y \cdot z\right)\right) + \color{blue}{2 \cdot x} \]

      4. associate-*r* N/A

         \[\leadsto \left(-9 \cdot t\right) \cdot \left(y \cdot z\right) + \color{blue}{2} \cdot x \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-9 \cdot t, \color{blue}{y \cdot z}, 2 \cdot x\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-9 \cdot t, \color{blue}{y} \cdot z, 2 \cdot x\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-9 \cdot t, z \cdot \color{blue}{y}, 2 \cdot x\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-9 \cdot t, z \cdot \color{blue}{y}, 2 \cdot x\right) \]

      9. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(-9 \cdot t, z \cdot y, x + x\right) \]

      10. lower-+.f64 80.5

          \[\leadsto \mathsf{fma}\left(-9 \cdot t, z \cdot y, x + x\right) \]

   4. Applied rewrites 80.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-9 \cdot t, z \cdot y, x + x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 83.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-9 \cdot t, z \cdot y, x + x\right)\\ t_2 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+69}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+93}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, 27, x + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (* -9.0 t) (* z y) (+ x x))) (t_2 (* (* (* y 9.0) z) t)))
   (if (<= t_2 -1e+69)
     t_1
     (if (<= t_2 1e+93) (fma (* b a) 27.0 (+ x x)) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((-9.0 * t), (z * y), (x + x));
	double t_2 = ((y * 9.0) * z) * t;
	double tmp;
	if (t_2 <= -1e+69) {
		tmp = t_1;
	} else if (t_2 <= 1e+93) {
		tmp = fma((b * a), 27.0, (x + x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = fma(Float64(-9.0 * t), Float64(z * y), Float64(x + x))
	t_2 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if (t_2 <= -1e+69)
		tmp = t_1;
	elseif (t_2 <= 1e+93)
		tmp = fma(Float64(b * a), 27.0, Float64(x + x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(-9.0 * t), $MachinePrecision] * N[(z * y), $MachinePrecision] + N[(x + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+69], t$95$1, If[LessEqual[t$95$2, 1e+93], N[(N[(b * a), $MachinePrecision] * 27.0 + N[(x + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -1.0000000000000001e69 or 1.00000000000000004e93 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 90.2%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{2 \cdot x - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   3. Step-by-step derivation

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

         \[\leadsto 2 \cdot x + \color{blue}{\left(\mathsf{neg}\left(9\right)\right) \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto 2 \cdot x + -9 \cdot \left(\color{blue}{t} \cdot \left(y \cdot z\right)\right) \]

      3. +-commutative N/A

         \[\leadsto -9 \cdot \left(t \cdot \left(y \cdot z\right)\right) + \color{blue}{2 \cdot x} \]

      4. associate-*r* N/A

         \[\leadsto \left(-9 \cdot t\right) \cdot \left(y \cdot z\right) + \color{blue}{2} \cdot x \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-9 \cdot t, \color{blue}{y \cdot z}, 2 \cdot x\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-9 \cdot t, \color{blue}{y} \cdot z, 2 \cdot x\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-9 \cdot t, z \cdot \color{blue}{y}, 2 \cdot x\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-9 \cdot t, z \cdot \color{blue}{y}, 2 \cdot x\right) \]

      9. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(-9 \cdot t, z \cdot y, x + x\right) \]

      10. lower-+.f64 80.7

          \[\leadsto \mathsf{fma}\left(-9 \cdot t, z \cdot y, x + x\right) \]

   4. Applied rewrites 80.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-9 \cdot t, z \cdot y, x + x\right)} \]

   if -1.0000000000000001e69 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 1.00000000000000004e93

   1. Initial program 99.1%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{2 \cdot x + 27 \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 27 \cdot \left(a \cdot b\right) + \color{blue}{2 \cdot x} \]

      2. *-commutative N/A

         \[\leadsto \left(a \cdot b\right) \cdot 27 + \color{blue}{2} \cdot x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot b, \color{blue}{27}, 2 \cdot x\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, 2 \cdot x\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, 2 \cdot x\right) \]

      6. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, x + x\right) \]

      7. lower-+.f64 88.6

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, x + x\right) \]

   4. Applied rewrites 88.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot a, 27, x + x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 82.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := -9 \cdot \left(\left(z \cdot y\right) \cdot t\right)\\ t_2 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+145}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+93}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, 27, x + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* -9.0 (* (* z y) t))) (t_2 (* (* (* y 9.0) z) t)))
   (if (<= t_2 -1e+145)
     t_1
     (if (<= t_2 1e+93) (fma (* b a) 27.0 (+ x x)) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = -9.0 * ((z * y) * t);
	double t_2 = ((y * 9.0) * z) * t;
	double tmp;
	if (t_2 <= -1e+145) {
		tmp = t_1;
	} else if (t_2 <= 1e+93) {
		tmp = fma((b * a), 27.0, (x + x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(-9.0 * Float64(Float64(z * y) * t))
	t_2 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if (t_2 <= -1e+145)
		tmp = t_1;
	elseif (t_2 <= 1e+93)
		tmp = fma(Float64(b * a), 27.0, Float64(x + x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(-9.0 * N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+145], t$95$1, If[LessEqual[t$95$2, 1e+93], N[(N[(b * a), $MachinePrecision] * 27.0 + N[(x + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -9.9999999999999999e144 or 1.00000000000000004e93 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 89.2%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{-9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -9 \cdot \color{blue}{\left(t \cdot \left(y \cdot z\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto -9 \cdot \left(\left(y \cdot z\right) \cdot \color{blue}{t}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto -9 \cdot \left(\left(y \cdot z\right) \cdot \color{blue}{t}\right) \]

      4. *-commutative N/A

         \[\leadsto -9 \cdot \left(\left(z \cdot y\right) \cdot t\right) \]

      5. lower-*.f64 75.2

         \[\leadsto -9 \cdot \left(\left(z \cdot y\right) \cdot t\right) \]

   4. Applied rewrites 75.2%

      \[\leadsto \color{blue}{-9 \cdot \left(\left(z \cdot y\right) \cdot t\right)} \]

   if -9.9999999999999999e144 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 1.00000000000000004e93

   1. Initial program 99.2%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{2 \cdot x + 27 \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 27 \cdot \left(a \cdot b\right) + \color{blue}{2 \cdot x} \]

      2. *-commutative N/A

         \[\leadsto \left(a \cdot b\right) \cdot 27 + \color{blue}{2} \cdot x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot b, \color{blue}{27}, 2 \cdot x\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, 2 \cdot x\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, 2 \cdot x\right) \]

      6. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, x + x\right) \]

      7. lower-+.f64 86.8

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, x + x\right) \]

   4. Applied rewrites 86.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot a, 27, x + x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 82.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := -9 \cdot \left(\left(z \cdot y\right) \cdot t\right)\\ t_2 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+145}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+93}:\\ \;\;\;\;\mathsf{fma}\left(a, 27 \cdot b, x + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* -9.0 (* (* z y) t))) (t_2 (* (* (* y 9.0) z) t)))
   (if (<= t_2 -1e+145)
     t_1
     (if (<= t_2 1e+93) (fma a (* 27.0 b) (+ x x)) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = -9.0 * ((z * y) * t);
	double t_2 = ((y * 9.0) * z) * t;
	double tmp;
	if (t_2 <= -1e+145) {
		tmp = t_1;
	} else if (t_2 <= 1e+93) {
		tmp = fma(a, (27.0 * b), (x + x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(-9.0 * Float64(Float64(z * y) * t))
	t_2 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if (t_2 <= -1e+145)
		tmp = t_1;
	elseif (t_2 <= 1e+93)
		tmp = fma(a, Float64(27.0 * b), Float64(x + x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(-9.0 * N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+145], t$95$1, If[LessEqual[t$95$2, 1e+93], N[(a * N[(27.0 * b), $MachinePrecision] + N[(x + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -9.9999999999999999e144 or 1.00000000000000004e93 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 89.2%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{-9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -9 \cdot \color{blue}{\left(t \cdot \left(y \cdot z\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto -9 \cdot \left(\left(y \cdot z\right) \cdot \color{blue}{t}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto -9 \cdot \left(\left(y \cdot z\right) \cdot \color{blue}{t}\right) \]

      4. *-commutative N/A

         \[\leadsto -9 \cdot \left(\left(z \cdot y\right) \cdot t\right) \]

      5. lower-*.f64 75.2

         \[\leadsto -9 \cdot \left(\left(z \cdot y\right) \cdot t\right) \]

   4. Applied rewrites 75.2%

      \[\leadsto \color{blue}{-9 \cdot \left(\left(z \cdot y\right) \cdot t\right)} \]

   if -9.9999999999999999e144 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 1.00000000000000004e93

   1. Initial program 99.2%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{2 \cdot x + 27 \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 27 \cdot \left(a \cdot b\right) + \color{blue}{2 \cdot x} \]

      2. *-commutative N/A

         \[\leadsto \left(a \cdot b\right) \cdot 27 + \color{blue}{2} \cdot x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot b, \color{blue}{27}, 2 \cdot x\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, 2 \cdot x\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, 2 \cdot x\right) \]

      6. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, x + x\right) \]

      7. lower-+.f64 86.8

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, x + x\right) \]

   4. Applied rewrites 86.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot a, 27, x + x\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(b \cdot a\right) \cdot 27 + \color{blue}{\left(x + x\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(b \cdot a\right) \cdot 27 + \left(x + x\right) \]

      3. lift-+.f64 N/A

         \[\leadsto \left(b \cdot a\right) \cdot 27 + \left(x + \color{blue}{x}\right) \]

      4. associate-*l* N/A

         \[\leadsto b \cdot \left(a \cdot 27\right) + \left(\color{blue}{x} + x\right) \]

      5. *-commutative N/A

         \[\leadsto b \cdot \left(27 \cdot a\right) + \left(x + x\right) \]

      6. *-commutative N/A

         \[\leadsto \left(27 \cdot a\right) \cdot b + \left(\color{blue}{x} + x\right) \]

      7. *-commutative N/A

         \[\leadsto \left(a \cdot 27\right) \cdot b + \left(x + x\right) \]

      8. associate-*l* N/A

         \[\leadsto a \cdot \left(27 \cdot b\right) + \left(\color{blue}{x} + x\right) \]

      9. count-2-rev N/A

         \[\leadsto a \cdot \left(27 \cdot b\right) + 2 \cdot \color{blue}{x} \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, \color{blue}{27 \cdot b}, 2 \cdot x\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, 27 \cdot \color{blue}{b}, 2 \cdot x\right) \]

      12. count-2-rev N/A

          \[\leadsto \mathsf{fma}\left(a, 27 \cdot b, x + x\right) \]

      13. lift-+.f64 86.7

          \[\leadsto \mathsf{fma}\left(a, 27 \cdot b, x + x\right) \]

   6. Applied rewrites 86.7%

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{27 \cdot b}, x + x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 82.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := -9 \cdot \left(\left(z \cdot y\right) \cdot t\right)\\ t_2 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+145}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+93}:\\ \;\;\;\;\mathsf{fma}\left(27 \cdot a, b, x\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* -9.0 (* (* z y) t))) (t_2 (* (* (* y 9.0) z) t)))
   (if (<= t_2 -1e+145)
     t_1
     (if (<= t_2 1e+93) (+ (fma (* 27.0 a) b x) x) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = -9.0 * ((z * y) * t);
	double t_2 = ((y * 9.0) * z) * t;
	double tmp;
	if (t_2 <= -1e+145) {
		tmp = t_1;
	} else if (t_2 <= 1e+93) {
		tmp = fma((27.0 * a), b, x) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(-9.0 * Float64(Float64(z * y) * t))
	t_2 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if (t_2 <= -1e+145)
		tmp = t_1;
	elseif (t_2 <= 1e+93)
		tmp = Float64(fma(Float64(27.0 * a), b, x) + x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(-9.0 * N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+145], t$95$1, If[LessEqual[t$95$2, 1e+93], N[(N[(N[(27.0 * a), $MachinePrecision] * b + x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -9.9999999999999999e144 or 1.00000000000000004e93 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 89.2%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{-9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -9 \cdot \color{blue}{\left(t \cdot \left(y \cdot z\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto -9 \cdot \left(\left(y \cdot z\right) \cdot \color{blue}{t}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto -9 \cdot \left(\left(y \cdot z\right) \cdot \color{blue}{t}\right) \]

      4. *-commutative N/A

         \[\leadsto -9 \cdot \left(\left(z \cdot y\right) \cdot t\right) \]

      5. lower-*.f64 75.2

         \[\leadsto -9 \cdot \left(\left(z \cdot y\right) \cdot t\right) \]

   4. Applied rewrites 75.2%

      \[\leadsto \color{blue}{-9 \cdot \left(\left(z \cdot y\right) \cdot t\right)} \]

   if -9.9999999999999999e144 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 1.00000000000000004e93

   1. Initial program 99.2%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{2 \cdot x + 27 \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 27 \cdot \left(a \cdot b\right) + \color{blue}{2 \cdot x} \]

      2. *-commutative N/A

         \[\leadsto \left(a \cdot b\right) \cdot 27 + \color{blue}{2} \cdot x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot b, \color{blue}{27}, 2 \cdot x\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, 2 \cdot x\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, 2 \cdot x\right) \]

      6. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, x + x\right) \]

      7. lower-+.f64 86.8

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, x + x\right) \]

   4. Applied rewrites 86.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot a, 27, x + x\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(b \cdot a\right) \cdot 27 + \color{blue}{\left(x + x\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(b \cdot a\right) \cdot 27 + \left(\color{blue}{x} + x\right) \]

      3. lift-+.f64 N/A

         \[\leadsto \left(b \cdot a\right) \cdot 27 + \left(x + \color{blue}{x}\right) \]

      4. associate-+r+ N/A

         \[\leadsto \left(\left(b \cdot a\right) \cdot 27 + x\right) + \color{blue}{x} \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\left(b \cdot a\right) \cdot 27 + x\right) + \color{blue}{x} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(b \cdot a\right) \cdot 27 + x\right) + x \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(b \cdot a\right) \cdot 27 + x\right) + x \]

      8. associate-*l* N/A

         \[\leadsto \left(b \cdot \left(a \cdot 27\right) + x\right) + x \]

      9. *-commutative N/A

         \[\leadsto \left(b \cdot \left(27 \cdot a\right) + x\right) + x \]

      10. *-commutative N/A

          \[\leadsto \left(\left(27 \cdot a\right) \cdot b + x\right) + x \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(27 \cdot a, b, x\right) + x \]

      12. lower-*.f64 86.7

          \[\leadsto \mathsf{fma}\left(27 \cdot a, b, x\right) + x \]

   6. Applied rewrites 86.7%

      \[\leadsto \mathsf{fma}\left(27 \cdot a, b, x\right) + \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 58.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := -9 \cdot \left(\left(z \cdot y\right) \cdot t\right)\\ t_2 := \left(a \cdot b\right) \cdot 27 + x\\ t_3 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-143}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-264}:\\ \;\;\;\;x + x\\ \mathbf{elif}\;t\_3 \leq 10^{+93}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* -9.0 (* (* z y) t)))
        (t_2 (+ (* (* a b) 27.0) x))
        (t_3 (* (* (* y 9.0) z) t)))
   (if (<= t_3 -5e+50)
     t_1
     (if (<= t_3 -1e-143)
       t_2
       (if (<= t_3 5e-264) (+ x x) (if (<= t_3 1e+93) t_2 t_1))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = -9.0 * ((z * y) * t);
	double t_2 = ((a * b) * 27.0) + x;
	double t_3 = ((y * 9.0) * z) * t;
	double tmp;
	if (t_3 <= -5e+50) {
		tmp = t_1;
	} else if (t_3 <= -1e-143) {
		tmp = t_2;
	} else if (t_3 <= 5e-264) {
		tmp = x + x;
	} else if (t_3 <= 1e+93) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b 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, b)
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), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (-9.0d0) * ((z * y) * t)
    t_2 = ((a * b) * 27.0d0) + x
    t_3 = ((y * 9.0d0) * z) * t
    if (t_3 <= (-5d+50)) then
        tmp = t_1
    else if (t_3 <= (-1d-143)) then
        tmp = t_2
    else if (t_3 <= 5d-264) then
        tmp = x + x
    else if (t_3 <= 1d+93) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b;
assert x < y && y < z && z < t && t < a && a < b;
public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = -9.0 * ((z * y) * t);
	double t_2 = ((a * b) * 27.0) + x;
	double t_3 = ((y * 9.0) * z) * t;
	double tmp;
	if (t_3 <= -5e+50) {
		tmp = t_1;
	} else if (t_3 <= -1e-143) {
		tmp = t_2;
	} else if (t_3 <= 5e-264) {
		tmp = x + x;
	} else if (t_3 <= 1e+93) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	t_1 = -9.0 * ((z * y) * t)
	t_2 = ((a * b) * 27.0) + x
	t_3 = ((y * 9.0) * z) * t
	tmp = 0
	if t_3 <= -5e+50:
		tmp = t_1
	elif t_3 <= -1e-143:
		tmp = t_2
	elif t_3 <= 5e-264:
		tmp = x + x
	elif t_3 <= 1e+93:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(-9.0 * Float64(Float64(z * y) * t))
	t_2 = Float64(Float64(Float64(a * b) * 27.0) + x)
	t_3 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if (t_3 <= -5e+50)
		tmp = t_1;
	elseif (t_3 <= -1e-143)
		tmp = t_2;
	elseif (t_3 <= 5e-264)
		tmp = Float64(x + x);
	elseif (t_3 <= 1e+93)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp_2 = code(x, y, z, t, a, b)
	t_1 = -9.0 * ((z * y) * t);
	t_2 = ((a * b) * 27.0) + x;
	t_3 = ((y * 9.0) * z) * t;
	tmp = 0.0;
	if (t_3 <= -5e+50)
		tmp = t_1;
	elseif (t_3 <= -1e-143)
		tmp = t_2;
	elseif (t_3 <= 5e-264)
		tmp = x + x;
	elseif (t_3 <= 1e+93)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(-9.0 * N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(a * b), $MachinePrecision] * 27.0), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+50], t$95$1, If[LessEqual[t$95$3, -1e-143], t$95$2, If[LessEqual[t$95$3, 5e-264], N[(x + x), $MachinePrecision], If[LessEqual[t$95$3, 1e+93], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -5e50 or 1.00000000000000004e93 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 90.5%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{-9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -9 \cdot \color{blue}{\left(t \cdot \left(y \cdot z\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto -9 \cdot \left(\left(y \cdot z\right) \cdot \color{blue}{t}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto -9 \cdot \left(\left(y \cdot z\right) \cdot \color{blue}{t}\right) \]

      4. *-commutative N/A

         \[\leadsto -9 \cdot \left(\left(z \cdot y\right) \cdot t\right) \]

      5. lower-*.f64 71.3

         \[\leadsto -9 \cdot \left(\left(z \cdot y\right) \cdot t\right) \]

   4. Applied rewrites 71.3%

      \[\leadsto \color{blue}{-9 \cdot \left(\left(z \cdot y\right) \cdot t\right)} \]

   if -5e50 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -9.9999999999999995e-144 or 5.0000000000000001e-264 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 1.00000000000000004e93

   1. Initial program 99.6%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{2 \cdot x + 27 \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 27 \cdot \left(a \cdot b\right) + \color{blue}{2 \cdot x} \]

      2. *-commutative N/A

         \[\leadsto \left(a \cdot b\right) \cdot 27 + \color{blue}{2} \cdot x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot b, \color{blue}{27}, 2 \cdot x\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, 2 \cdot x\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, 2 \cdot x\right) \]

      6. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, x + x\right) \]

      7. lower-+.f64 80.5

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, x + x\right) \]

   4. Applied rewrites 80.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot a, 27, x + x\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(b \cdot a\right) \cdot 27 + \color{blue}{\left(x + x\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(b \cdot a\right) \cdot 27 + \left(\color{blue}{x} + x\right) \]

      3. lift-+.f64 N/A

         \[\leadsto \left(b \cdot a\right) \cdot 27 + \left(x + \color{blue}{x}\right) \]

      4. associate-+r+ N/A

         \[\leadsto \left(\left(b \cdot a\right) \cdot 27 + x\right) + \color{blue}{x} \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\left(b \cdot a\right) \cdot 27 + x\right) + \color{blue}{x} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(b \cdot a\right) \cdot 27 + x\right) + x \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(b \cdot a\right) \cdot 27 + x\right) + x \]

      8. associate-*l* N/A

         \[\leadsto \left(b \cdot \left(a \cdot 27\right) + x\right) + x \]

      9. *-commutative N/A

         \[\leadsto \left(b \cdot \left(27 \cdot a\right) + x\right) + x \]

      10. *-commutative N/A

          \[\leadsto \left(\left(27 \cdot a\right) \cdot b + x\right) + x \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(27 \cdot a, b, x\right) + x \]

      12. lower-*.f64 80.4

          \[\leadsto \mathsf{fma}\left(27 \cdot a, b, x\right) + x \]

   6. Applied rewrites 80.4%

      \[\leadsto \mathsf{fma}\left(27 \cdot a, b, x\right) + \color{blue}{x} \]

   7. Taylor expanded in x around 0

      \[\leadsto 27 \cdot \left(a \cdot b\right) + x \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(a \cdot b\right) \cdot 27 + x \]

      2. lower-*.f64 N/A

         \[\leadsto \left(a \cdot b\right) \cdot 27 + x \]

      3. lower-*.f64 48.9

         \[\leadsto \left(a \cdot b\right) \cdot 27 + x \]

   9. Applied rewrites 48.9%

      \[\leadsto \left(a \cdot b\right) \cdot 27 + x \]

   if -9.9999999999999995e-144 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 5.0000000000000001e-264

   1. Initial program 98.6%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot x} \]
   3. Step-by-step derivation

      1. count-2-rev N/A

         \[\leadsto x + \color{blue}{x} \]

      2. lower-+.f64 50.5

         \[\leadsto x + \color{blue}{x} \]

   4. Applied rewrites 50.5%

      \[\leadsto \color{blue}{x + x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 52.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(a \cdot 27\right) \cdot b\\ t_2 := \left(a \cdot b\right) \cdot 27 + x\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+155}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;x + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* a 27.0) b)) (t_2 (+ (* (* a b) 27.0) x)))
   (if (<= t_1 -5e+155) t_2 (if (<= t_1 0.1) (+ x x) t_2))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * 27.0) * b;
	double t_2 = ((a * b) * 27.0) + x;
	double tmp;
	if (t_1 <= -5e+155) {
		tmp = t_2;
	} else if (t_1 <= 0.1) {
		tmp = x + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b 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, b)
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), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (a * 27.0d0) * b
    t_2 = ((a * b) * 27.0d0) + x
    if (t_1 <= (-5d+155)) then
        tmp = t_2
    else if (t_1 <= 0.1d0) then
        tmp = x + x
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b;
assert x < y && y < z && z < t && t < a && a < b;
public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * 27.0) * b;
	double t_2 = ((a * b) * 27.0) + x;
	double tmp;
	if (t_1 <= -5e+155) {
		tmp = t_2;
	} else if (t_1 <= 0.1) {
		tmp = x + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	t_1 = (a * 27.0) * b
	t_2 = ((a * b) * 27.0) + x
	tmp = 0
	if t_1 <= -5e+155:
		tmp = t_2
	elif t_1 <= 0.1:
		tmp = x + x
	else:
		tmp = t_2
	return tmp

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a * 27.0) * b)
	t_2 = Float64(Float64(Float64(a * b) * 27.0) + x)
	tmp = 0.0
	if (t_1 <= -5e+155)
		tmp = t_2;
	elseif (t_1 <= 0.1)
		tmp = Float64(x + x);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a * 27.0) * b;
	t_2 = ((a * b) * 27.0) + x;
	tmp = 0.0;
	if (t_1 <= -5e+155)
		tmp = t_2;
	elseif (t_1 <= 0.1)
		tmp = x + x;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(a * b), $MachinePrecision] * 27.0), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+155], t$95$2, If[LessEqual[t$95$1, 0.1], N[(x + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 a #s(literal 27 binary64)) b) < -4.9999999999999999e155 or 0.10000000000000001 < (*.f64 (*.f64 a #s(literal 27 binary64)) b)

   1. Initial program 94.3%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{2 \cdot x + 27 \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 27 \cdot \left(a \cdot b\right) + \color{blue}{2 \cdot x} \]

      2. *-commutative N/A

         \[\leadsto \left(a \cdot b\right) \cdot 27 + \color{blue}{2} \cdot x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot b, \color{blue}{27}, 2 \cdot x\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, 2 \cdot x\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, 2 \cdot x\right) \]

      6. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, x + x\right) \]

      7. lower-+.f64 78.5

         \[\leadsto \mathsf{fma}\left(b \cdot a, 27, x + x\right) \]

   4. Applied rewrites 78.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot a, 27, x + x\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(b \cdot a\right) \cdot 27 + \color{blue}{\left(x + x\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(b \cdot a\right) \cdot 27 + \left(\color{blue}{x} + x\right) \]

      3. lift-+.f64 N/A

         \[\leadsto \left(b \cdot a\right) \cdot 27 + \left(x + \color{blue}{x}\right) \]

      4. associate-+r+ N/A

         \[\leadsto \left(\left(b \cdot a\right) \cdot 27 + x\right) + \color{blue}{x} \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\left(b \cdot a\right) \cdot 27 + x\right) + \color{blue}{x} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(b \cdot a\right) \cdot 27 + x\right) + x \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(b \cdot a\right) \cdot 27 + x\right) + x \]

      8. associate-*l* N/A

         \[\leadsto \left(b \cdot \left(a \cdot 27\right) + x\right) + x \]

      9. *-commutative N/A

         \[\leadsto \left(b \cdot \left(27 \cdot a\right) + x\right) + x \]

      10. *-commutative N/A

          \[\leadsto \left(\left(27 \cdot a\right) \cdot b + x\right) + x \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(27 \cdot a, b, x\right) + x \]

      12. lower-*.f64 78.4

          \[\leadsto \mathsf{fma}\left(27 \cdot a, b, x\right) + x \]

   6. Applied rewrites 78.4%

      \[\leadsto \mathsf{fma}\left(27 \cdot a, b, x\right) + \color{blue}{x} \]

   7. Taylor expanded in x around 0

      \[\leadsto 27 \cdot \left(a \cdot b\right) + x \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(a \cdot b\right) \cdot 27 + x \]

      2. lower-*.f64 N/A

         \[\leadsto \left(a \cdot b\right) \cdot 27 + x \]

      3. lower-*.f64 68.3

         \[\leadsto \left(a \cdot b\right) \cdot 27 + x \]

   9. Applied rewrites 68.3%

      \[\leadsto \left(a \cdot b\right) \cdot 27 + x \]

   if -4.9999999999999999e155 < (*.f64 (*.f64 a #s(literal 27 binary64)) b) < 0.10000000000000001

   1. Initial program 96.2%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot x} \]
   3. Step-by-step derivation

      1. count-2-rev N/A

         \[\leadsto x + \color{blue}{x} \]

      2. lower-+.f64 42.0

         \[\leadsto x + \color{blue}{x} \]

   4. Applied rewrites 42.0%

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

Alternative 17: 51.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(a \cdot 27\right) \cdot b\\ t_2 := \left(b \cdot a\right) \cdot 27\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+155}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;x + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* a 27.0) b)) (t_2 (* (* b a) 27.0)))
   (if (<= t_1 -5e+155) t_2 (if (<= t_1 0.1) (+ x x) t_2))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * 27.0) * b;
	double t_2 = (b * a) * 27.0;
	double tmp;
	if (t_1 <= -5e+155) {
		tmp = t_2;
	} else if (t_1 <= 0.1) {
		tmp = x + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b 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, b)
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), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (a * 27.0d0) * b
    t_2 = (b * a) * 27.0d0
    if (t_1 <= (-5d+155)) then
        tmp = t_2
    else if (t_1 <= 0.1d0) then
        tmp = x + x
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b;
assert x < y && y < z && z < t && t < a && a < b;
public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * 27.0) * b;
	double t_2 = (b * a) * 27.0;
	double tmp;
	if (t_1 <= -5e+155) {
		tmp = t_2;
	} else if (t_1 <= 0.1) {
		tmp = x + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	t_1 = (a * 27.0) * b
	t_2 = (b * a) * 27.0
	tmp = 0
	if t_1 <= -5e+155:
		tmp = t_2
	elif t_1 <= 0.1:
		tmp = x + x
	else:
		tmp = t_2
	return tmp

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a * 27.0) * b)
	t_2 = Float64(Float64(b * a) * 27.0)
	tmp = 0.0
	if (t_1 <= -5e+155)
		tmp = t_2;
	elseif (t_1 <= 0.1)
		tmp = Float64(x + x);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a * 27.0) * b;
	t_2 = (b * a) * 27.0;
	tmp = 0.0;
	if (t_1 <= -5e+155)
		tmp = t_2;
	elseif (t_1 <= 0.1)
		tmp = x + x;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b * a), $MachinePrecision] * 27.0), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+155], t$95$2, If[LessEqual[t$95$1, 0.1], N[(x + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 a #s(literal 27 binary64)) b) < -4.9999999999999999e155 or 0.10000000000000001 < (*.f64 (*.f64 a #s(literal 27 binary64)) b)

   1. Initial program 94.3%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{27 \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(a \cdot b\right) \cdot \color{blue}{27} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(a \cdot b\right) \cdot \color{blue}{27} \]

      3. *-commutative N/A

         \[\leadsto \left(b \cdot a\right) \cdot 27 \]

      4. lower-*.f64 66.4

         \[\leadsto \left(b \cdot a\right) \cdot 27 \]

   4. Applied rewrites 66.4%

      \[\leadsto \color{blue}{\left(b \cdot a\right) \cdot 27} \]

   if -4.9999999999999999e155 < (*.f64 (*.f64 a #s(literal 27 binary64)) b) < 0.10000000000000001

   1. Initial program 96.2%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot x} \]
   3. Step-by-step derivation

      1. count-2-rev N/A

         \[\leadsto x + \color{blue}{x} \]

      2. lower-+.f64 42.0

         \[\leadsto x + \color{blue}{x} \]

   4. Applied rewrites 42.0%

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

Alternative 18: 51.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(a \cdot 27\right) \cdot b\\ t_2 := a \cdot \left(27 \cdot b\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+155}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;x + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* a 27.0) b)) (t_2 (* a (* 27.0 b))))
   (if (<= t_1 -5e+155) t_2 (if (<= t_1 0.1) (+ x x) t_2))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * 27.0) * b;
	double t_2 = a * (27.0 * b);
	double tmp;
	if (t_1 <= -5e+155) {
		tmp = t_2;
	} else if (t_1 <= 0.1) {
		tmp = x + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b 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, b)
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), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (a * 27.0d0) * b
    t_2 = a * (27.0d0 * b)
    if (t_1 <= (-5d+155)) then
        tmp = t_2
    else if (t_1 <= 0.1d0) then
        tmp = x + x
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b;
assert x < y && y < z && z < t && t < a && a < b;
public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * 27.0) * b;
	double t_2 = a * (27.0 * b);
	double tmp;
	if (t_1 <= -5e+155) {
		tmp = t_2;
	} else if (t_1 <= 0.1) {
		tmp = x + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	t_1 = (a * 27.0) * b
	t_2 = a * (27.0 * b)
	tmp = 0
	if t_1 <= -5e+155:
		tmp = t_2
	elif t_1 <= 0.1:
		tmp = x + x
	else:
		tmp = t_2
	return tmp

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a * 27.0) * b)
	t_2 = Float64(a * Float64(27.0 * b))
	tmp = 0.0
	if (t_1 <= -5e+155)
		tmp = t_2;
	elseif (t_1 <= 0.1)
		tmp = Float64(x + x);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a * 27.0) * b;
	t_2 = a * (27.0 * b);
	tmp = 0.0;
	if (t_1 <= -5e+155)
		tmp = t_2;
	elseif (t_1 <= 0.1)
		tmp = x + x;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[(27.0 * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+155], t$95$2, If[LessEqual[t$95$1, 0.1], N[(x + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 a #s(literal 27 binary64)) b) < -4.9999999999999999e155 or 0.10000000000000001 < (*.f64 (*.f64 a #s(literal 27 binary64)) b)

   1. Initial program 94.3%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{27 \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(a \cdot b\right) \cdot \color{blue}{27} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(a \cdot b\right) \cdot \color{blue}{27} \]

      3. *-commutative N/A

         \[\leadsto \left(b \cdot a\right) \cdot 27 \]

      4. lower-*.f64 66.4

         \[\leadsto \left(b \cdot a\right) \cdot 27 \]

   4. Applied rewrites 66.4%

      \[\leadsto \color{blue}{\left(b \cdot a\right) \cdot 27} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(b \cdot a\right) \cdot \color{blue}{27} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(b \cdot a\right) \cdot 27 \]

      3. *-commutative N/A

         \[\leadsto \left(a \cdot b\right) \cdot 27 \]

      4. *-commutative N/A

         \[\leadsto 27 \cdot \color{blue}{\left(a \cdot b\right)} \]

      5. associate-*r* N/A

         \[\leadsto \left(27 \cdot a\right) \cdot \color{blue}{b} \]

      6. *-commutative N/A

         \[\leadsto \left(a \cdot 27\right) \cdot b \]

      7. associate-*l* N/A

         \[\leadsto a \cdot \color{blue}{\left(27 \cdot b\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto a \cdot \color{blue}{\left(27 \cdot b\right)} \]

      9. lower-*.f64 66.3

         \[\leadsto a \cdot \left(27 \cdot \color{blue}{b}\right) \]

   6. Applied rewrites 66.3%

      \[\leadsto a \cdot \color{blue}{\left(27 \cdot b\right)} \]

   if -4.9999999999999999e155 < (*.f64 (*.f64 a #s(literal 27 binary64)) b) < 0.10000000000000001

   1. Initial program 96.2%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot x} \]
   3. Step-by-step derivation

      1. count-2-rev N/A

         \[\leadsto x + \color{blue}{x} \]

      2. lower-+.f64 42.0

         \[\leadsto x + \color{blue}{x} \]

   4. Applied rewrites 42.0%

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

Alternative 19: 31.3% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b 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, b)
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), intent (in) :: b
    code = x + x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.5%

   \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

2. Taylor expanded in x around inf

   \[\leadsto \color{blue}{2 \cdot x} \]
3. Step-by-step derivation

   1. count-2-rev N/A

      \[\leadsto x + \color{blue}{x} \]

   2. lower-+.f64 31.3

      \[\leadsto x + \color{blue}{x} \]

4. Applied rewrites 31.3%

   \[\leadsto \color{blue}{x + x} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025113 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, A"
  :precision binary64
  (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* (* a 27.0) b)))