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

Percentage Accurate: 95.4% → 97.0%

Time: 4.2s

Alternatives: 13

Speedup: 0.6×

• 27.2% 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: 95.4% 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: 97.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(-9 \cdot \left(z \cdot y\right), t, \mathsf{fma}\left(b \cdot a, 27, x + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(t \cdot y\right) \cdot z, -9, x + x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (if (<= (- (* x 2.0) (* (* (* y 9.0) z) t)) 2e+306)
   (fma (* -9.0 (* z y)) t (fma (* b a) 27.0 (+ x x)))
   (fma (* (* t y) z) -9.0 (+ x x))))

C

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 (((x * 2.0) - (((y * 9.0) * z) * t)) <= 2e+306) {
		tmp = fma((-9.0 * (z * y)), t, fma((b * a), 27.0, (x + x)));
	} else {
		tmp = fma(((t * y) * z), -9.0, (x + x));
	}
	return tmp;
}

Julia

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(x * 2.0) - Float64(Float64(Float64(y * 9.0) * z) * t)) <= 2e+306)
		tmp = fma(Float64(-9.0 * Float64(z * y)), t, fma(Float64(b * a), 27.0, Float64(x + x)));
	else
		tmp = fma(Float64(Float64(t * y) * 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.
code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(x * 2.0), $MachinePrecision] - N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], 2e+306], N[(N[(-9.0 * N[(z * y), $MachinePrecision]), $MachinePrecision] * t + N[(N[(b * a), $MachinePrecision] * 27.0 + N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t * y), $MachinePrecision] * z), $MachinePrecision] * -9.0 + N[(x + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)) < 2.00000000000000003e306

   1. Initial program 95.4%

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

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

      5. lift-*.f64 N/A

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

      6. associate-+l- N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      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--r- 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 95.8%

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

   if 2.00000000000000003e306 < (-.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t))

   1. Initial program 95.4%

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

   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 64.5

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

   4. Applied rewrites 64.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      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. count-2-rev N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 64.4

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

   6. Applied rewrites 64.4%

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

   7. Taylor expanded in a around 0

      \[\leadsto \color{blue}{2 \cdot x - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   8. 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

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 64.2

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

   9. Applied rewrites 64.2%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 62.7

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

   11. Applied rewrites 62.7%

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

Alternative 2: 86.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ t_2 := \left(b \cdot a\right) \cdot 27\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(t \cdot y\right) \cdot z, -9, x + x\right)\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+120}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, -9 \cdot t, t\_2\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+163}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, 27, x + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, y \cdot \left(-9 \cdot z\right), t\_2\right)\\ \end{array} \end{array} \]

FPCore

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)) (t_2 (* (* b a) 27.0)))
   (if (<= t_1 (- INFINITY))
     (fma (* (* t y) z) -9.0 (+ x x))
     (if (<= t_1 -1e+120)
       (fma (* z y) (* -9.0 t) t_2)
       (if (<= t_1 5e+163)
         (fma (* b a) 27.0 (+ x x))
         (fma t (* y (* -9.0 z)) t_2))))))

C

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 t_2 = (b * a) * 27.0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(((t * y) * z), -9.0, (x + x));
	} else if (t_1 <= -1e+120) {
		tmp = fma((z * y), (-9.0 * t), t_2);
	} else if (t_1 <= 5e+163) {
		tmp = fma((b * a), 27.0, (x + x));
	} else {
		tmp = fma(t, (y * (-9.0 * z)), t_2);
	}
	return tmp;
}

Julia

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)
	t_2 = Float64(Float64(b * a) * 27.0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(Float64(Float64(t * y) * z), -9.0, Float64(x + x));
	elseif (t_1 <= -1e+120)
		tmp = fma(Float64(z * y), Float64(-9.0 * t), t_2);
	elseif (t_1 <= 5e+163)
		tmp = fma(Float64(b * a), 27.0, Float64(x + x));
	else
		tmp = fma(t, Float64(y * Float64(-9.0 * z)), t_2);
	end
	return tmp
end

Wolfram

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]}, Block[{t$95$2 = N[(N[(b * a), $MachinePrecision] * 27.0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(t * y), $MachinePrecision] * z), $MachinePrecision] * -9.0 + N[(x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1e+120], N[(N[(z * y), $MachinePrecision] * N[(-9.0 * t), $MachinePrecision] + t$95$2), $MachinePrecision], If[LessEqual[t$95$1, 5e+163], N[(N[(b * a), $MachinePrecision] * 27.0 + N[(x + x), $MachinePrecision]), $MachinePrecision], N[(t * N[(y * N[(-9.0 * z), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -inf.0

   1. Initial program 95.4%

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

   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 64.5

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

   4. Applied rewrites 64.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      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. count-2-rev N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 64.4

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

   6. Applied rewrites 64.4%

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

   7. Taylor expanded in a around 0

      \[\leadsto \color{blue}{2 \cdot x - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   8. 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

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 64.2

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

   9. Applied rewrites 64.2%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 62.7

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

   11. Applied rewrites 62.7%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -9.9999999999999998e119

   1. Initial program 95.4%

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

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

      5. lift-*.f64 N/A

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

      6. associate-+l- N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      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--r- 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 93.2%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 64.9

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

   6. Applied rewrites 64.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 67.1

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 67.1

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

   8. Applied rewrites 67.1%

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

   if -9.9999999999999998e119 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 5e163

   1. Initial program 95.4%

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

   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 64.5

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

   4. Applied rewrites 64.5%

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

   if 5e163 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 95.4%

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

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

      5. lift-*.f64 N/A

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

      6. associate-+l- N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      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--r- 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 93.2%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 64.9

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

   6. Applied rewrites 64.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 67.2

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 67.2

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

   8. Applied rewrites 67.2%

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

Alternative 3: 86.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ t_2 := \mathsf{fma}\left(t, y \cdot \left(-9 \cdot z\right), \left(b \cdot a\right) \cdot 27\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(t \cdot y\right) \cdot z, -9, x + x\right)\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+120}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+163}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, 27, x + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* y 9.0) z) t))
        (t_2 (fma t (* y (* -9.0 z)) (* (* b a) 27.0))))
   (if (<= t_1 (- INFINITY))
     (fma (* (* t y) z) -9.0 (+ x x))
     (if (<= t_1 -1e+120)
       t_2
       (if (<= t_1 5e+163) (fma (* b a) 27.0 (+ x x)) t_2)))))

C

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 t_2 = fma(t, (y * (-9.0 * z)), ((b * a) * 27.0));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(((t * y) * z), -9.0, (x + x));
	} else if (t_1 <= -1e+120) {
		tmp = t_2;
	} else if (t_1 <= 5e+163) {
		tmp = fma((b * a), 27.0, (x + x));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

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)
	t_2 = fma(t, Float64(y * Float64(-9.0 * z)), Float64(Float64(b * a) * 27.0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(Float64(Float64(t * y) * z), -9.0, Float64(x + x));
	elseif (t_1 <= -1e+120)
		tmp = t_2;
	elseif (t_1 <= 5e+163)
		tmp = fma(Float64(b * a), 27.0, Float64(x + x));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

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]}, Block[{t$95$2 = N[(t * N[(y * N[(-9.0 * z), $MachinePrecision]), $MachinePrecision] + N[(N[(b * a), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(t * y), $MachinePrecision] * z), $MachinePrecision] * -9.0 + N[(x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1e+120], t$95$2, If[LessEqual[t$95$1, 5e+163], N[(N[(b * a), $MachinePrecision] * 27.0 + N[(x + x), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -inf.0

   1. Initial program 95.4%

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

   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 64.5

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

   4. Applied rewrites 64.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      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. count-2-rev N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 64.4

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

   6. Applied rewrites 64.4%

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

   7. Taylor expanded in a around 0

      \[\leadsto \color{blue}{2 \cdot x - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   8. 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

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 64.2

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

   9. Applied rewrites 64.2%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 62.7

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

   11. Applied rewrites 62.7%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -9.9999999999999998e119 or 5e163 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 95.4%

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

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

      5. lift-*.f64 N/A

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

      6. associate-+l- N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      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--r- 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 93.2%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 64.9

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

   6. Applied rewrites 64.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 67.2

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 67.2

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

   8. Applied rewrites 67.2%

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

   if -9.9999999999999998e119 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 5e163

   1. Initial program 95.4%

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

   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 64.5

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

   4. Applied rewrites 64.5%

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

Alternative 4: 86.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ t_2 := \left(z \cdot y\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(t \cdot y\right) \cdot z, -9, x + x\right)\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+120}:\\ \;\;\;\;\mathsf{fma}\left(-9, t\_2, \left(b \cdot a\right) \cdot 27\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, 27, x + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, -9, x + x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* y 9.0) z) t)) (t_2 (* (* z y) t)))
   (if (<= t_1 (- INFINITY))
     (fma (* (* t y) z) -9.0 (+ x x))
     (if (<= t_1 -1e+120)
       (fma -9.0 t_2 (* (* b a) 27.0))
       (if (<= t_1 4e+19)
         (fma (* b a) 27.0 (+ x x))
         (fma t_2 -9.0 (+ x x)))))))

C

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 t_2 = (z * y) * t;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(((t * y) * z), -9.0, (x + x));
	} else if (t_1 <= -1e+120) {
		tmp = fma(-9.0, t_2, ((b * a) * 27.0));
	} else if (t_1 <= 4e+19) {
		tmp = fma((b * a), 27.0, (x + x));
	} else {
		tmp = fma(t_2, -9.0, (x + x));
	}
	return tmp;
}

Julia

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)
	t_2 = Float64(Float64(z * y) * t)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(Float64(Float64(t * y) * z), -9.0, Float64(x + x));
	elseif (t_1 <= -1e+120)
		tmp = fma(-9.0, t_2, Float64(Float64(b * a) * 27.0));
	elseif (t_1 <= 4e+19)
		tmp = fma(Float64(b * a), 27.0, Float64(x + x));
	else
		tmp = fma(t_2, -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.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(t * y), $MachinePrecision] * z), $MachinePrecision] * -9.0 + N[(x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1e+120], N[(-9.0 * t$95$2 + N[(N[(b * a), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+19], N[(N[(b * a), $MachinePrecision] * 27.0 + N[(x + x), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * -9.0 + N[(x + x), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -inf.0

   1. Initial program 95.4%

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

   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 64.5

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

   4. Applied rewrites 64.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      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. count-2-rev N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 64.4

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

   6. Applied rewrites 64.4%

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

   7. Taylor expanded in a around 0

      \[\leadsto \color{blue}{2 \cdot x - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   8. 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

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 64.2

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

   9. Applied rewrites 64.2%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 62.7

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

   11. Applied rewrites 62.7%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -9.9999999999999998e119

   1. Initial program 95.4%

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

   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 66.9

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

   4. Applied rewrites 66.9%

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

   if -9.9999999999999998e119 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 4e19

   1. Initial program 95.4%

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

   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 64.5

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

   4. Applied rewrites 64.5%

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

   if 4e19 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 95.4%

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

   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 64.5

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

   4. Applied rewrites 64.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      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. count-2-rev N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 64.4

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

   6. Applied rewrites 64.4%

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

   7. Taylor expanded in a around 0

      \[\leadsto \color{blue}{2 \cdot x - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   8. 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

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 64.2

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

   9. Applied rewrites 64.2%

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

Alternative 5: 84.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+108}:\\ \;\;\;\;\mathsf{fma}\left(\left(t \cdot y\right) \cdot z, -9, x + x\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, 27, x + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(z \cdot y\right) \cdot t, -9, x + x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* y 9.0) z) t)))
   (if (<= t_1 -5e+108)
     (fma (* (* t y) z) -9.0 (+ x x))
     (if (<= t_1 4e+19)
       (fma (* b a) 27.0 (+ x x))
       (fma (* (* z y) t) -9.0 (+ x x))))))

C

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 <= -5e+108) {
		tmp = fma(((t * y) * z), -9.0, (x + x));
	} else if (t_1 <= 4e+19) {
		tmp = fma((b * a), 27.0, (x + x));
	} else {
		tmp = fma(((z * y) * t), -9.0, (x + x));
	}
	return tmp;
}

Julia

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 <= -5e+108)
		tmp = fma(Float64(Float64(t * y) * z), -9.0, Float64(x + x));
	elseif (t_1 <= 4e+19)
		tmp = fma(Float64(b * a), 27.0, Float64(x + x));
	else
		tmp = fma(Float64(Float64(z * y) * t), -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.
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, -5e+108], N[(N[(N[(t * y), $MachinePrecision] * z), $MachinePrecision] * -9.0 + N[(x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+19], N[(N[(b * a), $MachinePrecision] * 27.0 + N[(x + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] * -9.0 + N[(x + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -4.99999999999999991e108

   1. Initial program 95.4%

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

   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 64.5

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

   4. Applied rewrites 64.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      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. count-2-rev N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 64.4

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

   6. Applied rewrites 64.4%

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

   7. Taylor expanded in a around 0

      \[\leadsto \color{blue}{2 \cdot x - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   8. 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

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 64.2

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

   9. Applied rewrites 64.2%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 62.7

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

   11. Applied rewrites 62.7%

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

   if -4.99999999999999991e108 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 4e19

   1. Initial program 95.4%

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

   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 64.5

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

   4. Applied rewrites 64.5%

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

   if 4e19 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 95.4%

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

   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 64.5

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

   4. Applied rewrites 64.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      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. count-2-rev N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 64.4

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

   6. Applied rewrites 64.4%

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

   7. Taylor expanded in a around 0

      \[\leadsto \color{blue}{2 \cdot x - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   8. 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

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 64.2

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

   9. Applied rewrites 64.2%

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

Alternative 6: 84.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+108}:\\ \;\;\;\;\mathsf{fma}\left(\left(t \cdot y\right) \cdot z, -9, x + x\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, 27, x + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, t \cdot -9, x + x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* y 9.0) z) t)))
   (if (<= t_1 -5e+108)
     (fma (* (* t y) z) -9.0 (+ x x))
     (if (<= t_1 4e+19)
       (fma (* b a) 27.0 (+ x x))
       (fma (* z y) (* t -9.0) (+ x x))))))

C

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 <= -5e+108) {
		tmp = fma(((t * y) * z), -9.0, (x + x));
	} else if (t_1 <= 4e+19) {
		tmp = fma((b * a), 27.0, (x + x));
	} else {
		tmp = fma((z * y), (t * -9.0), (x + x));
	}
	return tmp;
}

Julia

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 <= -5e+108)
		tmp = fma(Float64(Float64(t * y) * z), -9.0, Float64(x + x));
	elseif (t_1 <= 4e+19)
		tmp = fma(Float64(b * a), 27.0, Float64(x + x));
	else
		tmp = fma(Float64(z * y), Float64(t * -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.
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, -5e+108], N[(N[(N[(t * y), $MachinePrecision] * z), $MachinePrecision] * -9.0 + N[(x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+19], N[(N[(b * a), $MachinePrecision] * 27.0 + N[(x + x), $MachinePrecision]), $MachinePrecision], N[(N[(z * y), $MachinePrecision] * N[(t * -9.0), $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) < -4.99999999999999991e108

   1. Initial program 95.4%

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

   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 64.5

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

   4. Applied rewrites 64.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      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. count-2-rev N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 64.4

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

   6. Applied rewrites 64.4%

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

   7. Taylor expanded in a around 0

      \[\leadsto \color{blue}{2 \cdot x - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   8. 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

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 64.2

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

   9. Applied rewrites 64.2%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 62.7

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

   11. Applied rewrites 62.7%

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

   if -4.99999999999999991e108 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 4e19

   1. Initial program 95.4%

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

   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 64.5

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

   4. Applied rewrites 64.5%

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

   if 4e19 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 95.4%

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

   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 64.5

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

   4. Applied rewrites 64.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      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. count-2-rev N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 64.4

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

   6. Applied rewrites 64.4%

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

   7. Taylor expanded in a around 0

      \[\leadsto \color{blue}{2 \cdot x - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   8. 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

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 64.2

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

   9. Applied rewrites 64.2%

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

       1. lift-+.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

       7. count-2-rev N/A

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

       8. lower-fma.f64 N/A

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

       9. *-commutative N/A

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

       10. lift-*.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. count-2-rev N/A

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

       13. lift-+.f64 64.2

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

   11. Applied rewrites 64.2%

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

Alternative 7: 84.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z \cdot y, t \cdot -9, x + x\right)\\ t_2 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+135}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, 27, x + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (* z y) (* t -9.0) (+ x x))) (t_2 (* (* (* y 9.0) z) t)))
   (if (<= t_2 -4e+135)
     t_1
     (if (<= t_2 4e+19) (fma (* b a) 27.0 (+ x x)) t_1))))

C

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((z * y), (t * -9.0), (x + x));
	double t_2 = ((y * 9.0) * z) * t;
	double tmp;
	if (t_2 <= -4e+135) {
		tmp = t_1;
	} else if (t_2 <= 4e+19) {
		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])
function code(x, y, z, t, a, b)
	t_1 = fma(Float64(z * y), Float64(t * -9.0), Float64(x + x))
	t_2 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if (t_2 <= -4e+135)
		tmp = t_1;
	elseif (t_2 <= 4e+19)
		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.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(z * y), $MachinePrecision] * N[(t * -9.0), $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, -4e+135], t$95$1, If[LessEqual[t$95$2, 4e+19], 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) < -3.99999999999999985e135 or 4e19 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 95.4%

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

   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 64.5

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

   4. Applied rewrites 64.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      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. count-2-rev N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 64.4

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

   6. Applied rewrites 64.4%

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

   7. Taylor expanded in a around 0

      \[\leadsto \color{blue}{2 \cdot x - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   8. 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

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 64.2

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

   9. Applied rewrites 64.2%

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

       1. lift-+.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

       7. count-2-rev N/A

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

       8. lower-fma.f64 N/A

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

       9. *-commutative N/A

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

       10. lift-*.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. count-2-rev N/A

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

       13. lift-+.f64 64.2

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

   11. Applied rewrites 64.2%

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

   if -3.99999999999999985e135 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 4e19

   1. Initial program 95.4%

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

   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 64.5

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

   4. Applied rewrites 64.5%

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

Alternative 8: 82.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(z \cdot y\right) \cdot t\right) \cdot -9\\ t_2 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+135}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+163}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, 27, x + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* z y) t) -9.0)) (t_2 (* (* (* y 9.0) z) t)))
   (if (<= t_2 -4e+135)
     t_1
     (if (<= t_2 5e+163) (fma (* b a) 27.0 (+ x x)) t_1))))

C

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 = ((z * y) * t) * -9.0;
	double t_2 = ((y * 9.0) * z) * t;
	double tmp;
	if (t_2 <= -4e+135) {
		tmp = t_1;
	} else if (t_2 <= 5e+163) {
		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])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(z * y) * t) * -9.0)
	t_2 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if (t_2 <= -4e+135)
		tmp = t_1;
	elseif (t_2 <= 5e+163)
		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.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] * -9.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+135], t$95$1, If[LessEqual[t$95$2, 5e+163], 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) < -3.99999999999999985e135 or 5e163 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 95.4%

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

   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 64.5

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

   4. Applied rewrites 64.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      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. count-2-rev N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 64.4

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

   6. Applied rewrites 64.4%

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

   7. Taylor expanded in a around 0

      \[\leadsto \color{blue}{2 \cdot x - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   8. 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

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 64.2

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

   9. Applied rewrites 64.2%

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

   10. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 36.0

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

   12. Applied rewrites 36.0%

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

   if -3.99999999999999985e135 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 5e163

   1. Initial program 95.4%

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

   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 64.5

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

   4. Applied rewrites 64.5%

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

Alternative 9: 82.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(z \cdot y\right) \cdot t\right) \cdot -9\\ t_2 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+135}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+163}:\\ \;\;\;\;\mathsf{fma}\left(27 \cdot a, b, x + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* z y) t) -9.0)) (t_2 (* (* (* y 9.0) z) t)))
   (if (<= t_2 -4e+135)
     t_1
     (if (<= t_2 5e+163) (fma (* 27.0 a) b (+ x x)) t_1))))

C

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 = ((z * y) * t) * -9.0;
	double t_2 = ((y * 9.0) * z) * t;
	double tmp;
	if (t_2 <= -4e+135) {
		tmp = t_1;
	} else if (t_2 <= 5e+163) {
		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])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(z * y) * t) * -9.0)
	t_2 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if (t_2 <= -4e+135)
		tmp = t_1;
	elseif (t_2 <= 5e+163)
		tmp = fma(Float64(27.0 * a), 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.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] * -9.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+135], t$95$1, If[LessEqual[t$95$2, 5e+163], N[(N[(27.0 * a), $MachinePrecision] * b + 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) < -3.99999999999999985e135 or 5e163 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 95.4%

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

   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 64.5

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

   4. Applied rewrites 64.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      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. count-2-rev N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 64.4

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

   6. Applied rewrites 64.4%

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

   7. Taylor expanded in a around 0

      \[\leadsto \color{blue}{2 \cdot x - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   8. 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

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 64.2

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

   9. Applied rewrites 64.2%

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

   10. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 36.0

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

   12. Applied rewrites 36.0%

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

   if -3.99999999999999985e135 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 5e163

   1. Initial program 95.4%

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

   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 64.5

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

   4. Applied rewrites 64.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      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. count-2-rev N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 64.4

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

   6. Applied rewrites 64.4%

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

Alternative 10: 56.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(z \cdot y\right) \cdot t\right) \cdot -9\\ t_2 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+135}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-304}:\\ \;\;\;\;\frac{x + x}{a} \cdot a\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+19}:\\ \;\;\;\;\left(27 \cdot a\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* z y) t) -9.0)) (t_2 (* (* (* y 9.0) z) t)))
   (if (<= t_2 -4e+135)
     t_1
     (if (<= t_2 -1e-304)
       (* (/ (+ x x) a) a)
       (if (<= t_2 4e+19) (* (* 27.0 a) b) t_1)))))

C

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 = ((z * y) * t) * -9.0;
	double t_2 = ((y * 9.0) * z) * t;
	double tmp;
	if (t_2 <= -4e+135) {
		tmp = t_1;
	} else if (t_2 <= -1e-304) {
		tmp = ((x + x) / a) * a;
	} else if (t_2 <= 4e+19) {
		tmp = (27.0 * a) * b;
	} 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.
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 = ((z * y) * t) * (-9.0d0)
    t_2 = ((y * 9.0d0) * z) * t
    if (t_2 <= (-4d+135)) then
        tmp = t_1
    else if (t_2 <= (-1d-304)) then
        tmp = ((x + x) / a) * a
    else if (t_2 <= 4d+19) then
        tmp = (27.0d0 * a) * b
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

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 = ((z * y) * t) * -9.0;
	double t_2 = ((y * 9.0) * z) * t;
	double tmp;
	if (t_2 <= -4e+135) {
		tmp = t_1;
	} else if (t_2 <= -1e-304) {
		tmp = ((x + x) / a) * a;
	} else if (t_2 <= 4e+19) {
		tmp = (27.0 * a) * b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	t_1 = ((z * y) * t) * -9.0
	t_2 = ((y * 9.0) * z) * t
	tmp = 0
	if t_2 <= -4e+135:
		tmp = t_1
	elif t_2 <= -1e-304:
		tmp = ((x + x) / a) * a
	elif t_2 <= 4e+19:
		tmp = (27.0 * a) * b
	else:
		tmp = t_1
	return tmp

Julia

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(z * y) * t) * -9.0)
	t_2 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if (t_2 <= -4e+135)
		tmp = t_1;
	elseif (t_2 <= -1e-304)
		tmp = Float64(Float64(Float64(x + x) / a) * a);
	elseif (t_2 <= 4e+19)
		tmp = Float64(Float64(27.0 * a) * b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

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 = ((z * y) * t) * -9.0;
	t_2 = ((y * 9.0) * z) * t;
	tmp = 0.0;
	if (t_2 <= -4e+135)
		tmp = t_1;
	elseif (t_2 <= -1e-304)
		tmp = ((x + x) / a) * a;
	elseif (t_2 <= 4e+19)
		tmp = (27.0 * a) * b;
	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.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] * -9.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+135], t$95$1, If[LessEqual[t$95$2, -1e-304], N[(N[(N[(x + x), $MachinePrecision] / a), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[t$95$2, 4e+19], N[(N[(27.0 * a), $MachinePrecision] * b), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -3.99999999999999985e135 or 4e19 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 95.4%

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

   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 64.5

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

   4. Applied rewrites 64.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      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. count-2-rev N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 64.4

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

   6. Applied rewrites 64.4%

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

   7. Taylor expanded in a around 0

      \[\leadsto \color{blue}{2 \cdot x - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   8. 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

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 64.2

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

   9. Applied rewrites 64.2%

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

   10. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 36.0

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

   12. Applied rewrites 36.0%

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

   if -3.99999999999999985e135 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -9.99999999999999971e-305

   1. Initial program 95.4%

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

   2. Taylor expanded in a around inf

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

   4. Applied rewrites 82.5%

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

   5. Taylor expanded in x around inf

      \[\leadsto \left(2 \cdot \frac{x}{a}\right) \cdot a \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot x}{a} \cdot a \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{2 \cdot x}{a} \cdot a \]

      3. count-2-rev N/A

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

      4. lift-+.f64 22.8

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

   7. Applied rewrites 22.8%

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

   if -9.99999999999999971e-305 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 4e19

   1. Initial program 95.4%

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

   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 64.5

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

   4. Applied rewrites 64.5%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 35.8

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

   7. Applied rewrites 35.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 35.8

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

   9. Applied rewrites 35.8%

      \[\leadsto \left(27 \cdot a\right) \cdot b \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 53.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(z \cdot y\right) \cdot t\right) \cdot -9\\ t_2 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+135}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+19}:\\ \;\;\;\;\left(a \cdot b\right) \cdot 27\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* z y) t) -9.0)) (t_2 (* (* (* y 9.0) z) t)))
   (if (<= t_2 -4e+135) t_1 (if (<= t_2 4e+19) (* (* a b) 27.0) t_1))))

C

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 = ((z * y) * t) * -9.0;
	double t_2 = ((y * 9.0) * z) * t;
	double tmp;
	if (t_2 <= -4e+135) {
		tmp = t_1;
	} else if (t_2 <= 4e+19) {
		tmp = (a * b) * 27.0;
	} 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.
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 = ((z * y) * t) * (-9.0d0)
    t_2 = ((y * 9.0d0) * z) * t
    if (t_2 <= (-4d+135)) then
        tmp = t_1
    else if (t_2 <= 4d+19) then
        tmp = (a * b) * 27.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

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 = ((z * y) * t) * -9.0;
	double t_2 = ((y * 9.0) * z) * t;
	double tmp;
	if (t_2 <= -4e+135) {
		tmp = t_1;
	} else if (t_2 <= 4e+19) {
		tmp = (a * b) * 27.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	t_1 = ((z * y) * t) * -9.0
	t_2 = ((y * 9.0) * z) * t
	tmp = 0
	if t_2 <= -4e+135:
		tmp = t_1
	elif t_2 <= 4e+19:
		tmp = (a * b) * 27.0
	else:
		tmp = t_1
	return tmp

Julia

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(z * y) * t) * -9.0)
	t_2 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if (t_2 <= -4e+135)
		tmp = t_1;
	elseif (t_2 <= 4e+19)
		tmp = Float64(Float64(a * b) * 27.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

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 = ((z * y) * t) * -9.0;
	t_2 = ((y * 9.0) * z) * t;
	tmp = 0.0;
	if (t_2 <= -4e+135)
		tmp = t_1;
	elseif (t_2 <= 4e+19)
		tmp = (a * b) * 27.0;
	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.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] * -9.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+135], t$95$1, If[LessEqual[t$95$2, 4e+19], N[(N[(a * b), $MachinePrecision] * 27.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -3.99999999999999985e135 or 4e19 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 95.4%

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

   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 64.5

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

   4. Applied rewrites 64.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      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. count-2-rev N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 64.4

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

   6. Applied rewrites 64.4%

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

   7. Taylor expanded in a around 0

      \[\leadsto \color{blue}{2 \cdot x - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   8. 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

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 64.2

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

   9. Applied rewrites 64.2%

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

   10. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 36.0

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

   12. Applied rewrites 36.0%

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

   if -3.99999999999999985e135 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 4e19

   1. Initial program 95.4%

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

   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 64.5

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

   4. Applied rewrites 64.5%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 35.8

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

   7. Applied rewrites 35.8%

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

Alternative 12: 35.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \left(a \cdot b\right) \cdot 27 \end{array} \]

FPCore

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 (* (* a b) 27.0))

C

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 (a * b) * 27.0;
}

Fortran

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 = (a * b) * 27.0d0
end function

Java

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 (a * b) * 27.0;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	return (a * b) * 27.0

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	return Float64(Float64(a * b) * 27.0)
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp = code(x, y, z, t, a, b)
	tmp = (a * b) * 27.0;
end

Wolfram

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[(N[(a * b), $MachinePrecision] * 27.0), $MachinePrecision]

Derivation

1. Initial program 95.4%

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

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 64.5

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

4. Applied rewrites 64.5%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 35.8

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

7. Applied rewrites 35.8%

   \[\leadsto \left(a \cdot b\right) \cdot \color{blue}{27} \]
8. Add Preprocessing

Alternative 13: 35.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \left(27 \cdot a\right) \cdot b \end{array} \]

FPCore

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 (* (* 27.0 a) b))

C

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 (27.0 * a) * b;
}

Fortran

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 = (27.0d0 * a) * b
end function

Java

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 (27.0 * a) * b;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	return (27.0 * a) * b

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	return Float64(Float64(27.0 * a) * b)
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp = code(x, y, z, t, a, b)
	tmp = (27.0 * a) * b;
end

Wolfram

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[(N[(27.0 * a), $MachinePrecision] * b), $MachinePrecision]

Derivation

1. Initial program 95.4%

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

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 64.5

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

4. Applied rewrites 64.5%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 35.8

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

7. Applied rewrites 35.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-*.f64 35.8

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

9. Applied rewrites 35.8%

   \[\leadsto \left(27 \cdot a\right) \cdot b \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025130 
(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)))