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

Percentage Accurate: 95.1% → 99.1%

Time: 6.8s

Alternatives: 14

Speedup: 0.9×

• 27.3% 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: 95.1% 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: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 10000:\\ \;\;\;\;\mathsf{fma}\left(-9 \cdot z, t \cdot y, \mathsf{fma}\left(b \cdot a, 27, 2 \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot 27, a, \left(x + x\right) - t \cdot \left(z \cdot \left(9 \cdot y\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (if (<= t 10000.0)
   (fma (* -9.0 z) (* t y) (fma (* b a) 27.0 (* 2.0 x)))
   (fma (* b 27.0) a (- (+ x x) (* t (* z (* 9.0 y)))))))

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 (t <= 10000.0) {
		tmp = fma((-9.0 * z), (t * y), fma((b * a), 27.0, (2.0 * x)));
	} else {
		tmp = fma((b * 27.0), a, ((x + x) - (t * (z * (9.0 * y)))));
	}
	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 (t <= 10000.0)
		tmp = fma(Float64(-9.0 * z), Float64(t * y), fma(Float64(b * a), 27.0, Float64(2.0 * x)));
	else
		tmp = fma(Float64(b * 27.0), a, Float64(Float64(x + x) - Float64(t * Float64(z * Float64(9.0 * y)))));
	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[t, 10000.0], N[(N[(-9.0 * z), $MachinePrecision] * N[(t * y), $MachinePrecision] + N[(N[(b * a), $MachinePrecision] * 27.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * 27.0), $MachinePrecision] * a + N[(N[(x + x), $MachinePrecision] - N[(t * N[(z * N[(9.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1e4

   1. Initial program 93.1%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

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

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

      9. lower-fma.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

   5. Applied rewrites 97.3%

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

   if 1e4 < t

   1. Initial program 95.0%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   2. Add Preprocessing
   3. 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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 98.3

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 98.3

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 98.3

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 98.3

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 98.3

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

   4. Applied rewrites 98.3%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 98.3

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

   6. Applied rewrites 98.3%

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

Alternative 2: 86.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-9 \cdot z, y \cdot t, \left(a \cdot 27\right) \cdot b\right)\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+94} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+100}\right):\\ \;\;\;\;\mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(b \cdot a\right) \cdot 27\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 27, b, x + x\right)\\ \end{array} \end{array} \]

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 (- INFINITY))
     (fma (* -9.0 z) (* y t) (* (* a 27.0) b))
     (if (or (<= t_1 -2e+94) (not (<= t_1 2e+100)))
       (fma -9.0 (* (* z y) t) (* (* b a) 27.0))
       (fma (* a 27.0) b (+ 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 <= -((double) INFINITY)) {
		tmp = fma((-9.0 * z), (y * t), ((a * 27.0) * b));
	} else if ((t_1 <= -2e+94) || !(t_1 <= 2e+100)) {
		tmp = fma(-9.0, ((z * y) * t), ((b * a) * 27.0));
	} else {
		tmp = fma((a * 27.0), b, (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 <= Float64(-Inf))
		tmp = fma(Float64(-9.0 * z), Float64(y * t), Float64(Float64(a * 27.0) * b));
	elseif ((t_1 <= -2e+94) || !(t_1 <= 2e+100))
		tmp = fma(-9.0, Float64(Float64(z * y) * t), Float64(Float64(b * a) * 27.0));
	else
		tmp = fma(Float64(a * 27.0), b, 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, (-Infinity)], N[(N[(-9.0 * z), $MachinePrecision] * N[(y * t), $MachinePrecision] + N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$1, -2e+94], N[Not[LessEqual[t$95$1, 2e+100]], $MachinePrecision]], N[(-9.0 * N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] + N[(N[(b * a), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * 27.0), $MachinePrecision] * b + N[(x + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 66.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{27 \cdot \left(a \cdot b\right) - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   4. 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 66.0

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

   5. Applied rewrites 66.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 93.5

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

   7. Applied rewrites 93.5%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -2e94 or 2.00000000000000003e100 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 94.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{27 \cdot \left(a \cdot b\right) - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   4. 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 86.3

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

   5. Applied rewrites 86.3%

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

   if -2e94 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 2.00000000000000003e100

   1. Initial program 98.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 89.0

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

   5. Applied rewrites 89.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 89.0

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 89.0

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

   7. Applied rewrites 89.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 89.0

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

   9. Applied rewrites 89.0%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-9 \cdot z, y \cdot t, \left(a \cdot 27\right) \cdot b\right)\\ \mathbf{elif}\;\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \leq -2 \cdot 10^{+94} \lor \neg \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \leq 2 \cdot 10^{+100}\right):\\ \;\;\;\;\mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(b \cdot a\right) \cdot 27\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 27, b, x + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(t \cdot y\right) \cdot z, -9, x\right) + x\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+94} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+100}\right):\\ \;\;\;\;\mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(b \cdot a\right) \cdot 27\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 27, b, x + x\right)\\ \end{array} \end{array} \]

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 (- INFINITY))
     (+ (fma (* (* t y) z) -9.0 x) x)
     (if (or (<= t_1 -2e+94) (not (<= t_1 2e+100)))
       (fma -9.0 (* (* z y) t) (* (* b a) 27.0))
       (fma (* a 27.0) b (+ 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 <= -((double) INFINITY)) {
		tmp = fma(((t * y) * z), -9.0, x) + x;
	} else if ((t_1 <= -2e+94) || !(t_1 <= 2e+100)) {
		tmp = fma(-9.0, ((z * y) * t), ((b * a) * 27.0));
	} else {
		tmp = fma((a * 27.0), b, (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 <= Float64(-Inf))
		tmp = Float64(fma(Float64(Float64(t * y) * z), -9.0, x) + x);
	elseif ((t_1 <= -2e+94) || !(t_1 <= 2e+100))
		tmp = fma(-9.0, Float64(Float64(z * y) * t), Float64(Float64(b * a) * 27.0));
	else
		tmp = fma(Float64(a * 27.0), b, 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, (-Infinity)], N[(N[(N[(N[(t * y), $MachinePrecision] * z), $MachinePrecision] * -9.0 + x), $MachinePrecision] + x), $MachinePrecision], If[Or[LessEqual[t$95$1, -2e+94], N[Not[LessEqual[t$95$1, 2e+100]], $MachinePrecision]], N[(-9.0 * N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] + N[(N[(b * a), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * 27.0), $MachinePrecision] * b + N[(x + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 66.0%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   2. Add Preprocessing
   3. 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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 69.5

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 69.5

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 69.5

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 69.5

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 69.5

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

   4. Applied rewrites 69.5%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

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

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

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

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

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

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

   7. Applied rewrites 89.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 89.2

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 89.2

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

   9. Applied rewrites 89.2%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -2e94 or 2.00000000000000003e100 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 94.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{27 \cdot \left(a \cdot b\right) - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   4. 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 86.3

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

   5. Applied rewrites 86.3%

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

   if -2e94 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 2.00000000000000003e100

   1. Initial program 98.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 89.0

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

   5. Applied rewrites 89.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 89.0

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 89.0

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

   7. Applied rewrites 89.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 89.0

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

   9. Applied rewrites 89.0%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(t \cdot y\right) \cdot z, -9, x\right) + x\\ \mathbf{elif}\;\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \leq -2 \cdot 10^{+94} \lor \neg \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \leq 2 \cdot 10^{+100}\right):\\ \;\;\;\;\mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(b \cdot a\right) \cdot 27\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 27, b, x + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(a \cdot 27\right) \cdot b\\ t_2 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+295}:\\ \;\;\;\;\mathsf{fma}\left(-9 \cdot z, y \cdot t, t\_1\right)\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+94}:\\ \;\;\;\;t\_1 - t\_2\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+100}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 27, b, x + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(b \cdot a\right) \cdot 27\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* a 27.0) b)) (t_2 (* (* (* y 9.0) z) t)))
   (if (<= t_2 -5e+295)
     (fma (* -9.0 z) (* y t) t_1)
     (if (<= t_2 -2e+94)
       (- t_1 t_2)
       (if (<= t_2 2e+100)
         (fma (* a 27.0) b (+ x x))
         (fma -9.0 (* (* z y) t) (* (* b a) 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) {
	double t_1 = (a * 27.0) * b;
	double t_2 = ((y * 9.0) * z) * t;
	double tmp;
	if (t_2 <= -5e+295) {
		tmp = fma((-9.0 * z), (y * t), t_1);
	} else if (t_2 <= -2e+94) {
		tmp = t_1 - t_2;
	} else if (t_2 <= 2e+100) {
		tmp = fma((a * 27.0), b, (x + x));
	} else {
		tmp = fma(-9.0, ((z * y) * t), ((b * a) * 27.0));
	}
	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(a * 27.0) * b)
	t_2 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if (t_2 <= -5e+295)
		tmp = fma(Float64(-9.0 * z), Float64(y * t), t_1);
	elseif (t_2 <= -2e+94)
		tmp = Float64(t_1 - t_2);
	elseif (t_2 <= 2e+100)
		tmp = fma(Float64(a * 27.0), b, Float64(x + x));
	else
		tmp = fma(-9.0, Float64(Float64(z * y) * t), Float64(Float64(b * a) * 27.0));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+295], N[(N[(-9.0 * z), $MachinePrecision] * N[(y * t), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[t$95$2, -2e+94], N[(t$95$1 - t$95$2), $MachinePrecision], If[LessEqual[t$95$2, 2e+100], N[(N[(a * 27.0), $MachinePrecision] * b + N[(x + x), $MachinePrecision]), $MachinePrecision], N[(-9.0 * N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] + N[(N[(b * a), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -4.99999999999999991e295

   1. Initial program 68.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{27 \cdot \left(a \cdot b\right) - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   4. 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 68.2

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

   5. Applied rewrites 68.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 91.1

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

   7. Applied rewrites 91.1%

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

   if -4.99999999999999991e295 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -2e94

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{27 \cdot \left(a \cdot b\right) - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   4. 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 88.0

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

   5. Applied rewrites 88.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

      17. associate-*l* N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

      20. *-commutative N/A

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lift-*.f64 N/A

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

      24. lower--.f64 87.9

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

   7. Applied rewrites 87.9%

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

   if -2e94 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 2.00000000000000003e100

   1. Initial program 98.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 89.0

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

   5. Applied rewrites 89.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 89.0

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 89.0

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

   7. Applied rewrites 89.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 89.0

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

   9. Applied rewrites 89.0%

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

   if 2.00000000000000003e100 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 91.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{27 \cdot \left(a \cdot b\right) - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   4. 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 84.7

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

   5. Applied rewrites 84.7%

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

Alternative 5: 81.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+145} \lor \neg \left(t\_1 \leq 4 \cdot 10^{+295}\right):\\ \;\;\;\;-9 \cdot \left(\left(z \cdot t\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 27, b, x + x\right)\\ \end{array} \end{array} \]

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 (or (<= t_1 -1e+145) (not (<= t_1 4e+295)))
     (* -9.0 (* (* z t) y))
     (fma (* a 27.0) b (+ 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 <= -1e+145) || !(t_1 <= 4e+295)) {
		tmp = -9.0 * ((z * t) * y);
	} else {
		tmp = fma((a * 27.0), b, (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 <= -1e+145) || !(t_1 <= 4e+295))
		tmp = Float64(-9.0 * Float64(Float64(z * t) * y));
	else
		tmp = fma(Float64(a * 27.0), b, 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[Or[LessEqual[t$95$1, -1e+145], N[Not[LessEqual[t$95$1, 4e+295]], $MachinePrecision]], N[(-9.0 * N[(N[(z * t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[(a * 27.0), $MachinePrecision] * b + N[(x + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -9.9999999999999999e144 or 3.9999999999999999e295 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 80.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 75.2

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

   5. Applied rewrites 75.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 80.8

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

   7. Applied rewrites 80.8%

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

   if -9.9999999999999999e144 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 3.9999999999999999e295

   1. Initial program 98.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 83.9

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

   5. Applied rewrites 83.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 83.9

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 83.9

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

   7. Applied rewrites 83.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 83.9

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

   9. Applied rewrites 83.9%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \leq -1 \cdot 10^{+145} \lor \neg \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \leq 4 \cdot 10^{+295}\right):\\ \;\;\;\;-9 \cdot \left(\left(z \cdot t\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 27, b, x + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(-9 \cdot y, t \cdot z, 2 \cdot x\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 27, b, x + x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot z\right) \cdot \left(t \cdot -9\right)\\ \end{array} \end{array} \]

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-58)
     (fma (* -9.0 y) (* t z) (* 2.0 x))
     (if (<= t_1 5e+101) (fma (* a 27.0) b (+ x x)) (* (* y z) (* t -9.0))))))

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) {
	double t_1 = ((y * 9.0) * z) * t;
	double tmp;
	if (t_1 <= -5e-58) {
		tmp = fma((-9.0 * y), (t * z), (2.0 * x));
	} else if (t_1 <= 5e+101) {
		tmp = fma((a * 27.0), b, (x + x));
	} else {
		tmp = (y * z) * (t * -9.0);
	}
	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-58)
		tmp = fma(Float64(-9.0 * y), Float64(t * z), Float64(2.0 * x));
	elseif (t_1 <= 5e+101)
		tmp = fma(Float64(a * 27.0), b, Float64(x + x));
	else
		tmp = Float64(Float64(y * z) * Float64(t * -9.0));
	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-58], N[(N[(-9.0 * y), $MachinePrecision] * N[(t * z), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+101], N[(N[(a * 27.0), $MachinePrecision] * b + N[(x + x), $MachinePrecision]), $MachinePrecision], N[(N[(y * z), $MachinePrecision] * N[(t * -9.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -4.99999999999999977e-58

   1. Initial program 87.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 40.9

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

   5. Applied rewrites 40.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 40.9

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 40.9

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

   7. Applied rewrites 40.9%

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

   8. Taylor expanded in a around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

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

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

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

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(y \cdot 9\right)\right) \cdot \left(t \cdot z\right) + \left(\mathsf{neg}\left(-1 \cdot x\right)\right) \cdot 2 \]

      17. distribute-lft-neg-out N/A

          \[\leadsto \left(\mathsf{neg}\left(y \cdot 9\right)\right) \cdot \left(t \cdot z\right) + \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right) \cdot 2 \]

      18. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(y \cdot 9\right)\right) \cdot \left(t \cdot z\right) + \left(1 \cdot x\right) \cdot 2 \]

      19. *-lft-identity N/A

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

      20. *-commutative N/A

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

   10. Applied rewrites 72.2%

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

   if -4.99999999999999977e-58 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 4.99999999999999989e101

   1. Initial program 97.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 93.4

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

   5. Applied rewrites 93.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 93.4

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 93.4

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

   7. Applied rewrites 93.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 93.4

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

   9. Applied rewrites 93.4%

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

   if 4.99999999999999989e101 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 91.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 75.2

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

   5. Applied rewrites 75.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 75.2

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

   7. Applied rewrites 75.2%

      \[\leadsto \left(y \cdot z\right) \cdot \color{blue}{\left(t \cdot -9\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \leq -5 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(-9 \cdot y, t \cdot z, 2 \cdot x\right)\\ \mathbf{elif}\;\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \leq 5 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 27, b, x + x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot z\right) \cdot \left(t \cdot -9\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+182}:\\ \;\;\;\;\mathsf{fma}\left(\left(t \cdot y\right) \cdot z, -9, x\right) + x\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 27, b, x + x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot z\right) \cdot \left(t \cdot -9\right)\\ \end{array} \end{array} \]

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 -4e+182)
     (+ (fma (* (* t y) z) -9.0 x) x)
     (if (<= t_1 5e+101) (fma (* a 27.0) b (+ x x)) (* (* y z) (* t -9.0))))))

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) {
	double t_1 = ((y * 9.0) * z) * t;
	double tmp;
	if (t_1 <= -4e+182) {
		tmp = fma(((t * y) * z), -9.0, x) + x;
	} else if (t_1 <= 5e+101) {
		tmp = fma((a * 27.0), b, (x + x));
	} else {
		tmp = (y * z) * (t * -9.0);
	}
	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 <= -4e+182)
		tmp = Float64(fma(Float64(Float64(t * y) * z), -9.0, x) + x);
	elseif (t_1 <= 5e+101)
		tmp = fma(Float64(a * 27.0), b, Float64(x + x));
	else
		tmp = Float64(Float64(y * z) * Float64(t * -9.0));
	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, -4e+182], N[(N[(N[(N[(t * y), $MachinePrecision] * z), $MachinePrecision] * -9.0 + x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 5e+101], N[(N[(a * 27.0), $MachinePrecision] * b + N[(x + x), $MachinePrecision]), $MachinePrecision], N[(N[(y * z), $MachinePrecision] * N[(t * -9.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -4.0000000000000003e182

   1. Initial program 75.3%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   2. Add Preprocessing
   3. 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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 77.9

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 77.9

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 77.9

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 77.9

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 77.9

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

   4. Applied rewrites 77.9%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

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

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

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

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

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

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

   7. Applied rewrites 79.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 79.7

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 79.7

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

   9. Applied rewrites 79.7%

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

   if -4.0000000000000003e182 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 4.99999999999999989e101

   1. Initial program 98.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 86.6

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

   5. Applied rewrites 86.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 86.6

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 86.6

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

   7. Applied rewrites 86.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 86.6

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

   9. Applied rewrites 86.6%

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

   if 4.99999999999999989e101 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 91.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 75.2

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

   5. Applied rewrites 75.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 75.2

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

   7. Applied rewrites 75.2%

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

Alternative 8: 82.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+182}:\\ \;\;\;\;z \cdot \left(\left(y \cdot t\right) \cdot -9\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 27, b, x + x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot z\right) \cdot \left(t \cdot -9\right)\\ \end{array} \end{array} \]

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 -4e+182)
     (* z (* (* y t) -9.0))
     (if (<= t_1 5e+101) (fma (* a 27.0) b (+ x x)) (* (* y z) (* t -9.0))))))

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) {
	double t_1 = ((y * 9.0) * z) * t;
	double tmp;
	if (t_1 <= -4e+182) {
		tmp = z * ((y * t) * -9.0);
	} else if (t_1 <= 5e+101) {
		tmp = fma((a * 27.0), b, (x + x));
	} else {
		tmp = (y * z) * (t * -9.0);
	}
	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 <= -4e+182)
		tmp = Float64(z * Float64(Float64(y * t) * -9.0));
	elseif (t_1 <= 5e+101)
		tmp = fma(Float64(a * 27.0), b, Float64(x + x));
	else
		tmp = Float64(Float64(y * z) * Float64(t * -9.0));
	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, -4e+182], N[(z * N[(N[(y * t), $MachinePrecision] * -9.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+101], N[(N[(a * 27.0), $MachinePrecision] * b + N[(x + x), $MachinePrecision]), $MachinePrecision], N[(N[(y * z), $MachinePrecision] * N[(t * -9.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -4.0000000000000003e182

   1. Initial program 75.3%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 70.2

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

   5. Applied rewrites 70.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 75.3

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

   7. Applied rewrites 75.3%

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

   if -4.0000000000000003e182 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 4.99999999999999989e101

   1. Initial program 98.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 86.6

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

   5. Applied rewrites 86.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 86.6

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 86.6

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

   7. Applied rewrites 86.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 86.6

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

   9. Applied rewrites 86.6%

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

   if 4.99999999999999989e101 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 91.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 75.2

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

   5. Applied rewrites 75.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 75.2

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

   7. Applied rewrites 75.2%

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

Alternative 9: 82.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+182}:\\ \;\;\;\;z \cdot \left(\left(y \cdot t\right) \cdot -9\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 27, b, x + x\right)\\ \mathbf{else}:\\ \;\;\;\;-9 \cdot \left(\left(z \cdot y\right) \cdot t\right)\\ \end{array} \end{array} \]

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 -4e+182)
     (* z (* (* y t) -9.0))
     (if (<= t_1 5e+101) (fma (* a 27.0) b (+ x x)) (* -9.0 (* (* z y) t))))))

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 <= -4e+182) {
		tmp = z * ((y * t) * -9.0);
	} else if (t_1 <= 5e+101) {
		tmp = fma((a * 27.0), b, (x + x));
	} else {
		tmp = -9.0 * ((z * y) * t);
	}
	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 <= -4e+182)
		tmp = Float64(z * Float64(Float64(y * t) * -9.0));
	elseif (t_1 <= 5e+101)
		tmp = fma(Float64(a * 27.0), b, Float64(x + x));
	else
		tmp = Float64(-9.0 * Float64(Float64(z * y) * t));
	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, -4e+182], N[(z * N[(N[(y * t), $MachinePrecision] * -9.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+101], N[(N[(a * 27.0), $MachinePrecision] * b + N[(x + x), $MachinePrecision]), $MachinePrecision], N[(-9.0 * N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -4.0000000000000003e182

   1. Initial program 75.3%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 70.2

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

   5. Applied rewrites 70.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 75.3

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

   7. Applied rewrites 75.3%

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

   if -4.0000000000000003e182 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 4.99999999999999989e101

   1. Initial program 98.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 86.6

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

   5. Applied rewrites 86.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 86.6

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 86.6

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

   7. Applied rewrites 86.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 86.6

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

   9. Applied rewrites 86.6%

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

   if 4.99999999999999989e101 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 91.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 75.2

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

   5. Applied rewrites 75.2%

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

Alternative 10: 82.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+145}:\\ \;\;\;\;-9 \cdot \left(\left(z \cdot t\right) \cdot y\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 27, b, x + x\right)\\ \mathbf{else}:\\ \;\;\;\;-9 \cdot \left(\left(z \cdot y\right) \cdot t\right)\\ \end{array} \end{array} \]

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 -1e+145)
     (* -9.0 (* (* z t) y))
     (if (<= t_1 5e+101) (fma (* a 27.0) b (+ x x)) (* -9.0 (* (* z y) t))))))

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 <= -1e+145) {
		tmp = -9.0 * ((z * t) * y);
	} else if (t_1 <= 5e+101) {
		tmp = fma((a * 27.0), b, (x + x));
	} else {
		tmp = -9.0 * ((z * y) * t);
	}
	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 <= -1e+145)
		tmp = Float64(-9.0 * Float64(Float64(z * t) * y));
	elseif (t_1 <= 5e+101)
		tmp = fma(Float64(a * 27.0), b, Float64(x + x));
	else
		tmp = Float64(-9.0 * Float64(Float64(z * y) * t));
	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, -1e+145], N[(-9.0 * N[(N[(z * t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+101], N[(N[(a * 27.0), $MachinePrecision] * b + N[(x + x), $MachinePrecision]), $MachinePrecision], N[(-9.0 * N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 78.1%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 66.9

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

   5. Applied rewrites 66.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 74.0

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

   7. Applied rewrites 74.0%

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

   if -9.9999999999999999e144 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 4.99999999999999989e101

   1. Initial program 98.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 87.4

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

   5. Applied rewrites 87.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 87.4

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 87.4

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

   7. Applied rewrites 87.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 87.4

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

   9. Applied rewrites 87.4%

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

   if 4.99999999999999989e101 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 91.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 75.2

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

   5. Applied rewrites 75.2%

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

Alternative 11: 52.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(a \cdot 27\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+140} \lor \neg \left(t\_1 \leq 4 \cdot 10^{+65}\right):\\ \;\;\;\;\left(27 \cdot a\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;x + x\\ \end{array} \end{array} \]

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 (* (* a 27.0) b)))
   (if (or (<= t_1 -1e+140) (not (<= t_1 4e+65))) (* (* 27.0 a) b) (+ 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 = (a * 27.0) * b;
	double tmp;
	if ((t_1 <= -1e+140) || !(t_1 <= 4e+65)) {
		tmp = (27.0 * a) * b;
	} else {
		tmp = x + x;
	}
	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) :: tmp
    t_1 = (a * 27.0d0) * b
    if ((t_1 <= (-1d+140)) .or. (.not. (t_1 <= 4d+65))) then
        tmp = (27.0d0 * a) * b
    else
        tmp = x + x
    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 = (a * 27.0) * b;
	double tmp;
	if ((t_1 <= -1e+140) || !(t_1 <= 4e+65)) {
		tmp = (27.0 * a) * b;
	} else {
		tmp = x + x;
	}
	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 = (a * 27.0) * b
	tmp = 0
	if (t_1 <= -1e+140) or not (t_1 <= 4e+65):
		tmp = (27.0 * a) * b
	else:
		tmp = 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(a * 27.0) * b)
	tmp = 0.0
	if ((t_1 <= -1e+140) || !(t_1 <= 4e+65))
		tmp = Float64(Float64(27.0 * a) * b);
	else
		tmp = Float64(x + x);
	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 = (a * 27.0) * b;
	tmp = 0.0;
	if ((t_1 <= -1e+140) || ~((t_1 <= 4e+65)))
		tmp = (27.0 * a) * b;
	else
		tmp = x + x;
	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[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e+140], N[Not[LessEqual[t$95$1, 4e+65]], $MachinePrecision]], N[(N[(27.0 * a), $MachinePrecision] * b), $MachinePrecision], N[(x + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 a #s(literal 27 binary64)) b) < -1.00000000000000006e140 or 4e65 < (*.f64 (*.f64 a #s(literal 27 binary64)) b)

   1. Initial program 90.9%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 76.0

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

   5. Applied rewrites 76.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 76.2

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 76.2

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

   7. Applied rewrites 76.2%

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

   if -1.00000000000000006e140 < (*.f64 (*.f64 a #s(literal 27 binary64)) b) < 4e65

   1. Initial program 94.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 46.4

         \[\leadsto 2 \cdot \color{blue}{x} \]

   5. Applied rewrites 46.4%

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

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{x} \]

      2. count-2-rev N/A

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

      3. lower-+.f64 46.4

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

   7. Applied rewrites 46.4%

      \[\leadsto x + \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a \cdot 27\right) \cdot b \leq -1 \cdot 10^{+140} \lor \neg \left(\left(a \cdot 27\right) \cdot b \leq 4 \cdot 10^{+65}\right):\\ \;\;\;\;\left(27 \cdot a\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;x + x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 93.5% accurate, 1.1× speedup

Math

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

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

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	return fma(Float64(-9.0 * z), Float64(t * y), fma(Float64(b * a), 27.0, Float64(2.0 * x)))
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[(-9.0 * z), $MachinePrecision] * N[(t * y), $MachinePrecision] + N[(N[(b * a), $MachinePrecision] * 27.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.6%

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

3. Taylor expanded in x around 0

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

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

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. metadata-eval N/A

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

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

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

   9. lower-fma.f64 N/A

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

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

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

   11. metadata-eval N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. +-commutative N/A

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

   15. *-commutative N/A

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

5. Applied rewrites 93.8%

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

Alternative 13: 64.8% accurate, 2.5× speedup

Math

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

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 (fma (* a 27.0) b (+ 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) {
	return fma((a * 27.0), b, (x + x));
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	return fma(Float64(a * 27.0), b, Float64(x + x))
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 * 27.0), $MachinePrecision] * b + N[(x + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.6%

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

3. Taylor expanded in x around inf

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

   1. lower-*.f64 67.3

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

5. Applied rewrites 67.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 67.3

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 67.3

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

7. Applied rewrites 67.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f64 67.3

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

9. Applied rewrites 67.3%

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

Alternative 14: 30.7% accurate, 9.3× speedup

Math

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

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b) :precision binary64 (+ 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) {
	return x + x;
}

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 = x + x
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 x + x;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.6%

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

3. Taylor expanded in x around inf

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

   1. lower-*.f64 34.4

      \[\leadsto 2 \cdot \color{blue}{x} \]

5. Applied rewrites 34.4%

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

   1. lift-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{x} \]

   2. count-2-rev N/A

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

   3. lower-+.f64 34.4

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

7. Applied rewrites 34.4%

   \[\leadsto x + \color{blue}{x} \]
8. Add Preprocessing

Developer Target 1: 94.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y < 7.590524218811189 \cdot 10^{-161}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - 9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (< y 7.590524218811189e-161)
   (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* a (* 27.0 b)))
   (+ (- (* x 2.0) (* 9.0 (* y (* t z)))) (* (* a 27.0) b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y < 7.590524218811189e-161) {
		tmp = ((x * 2.0) - (((y * 9.0) * z) * t)) + (a * (27.0 * b));
	} else {
		tmp = ((x * 2.0) - (9.0 * (y * (t * z)))) + ((a * 27.0) * b);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, 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) :: tmp
    if (y < 7.590524218811189d-161) then
        tmp = ((x * 2.0d0) - (((y * 9.0d0) * z) * t)) + (a * (27.0d0 * b))
    else
        tmp = ((x * 2.0d0) - (9.0d0 * (y * (t * z)))) + ((a * 27.0d0) * b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y < 7.590524218811189e-161) {
		tmp = ((x * 2.0) - (((y * 9.0) * z) * t)) + (a * (27.0 * b));
	} else {
		tmp = ((x * 2.0) - (9.0 * (y * (t * z)))) + ((a * 27.0) * b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y < 7.590524218811189e-161:
		tmp = ((x * 2.0) - (((y * 9.0) * z) * t)) + (a * (27.0 * b))
	else:
		tmp = ((x * 2.0) - (9.0 * (y * (t * z)))) + ((a * 27.0) * b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y < 7.590524218811189e-161)
		tmp = Float64(Float64(Float64(x * 2.0) - Float64(Float64(Float64(y * 9.0) * z) * t)) + Float64(a * Float64(27.0 * b)));
	else
		tmp = Float64(Float64(Float64(x * 2.0) - Float64(9.0 * Float64(y * Float64(t * z)))) + Float64(Float64(a * 27.0) * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y < 7.590524218811189e-161)
		tmp = ((x * 2.0) - (((y * 9.0) * z) * t)) + (a * (27.0 * b));
	else
		tmp = ((x * 2.0) - (9.0 * (y * (t * z)))) + ((a * 27.0) * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Less[y, 7.590524218811189e-161], N[(N[(N[(x * 2.0), $MachinePrecision] - N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(27.0 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * 2.0), $MachinePrecision] - N[(9.0 * N[(y * N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2025026 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y 7590524218811189/100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (- (* x 2) (* (* (* y 9) z) t)) (* a (* 27 b))) (+ (- (* x 2) (* 9 (* y (* t z)))) (* (* a 27) b))))

  (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* (* a 27.0) b)))