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

Percentage Accurate: 95.7% → 98.8%

Time: 3.7s

Alternatives: 16

Speedup: 0.9×

• 27.8% 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: 95.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((x * 2.0d0) - (((y * 9.0d0) * z) * t)) + ((a * 27.0d0) * b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 4.2 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(-9 \cdot \left(t \cdot z\right), y, \mathsf{fma}\left(b \cdot a, 27, x + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot 27, a, \mathsf{fma}\left(-9 \cdot t, z \cdot y, x + x\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (if (<= z 4.2e+14)
   (fma (* -9.0 (* t z)) y (fma (* b a) 27.0 (+ x x)))
   (fma (* b 27.0) a (fma (* -9.0 t) (* z y) (+ x x)))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= 4.2e+14) {
		tmp = fma((-9.0 * (t * z)), y, fma((b * a), 27.0, (x + x)));
	} else {
		tmp = fma((b * 27.0), a, fma((-9.0 * t), (z * y), (x + x)));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= 4.2e+14)
		tmp = fma(Float64(-9.0 * Float64(t * z)), y, fma(Float64(b * a), 27.0, Float64(x + x)));
	else
		tmp = fma(Float64(b * 27.0), a, fma(Float64(-9.0 * t), Float64(z * y), Float64(x + x)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, 4.2e+14], N[(N[(-9.0 * N[(t * z), $MachinePrecision]), $MachinePrecision] * y + N[(N[(b * a), $MachinePrecision] * 27.0 + N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * 27.0), $MachinePrecision] * a + N[(N[(-9.0 * t), $MachinePrecision] * N[(z * y), $MachinePrecision] + N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.2e14

   1. Initial program 95.7%

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

   3. Applied rewrites 96.0%

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

   if 4.2e14 < z

   1. Initial program 95.7%

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

      1. lift-+.f64 N/A

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

      2. +-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 96.3

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

      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. sub-negate-rev N/A

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

      12. sub-flip N/A

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

      13. distribute-neg-in N/A

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

      14. lift-*.f64 N/A

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

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

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

      16. add-flip N/A

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

      17. remove-double-neg N/A

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

      18. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(b \cdot 27, a, \left(\mathsf{neg}\left(\left(y \cdot 9\right) \cdot z\right)\right) \cdot t - \color{blue}{1 \cdot \left(\mathsf{neg}\left(x \cdot 2\right)\right)}\right) \]

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

          \[\leadsto \mathsf{fma}\left(b \cdot 27, a, \color{blue}{\left(\mathsf{neg}\left(\left(y \cdot 9\right) \cdot z\right)\right) \cdot t + \left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(x \cdot 2\right)\right)}\right) \]

   3. Applied rewrites 96.4%

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

Alternative 2: 98.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(-9 \cdot \left(t \cdot z\right), y, \mathsf{fma}\left(b \cdot a, 27, x + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot t\right) \cdot -9, z, \mathsf{fma}\left(27 \cdot b, a, x\right)\right) + x\\ \end{array} \end{array} \]

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
 (if (<= y -4.4e-46)
   (fma (* -9.0 (* t z)) y (fma (* b a) 27.0 (+ x x)))
   (+ (fma (* (* y t) -9.0) z (fma (* 27.0 b) a x)) x)))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -4.4e-46) {
		tmp = fma((-9.0 * (t * z)), y, fma((b * a), 27.0, (x + x)));
	} else {
		tmp = fma(((y * t) * -9.0), z, fma((27.0 * b), a, x)) + x;
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -4.4e-46)
		tmp = fma(Float64(-9.0 * Float64(t * z)), y, fma(Float64(b * a), 27.0, Float64(x + x)));
	else
		tmp = Float64(fma(Float64(Float64(y * t) * -9.0), z, fma(Float64(27.0 * b), a, x)) + x);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -4.4e-46], N[(N[(-9.0 * N[(t * z), $MachinePrecision]), $MachinePrecision] * y + N[(N[(b * a), $MachinePrecision] * 27.0 + N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y * t), $MachinePrecision] * -9.0), $MachinePrecision] * z + N[(N[(27.0 * b), $MachinePrecision] * a + x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.4000000000000002e-46

   1. Initial program 95.7%

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

   3. Applied rewrites 96.0%

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

   if -4.4000000000000002e-46 < y

   1. Initial program 95.7%

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

   3. Applied rewrites 96.0%

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

   5. Applied rewrites 93.3%

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

Alternative 3: 98.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 4 \cdot 10^{-139}:\\ \;\;\;\;\mathsf{fma}\left(\left(z \cdot t\right) \cdot y, -9, \mathsf{fma}\left(27, a \cdot b, x\right)\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-9 \cdot t, z \cdot y, \mathsf{fma}\left(b \cdot a, 27, x\right)\right) + x\\ \end{array} \end{array} \]

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
 (if (<= z 4e-139)
   (+ (fma (* (* z t) y) -9.0 (fma 27.0 (* a b) x)) x)
   (+ (fma (* -9.0 t) (* z y) (fma (* b a) 27.0 x)) x)))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= 4e-139) {
		tmp = fma(((z * t) * y), -9.0, fma(27.0, (a * b), x)) + x;
	} else {
		tmp = fma((-9.0 * t), (z * y), fma((b * a), 27.0, x)) + x;
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= 4e-139)
		tmp = Float64(fma(Float64(Float64(z * t) * y), -9.0, fma(27.0, Float64(a * b), x)) + x);
	else
		tmp = Float64(fma(Float64(-9.0 * t), Float64(z * y), fma(Float64(b * a), 27.0, x)) + x);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, 4e-139], N[(N[(N[(N[(z * t), $MachinePrecision] * y), $MachinePrecision] * -9.0 + N[(27.0 * N[(a * b), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(-9.0 * t), $MachinePrecision] * N[(z * y), $MachinePrecision] + N[(N[(b * a), $MachinePrecision] * 27.0 + x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.00000000000000012e-139

   1. Initial program 95.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. count-2-rev N/A

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

      7. associate--l+ N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

   3. Applied rewrites 96.1%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. sub-flip N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. remove-double-neg N/A

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

      15. lower-fma.f64 95.7

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 95.7

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

      19. lift-fma.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 95.7

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lift-*.f64 95.7

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

   5. Applied rewrites 95.7%

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

   if 4.00000000000000012e-139 < z

   1. Initial program 95.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. count-2-rev N/A

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

      7. associate--l+ N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

   3. Applied rewrites 96.1%

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

Alternative 4: 98.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;\left(y \cdot 9\right) \cdot z \leq 4 \cdot 10^{+125}:\\ \;\;\;\;\mathsf{fma}\left(-9 \cdot t, z \cdot y, \mathsf{fma}\left(b \cdot a, 27, x\right)\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot t\right) \cdot -9, z, \mathsf{fma}\left(27 \cdot b, a, x\right)\right) + x\\ \end{array} \end{array} \]

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
 (if (<= (* (* y 9.0) z) 4e+125)
   (+ (fma (* -9.0 t) (* z y) (fma (* b a) 27.0 x)) x)
   (+ (fma (* (* y t) -9.0) z (fma (* 27.0 b) a x)) x)))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((y * 9.0) * z) <= 4e+125) {
		tmp = fma((-9.0 * t), (z * y), fma((b * a), 27.0, x)) + x;
	} else {
		tmp = fma(((y * t) * -9.0), z, fma((27.0 * b), a, x)) + x;
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(y * 9.0) * z) <= 4e+125)
		tmp = Float64(fma(Float64(-9.0 * t), Float64(z * y), fma(Float64(b * a), 27.0, x)) + x);
	else
		tmp = Float64(fma(Float64(Float64(y * t) * -9.0), z, fma(Float64(27.0 * b), a, x)) + x);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision], 4e+125], N[(N[(N[(-9.0 * t), $MachinePrecision] * N[(z * y), $MachinePrecision] + N[(N[(b * a), $MachinePrecision] * 27.0 + x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(N[(y * t), $MachinePrecision] * -9.0), $MachinePrecision] * z + N[(N[(27.0 * b), $MachinePrecision] * a + x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y #s(literal 9 binary64)) z) < 3.9999999999999997e125

   1. Initial program 95.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. count-2-rev N/A

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

      7. associate--l+ N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

   3. Applied rewrites 96.1%

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

   if 3.9999999999999997e125 < (*.f64 (*.f64 y #s(literal 9 binary64)) z)

   1. Initial program 95.7%

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

   3. Applied rewrites 96.0%

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

   5. Applied rewrites 93.3%

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

Alternative 5: 97.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, -9 \cdot y, x\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-9 \cdot t, z \cdot y, \mathsf{fma}\left(b \cdot a, 27, x\right)\right) + x\\ \end{array} \end{array} \]

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
 (if (<= z -3e+85)
   (+ (fma (* t z) (* -9.0 y) x) x)
   (+ (fma (* -9.0 t) (* z y) (fma (* b a) 27.0 x)) x)))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -3e+85) {
		tmp = fma((t * z), (-9.0 * y), x) + x;
	} else {
		tmp = fma((-9.0 * t), (z * y), fma((b * a), 27.0, x)) + x;
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -3e+85)
		tmp = Float64(fma(Float64(t * z), Float64(-9.0 * y), x) + x);
	else
		tmp = Float64(fma(Float64(-9.0 * t), Float64(z * y), fma(Float64(b * a), 27.0, x)) + x);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -3e+85], N[(N[(N[(t * z), $MachinePrecision] * N[(-9.0 * y), $MachinePrecision] + x), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(-9.0 * t), $MachinePrecision] * N[(z * y), $MachinePrecision] + N[(N[(b * a), $MachinePrecision] * 27.0 + x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3e85

   1. Initial program 95.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. count-2-rev N/A

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

      7. associate--l+ N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

   3. Applied rewrites 96.1%

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

   4. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 64.4

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

   6. Applied rewrites 64.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. sub-flip N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. remove-double-neg N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 64.2

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

   8. Applied rewrites 64.2%

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

   if -3e85 < z

   1. Initial program 95.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. count-2-rev N/A

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

      7. associate--l+ N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

   3. Applied rewrites 96.1%

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

Alternative 6: 88.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(\left(-9 \cdot z\right) \cdot t, y, x\right) + x\\ \mathbf{elif}\;t\_1 \leq -1.56 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(27 \cdot b, a, \left(\left(y \cdot z\right) \cdot t\right) \cdot -9\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+117}:\\ \;\;\;\;\mathsf{fma}\left(27, a \cdot b, x\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot t, y \cdot -9, \left(27 \cdot a\right) \cdot b\right)\\ \end{array} \end{array} \]

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+306)
     (+ (fma (* (* -9.0 z) t) y x) x)
     (if (<= t_1 -1.56e+82)
       (fma (* 27.0 b) a (* (* (* y z) t) -9.0))
       (if (<= t_1 2e+117)
         (+ (fma 27.0 (* a b) x) x)
         (fma (* z t) (* y -9.0) (* (* 27.0 a) b)))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((y * 9.0) * z) * t;
	double tmp;
	if (t_1 <= -5e+306) {
		tmp = fma(((-9.0 * z) * t), y, x) + x;
	} else if (t_1 <= -1.56e+82) {
		tmp = fma((27.0 * b), a, (((y * z) * t) * -9.0));
	} else if (t_1 <= 2e+117) {
		tmp = fma(27.0, (a * b), x) + x;
	} else {
		tmp = fma((z * t), (y * -9.0), ((27.0 * a) * b));
	}
	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+306)
		tmp = Float64(fma(Float64(Float64(-9.0 * z) * t), y, x) + x);
	elseif (t_1 <= -1.56e+82)
		tmp = fma(Float64(27.0 * b), a, Float64(Float64(Float64(y * z) * t) * -9.0));
	elseif (t_1 <= 2e+117)
		tmp = Float64(fma(27.0, Float64(a * b), x) + x);
	else
		tmp = fma(Float64(z * t), Float64(y * -9.0), Float64(Float64(27.0 * a) * b));
	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+306], N[(N[(N[(N[(-9.0 * z), $MachinePrecision] * t), $MachinePrecision] * y + x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, -1.56e+82], N[(N[(27.0 * b), $MachinePrecision] * a + N[(N[(N[(y * z), $MachinePrecision] * t), $MachinePrecision] * -9.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+117], N[(N[(27.0 * N[(a * b), $MachinePrecision] + x), $MachinePrecision] + x), $MachinePrecision], N[(N[(z * t), $MachinePrecision] * N[(y * -9.0), $MachinePrecision] + N[(N[(27.0 * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -4.99999999999999993e306

   1. Initial program 95.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. count-2-rev N/A

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

      7. associate--l+ N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

   3. Applied rewrites 96.1%

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

   4. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 64.4

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

   6. Applied rewrites 64.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. sub-flip N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. associate-*l* N/A

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

      20. remove-double-neg N/A

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

      21. lower-fma.f64 N/A

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

   8. Applied rewrites 64.3%

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

   if -4.99999999999999993e306 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -1.56e82

   1. Initial program 95.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 66.6

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

   4. Applied rewrites 66.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. remove-double-neg N/A

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

      12. lower-fma.f64 67.2

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 67.2

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 67.2

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

   6. Applied rewrites 67.2%

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

   if -1.56e82 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 2.0000000000000001e117

   1. Initial program 95.7%

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

   2. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 65.2

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

   4. Applied rewrites 65.2%

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

      1. lift-fma.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-fma.f64 65.2

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 65.2

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*l* N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. *-lft-identity N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + 1 \cdot \color{blue}{\left(x + x\right)} \]

      24. lift-+.f64 N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + 1 \cdot \left(x + \color{blue}{x}\right) \]

      25. remove-double-neg N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right) \cdot \left(\color{blue}{x} + x\right) \]

      26. lift-+.f64 N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right) \cdot \left(x + \color{blue}{x}\right) \]

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

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(x + x\right)\right)\right) \]

      28. distribute-rgt-neg-in N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\mathsf{neg}\left(1\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\left(x + x\right)\right)\right)} \]

   6. Applied rewrites 65.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-+r+ N/A

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

      10. lift-fma.f64 N/A

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

      11. lower-+.f64 65.2

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

      12. lift-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 65.2

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 65.2

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

   8. Applied rewrites 65.2%

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

   if 2.0000000000000001e117 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 95.7%

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

   3. Applied rewrites 96.0%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 66.8

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

   6. Applied rewrites 66.8%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lift-*.f64 N/A

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

      17. sub-flip N/A

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

   8. Applied rewrites 66.6%

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

Alternative 7: 88.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(\left(-9 \cdot z\right) \cdot t, y, x\right) + x\\ \mathbf{elif}\;t\_1 \leq -1.56 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(27 \cdot b, a, \left(\left(y \cdot z\right) \cdot t\right) \cdot -9\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+117}:\\ \;\;\;\;\mathsf{fma}\left(27, a \cdot b, x\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-9 \cdot \left(t \cdot z\right), y, 27 \cdot \left(a \cdot b\right)\right)\\ \end{array} \end{array} \]

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
 (let* ((t_1 (* (* (* y 9.0) z) t)))
   (if (<= t_1 -5e+306)
     (+ (fma (* (* -9.0 z) t) y x) x)
     (if (<= t_1 -1.56e+82)
       (fma (* 27.0 b) a (* (* (* y z) t) -9.0))
       (if (<= t_1 2e+117)
         (+ (fma 27.0 (* a b) x) x)
         (fma (* -9.0 (* t z)) y (* 27.0 (* a b))))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((y * 9.0) * z) * t;
	double tmp;
	if (t_1 <= -5e+306) {
		tmp = fma(((-9.0 * z) * t), y, x) + x;
	} else if (t_1 <= -1.56e+82) {
		tmp = fma((27.0 * b), a, (((y * z) * t) * -9.0));
	} else if (t_1 <= 2e+117) {
		tmp = fma(27.0, (a * b), x) + x;
	} else {
		tmp = fma((-9.0 * (t * z)), y, (27.0 * (a * b)));
	}
	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+306)
		tmp = Float64(fma(Float64(Float64(-9.0 * z) * t), y, x) + x);
	elseif (t_1 <= -1.56e+82)
		tmp = fma(Float64(27.0 * b), a, Float64(Float64(Float64(y * z) * t) * -9.0));
	elseif (t_1 <= 2e+117)
		tmp = Float64(fma(27.0, Float64(a * b), x) + x);
	else
		tmp = fma(Float64(-9.0 * Float64(t * z)), y, Float64(27.0 * Float64(a * b)));
	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+306], N[(N[(N[(N[(-9.0 * z), $MachinePrecision] * t), $MachinePrecision] * y + x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, -1.56e+82], N[(N[(27.0 * b), $MachinePrecision] * a + N[(N[(N[(y * z), $MachinePrecision] * t), $MachinePrecision] * -9.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+117], N[(N[(27.0 * N[(a * b), $MachinePrecision] + x), $MachinePrecision] + x), $MachinePrecision], N[(N[(-9.0 * N[(t * z), $MachinePrecision]), $MachinePrecision] * y + N[(27.0 * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -4.99999999999999993e306

   1. Initial program 95.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. count-2-rev N/A

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

      7. associate--l+ N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

   3. Applied rewrites 96.1%

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

   4. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 64.4

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

   6. Applied rewrites 64.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. sub-flip N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. associate-*l* N/A

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

      20. remove-double-neg N/A

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

      21. lower-fma.f64 N/A

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

   8. Applied rewrites 64.3%

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

   if -4.99999999999999993e306 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -1.56e82

   1. Initial program 95.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 66.6

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

   4. Applied rewrites 66.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. remove-double-neg N/A

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

      12. lower-fma.f64 67.2

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 67.2

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 67.2

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

   6. Applied rewrites 67.2%

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

   if -1.56e82 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 2.0000000000000001e117

   1. Initial program 95.7%

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

   2. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 65.2

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

   4. Applied rewrites 65.2%

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

      1. lift-fma.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-fma.f64 65.2

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 65.2

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*l* N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. *-lft-identity N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + 1 \cdot \color{blue}{\left(x + x\right)} \]

      24. lift-+.f64 N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + 1 \cdot \left(x + \color{blue}{x}\right) \]

      25. remove-double-neg N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right) \cdot \left(\color{blue}{x} + x\right) \]

      26. lift-+.f64 N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right) \cdot \left(x + \color{blue}{x}\right) \]

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

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(x + x\right)\right)\right) \]

      28. distribute-rgt-neg-in N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\mathsf{neg}\left(1\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\left(x + x\right)\right)\right)} \]

   6. Applied rewrites 65.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-+r+ N/A

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

      10. lift-fma.f64 N/A

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

      11. lower-+.f64 65.2

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

      12. lift-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 65.2

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 65.2

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

   8. Applied rewrites 65.2%

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

   if 2.0000000000000001e117 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 95.7%

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

   3. Applied rewrites 96.0%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 66.8

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

   6. Applied rewrites 66.8%

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

Alternative 8: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(a \cdot 27\right) \cdot b\\ t_2 := \mathsf{fma}\left(-9 \cdot \left(t \cdot z\right), y, 27 \cdot \left(a \cdot b\right)\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+137}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+48}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, -9 \cdot y, x\right) + x\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+170}:\\ \;\;\;\;\mathsf{fma}\left(27 \cdot b, a, x + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* a 27.0) b)) (t_2 (fma (* -9.0 (* t z)) y (* 27.0 (* a b)))))
   (if (<= t_1 -4e+137)
     t_2
     (if (<= t_1 5e+48)
       (+ (fma (* t z) (* -9.0 y) x) x)
       (if (<= t_1 4e+170) (fma (* 27.0 b) a (+ x x)) t_2)))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * 27.0) * b;
	double t_2 = fma((-9.0 * (t * z)), y, (27.0 * (a * b)));
	double tmp;
	if (t_1 <= -4e+137) {
		tmp = t_2;
	} else if (t_1 <= 5e+48) {
		tmp = fma((t * z), (-9.0 * y), x) + x;
	} else if (t_1 <= 4e+170) {
		tmp = fma((27.0 * b), a, (x + x));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a * 27.0) * b)
	t_2 = fma(Float64(-9.0 * Float64(t * z)), y, Float64(27.0 * Float64(a * b)))
	tmp = 0.0
	if (t_1 <= -4e+137)
		tmp = t_2;
	elseif (t_1 <= 5e+48)
		tmp = Float64(fma(Float64(t * z), Float64(-9.0 * y), x) + x);
	elseif (t_1 <= 4e+170)
		tmp = fma(Float64(27.0 * b), a, Float64(x + x));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-9.0 * N[(t * z), $MachinePrecision]), $MachinePrecision] * y + N[(27.0 * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+137], t$95$2, If[LessEqual[t$95$1, 5e+48], N[(N[(N[(t * z), $MachinePrecision] * N[(-9.0 * y), $MachinePrecision] + x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 4e+170], N[(N[(27.0 * b), $MachinePrecision] * a + N[(x + x), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 a #s(literal 27 binary64)) b) < -4.0000000000000001e137 or 4.00000000000000014e170 < (*.f64 (*.f64 a #s(literal 27 binary64)) b)

   1. Initial program 95.7%

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

   3. Applied rewrites 96.0%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 66.8

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

   6. Applied rewrites 66.8%

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

   if -4.0000000000000001e137 < (*.f64 (*.f64 a #s(literal 27 binary64)) b) < 4.99999999999999973e48

   1. Initial program 95.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. count-2-rev N/A

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

      7. associate--l+ N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

   3. Applied rewrites 96.1%

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

   4. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 64.4

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

   6. Applied rewrites 64.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. sub-flip N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. remove-double-neg N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 64.2

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

   8. Applied rewrites 64.2%

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

   if 4.99999999999999973e48 < (*.f64 (*.f64 a #s(literal 27 binary64)) b) < 4.00000000000000014e170

   1. Initial program 95.7%

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

   2. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 65.2

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

   4. Applied rewrites 65.2%

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

      1. lift-fma.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-fma.f64 65.2

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 65.2

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*l* N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. *-lft-identity N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + 1 \cdot \color{blue}{\left(x + x\right)} \]

      24. lift-+.f64 N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + 1 \cdot \left(x + \color{blue}{x}\right) \]

      25. remove-double-neg N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right) \cdot \left(\color{blue}{x} + x\right) \]

      26. lift-+.f64 N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right) \cdot \left(x + \color{blue}{x}\right) \]

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

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(x + x\right)\right)\right) \]

      28. distribute-rgt-neg-in N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\mathsf{neg}\left(1\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\left(x + x\right)\right)\right)} \]

   6. Applied rewrites 65.1%

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

Alternative 9: 85.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+137}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot t, -9 \cdot z, x\right) + x\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(27, a \cdot b, x\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-9 \cdot z\right) \cdot t, y, x\right) + x\\ \end{array} \end{array} \]

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 (* (* (* y 9.0) z) t)))
   (if (<= t_1 -5e+137)
     (+ (fma (* y t) (* -9.0 z) x) x)
     (if (<= t_1 2e-18)
       (+ (fma 27.0 (* a b) x) x)
       (+ (fma (* (* -9.0 z) t) y x) x)))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((y * 9.0) * z) * t;
	double tmp;
	if (t_1 <= -5e+137) {
		tmp = fma((y * t), (-9.0 * z), x) + x;
	} else if (t_1 <= 2e-18) {
		tmp = fma(27.0, (a * b), x) + x;
	} else {
		tmp = fma(((-9.0 * z) * t), y, x) + x;
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if (t_1 <= -5e+137)
		tmp = Float64(fma(Float64(y * t), Float64(-9.0 * z), x) + x);
	elseif (t_1 <= 2e-18)
		tmp = Float64(fma(27.0, Float64(a * b), x) + x);
	else
		tmp = Float64(fma(Float64(Float64(-9.0 * z) * t), y, x) + x);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+137], N[(N[(N[(y * t), $MachinePrecision] * N[(-9.0 * z), $MachinePrecision] + x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e-18], N[(N[(27.0 * N[(a * b), $MachinePrecision] + x), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(N[(-9.0 * z), $MachinePrecision] * t), $MachinePrecision] * y + x), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -5.0000000000000002e137

   1. Initial program 95.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. count-2-rev N/A

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

      7. associate--l+ N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

   3. Applied rewrites 96.1%

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

   4. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 64.4

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

   6. Applied rewrites 64.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. sub-flip N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. associate-*l* N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. remove-double-neg N/A

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

      25. lower-fma.f64 62.4

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

   8. Applied rewrites 62.4%

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

   if -5.0000000000000002e137 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 2.0000000000000001e-18

   1. Initial program 95.7%

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

   2. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 65.2

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

   4. Applied rewrites 65.2%

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

      1. lift-fma.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-fma.f64 65.2

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 65.2

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*l* N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. *-lft-identity N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + 1 \cdot \color{blue}{\left(x + x\right)} \]

      24. lift-+.f64 N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + 1 \cdot \left(x + \color{blue}{x}\right) \]

      25. remove-double-neg N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right) \cdot \left(\color{blue}{x} + x\right) \]

      26. lift-+.f64 N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right) \cdot \left(x + \color{blue}{x}\right) \]

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

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(x + x\right)\right)\right) \]

      28. distribute-rgt-neg-in N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\mathsf{neg}\left(1\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\left(x + x\right)\right)\right)} \]

   6. Applied rewrites 65.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-+r+ N/A

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

      10. lift-fma.f64 N/A

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

      11. lower-+.f64 65.2

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

      12. lift-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 65.2

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 65.2

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

   8. Applied rewrites 65.2%

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

   if 2.0000000000000001e-18 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 95.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. count-2-rev N/A

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

      7. associate--l+ N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

   3. Applied rewrites 96.1%

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

   4. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 64.4

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

   6. Applied rewrites 64.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. sub-flip N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. associate-*l* N/A

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

      20. remove-double-neg N/A

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

      21. lower-fma.f64 N/A

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

   8. Applied rewrites 64.3%

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

Alternative 10: 84.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+137}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot t, -9 \cdot z, x\right) + x\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(27, a \cdot b, x\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, -9 \cdot y, x\right) + x\\ \end{array} \end{array} \]

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 (* (* (* y 9.0) z) t)))
   (if (<= t_1 -5e+137)
     (+ (fma (* y t) (* -9.0 z) x) x)
     (if (<= t_1 2e-18)
       (+ (fma 27.0 (* a b) x) x)
       (+ (fma (* t z) (* -9.0 y) x) x)))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((y * 9.0) * z) * t;
	double tmp;
	if (t_1 <= -5e+137) {
		tmp = fma((y * t), (-9.0 * z), x) + x;
	} else if (t_1 <= 2e-18) {
		tmp = fma(27.0, (a * b), x) + x;
	} else {
		tmp = fma((t * z), (-9.0 * y), x) + x;
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if (t_1 <= -5e+137)
		tmp = Float64(fma(Float64(y * t), Float64(-9.0 * z), x) + x);
	elseif (t_1 <= 2e-18)
		tmp = Float64(fma(27.0, Float64(a * b), x) + x);
	else
		tmp = Float64(fma(Float64(t * z), Float64(-9.0 * y), x) + x);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+137], N[(N[(N[(y * t), $MachinePrecision] * N[(-9.0 * z), $MachinePrecision] + x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e-18], N[(N[(27.0 * N[(a * b), $MachinePrecision] + x), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(t * z), $MachinePrecision] * N[(-9.0 * y), $MachinePrecision] + x), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -5.0000000000000002e137

   1. Initial program 95.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. count-2-rev N/A

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

      7. associate--l+ N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

   3. Applied rewrites 96.1%

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

   4. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 64.4

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

   6. Applied rewrites 64.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. sub-flip N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. associate-*l* N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. remove-double-neg N/A

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

      25. lower-fma.f64 62.4

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

   8. Applied rewrites 62.4%

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

   if -5.0000000000000002e137 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 2.0000000000000001e-18

   1. Initial program 95.7%

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

   2. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 65.2

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

   4. Applied rewrites 65.2%

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

      1. lift-fma.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-fma.f64 65.2

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 65.2

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*l* N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. *-lft-identity N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + 1 \cdot \color{blue}{\left(x + x\right)} \]

      24. lift-+.f64 N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + 1 \cdot \left(x + \color{blue}{x}\right) \]

      25. remove-double-neg N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right) \cdot \left(\color{blue}{x} + x\right) \]

      26. lift-+.f64 N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right) \cdot \left(x + \color{blue}{x}\right) \]

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

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(x + x\right)\right)\right) \]

      28. distribute-rgt-neg-in N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\mathsf{neg}\left(1\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\left(x + x\right)\right)\right)} \]

   6. Applied rewrites 65.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-+r+ N/A

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

      10. lift-fma.f64 N/A

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

      11. lower-+.f64 65.2

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

      12. lift-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 65.2

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 65.2

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

   8. Applied rewrites 65.2%

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

   if 2.0000000000000001e-18 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 95.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. count-2-rev N/A

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

      7. associate--l+ N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

   3. Applied rewrites 96.1%

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

   4. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 64.4

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

   6. Applied rewrites 64.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. sub-flip N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. remove-double-neg N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 64.2

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

   8. Applied rewrites 64.2%

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

Alternative 11: 83.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(a \cdot 27\right) \cdot b\\ t_2 := \mathsf{fma}\left(27, a \cdot b, x\right) + x\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+33}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+48}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, -9 \cdot y, x\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* a 27.0) b)) (t_2 (+ (fma 27.0 (* a b) x) x)))
   (if (<= t_1 -2e+33)
     t_2
     (if (<= t_1 5e+48) (+ (fma (* t z) (* -9.0 y) x) x) t_2))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * 27.0) * b;
	double t_2 = fma(27.0, (a * b), x) + x;
	double tmp;
	if (t_1 <= -2e+33) {
		tmp = t_2;
	} else if (t_1 <= 5e+48) {
		tmp = fma((t * z), (-9.0 * y), x) + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a * 27.0) * b)
	t_2 = Float64(fma(27.0, Float64(a * b), x) + x)
	tmp = 0.0
	if (t_1 <= -2e+33)
		tmp = t_2;
	elseif (t_1 <= 5e+48)
		tmp = Float64(fma(Float64(t * z), Float64(-9.0 * y), x) + x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$2 = N[(N[(27.0 * N[(a * b), $MachinePrecision] + x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+33], t$95$2, If[LessEqual[t$95$1, 5e+48], N[(N[(N[(t * z), $MachinePrecision] * N[(-9.0 * y), $MachinePrecision] + x), $MachinePrecision] + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 a #s(literal 27 binary64)) b) < -1.9999999999999999e33 or 4.99999999999999973e48 < (*.f64 (*.f64 a #s(literal 27 binary64)) b)

   1. Initial program 95.7%

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

   2. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 65.2

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

   4. Applied rewrites 65.2%

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

      1. lift-fma.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-fma.f64 65.2

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 65.2

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*l* N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. *-lft-identity N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + 1 \cdot \color{blue}{\left(x + x\right)} \]

      24. lift-+.f64 N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + 1 \cdot \left(x + \color{blue}{x}\right) \]

      25. remove-double-neg N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right) \cdot \left(\color{blue}{x} + x\right) \]

      26. lift-+.f64 N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right) \cdot \left(x + \color{blue}{x}\right) \]

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

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(x + x\right)\right)\right) \]

      28. distribute-rgt-neg-in N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\mathsf{neg}\left(1\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\left(x + x\right)\right)\right)} \]

   6. Applied rewrites 65.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-+r+ N/A

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

      10. lift-fma.f64 N/A

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

      11. lower-+.f64 65.2

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

      12. lift-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 65.2

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 65.2

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

   8. Applied rewrites 65.2%

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

   if -1.9999999999999999e33 < (*.f64 (*.f64 a #s(literal 27 binary64)) b) < 4.99999999999999973e48

   1. Initial program 95.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. count-2-rev N/A

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

      7. associate--l+ N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

   3. Applied rewrites 96.1%

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

   4. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 64.4

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

   6. Applied rewrites 64.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

         \[\leadsto \left(-9 \cdot \left(t \cdot \left(y \cdot z\right)\right) - \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) + x \]

      4. lift-*.f64 N/A

         \[\leadsto \left(-9 \cdot \left(t \cdot \left(y \cdot z\right)\right) - \left(\mathsf{neg}\left(\color{blue}{x}\right)\right)\right) + x \]

      5. *-commutative N/A

         \[\leadsto \left(\left(t \cdot \left(y \cdot z\right)\right) \cdot -9 - \left(\mathsf{neg}\left(\color{blue}{x}\right)\right)\right) + x \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(t \cdot \left(y \cdot z\right)\right) \cdot -9 - \left(\mathsf{neg}\left(x\right)\right)\right) + x \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(t \cdot \left(y \cdot z\right)\right) \cdot -9 - \left(\mathsf{neg}\left(x\right)\right)\right) + x \]

      8. *-commutative N/A

         \[\leadsto \left(\left(t \cdot \left(z \cdot y\right)\right) \cdot -9 - \left(\mathsf{neg}\left(x\right)\right)\right) + x \]

      9. associate-*l* N/A

         \[\leadsto \left(\left(\left(t \cdot z\right) \cdot y\right) \cdot -9 - \left(\mathsf{neg}\left(x\right)\right)\right) + x \]

      10. *-commutative N/A

          \[\leadsto \left(\left(\left(z \cdot t\right) \cdot y\right) \cdot -9 - \left(\mathsf{neg}\left(x\right)\right)\right) + x \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(z \cdot t\right) \cdot y\right) \cdot -9 - \left(\mathsf{neg}\left(x\right)\right)\right) + x \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(z \cdot t\right) \cdot y\right) \cdot -9 - \left(\mathsf{neg}\left(x\right)\right)\right) + x \]

      13. sub-flip N/A

          \[\leadsto \left(\left(\left(z \cdot t\right) \cdot y\right) \cdot -9 + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)}\right) + x \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(z \cdot t\right) \cdot y\right) \cdot -9 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{x}\right)\right)\right)\right)\right) + x \]

      15. associate-*l* N/A

          \[\leadsto \left(\left(z \cdot t\right) \cdot \left(y \cdot -9\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)\right)\right) + x \]

      16. remove-double-neg N/A

          \[\leadsto \left(\left(z \cdot t\right) \cdot \left(y \cdot -9\right) + x\right) + x \]

      17. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(z \cdot t, \color{blue}{y \cdot -9}, x\right) + x \]

      18. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(z \cdot t, \color{blue}{y} \cdot -9, x\right) + x \]

      19. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t \cdot z, \color{blue}{y} \cdot -9, x\right) + x \]

      20. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t \cdot z, \color{blue}{y} \cdot -9, x\right) + x \]

      21. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t \cdot z, -9 \cdot \color{blue}{y}, x\right) + x \]

      22. lower-*.f64 64.2

          \[\leadsto \mathsf{fma}\left(t \cdot z, -9 \cdot \color{blue}{y}, x\right) + x \]

   8. Applied rewrites 64.2%

      \[\leadsto \mathsf{fma}\left(t \cdot z, \color{blue}{-9 \cdot y}, x\right) + x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 82.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := -9 \cdot \left(t \cdot \left(y \cdot z\right)\right) + x\\ t_2 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+137}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+163}:\\ \;\;\;\;\mathsf{fma}\left(27, a \cdot b, x\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* -9.0 (* t (* y z))) x)) (t_2 (* (* (* y 9.0) z) t)))
   (if (<= t_2 -5e+137)
     t_1
     (if (<= t_2 4e+163) (+ (fma 27.0 (* a b) x) x) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (-9.0 * (t * (y * z))) + x;
	double t_2 = ((y * 9.0) * z) * t;
	double tmp;
	if (t_2 <= -5e+137) {
		tmp = t_1;
	} else if (t_2 <= 4e+163) {
		tmp = fma(27.0, (a * b), x) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(-9.0 * Float64(t * Float64(y * z))) + x)
	t_2 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if (t_2 <= -5e+137)
		tmp = t_1;
	elseif (t_2 <= 4e+163)
		tmp = Float64(fma(27.0, Float64(a * b), x) + x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(-9.0 * N[(t * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+137], t$95$1, If[LessEqual[t$95$2, 4e+163], N[(N[(27.0 * N[(a * b), $MachinePrecision] + x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -5.0000000000000002e137 or 3.9999999999999998e163 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 95.7%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)} + \left(a \cdot 27\right) \cdot b \]

      3. associate-+l- N/A

         \[\leadsto \color{blue}{x \cdot 2 - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot 2} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{2 \cdot x} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      6. count-2-rev N/A

         \[\leadsto \color{blue}{\left(x + x\right)} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      7. associate--l+ N/A

         \[\leadsto \color{blue}{x + \left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto \color{blue}{\left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)\right) + x} \]

      9. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)\right) + x} \]

   3. Applied rewrites 96.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-9 \cdot t, z \cdot y, \mathsf{fma}\left(b \cdot a, 27, x\right)\right) + x} \]

   4. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(x + -9 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right)} + x \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \left(x + \color{blue}{-9 \cdot \left(t \cdot \left(y \cdot z\right)\right)}\right) + x \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x + -9 \cdot \color{blue}{\left(t \cdot \left(y \cdot z\right)\right)}\right) + x \]

      3. lower-*.f64 N/A

         \[\leadsto \left(x + -9 \cdot \left(t \cdot \color{blue}{\left(y \cdot z\right)}\right)\right) + x \]

      4. lower-*.f64 64.4

         \[\leadsto \left(x + -9 \cdot \left(t \cdot \left(y \cdot \color{blue}{z}\right)\right)\right) + x \]

   6. Applied rewrites 64.4%

      \[\leadsto \color{blue}{\left(x + -9 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right)} + x \]

   7. Taylor expanded in x around 0

      \[\leadsto -9 \cdot \color{blue}{\left(t \cdot \left(y \cdot z\right)\right)} + x \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -9 \cdot \left(t \cdot \color{blue}{\left(y \cdot z\right)}\right) + x \]

      2. lower-*.f64 N/A

         \[\leadsto -9 \cdot \left(t \cdot \left(y \cdot \color{blue}{z}\right)\right) + x \]

      3. lower-*.f64 40.5

         \[\leadsto -9 \cdot \left(t \cdot \left(y \cdot z\right)\right) + x \]

   9. Applied rewrites 40.5%

      \[\leadsto -9 \cdot \color{blue}{\left(t \cdot \left(y \cdot z\right)\right)} + x \]

   if -5.0000000000000002e137 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 3.9999999999999998e163

   1. Initial program 95.7%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{2 \cdot x + 27 \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, \color{blue}{x}, 27 \cdot \left(a \cdot b\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right)\right) \]

      3. lower-*.f64 65.2

         \[\leadsto \mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right)\right) \]

   4. Applied rewrites 65.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto 2 \cdot x + \color{blue}{27 \cdot \left(a \cdot b\right)} \]

      2. count-2-rev N/A

         \[\leadsto \left(x + x\right) + \color{blue}{27} \cdot \left(a \cdot b\right) \]

      3. lift-+.f64 N/A

         \[\leadsto \left(x + x\right) + \color{blue}{27} \cdot \left(a \cdot b\right) \]

      4. +-commutative N/A

         \[\leadsto 27 \cdot \left(a \cdot b\right) + \color{blue}{\left(x + x\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto 27 \cdot \left(a \cdot b\right) + \left(\color{blue}{x} + x\right) \]

      6. *-commutative N/A

         \[\leadsto \left(a \cdot b\right) \cdot 27 + \left(\color{blue}{x} + x\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(a \cdot b\right) \cdot 27 + \left(x + x\right) \]

      8. *-commutative N/A

         \[\leadsto \left(b \cdot a\right) \cdot 27 + \left(x + x\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(b \cdot a\right) \cdot 27 + \left(x + x\right) \]

      10. lift-fma.f64 65.2

          \[\leadsto \mathsf{fma}\left(b \cdot a, \color{blue}{27}, x + x\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot a, 27, x + x\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(a \cdot b, 27, x + x\right) \]

      13. lift-*.f64 65.2

          \[\leadsto \mathsf{fma}\left(a \cdot b, 27, x + x\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \left(a \cdot b\right) \cdot 27 + \color{blue}{\left(x + x\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \left(a \cdot b\right) \cdot 27 + \left(x + x\right) \]

      16. *-commutative N/A

          \[\leadsto \left(b \cdot a\right) \cdot 27 + \left(x + x\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left(b \cdot a\right) \cdot 27 + \left(x + x\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \left(b \cdot a\right) \cdot 27 + \left(x + x\right) \]

      19. associate-*l* N/A

          \[\leadsto b \cdot \left(a \cdot 27\right) + \left(\color{blue}{x} + x\right) \]

      20. lift-*.f64 N/A

          \[\leadsto b \cdot \left(a \cdot 27\right) + \left(x + x\right) \]

      21. *-commutative N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\color{blue}{x} + x\right) \]

      22. lift-*.f64 N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\color{blue}{x} + x\right) \]

      23. *-lft-identity N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + 1 \cdot \color{blue}{\left(x + x\right)} \]

      24. lift-+.f64 N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + 1 \cdot \left(x + \color{blue}{x}\right) \]

      25. remove-double-neg N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right) \cdot \left(\color{blue}{x} + x\right) \]

      26. lift-+.f64 N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right) \cdot \left(x + \color{blue}{x}\right) \]

      27. distribute-lft-neg-out N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(x + x\right)\right)\right) \]

      28. distribute-rgt-neg-in N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\mathsf{neg}\left(1\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\left(x + x\right)\right)\right)} \]

   6. Applied rewrites 65.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(27 \cdot b, a, x + x\right)} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(27 \cdot b\right) \cdot a + \color{blue}{\left(x + x\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(27 \cdot b\right) \cdot a + \left(x + x\right) \]

      3. associate-*l* N/A

         \[\leadsto 27 \cdot \left(b \cdot a\right) + \left(\color{blue}{x} + x\right) \]

      4. lift-*.f64 N/A

         \[\leadsto 27 \cdot \left(b \cdot a\right) + \left(x + x\right) \]

      5. *-commutative N/A

         \[\leadsto \left(b \cdot a\right) \cdot 27 + \left(\color{blue}{x} + x\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(b \cdot a\right) \cdot 27 + \left(x + x\right) \]

      7. lift-+.f64 N/A

         \[\leadsto \left(b \cdot a\right) \cdot 27 + \left(x + \color{blue}{x}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(b \cdot a\right) \cdot 27 + \left(x + x\right) \]

      9. associate-+r+ N/A

         \[\leadsto \left(\left(b \cdot a\right) \cdot 27 + x\right) + \color{blue}{x} \]

      10. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot a, 27, x\right) + x \]

      11. lower-+.f64 65.2

          \[\leadsto \mathsf{fma}\left(b \cdot a, 27, x\right) + \color{blue}{x} \]

      12. lift-fma.f64 N/A

          \[\leadsto \left(\left(b \cdot a\right) \cdot 27 + x\right) + x \]

      13. *-commutative N/A

          \[\leadsto \left(27 \cdot \left(b \cdot a\right) + x\right) + x \]

      14. lower-fma.f64 65.2

          \[\leadsto \mathsf{fma}\left(27, b \cdot a, x\right) + x \]

      15. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(27, b \cdot a, x\right) + x \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(27, a \cdot b, x\right) + x \]

      17. lift-*.f64 65.2

          \[\leadsto \mathsf{fma}\left(27, a \cdot b, x\right) + x \]

   8. Applied rewrites 65.2%

      \[\leadsto \mathsf{fma}\left(27, a \cdot b, x\right) + \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 65.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \mathsf{fma}\left(27, a \cdot b, x\right) + x \end{array} \]

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 (+ (fma 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) {
	return fma(27.0, (a * 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 Float64(fma(27.0, Float64(a * b), 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[(27.0 * N[(a * b), $MachinePrecision] + x), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 95.7%

   \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{2 \cdot x + 27 \cdot \left(a \cdot b\right)} \]
3. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(2, \color{blue}{x}, 27 \cdot \left(a \cdot b\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right)\right) \]

   3. lower-*.f64 65.2

      \[\leadsto \mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right)\right) \]

4. Applied rewrites 65.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right)\right)} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto 2 \cdot x + \color{blue}{27 \cdot \left(a \cdot b\right)} \]

   2. count-2-rev N/A

      \[\leadsto \left(x + x\right) + \color{blue}{27} \cdot \left(a \cdot b\right) \]

   3. lift-+.f64 N/A

      \[\leadsto \left(x + x\right) + \color{blue}{27} \cdot \left(a \cdot b\right) \]

   4. +-commutative N/A

      \[\leadsto 27 \cdot \left(a \cdot b\right) + \color{blue}{\left(x + x\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto 27 \cdot \left(a \cdot b\right) + \left(\color{blue}{x} + x\right) \]

   6. *-commutative N/A

      \[\leadsto \left(a \cdot b\right) \cdot 27 + \left(\color{blue}{x} + x\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \left(a \cdot b\right) \cdot 27 + \left(x + x\right) \]

   8. *-commutative N/A

      \[\leadsto \left(b \cdot a\right) \cdot 27 + \left(x + x\right) \]

   9. lift-*.f64 N/A

      \[\leadsto \left(b \cdot a\right) \cdot 27 + \left(x + x\right) \]

   10. lift-fma.f64 65.2

       \[\leadsto \mathsf{fma}\left(b \cdot a, \color{blue}{27}, x + x\right) \]

   11. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(b \cdot a, 27, x + x\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(a \cdot b, 27, x + x\right) \]

   13. lift-*.f64 65.2

       \[\leadsto \mathsf{fma}\left(a \cdot b, 27, x + x\right) \]

   14. lower-fma.f64 N/A

       \[\leadsto \left(a \cdot b\right) \cdot 27 + \color{blue}{\left(x + x\right)} \]

   15. lift-*.f64 N/A

       \[\leadsto \left(a \cdot b\right) \cdot 27 + \left(x + x\right) \]

   16. *-commutative N/A

       \[\leadsto \left(b \cdot a\right) \cdot 27 + \left(x + x\right) \]

   17. lift-*.f64 N/A

       \[\leadsto \left(b \cdot a\right) \cdot 27 + \left(x + x\right) \]

   18. lift-*.f64 N/A

       \[\leadsto \left(b \cdot a\right) \cdot 27 + \left(x + x\right) \]

   19. associate-*l* N/A

       \[\leadsto b \cdot \left(a \cdot 27\right) + \left(\color{blue}{x} + x\right) \]

   20. lift-*.f64 N/A

       \[\leadsto b \cdot \left(a \cdot 27\right) + \left(x + x\right) \]

   21. *-commutative N/A

       \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\color{blue}{x} + x\right) \]

   22. lift-*.f64 N/A

       \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\color{blue}{x} + x\right) \]

   23. *-lft-identity N/A

       \[\leadsto \left(a \cdot 27\right) \cdot b + 1 \cdot \color{blue}{\left(x + x\right)} \]

   24. lift-+.f64 N/A

       \[\leadsto \left(a \cdot 27\right) \cdot b + 1 \cdot \left(x + \color{blue}{x}\right) \]

   25. remove-double-neg N/A

       \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right) \cdot \left(\color{blue}{x} + x\right) \]

   26. lift-+.f64 N/A

       \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right) \cdot \left(x + \color{blue}{x}\right) \]

   27. distribute-lft-neg-out N/A

       \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(x + x\right)\right)\right) \]

   28. distribute-rgt-neg-in N/A

       \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\mathsf{neg}\left(1\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\left(x + x\right)\right)\right)} \]

6. Applied rewrites 65.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(27 \cdot b, a, x + x\right)} \]
7. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto \left(27 \cdot b\right) \cdot a + \color{blue}{\left(x + x\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(27 \cdot b\right) \cdot a + \left(x + x\right) \]

   3. associate-*l* N/A

      \[\leadsto 27 \cdot \left(b \cdot a\right) + \left(\color{blue}{x} + x\right) \]

   4. lift-*.f64 N/A

      \[\leadsto 27 \cdot \left(b \cdot a\right) + \left(x + x\right) \]

   5. *-commutative N/A

      \[\leadsto \left(b \cdot a\right) \cdot 27 + \left(\color{blue}{x} + x\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \left(b \cdot a\right) \cdot 27 + \left(x + x\right) \]

   7. lift-+.f64 N/A

      \[\leadsto \left(b \cdot a\right) \cdot 27 + \left(x + \color{blue}{x}\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \left(b \cdot a\right) \cdot 27 + \left(x + x\right) \]

   9. associate-+r+ N/A

      \[\leadsto \left(\left(b \cdot a\right) \cdot 27 + x\right) + \color{blue}{x} \]

   10. lift-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(b \cdot a, 27, x\right) + x \]

   11. lower-+.f64 65.2

       \[\leadsto \mathsf{fma}\left(b \cdot a, 27, x\right) + \color{blue}{x} \]

   12. lift-fma.f64 N/A

       \[\leadsto \left(\left(b \cdot a\right) \cdot 27 + x\right) + x \]

   13. *-commutative N/A

       \[\leadsto \left(27 \cdot \left(b \cdot a\right) + x\right) + x \]

   14. lower-fma.f64 65.2

       \[\leadsto \mathsf{fma}\left(27, b \cdot a, x\right) + x \]

   15. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(27, b \cdot a, x\right) + x \]

   16. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(27, a \cdot b, x\right) + x \]

   17. lift-*.f64 65.2

       \[\leadsto \mathsf{fma}\left(27, a \cdot b, x\right) + x \]

8. Applied rewrites 65.2%

   \[\leadsto \mathsf{fma}\left(27, a \cdot b, x\right) + \color{blue}{x} \]
9. Add Preprocessing

Alternative 14: 54.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(a \cdot 27\right) \cdot b\\ t_2 := 27 \cdot \left(a \cdot b\right) + x\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+33}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+94}:\\ \;\;\;\;2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* a 27.0) b)) (t_2 (+ (* 27.0 (* a b)) x)))
   (if (<= t_1 -2e+33) t_2 (if (<= t_1 5e+94) (* 2.0 x) t_2))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * 27.0) * b;
	double t_2 = (27.0 * (a * b)) + x;
	double tmp;
	if (t_1 <= -2e+33) {
		tmp = t_2;
	} else if (t_1 <= 5e+94) {
		tmp = 2.0 * x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (a * 27.0d0) * b
    t_2 = (27.0d0 * (a * b)) + x
    if (t_1 <= (-2d+33)) then
        tmp = t_2
    else if (t_1 <= 5d+94) then
        tmp = 2.0d0 * x
    else
        tmp = t_2
    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 t_2 = (27.0 * (a * b)) + x;
	double tmp;
	if (t_1 <= -2e+33) {
		tmp = t_2;
	} else if (t_1 <= 5e+94) {
		tmp = 2.0 * x;
	} else {
		tmp = t_2;
	}
	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
	t_2 = (27.0 * (a * b)) + x
	tmp = 0
	if t_1 <= -2e+33:
		tmp = t_2
	elif t_1 <= 5e+94:
		tmp = 2.0 * x
	else:
		tmp = t_2
	return tmp

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a * 27.0) * b)
	t_2 = Float64(Float64(27.0 * Float64(a * b)) + x)
	tmp = 0.0
	if (t_1 <= -2e+33)
		tmp = t_2;
	elseif (t_1 <= 5e+94)
		tmp = Float64(2.0 * x);
	else
		tmp = t_2;
	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;
	t_2 = (27.0 * (a * b)) + x;
	tmp = 0.0;
	if (t_1 <= -2e+33)
		tmp = t_2;
	elseif (t_1 <= 5e+94)
		tmp = 2.0 * x;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
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[(27.0 * N[(a * b), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+33], t$95$2, If[LessEqual[t$95$1, 5e+94], N[(2.0 * x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 a #s(literal 27 binary64)) b) < -1.9999999999999999e33 or 5.0000000000000001e94 < (*.f64 (*.f64 a #s(literal 27 binary64)) b)

   1. Initial program 95.7%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{2 \cdot x + 27 \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, \color{blue}{x}, 27 \cdot \left(a \cdot b\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right)\right) \]

      3. lower-*.f64 65.2

         \[\leadsto \mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right)\right) \]

   4. Applied rewrites 65.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto 2 \cdot x + \color{blue}{27 \cdot \left(a \cdot b\right)} \]

      2. count-2-rev N/A

         \[\leadsto \left(x + x\right) + \color{blue}{27} \cdot \left(a \cdot b\right) \]

      3. lift-+.f64 N/A

         \[\leadsto \left(x + x\right) + \color{blue}{27} \cdot \left(a \cdot b\right) \]

      4. +-commutative N/A

         \[\leadsto 27 \cdot \left(a \cdot b\right) + \color{blue}{\left(x + x\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto 27 \cdot \left(a \cdot b\right) + \left(\color{blue}{x} + x\right) \]

      6. *-commutative N/A

         \[\leadsto \left(a \cdot b\right) \cdot 27 + \left(\color{blue}{x} + x\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(a \cdot b\right) \cdot 27 + \left(x + x\right) \]

      8. *-commutative N/A

         \[\leadsto \left(b \cdot a\right) \cdot 27 + \left(x + x\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(b \cdot a\right) \cdot 27 + \left(x + x\right) \]

      10. lift-fma.f64 65.2

          \[\leadsto \mathsf{fma}\left(b \cdot a, \color{blue}{27}, x + x\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot a, 27, x + x\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(a \cdot b, 27, x + x\right) \]

      13. lift-*.f64 65.2

          \[\leadsto \mathsf{fma}\left(a \cdot b, 27, x + x\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \left(a \cdot b\right) \cdot 27 + \color{blue}{\left(x + x\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \left(a \cdot b\right) \cdot 27 + \left(x + x\right) \]

      16. *-commutative N/A

          \[\leadsto \left(b \cdot a\right) \cdot 27 + \left(x + x\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left(b \cdot a\right) \cdot 27 + \left(x + x\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \left(b \cdot a\right) \cdot 27 + \left(x + x\right) \]

      19. associate-*l* N/A

          \[\leadsto b \cdot \left(a \cdot 27\right) + \left(\color{blue}{x} + x\right) \]

      20. lift-*.f64 N/A

          \[\leadsto b \cdot \left(a \cdot 27\right) + \left(x + x\right) \]

      21. *-commutative N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\color{blue}{x} + x\right) \]

      22. lift-*.f64 N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\color{blue}{x} + x\right) \]

      23. *-lft-identity N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + 1 \cdot \color{blue}{\left(x + x\right)} \]

      24. lift-+.f64 N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + 1 \cdot \left(x + \color{blue}{x}\right) \]

      25. remove-double-neg N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right) \cdot \left(\color{blue}{x} + x\right) \]

      26. lift-+.f64 N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right) \cdot \left(x + \color{blue}{x}\right) \]

      27. distribute-lft-neg-out N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(x + x\right)\right)\right) \]

      28. distribute-rgt-neg-in N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\mathsf{neg}\left(1\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\left(x + x\right)\right)\right)} \]

   6. Applied rewrites 65.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(27 \cdot b, a, x + x\right)} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(27 \cdot b\right) \cdot a + \color{blue}{\left(x + x\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(27 \cdot b\right) \cdot a + \left(x + x\right) \]

      3. associate-*l* N/A

         \[\leadsto 27 \cdot \left(b \cdot a\right) + \left(\color{blue}{x} + x\right) \]

      4. lift-*.f64 N/A

         \[\leadsto 27 \cdot \left(b \cdot a\right) + \left(x + x\right) \]

      5. *-commutative N/A

         \[\leadsto \left(b \cdot a\right) \cdot 27 + \left(\color{blue}{x} + x\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(b \cdot a\right) \cdot 27 + \left(x + x\right) \]

      7. lift-+.f64 N/A

         \[\leadsto \left(b \cdot a\right) \cdot 27 + \left(x + \color{blue}{x}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(b \cdot a\right) \cdot 27 + \left(x + x\right) \]

      9. associate-+r+ N/A

         \[\leadsto \left(\left(b \cdot a\right) \cdot 27 + x\right) + \color{blue}{x} \]

      10. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot a, 27, x\right) + x \]

      11. lower-+.f64 65.2

          \[\leadsto \mathsf{fma}\left(b \cdot a, 27, x\right) + \color{blue}{x} \]

      12. lift-fma.f64 N/A

          \[\leadsto \left(\left(b \cdot a\right) \cdot 27 + x\right) + x \]

      13. *-commutative N/A

          \[\leadsto \left(27 \cdot \left(b \cdot a\right) + x\right) + x \]

      14. lower-fma.f64 65.2

          \[\leadsto \mathsf{fma}\left(27, b \cdot a, x\right) + x \]

      15. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(27, b \cdot a, x\right) + x \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(27, a \cdot b, x\right) + x \]

      17. lift-*.f64 65.2

          \[\leadsto \mathsf{fma}\left(27, a \cdot b, x\right) + x \]

   8. Applied rewrites 65.2%

      \[\leadsto \mathsf{fma}\left(27, a \cdot b, x\right) + \color{blue}{x} \]

   9. Taylor expanded in x around 0

      \[\leadsto 27 \cdot \left(a \cdot b\right) + x \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 27 \cdot \left(a \cdot b\right) + x \]

       2. lower-*.f64 40.7

          \[\leadsto 27 \cdot \left(a \cdot b\right) + x \]

   11. Applied rewrites 40.7%

       \[\leadsto 27 \cdot \left(a \cdot b\right) + x \]

   if -1.9999999999999999e33 < (*.f64 (*.f64 a #s(literal 27 binary64)) b) < 5.0000000000000001e94

   1. Initial program 95.7%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)} + \left(a \cdot 27\right) \cdot b \]

      3. associate-+l- N/A

         \[\leadsto \color{blue}{x \cdot 2 - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot 2} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{2 \cdot x} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      6. count-2-rev N/A

         \[\leadsto \color{blue}{\left(x + x\right)} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      7. associate--l+ N/A

         \[\leadsto \color{blue}{x + \left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto \color{blue}{\left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)\right) + x} \]

      9. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)\right) + x} \]

   3. Applied rewrites 96.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-9 \cdot t, z \cdot y, \mathsf{fma}\left(b \cdot a, 27, x\right)\right) + x} \]
   4. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\left(-9 \cdot t\right) \cdot \left(z \cdot y\right) + \mathsf{fma}\left(b \cdot a, 27, x\right)\right)} + x \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(\left(-9 \cdot t\right) \cdot \left(z \cdot y\right) - \left(\mathsf{neg}\left(\mathsf{fma}\left(b \cdot a, 27, x\right)\right)\right)\right)} + x \]

      3. sub-flip N/A

         \[\leadsto \color{blue}{\left(\left(-9 \cdot t\right) \cdot \left(z \cdot y\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{fma}\left(b \cdot a, 27, x\right)\right)\right)\right)\right)\right)} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(-9 \cdot t\right) \cdot \color{blue}{\left(z \cdot y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{fma}\left(b \cdot a, 27, x\right)\right)\right)\right)\right)\right) + x \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\left(-9 \cdot t\right) \cdot z\right) \cdot y} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{fma}\left(b \cdot a, 27, x\right)\right)\right)\right)\right)\right) + x \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(-9 \cdot t\right)} \cdot z\right) \cdot y + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{fma}\left(b \cdot a, 27, x\right)\right)\right)\right)\right)\right) + x \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-9 \cdot \left(t \cdot z\right)\right)} \cdot y + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{fma}\left(b \cdot a, 27, x\right)\right)\right)\right)\right)\right) + x \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(-9 \cdot \color{blue}{\left(t \cdot z\right)}\right) \cdot y + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{fma}\left(b \cdot a, 27, x\right)\right)\right)\right)\right)\right) + x \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(-9 \cdot \left(t \cdot z\right)\right)} \cdot y + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{fma}\left(b \cdot a, 27, x\right)\right)\right)\right)\right)\right) + x \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(-9 \cdot \left(t \cdot z\right)\right)} \cdot y + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{fma}\left(b \cdot a, 27, x\right)\right)\right)\right)\right)\right) + x \]

      11. associate-*l* N/A

          \[\leadsto \left(\color{blue}{-9 \cdot \left(\left(t \cdot z\right) \cdot y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{fma}\left(b \cdot a, 27, x\right)\right)\right)\right)\right)\right) + x \]

      12. lift-*.f64 N/A

          \[\leadsto \left(-9 \cdot \color{blue}{\left(\left(t \cdot z\right) \cdot y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{fma}\left(b \cdot a, 27, x\right)\right)\right)\right)\right)\right) + x \]

      13. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\left(t \cdot z\right) \cdot y\right) \cdot -9} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{fma}\left(b \cdot a, 27, x\right)\right)\right)\right)\right)\right) + x \]

      14. remove-double-neg N/A

          \[\leadsto \left(\left(\left(t \cdot z\right) \cdot y\right) \cdot -9 + \color{blue}{\mathsf{fma}\left(b \cdot a, 27, x\right)}\right) + x \]

      15. lower-fma.f64 95.7

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(t \cdot z\right) \cdot y, -9, \mathsf{fma}\left(b \cdot a, 27, x\right)\right)} + x \]

      16. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(t \cdot z\right)} \cdot y, -9, \mathsf{fma}\left(b \cdot a, 27, x\right)\right) + x \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(z \cdot t\right)} \cdot y, -9, \mathsf{fma}\left(b \cdot a, 27, x\right)\right) + x \]

      18. lower-*.f64 95.7

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(z \cdot t\right)} \cdot y, -9, \mathsf{fma}\left(b \cdot a, 27, x\right)\right) + x \]

      19. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(z \cdot t\right) \cdot y, -9, \color{blue}{\left(b \cdot a\right) \cdot 27 + x}\right) + x \]

      20. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(z \cdot t\right) \cdot y, -9, \color{blue}{27 \cdot \left(b \cdot a\right)} + x\right) + x \]

      21. lower-fma.f64 95.7

          \[\leadsto \mathsf{fma}\left(\left(z \cdot t\right) \cdot y, -9, \color{blue}{\mathsf{fma}\left(27, b \cdot a, x\right)}\right) + x \]

      22. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(z \cdot t\right) \cdot y, -9, \mathsf{fma}\left(27, \color{blue}{b \cdot a}, x\right)\right) + x \]

      23. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(z \cdot t\right) \cdot y, -9, \mathsf{fma}\left(27, \color{blue}{a \cdot b}, x\right)\right) + x \]

      24. lift-*.f64 95.7

          \[\leadsto \mathsf{fma}\left(\left(z \cdot t\right) \cdot y, -9, \mathsf{fma}\left(27, \color{blue}{a \cdot b}, x\right)\right) + x \]

   5. Applied rewrites 95.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(z \cdot t\right) \cdot y, -9, \mathsf{fma}\left(27, a \cdot b, x\right)\right)} + x \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot x} \]
   7. Step-by-step derivation

      1. lower-*.f64 31.3

         \[\leadsto 2 \cdot \color{blue}{x} \]

   8. Applied rewrites 31.3%

      \[\leadsto \color{blue}{2 \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 52.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(a \cdot 27\right) \cdot b\\ t_2 := a \cdot \left(27 \cdot b\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+172}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+94}:\\ \;\;\;\;2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* a 27.0) b)) (t_2 (* a (* 27.0 b))))
   (if (<= t_1 -4e+172) t_2 (if (<= t_1 5e+94) (* 2.0 x) t_2))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * 27.0) * b;
	double t_2 = a * (27.0 * b);
	double tmp;
	if (t_1 <= -4e+172) {
		tmp = t_2;
	} else if (t_1 <= 5e+94) {
		tmp = 2.0 * x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (a * 27.0d0) * b
    t_2 = a * (27.0d0 * b)
    if (t_1 <= (-4d+172)) then
        tmp = t_2
    else if (t_1 <= 5d+94) then
        tmp = 2.0d0 * x
    else
        tmp = t_2
    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 t_2 = a * (27.0 * b);
	double tmp;
	if (t_1 <= -4e+172) {
		tmp = t_2;
	} else if (t_1 <= 5e+94) {
		tmp = 2.0 * x;
	} else {
		tmp = t_2;
	}
	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
	t_2 = a * (27.0 * b)
	tmp = 0
	if t_1 <= -4e+172:
		tmp = t_2
	elif t_1 <= 5e+94:
		tmp = 2.0 * x
	else:
		tmp = t_2
	return tmp

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a * 27.0) * b)
	t_2 = Float64(a * Float64(27.0 * b))
	tmp = 0.0
	if (t_1 <= -4e+172)
		tmp = t_2;
	elseif (t_1 <= 5e+94)
		tmp = Float64(2.0 * x);
	else
		tmp = t_2;
	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;
	t_2 = a * (27.0 * b);
	tmp = 0.0;
	if (t_1 <= -4e+172)
		tmp = t_2;
	elseif (t_1 <= 5e+94)
		tmp = 2.0 * x;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[(27.0 * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+172], t$95$2, If[LessEqual[t$95$1, 5e+94], N[(2.0 * x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 a #s(literal 27 binary64)) b) < -4.0000000000000003e172 or 5.0000000000000001e94 < (*.f64 (*.f64 a #s(literal 27 binary64)) b)

   1. Initial program 95.7%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{2 \cdot x + 27 \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, \color{blue}{x}, 27 \cdot \left(a \cdot b\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right)\right) \]

      3. lower-*.f64 65.2

         \[\leadsto \mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right)\right) \]

   4. Applied rewrites 65.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto 2 \cdot x + \color{blue}{27 \cdot \left(a \cdot b\right)} \]

      2. count-2-rev N/A

         \[\leadsto \left(x + x\right) + \color{blue}{27} \cdot \left(a \cdot b\right) \]

      3. lift-+.f64 N/A

         \[\leadsto \left(x + x\right) + \color{blue}{27} \cdot \left(a \cdot b\right) \]

      4. +-commutative N/A

         \[\leadsto 27 \cdot \left(a \cdot b\right) + \color{blue}{\left(x + x\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto 27 \cdot \left(a \cdot b\right) + \left(\color{blue}{x} + x\right) \]

      6. *-commutative N/A

         \[\leadsto \left(a \cdot b\right) \cdot 27 + \left(\color{blue}{x} + x\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(a \cdot b\right) \cdot 27 + \left(x + x\right) \]

      8. *-commutative N/A

         \[\leadsto \left(b \cdot a\right) \cdot 27 + \left(x + x\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(b \cdot a\right) \cdot 27 + \left(x + x\right) \]

      10. lift-fma.f64 65.2

          \[\leadsto \mathsf{fma}\left(b \cdot a, \color{blue}{27}, x + x\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot a, 27, x + x\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(a \cdot b, 27, x + x\right) \]

      13. lift-*.f64 65.2

          \[\leadsto \mathsf{fma}\left(a \cdot b, 27, x + x\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \left(a \cdot b\right) \cdot 27 + \color{blue}{\left(x + x\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \left(a \cdot b\right) \cdot 27 + \left(x + x\right) \]

      16. *-commutative N/A

          \[\leadsto \left(b \cdot a\right) \cdot 27 + \left(x + x\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left(b \cdot a\right) \cdot 27 + \left(x + x\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \left(b \cdot a\right) \cdot 27 + \left(x + x\right) \]

      19. associate-*l* N/A

          \[\leadsto b \cdot \left(a \cdot 27\right) + \left(\color{blue}{x} + x\right) \]

      20. lift-*.f64 N/A

          \[\leadsto b \cdot \left(a \cdot 27\right) + \left(x + x\right) \]

      21. *-commutative N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\color{blue}{x} + x\right) \]

      22. lift-*.f64 N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\color{blue}{x} + x\right) \]

      23. *-lft-identity N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + 1 \cdot \color{blue}{\left(x + x\right)} \]

      24. lift-+.f64 N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + 1 \cdot \left(x + \color{blue}{x}\right) \]

      25. remove-double-neg N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right) \cdot \left(\color{blue}{x} + x\right) \]

      26. lift-+.f64 N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right) \cdot \left(x + \color{blue}{x}\right) \]

      27. distribute-lft-neg-out N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(x + x\right)\right)\right) \]

      28. distribute-rgt-neg-in N/A

          \[\leadsto \left(a \cdot 27\right) \cdot b + \left(\mathsf{neg}\left(1\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\left(x + x\right)\right)\right)} \]

   6. Applied rewrites 65.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(27 \cdot b, a, x + x\right)} \]

   7. Taylor expanded in a around inf

      \[\leadsto a \cdot \color{blue}{\left(2 \cdot \frac{x}{a} + 27 \cdot b\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto a \cdot \left(2 \cdot \frac{x}{a} + \color{blue}{27 \cdot b}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto a \cdot \mathsf{fma}\left(2, \frac{x}{\color{blue}{a}}, 27 \cdot b\right) \]

      3. lower-/.f64 N/A

         \[\leadsto a \cdot \mathsf{fma}\left(2, \frac{x}{a}, 27 \cdot b\right) \]

      4. lower-*.f64 57.3

         \[\leadsto a \cdot \mathsf{fma}\left(2, \frac{x}{a}, 27 \cdot b\right) \]

   9. Applied rewrites 57.3%

      \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(2, \frac{x}{a}, 27 \cdot b\right)} \]

   10. Taylor expanded in x around 0

       \[\leadsto a \cdot \left(27 \cdot b\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 35.7

          \[\leadsto a \cdot \left(27 \cdot b\right) \]

   12. Applied rewrites 35.7%

       \[\leadsto a \cdot \left(27 \cdot b\right) \]

   if -4.0000000000000003e172 < (*.f64 (*.f64 a #s(literal 27 binary64)) b) < 5.0000000000000001e94

   1. Initial program 95.7%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)} + \left(a \cdot 27\right) \cdot b \]

      3. associate-+l- N/A

         \[\leadsto \color{blue}{x \cdot 2 - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot 2} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{2 \cdot x} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      6. count-2-rev N/A

         \[\leadsto \color{blue}{\left(x + x\right)} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      7. associate--l+ N/A

         \[\leadsto \color{blue}{x + \left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto \color{blue}{\left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)\right) + x} \]

      9. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)\right) + x} \]

   3. Applied rewrites 96.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-9 \cdot t, z \cdot y, \mathsf{fma}\left(b \cdot a, 27, x\right)\right) + x} \]
   4. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\left(-9 \cdot t\right) \cdot \left(z \cdot y\right) + \mathsf{fma}\left(b \cdot a, 27, x\right)\right)} + x \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(\left(-9 \cdot t\right) \cdot \left(z \cdot y\right) - \left(\mathsf{neg}\left(\mathsf{fma}\left(b \cdot a, 27, x\right)\right)\right)\right)} + x \]

      3. sub-flip N/A

         \[\leadsto \color{blue}{\left(\left(-9 \cdot t\right) \cdot \left(z \cdot y\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{fma}\left(b \cdot a, 27, x\right)\right)\right)\right)\right)\right)} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(-9 \cdot t\right) \cdot \color{blue}{\left(z \cdot y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{fma}\left(b \cdot a, 27, x\right)\right)\right)\right)\right)\right) + x \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\left(-9 \cdot t\right) \cdot z\right) \cdot y} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{fma}\left(b \cdot a, 27, x\right)\right)\right)\right)\right)\right) + x \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(-9 \cdot t\right)} \cdot z\right) \cdot y + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{fma}\left(b \cdot a, 27, x\right)\right)\right)\right)\right)\right) + x \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-9 \cdot \left(t \cdot z\right)\right)} \cdot y + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{fma}\left(b \cdot a, 27, x\right)\right)\right)\right)\right)\right) + x \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(-9 \cdot \color{blue}{\left(t \cdot z\right)}\right) \cdot y + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{fma}\left(b \cdot a, 27, x\right)\right)\right)\right)\right)\right) + x \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(-9 \cdot \left(t \cdot z\right)\right)} \cdot y + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{fma}\left(b \cdot a, 27, x\right)\right)\right)\right)\right)\right) + x \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(-9 \cdot \left(t \cdot z\right)\right)} \cdot y + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{fma}\left(b \cdot a, 27, x\right)\right)\right)\right)\right)\right) + x \]

      11. associate-*l* N/A

          \[\leadsto \left(\color{blue}{-9 \cdot \left(\left(t \cdot z\right) \cdot y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{fma}\left(b \cdot a, 27, x\right)\right)\right)\right)\right)\right) + x \]

      12. lift-*.f64 N/A

          \[\leadsto \left(-9 \cdot \color{blue}{\left(\left(t \cdot z\right) \cdot y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{fma}\left(b \cdot a, 27, x\right)\right)\right)\right)\right)\right) + x \]

      13. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\left(t \cdot z\right) \cdot y\right) \cdot -9} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{fma}\left(b \cdot a, 27, x\right)\right)\right)\right)\right)\right) + x \]

      14. remove-double-neg N/A

          \[\leadsto \left(\left(\left(t \cdot z\right) \cdot y\right) \cdot -9 + \color{blue}{\mathsf{fma}\left(b \cdot a, 27, x\right)}\right) + x \]

      15. lower-fma.f64 95.7

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(t \cdot z\right) \cdot y, -9, \mathsf{fma}\left(b \cdot a, 27, x\right)\right)} + x \]

      16. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(t \cdot z\right)} \cdot y, -9, \mathsf{fma}\left(b \cdot a, 27, x\right)\right) + x \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(z \cdot t\right)} \cdot y, -9, \mathsf{fma}\left(b \cdot a, 27, x\right)\right) + x \]

      18. lower-*.f64 95.7

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(z \cdot t\right)} \cdot y, -9, \mathsf{fma}\left(b \cdot a, 27, x\right)\right) + x \]

      19. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(z \cdot t\right) \cdot y, -9, \color{blue}{\left(b \cdot a\right) \cdot 27 + x}\right) + x \]

      20. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(z \cdot t\right) \cdot y, -9, \color{blue}{27 \cdot \left(b \cdot a\right)} + x\right) + x \]

      21. lower-fma.f64 95.7

          \[\leadsto \mathsf{fma}\left(\left(z \cdot t\right) \cdot y, -9, \color{blue}{\mathsf{fma}\left(27, b \cdot a, x\right)}\right) + x \]

      22. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(z \cdot t\right) \cdot y, -9, \mathsf{fma}\left(27, \color{blue}{b \cdot a}, x\right)\right) + x \]

      23. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(z \cdot t\right) \cdot y, -9, \mathsf{fma}\left(27, \color{blue}{a \cdot b}, x\right)\right) + x \]

      24. lift-*.f64 95.7

          \[\leadsto \mathsf{fma}\left(\left(z \cdot t\right) \cdot y, -9, \mathsf{fma}\left(27, \color{blue}{a \cdot b}, x\right)\right) + x \]

   5. Applied rewrites 95.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(z \cdot t\right) \cdot y, -9, \mathsf{fma}\left(27, a \cdot b, x\right)\right)} + x \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot x} \]
   7. Step-by-step derivation

      1. lower-*.f64 31.3

         \[\leadsto 2 \cdot \color{blue}{x} \]

   8. Applied rewrites 31.3%

      \[\leadsto \color{blue}{2 \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 31.3% accurate, 6.1× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ 2 \cdot x \end{array} \]

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 (* 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 2.0 * 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 = 2.0d0 * 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 2.0 * x;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	return 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 Float64(2.0 * 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 = 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[(2.0 * x), $MachinePrecision]

Derivation

1. Initial program 95.7%

   \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b} \]

   2. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)} + \left(a \cdot 27\right) \cdot b \]

   3. associate-+l- N/A

      \[\leadsto \color{blue}{x \cdot 2 - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot 2} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{2 \cdot x} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

   6. count-2-rev N/A

      \[\leadsto \color{blue}{\left(x + x\right)} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

   7. associate--l+ N/A

      \[\leadsto \color{blue}{x + \left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)\right)} \]

   8. +-commutative N/A

      \[\leadsto \color{blue}{\left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)\right) + x} \]

   9. lower-+.f64 N/A

      \[\leadsto \color{blue}{\left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)\right) + x} \]

3. Applied rewrites 96.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-9 \cdot t, z \cdot y, \mathsf{fma}\left(b \cdot a, 27, x\right)\right) + x} \]
4. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto \color{blue}{\left(\left(-9 \cdot t\right) \cdot \left(z \cdot y\right) + \mathsf{fma}\left(b \cdot a, 27, x\right)\right)} + x \]

   2. add-flip N/A

      \[\leadsto \color{blue}{\left(\left(-9 \cdot t\right) \cdot \left(z \cdot y\right) - \left(\mathsf{neg}\left(\mathsf{fma}\left(b \cdot a, 27, x\right)\right)\right)\right)} + x \]

   3. sub-flip N/A

      \[\leadsto \color{blue}{\left(\left(-9 \cdot t\right) \cdot \left(z \cdot y\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{fma}\left(b \cdot a, 27, x\right)\right)\right)\right)\right)\right)} + x \]

   4. lift-*.f64 N/A

      \[\leadsto \left(\left(-9 \cdot t\right) \cdot \color{blue}{\left(z \cdot y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{fma}\left(b \cdot a, 27, x\right)\right)\right)\right)\right)\right) + x \]

   5. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(\left(-9 \cdot t\right) \cdot z\right) \cdot y} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{fma}\left(b \cdot a, 27, x\right)\right)\right)\right)\right)\right) + x \]

   6. lift-*.f64 N/A

      \[\leadsto \left(\left(\color{blue}{\left(-9 \cdot t\right)} \cdot z\right) \cdot y + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{fma}\left(b \cdot a, 27, x\right)\right)\right)\right)\right)\right) + x \]

   7. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(-9 \cdot \left(t \cdot z\right)\right)} \cdot y + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{fma}\left(b \cdot a, 27, x\right)\right)\right)\right)\right)\right) + x \]

   8. lift-*.f64 N/A

      \[\leadsto \left(\left(-9 \cdot \color{blue}{\left(t \cdot z\right)}\right) \cdot y + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{fma}\left(b \cdot a, 27, x\right)\right)\right)\right)\right)\right) + x \]

   9. lift-*.f64 N/A

      \[\leadsto \left(\color{blue}{\left(-9 \cdot \left(t \cdot z\right)\right)} \cdot y + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{fma}\left(b \cdot a, 27, x\right)\right)\right)\right)\right)\right) + x \]

   10. lift-*.f64 N/A

       \[\leadsto \left(\color{blue}{\left(-9 \cdot \left(t \cdot z\right)\right)} \cdot y + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{fma}\left(b \cdot a, 27, x\right)\right)\right)\right)\right)\right) + x \]

   11. associate-*l* N/A

       \[\leadsto \left(\color{blue}{-9 \cdot \left(\left(t \cdot z\right) \cdot y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{fma}\left(b \cdot a, 27, x\right)\right)\right)\right)\right)\right) + x \]

   12. lift-*.f64 N/A

       \[\leadsto \left(-9 \cdot \color{blue}{\left(\left(t \cdot z\right) \cdot y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{fma}\left(b \cdot a, 27, x\right)\right)\right)\right)\right)\right) + x \]

   13. *-commutative N/A

       \[\leadsto \left(\color{blue}{\left(\left(t \cdot z\right) \cdot y\right) \cdot -9} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{fma}\left(b \cdot a, 27, x\right)\right)\right)\right)\right)\right) + x \]

   14. remove-double-neg N/A

       \[\leadsto \left(\left(\left(t \cdot z\right) \cdot y\right) \cdot -9 + \color{blue}{\mathsf{fma}\left(b \cdot a, 27, x\right)}\right) + x \]

   15. lower-fma.f64 95.7

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(t \cdot z\right) \cdot y, -9, \mathsf{fma}\left(b \cdot a, 27, x\right)\right)} + x \]

   16. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(t \cdot z\right)} \cdot y, -9, \mathsf{fma}\left(b \cdot a, 27, x\right)\right) + x \]

   17. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(z \cdot t\right)} \cdot y, -9, \mathsf{fma}\left(b \cdot a, 27, x\right)\right) + x \]

   18. lower-*.f64 95.7

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(z \cdot t\right)} \cdot y, -9, \mathsf{fma}\left(b \cdot a, 27, x\right)\right) + x \]

   19. lift-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(z \cdot t\right) \cdot y, -9, \color{blue}{\left(b \cdot a\right) \cdot 27 + x}\right) + x \]

   20. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\left(z \cdot t\right) \cdot y, -9, \color{blue}{27 \cdot \left(b \cdot a\right)} + x\right) + x \]

   21. lower-fma.f64 95.7

       \[\leadsto \mathsf{fma}\left(\left(z \cdot t\right) \cdot y, -9, \color{blue}{\mathsf{fma}\left(27, b \cdot a, x\right)}\right) + x \]

   22. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(z \cdot t\right) \cdot y, -9, \mathsf{fma}\left(27, \color{blue}{b \cdot a}, x\right)\right) + x \]

   23. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\left(z \cdot t\right) \cdot y, -9, \mathsf{fma}\left(27, \color{blue}{a \cdot b}, x\right)\right) + x \]

   24. lift-*.f64 95.7

       \[\leadsto \mathsf{fma}\left(\left(z \cdot t\right) \cdot y, -9, \mathsf{fma}\left(27, \color{blue}{a \cdot b}, x\right)\right) + x \]

5. Applied rewrites 95.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(z \cdot t\right) \cdot y, -9, \mathsf{fma}\left(27, a \cdot b, x\right)\right)} + x \]

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{2 \cdot x} \]
7. Step-by-step derivation

   1. lower-*.f64 31.3

      \[\leadsto 2 \cdot \color{blue}{x} \]

8. Applied rewrites 31.3%

   \[\leadsto \color{blue}{2 \cdot x} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025156 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, A"
  :precision binary64
  (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* (* a 27.0) b)))