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

Percentage Accurate: 95.3% → 98.5%

Time: 18.5s

Alternatives: 16

Speedup: 0.9×

• 27.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a, b)
    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: 95.3% 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

real(8) function code(x, y, z, t, a, b)
    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: 98.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{-76}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, y \cdot -9, \mathsf{fma}\left(27 \cdot b, a, x \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(27 \cdot b, a, \mathsf{fma}\left(\left(y \cdot -9\right) \cdot t, z, x \cdot 2\right)\right)\\ \end{array} \end{array} \]

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 5e-76)
   (fma (* t z) (* y -9.0) (fma (* 27.0 b) a (* x 2.0)))
   (fma (* 27.0 b) a (fma (* (* y -9.0) t) z (* x 2.0)))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= 5e-76) {
		tmp = fma((t * z), (y * -9.0), fma((27.0 * b), a, (x * 2.0)));
	} else {
		tmp = fma((27.0 * b), a, fma(((y * -9.0) * t), z, (x * 2.0)));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= 5e-76)
		tmp = fma(Float64(t * z), Float64(y * -9.0), fma(Float64(27.0 * b), a, Float64(x * 2.0)));
	else
		tmp = fma(Float64(27.0 * b), a, fma(Float64(Float64(y * -9.0) * t), z, Float64(x * 2.0)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, 5e-76], N[(N[(t * z), $MachinePrecision] * N[(y * -9.0), $MachinePrecision] + N[(N[(27.0 * b), $MachinePrecision] * a + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(27.0 * b), $MachinePrecision] * a + N[(N[(N[(y * -9.0), $MachinePrecision] * t), $MachinePrecision] * z + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.9999999999999998e-76

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. 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. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

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

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

      18. lower-*.f64 N/A

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

      19. metadata-eval N/A

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

   4. Applied rewrites 94.3%

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

   if 4.9999999999999998e-76 < z

   1. Initial program 97.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 98.5

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

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

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

      17. *-commutative N/A

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

      18. associate-*r* N/A

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

      19. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{-76}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, y \cdot -9, \mathsf{fma}\left(27 \cdot b, a, x \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(27 \cdot b, a, \mathsf{fma}\left(\left(y \cdot -9\right) \cdot t, z, x \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 56.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := x \cdot 2 - \left(\left(9 \cdot y\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+306}:\\ \;\;\;\;\left(\left(-9 \cdot t\right) \cdot z\right) \cdot y\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-60}:\\ \;\;\;\;x \cdot 2\\ \mathbf{elif}\;t\_1 \leq 10^{+200}:\\ \;\;\;\;\left(a \cdot b\right) \cdot 27\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+287}:\\ \;\;\;\;x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(-9 \cdot \left(t \cdot z\right)\right) \cdot y\\ \end{array} \end{array} \]

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 (- (* x 2.0) (* (* (* 9.0 y) z) t))))
   (if (<= t_1 -2e+306)
     (* (* (* -9.0 t) z) y)
     (if (<= t_1 -1e-60)
       (* x 2.0)
       (if (<= t_1 1e+200)
         (* (* a b) 27.0)
         (if (<= t_1 2e+287) (* x 2.0) (* (* -9.0 (* t z)) y)))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * 2.0) - (((9.0 * y) * z) * t);
	double tmp;
	if (t_1 <= -2e+306) {
		tmp = ((-9.0 * t) * z) * y;
	} else if (t_1 <= -1e-60) {
		tmp = x * 2.0;
	} else if (t_1 <= 1e+200) {
		tmp = (a * b) * 27.0;
	} else if (t_1 <= 2e+287) {
		tmp = x * 2.0;
	} else {
		tmp = (-9.0 * (t * z)) * y;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * 2.0d0) - (((9.0d0 * y) * z) * t)
    if (t_1 <= (-2d+306)) then
        tmp = (((-9.0d0) * t) * z) * y
    else if (t_1 <= (-1d-60)) then
        tmp = x * 2.0d0
    else if (t_1 <= 1d+200) then
        tmp = (a * b) * 27.0d0
    else if (t_1 <= 2d+287) then
        tmp = x * 2.0d0
    else
        tmp = ((-9.0d0) * (t * z)) * y
    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 = (x * 2.0) - (((9.0 * y) * z) * t);
	double tmp;
	if (t_1 <= -2e+306) {
		tmp = ((-9.0 * t) * z) * y;
	} else if (t_1 <= -1e-60) {
		tmp = x * 2.0;
	} else if (t_1 <= 1e+200) {
		tmp = (a * b) * 27.0;
	} else if (t_1 <= 2e+287) {
		tmp = x * 2.0;
	} else {
		tmp = (-9.0 * (t * z)) * y;
	}
	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 = (x * 2.0) - (((9.0 * y) * z) * t)
	tmp = 0
	if t_1 <= -2e+306:
		tmp = ((-9.0 * t) * z) * y
	elif t_1 <= -1e-60:
		tmp = x * 2.0
	elif t_1 <= 1e+200:
		tmp = (a * b) * 27.0
	elif t_1 <= 2e+287:
		tmp = x * 2.0
	else:
		tmp = (-9.0 * (t * z)) * y
	return tmp

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * 2.0) - Float64(Float64(Float64(9.0 * y) * z) * t))
	tmp = 0.0
	if (t_1 <= -2e+306)
		tmp = Float64(Float64(Float64(-9.0 * t) * z) * y);
	elseif (t_1 <= -1e-60)
		tmp = Float64(x * 2.0);
	elseif (t_1 <= 1e+200)
		tmp = Float64(Float64(a * b) * 27.0);
	elseif (t_1 <= 2e+287)
		tmp = Float64(x * 2.0);
	else
		tmp = Float64(Float64(-9.0 * Float64(t * z)) * y);
	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 = (x * 2.0) - (((9.0 * y) * z) * t);
	tmp = 0.0;
	if (t_1 <= -2e+306)
		tmp = ((-9.0 * t) * z) * y;
	elseif (t_1 <= -1e-60)
		tmp = x * 2.0;
	elseif (t_1 <= 1e+200)
		tmp = (a * b) * 27.0;
	elseif (t_1 <= 2e+287)
		tmp = x * 2.0;
	else
		tmp = (-9.0 * (t * z)) * y;
	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[(x * 2.0), $MachinePrecision] - N[(N[(N[(9.0 * y), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+306], N[(N[(N[(-9.0 * t), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$1, -1e-60], N[(x * 2.0), $MachinePrecision], If[LessEqual[t$95$1, 1e+200], N[(N[(a * b), $MachinePrecision] * 27.0), $MachinePrecision], If[LessEqual[t$95$1, 2e+287], N[(x * 2.0), $MachinePrecision], N[(N[(-9.0 * N[(t * z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 93.9%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 94.2%

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

   6. Taylor expanded in y around inf

      \[\leadsto \left(-9 \cdot \left(t \cdot z\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 94.2%

      \[\leadsto \left(\left(z \cdot t\right) \cdot -9\right) \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 94.2%

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

   if -2.00000000000000003e306 < (-.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)) < -9.9999999999999997e-61 or 9.9999999999999997e199 < (-.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)) < 2.0000000000000002e287

   1. Initial program 99.8%

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

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.8

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l* N/A

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

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

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

      18. *-commutative N/A

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

      19. associate-*r* N/A

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

      20. lower-fma.f64 N/A

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

   4. Applied rewrites 93.7%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.8

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 51.0

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

   9. Applied rewrites 51.0%

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

   if -9.9999999999999997e-61 < (-.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)) < 9.9999999999999997e199

   1. Initial program 98.6%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 84.1

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

   5. Applied rewrites 84.1%

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

   7. Applied rewrites 84.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 62.2%

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

   if 2.0000000000000002e287 < (-.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t))

   1. Initial program 79.7%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 94.1%

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

   6. Taylor expanded in y around inf

      \[\leadsto \left(-9 \cdot \left(t \cdot z\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 82.4%

      \[\leadsto \left(\left(z \cdot t\right) \cdot -9\right) \cdot y \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 - \left(\left(9 \cdot y\right) \cdot z\right) \cdot t \leq -2 \cdot 10^{+306}:\\ \;\;\;\;\left(\left(-9 \cdot t\right) \cdot z\right) \cdot y\\ \mathbf{elif}\;x \cdot 2 - \left(\left(9 \cdot y\right) \cdot z\right) \cdot t \leq -1 \cdot 10^{-60}:\\ \;\;\;\;x \cdot 2\\ \mathbf{elif}\;x \cdot 2 - \left(\left(9 \cdot y\right) \cdot z\right) \cdot t \leq 10^{+200}:\\ \;\;\;\;\left(a \cdot b\right) \cdot 27\\ \mathbf{elif}\;x \cdot 2 - \left(\left(9 \cdot y\right) \cdot z\right) \cdot t \leq 2 \cdot 10^{+287}:\\ \;\;\;\;x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(-9 \cdot \left(t \cdot z\right)\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(-9 \cdot t\right) \cdot z\right) \cdot y\\ t_2 := x \cdot 2 - \left(\left(9 \cdot y\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+306}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-60}:\\ \;\;\;\;x \cdot 2\\ \mathbf{elif}\;t\_2 \leq 10^{+200}:\\ \;\;\;\;\left(a \cdot b\right) \cdot 27\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+287}:\\ \;\;\;\;x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* -9.0 t) z) y)) (t_2 (- (* x 2.0) (* (* (* 9.0 y) z) t))))
   (if (<= t_2 -2e+306)
     t_1
     (if (<= t_2 -1e-60)
       (* x 2.0)
       (if (<= t_2 1e+200)
         (* (* a b) 27.0)
         (if (<= t_2 2e+287) (* x 2.0) t_1))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((-9.0 * t) * z) * y;
	double t_2 = (x * 2.0) - (((9.0 * y) * z) * t);
	double tmp;
	if (t_2 <= -2e+306) {
		tmp = t_1;
	} else if (t_2 <= -1e-60) {
		tmp = x * 2.0;
	} else if (t_2 <= 1e+200) {
		tmp = (a * b) * 27.0;
	} else if (t_2 <= 2e+287) {
		tmp = x * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b)
    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 = (((-9.0d0) * t) * z) * y
    t_2 = (x * 2.0d0) - (((9.0d0 * y) * z) * t)
    if (t_2 <= (-2d+306)) then
        tmp = t_1
    else if (t_2 <= (-1d-60)) then
        tmp = x * 2.0d0
    else if (t_2 <= 1d+200) then
        tmp = (a * b) * 27.0d0
    else if (t_2 <= 2d+287) then
        tmp = x * 2.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b;
public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((-9.0 * t) * z) * y;
	double t_2 = (x * 2.0) - (((9.0 * y) * z) * t);
	double tmp;
	if (t_2 <= -2e+306) {
		tmp = t_1;
	} else if (t_2 <= -1e-60) {
		tmp = x * 2.0;
	} else if (t_2 <= 1e+200) {
		tmp = (a * b) * 27.0;
	} else if (t_2 <= 2e+287) {
		tmp = x * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	t_1 = ((-9.0 * t) * z) * y
	t_2 = (x * 2.0) - (((9.0 * y) * z) * t)
	tmp = 0
	if t_2 <= -2e+306:
		tmp = t_1
	elif t_2 <= -1e-60:
		tmp = x * 2.0
	elif t_2 <= 1e+200:
		tmp = (a * b) * 27.0
	elif t_2 <= 2e+287:
		tmp = x * 2.0
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(-9.0 * t) * z) * y)
	t_2 = Float64(Float64(x * 2.0) - Float64(Float64(Float64(9.0 * y) * z) * t))
	tmp = 0.0
	if (t_2 <= -2e+306)
		tmp = t_1;
	elseif (t_2 <= -1e-60)
		tmp = Float64(x * 2.0);
	elseif (t_2 <= 1e+200)
		tmp = Float64(Float64(a * b) * 27.0);
	elseif (t_2 <= 2e+287)
		tmp = Float64(x * 2.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp_2 = code(x, y, z, t, a, b)
	t_1 = ((-9.0 * t) * z) * y;
	t_2 = (x * 2.0) - (((9.0 * y) * z) * t);
	tmp = 0.0;
	if (t_2 <= -2e+306)
		tmp = t_1;
	elseif (t_2 <= -1e-60)
		tmp = x * 2.0;
	elseif (t_2 <= 1e+200)
		tmp = (a * b) * 27.0;
	elseif (t_2 <= 2e+287)
		tmp = x * 2.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(-9.0 * t), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * 2.0), $MachinePrecision] - N[(N[(N[(9.0 * y), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+306], t$95$1, If[LessEqual[t$95$2, -1e-60], N[(x * 2.0), $MachinePrecision], If[LessEqual[t$95$2, 1e+200], N[(N[(a * b), $MachinePrecision] * 27.0), $MachinePrecision], If[LessEqual[t$95$2, 2e+287], N[(x * 2.0), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)) < -2.00000000000000003e306 or 2.0000000000000002e287 < (-.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t))

   1. Initial program 86.7%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 94.1%

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

   6. Taylor expanded in y around inf

      \[\leadsto \left(-9 \cdot \left(t \cdot z\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 88.2%

      \[\leadsto \left(\left(z \cdot t\right) \cdot -9\right) \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 88.2%

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

   if -2.00000000000000003e306 < (-.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)) < -9.9999999999999997e-61 or 9.9999999999999997e199 < (-.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)) < 2.0000000000000002e287

   1. Initial program 99.8%

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

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.8

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l* N/A

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

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

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

      18. *-commutative N/A

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

      19. associate-*r* N/A

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

      20. lower-fma.f64 N/A

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

   4. Applied rewrites 93.7%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.8

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 51.0

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

   9. Applied rewrites 51.0%

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

   if -9.9999999999999997e-61 < (-.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)) < 9.9999999999999997e199

   1. Initial program 98.6%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 84.1

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

   5. Applied rewrites 84.1%

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

   7. Applied rewrites 84.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 62.2%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 - \left(\left(9 \cdot y\right) \cdot z\right) \cdot t \leq -2 \cdot 10^{+306}:\\ \;\;\;\;\left(\left(-9 \cdot t\right) \cdot z\right) \cdot y\\ \mathbf{elif}\;x \cdot 2 - \left(\left(9 \cdot y\right) \cdot z\right) \cdot t \leq -1 \cdot 10^{-60}:\\ \;\;\;\;x \cdot 2\\ \mathbf{elif}\;x \cdot 2 - \left(\left(9 \cdot y\right) \cdot z\right) \cdot t \leq 10^{+200}:\\ \;\;\;\;\left(a \cdot b\right) \cdot 27\\ \mathbf{elif}\;x \cdot 2 - \left(\left(9 \cdot y\right) \cdot z\right) \cdot t \leq 2 \cdot 10^{+287}:\\ \;\;\;\;x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-9 \cdot t\right) \cdot z\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\left(y \cdot z\right) \cdot -9, t, x \cdot 2\right)\\ t_2 := \left(\left(9 \cdot y\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+73}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 27, b, x \cdot 2\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+176}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot b, 27, \left(\left(y \cdot -9\right) \cdot t\right) \cdot z\right)\\ \end{array} \end{array} \]

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 (fma (* (* y z) -9.0) t (* x 2.0))) (t_2 (* (* (* 9.0 y) z) t)))
   (if (<= t_2 -5e+73)
     t_1
     (if (<= t_2 4e+18)
       (fma (* a 27.0) b (* x 2.0))
       (if (<= t_2 1e+176) t_1 (fma (* a b) 27.0 (* (* (* y -9.0) t) z)))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(((y * z) * -9.0), t, (x * 2.0));
	double t_2 = ((9.0 * y) * z) * t;
	double tmp;
	if (t_2 <= -5e+73) {
		tmp = t_1;
	} else if (t_2 <= 4e+18) {
		tmp = fma((a * 27.0), b, (x * 2.0));
	} else if (t_2 <= 1e+176) {
		tmp = t_1;
	} else {
		tmp = fma((a * b), 27.0, (((y * -9.0) * t) * z));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = fma(Float64(Float64(y * z) * -9.0), t, Float64(x * 2.0))
	t_2 = Float64(Float64(Float64(9.0 * y) * z) * t)
	tmp = 0.0
	if (t_2 <= -5e+73)
		tmp = t_1;
	elseif (t_2 <= 4e+18)
		tmp = fma(Float64(a * 27.0), b, Float64(x * 2.0));
	elseif (t_2 <= 1e+176)
		tmp = t_1;
	else
		tmp = fma(Float64(a * b), 27.0, Float64(Float64(Float64(y * -9.0) * t) * z));
	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 * z), $MachinePrecision] * -9.0), $MachinePrecision] * t + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(9.0 * y), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+73], t$95$1, If[LessEqual[t$95$2, 4e+18], N[(N[(a * 27.0), $MachinePrecision] * b + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+176], t$95$1, N[(N[(a * b), $MachinePrecision] * 27.0 + N[(N[(N[(y * -9.0), $MachinePrecision] * t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -4.99999999999999976e73 or 4e18 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 1e176

   1. Initial program 91.2%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 74.1

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

   5. Applied rewrites 74.1%

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

   6. Taylor expanded in a around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 79.1

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

   8. Applied rewrites 79.1%

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

   if -4.99999999999999976e73 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 4e18

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 93.9

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

   5. Applied rewrites 93.9%

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

   7. Applied rewrites 93.9%

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

   9. Applied rewrites 93.9%

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

   if 1e176 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 95.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.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

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 95.1

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

   5. Applied rewrites 95.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 95.2

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 95.2

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

   7. Applied rewrites 90.5%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(9 \cdot y\right) \cdot z\right) \cdot t \leq -5 \cdot 10^{+73}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot z\right) \cdot -9, t, x \cdot 2\right)\\ \mathbf{elif}\;\left(\left(9 \cdot y\right) \cdot z\right) \cdot t \leq 4 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 27, b, x \cdot 2\right)\\ \mathbf{elif}\;\left(\left(9 \cdot y\right) \cdot z\right) \cdot t \leq 10^{+176}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot z\right) \cdot -9, t, x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot b, 27, \left(\left(y \cdot -9\right) \cdot t\right) \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t \cdot z, y \cdot -9, \left(a \cdot b\right) \cdot 27\right)\\ t_2 := \left(\left(9 \cdot y\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+199}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 27, b, x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (* t z) (* y -9.0) (* (* a b) 27.0)))
        (t_2 (* (* (* 9.0 y) z) t)))
   (if (<= t_2 -5e+199)
     t_1
     (if (<= t_2 1e+30) (fma (* a 27.0) b (* x 2.0)) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((t * z), (y * -9.0), ((a * b) * 27.0));
	double t_2 = ((9.0 * y) * z) * t;
	double tmp;
	if (t_2 <= -5e+199) {
		tmp = t_1;
	} else if (t_2 <= 1e+30) {
		tmp = fma((a * 27.0), b, (x * 2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = fma(Float64(t * z), Float64(y * -9.0), Float64(Float64(a * b) * 27.0))
	t_2 = Float64(Float64(Float64(9.0 * y) * z) * t)
	tmp = 0.0
	if (t_2 <= -5e+199)
		tmp = t_1;
	elseif (t_2 <= 1e+30)
		tmp = fma(Float64(a * 27.0), b, Float64(x * 2.0));
	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[(t * z), $MachinePrecision] * N[(y * -9.0), $MachinePrecision] + N[(N[(a * b), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(9.0 * y), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+199], t$95$1, If[LessEqual[t$95$2, 1e+30], N[(N[(a * 27.0), $MachinePrecision] * b + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -4.9999999999999998e199 or 1e30 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 91.9%

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

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

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

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

      18. lower-*.f64 N/A

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

      19. metadata-eval N/A

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

   4. Applied rewrites 88.5%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 82.7

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

   7. Applied rewrites 82.7%

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

   if -4.9999999999999998e199 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 1e30

   1. Initial program 99.2%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 90.2

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

   5. Applied rewrites 90.2%

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

   7. Applied rewrites 90.2%

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

   9. Applied rewrites 90.2%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(9 \cdot y\right) \cdot z\right) \cdot t \leq -5 \cdot 10^{+199}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, y \cdot -9, \left(a \cdot b\right) \cdot 27\right)\\ \mathbf{elif}\;\left(\left(9 \cdot y\right) \cdot z\right) \cdot t \leq 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 27, b, x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, y \cdot -9, \left(a \cdot b\right) \cdot 27\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(9 \cdot y\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+73}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot z\right) \cdot -9, t, x \cdot 2\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 27, b, x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot b, 27, \left(\left(-9 \cdot z\right) \cdot y\right) \cdot t\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* 9.0 y) z) t)))
   (if (<= t_1 -5e+73)
     (fma (* (* y z) -9.0) t (* x 2.0))
     (if (<= t_1 1e+30)
       (fma (* a 27.0) b (* x 2.0))
       (fma (* a b) 27.0 (* (* (* -9.0 z) y) t))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((9.0 * y) * z) * t;
	double tmp;
	if (t_1 <= -5e+73) {
		tmp = fma(((y * z) * -9.0), t, (x * 2.0));
	} else if (t_1 <= 1e+30) {
		tmp = fma((a * 27.0), b, (x * 2.0));
	} else {
		tmp = fma((a * b), 27.0, (((-9.0 * z) * y) * t));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(9.0 * y) * z) * t)
	tmp = 0.0
	if (t_1 <= -5e+73)
		tmp = fma(Float64(Float64(y * z) * -9.0), t, Float64(x * 2.0));
	elseif (t_1 <= 1e+30)
		tmp = fma(Float64(a * 27.0), b, Float64(x * 2.0));
	else
		tmp = fma(Float64(a * b), 27.0, Float64(Float64(Float64(-9.0 * z) * y) * t));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(9.0 * y), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+73], N[(N[(N[(y * z), $MachinePrecision] * -9.0), $MachinePrecision] * t + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+30], N[(N[(a * 27.0), $MachinePrecision] * b + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * b), $MachinePrecision] * 27.0 + N[(N[(N[(-9.0 * z), $MachinePrecision] * y), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -4.99999999999999976e73

   1. Initial program 88.7%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 77.8

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

   5. Applied rewrites 77.8%

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

   6. Taylor expanded in a around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 77.4

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

   8. Applied rewrites 77.4%

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

   if -4.99999999999999976e73 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 1e30

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 94.0

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

   5. Applied rewrites 94.0%

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

   7. Applied rewrites 94.0%

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

   9. Applied rewrites 94.0%

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

   if 1e30 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 96.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.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

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 89.0

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

   5. Applied rewrites 89.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 89.0

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 89.0

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

   7. Applied rewrites 85.5%

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

   9. Applied rewrites 89.0%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(9 \cdot y\right) \cdot z\right) \cdot t \leq -5 \cdot 10^{+73}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot z\right) \cdot -9, t, x \cdot 2\right)\\ \mathbf{elif}\;\left(\left(9 \cdot y\right) \cdot z\right) \cdot t \leq 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 27, b, x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot b, 27, \left(\left(-9 \cdot z\right) \cdot y\right) \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(9 \cdot y\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+73}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot z\right) \cdot -9, t, x \cdot 2\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 27, b, x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-9, \left(y \cdot z\right) \cdot t, \left(a \cdot b\right) \cdot 27\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* 9.0 y) z) t)))
   (if (<= t_1 -5e+73)
     (fma (* (* y z) -9.0) t (* x 2.0))
     (if (<= t_1 1e+30)
       (fma (* a 27.0) b (* x 2.0))
       (fma -9.0 (* (* y z) t) (* (* a b) 27.0))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((9.0 * y) * z) * t;
	double tmp;
	if (t_1 <= -5e+73) {
		tmp = fma(((y * z) * -9.0), t, (x * 2.0));
	} else if (t_1 <= 1e+30) {
		tmp = fma((a * 27.0), b, (x * 2.0));
	} else {
		tmp = fma(-9.0, ((y * z) * t), ((a * b) * 27.0));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(9.0 * y) * z) * t)
	tmp = 0.0
	if (t_1 <= -5e+73)
		tmp = fma(Float64(Float64(y * z) * -9.0), t, Float64(x * 2.0));
	elseif (t_1 <= 1e+30)
		tmp = fma(Float64(a * 27.0), b, Float64(x * 2.0));
	else
		tmp = fma(-9.0, Float64(Float64(y * z) * t), Float64(Float64(a * b) * 27.0));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(9.0 * y), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+73], N[(N[(N[(y * z), $MachinePrecision] * -9.0), $MachinePrecision] * t + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+30], N[(N[(a * 27.0), $MachinePrecision] * b + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], N[(-9.0 * N[(N[(y * z), $MachinePrecision] * t), $MachinePrecision] + N[(N[(a * b), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -4.99999999999999976e73

   1. Initial program 88.7%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 77.8

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

   5. Applied rewrites 77.8%

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

   6. Taylor expanded in a around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 77.4

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

   8. Applied rewrites 77.4%

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

   if -4.99999999999999976e73 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 1e30

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 94.0

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

   5. Applied rewrites 94.0%

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

   7. Applied rewrites 94.0%

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

   9. Applied rewrites 94.0%

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

   if 1e30 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 96.8%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 89.0

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

   5. Applied rewrites 89.0%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(9 \cdot y\right) \cdot z\right) \cdot t \leq -5 \cdot 10^{+73}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot z\right) \cdot -9, t, x \cdot 2\right)\\ \mathbf{elif}\;\left(\left(9 \cdot y\right) \cdot z\right) \cdot t \leq 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 27, b, x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-9, \left(y \cdot z\right) \cdot t, \left(a \cdot b\right) \cdot 27\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\left(y \cdot z\right) \cdot -9, t, x \cdot 2\right)\\ t_2 := \left(\left(9 \cdot y\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+73}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 27, b, x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (* (* y z) -9.0) t (* x 2.0))) (t_2 (* (* (* 9.0 y) z) t)))
   (if (<= t_2 -5e+73)
     t_1
     (if (<= t_2 4e+18) (fma (* a 27.0) b (* x 2.0)) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(((y * z) * -9.0), t, (x * 2.0));
	double t_2 = ((9.0 * y) * z) * t;
	double tmp;
	if (t_2 <= -5e+73) {
		tmp = t_1;
	} else if (t_2 <= 4e+18) {
		tmp = fma((a * 27.0), b, (x * 2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = fma(Float64(Float64(y * z) * -9.0), t, Float64(x * 2.0))
	t_2 = Float64(Float64(Float64(9.0 * y) * z) * t)
	tmp = 0.0
	if (t_2 <= -5e+73)
		tmp = t_1;
	elseif (t_2 <= 4e+18)
		tmp = fma(Float64(a * 27.0), b, Float64(x * 2.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y * z), $MachinePrecision] * -9.0), $MachinePrecision] * t + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(9.0 * y), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+73], t$95$1, If[LessEqual[t$95$2, 4e+18], N[(N[(a * 27.0), $MachinePrecision] * b + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -4.99999999999999976e73 or 4e18 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 93.1%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 82.4

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

   5. Applied rewrites 82.4%

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

   6. Taylor expanded in a around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 83.0

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

   8. Applied rewrites 83.0%

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

   if -4.99999999999999976e73 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 4e18

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 93.9

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

   5. Applied rewrites 93.9%

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

   7. Applied rewrites 93.9%

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

   9. Applied rewrites 93.9%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(9 \cdot y\right) \cdot z\right) \cdot t \leq -5 \cdot 10^{+73}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot z\right) \cdot -9, t, x \cdot 2\right)\\ \mathbf{elif}\;\left(\left(9 \cdot y\right) \cdot z\right) \cdot t \leq 4 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 27, b, x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot z\right) \cdot -9, t, x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(9 \cdot y\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+199}:\\ \;\;\;\;\left(-9 \cdot \left(t \cdot z\right)\right) \cdot y\\ \mathbf{elif}\;t\_1 \leq 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 27, b, x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-9 \cdot t\right) \cdot z\right) \cdot y\\ \end{array} \end{array} \]

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 y) z) t)))
   (if (<= t_1 -5e+199)
     (* (* -9.0 (* t z)) y)
     (if (<= t_1 1e+30) (fma (* a 27.0) b (* x 2.0)) (* (* (* -9.0 t) z) y)))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((9.0 * y) * z) * t;
	double tmp;
	if (t_1 <= -5e+199) {
		tmp = (-9.0 * (t * z)) * y;
	} else if (t_1 <= 1e+30) {
		tmp = fma((a * 27.0), b, (x * 2.0));
	} else {
		tmp = ((-9.0 * t) * z) * y;
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(9.0 * y) * z) * t)
	tmp = 0.0
	if (t_1 <= -5e+199)
		tmp = Float64(Float64(-9.0 * Float64(t * z)) * y);
	elseif (t_1 <= 1e+30)
		tmp = fma(Float64(a * 27.0), b, Float64(x * 2.0));
	else
		tmp = Float64(Float64(Float64(-9.0 * t) * z) * y);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(9.0 * y), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+199], N[(N[(-9.0 * N[(t * z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$1, 1e+30], N[(N[(a * 27.0), $MachinePrecision] * b + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-9.0 * t), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -4.9999999999999998e199

   1. Initial program 84.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 84.0%

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

   6. Taylor expanded in y around inf

      \[\leadsto \left(-9 \cdot \left(t \cdot z\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 77.4%

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

   if -4.9999999999999998e199 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 1e30

   1. Initial program 99.2%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 90.2

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

   5. Applied rewrites 90.2%

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

   7. Applied rewrites 90.2%

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

   9. Applied rewrites 90.2%

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

   if 1e30 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 96.8%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 87.1%

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

   6. Taylor expanded in y around inf

      \[\leadsto \left(-9 \cdot \left(t \cdot z\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 75.2%

      \[\leadsto \left(\left(z \cdot t\right) \cdot -9\right) \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 75.3%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(9 \cdot y\right) \cdot z\right) \cdot t \leq -5 \cdot 10^{+199}:\\ \;\;\;\;\left(-9 \cdot \left(t \cdot z\right)\right) \cdot y\\ \mathbf{elif}\;\left(\left(9 \cdot y\right) \cdot z\right) \cdot t \leq 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 27, b, x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-9 \cdot t\right) \cdot z\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(9 \cdot y\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+199}:\\ \;\;\;\;\left(-9 \cdot \left(t \cdot z\right)\right) \cdot y\\ \mathbf{elif}\;t\_1 \leq 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(a \cdot 27\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-9 \cdot t\right) \cdot z\right) \cdot y\\ \end{array} \end{array} \]

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 y) z) t)))
   (if (<= t_1 -5e+199)
     (* (* -9.0 (* t z)) y)
     (if (<= t_1 1e+30) (fma 2.0 x (* (* a 27.0) b)) (* (* (* -9.0 t) z) y)))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((9.0 * y) * z) * t;
	double tmp;
	if (t_1 <= -5e+199) {
		tmp = (-9.0 * (t * z)) * y;
	} else if (t_1 <= 1e+30) {
		tmp = fma(2.0, x, ((a * 27.0) * b));
	} else {
		tmp = ((-9.0 * t) * z) * y;
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(9.0 * y) * z) * t)
	tmp = 0.0
	if (t_1 <= -5e+199)
		tmp = Float64(Float64(-9.0 * Float64(t * z)) * y);
	elseif (t_1 <= 1e+30)
		tmp = fma(2.0, x, Float64(Float64(a * 27.0) * b));
	else
		tmp = Float64(Float64(Float64(-9.0 * t) * z) * y);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(9.0 * y), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+199], N[(N[(-9.0 * N[(t * z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$1, 1e+30], N[(2.0 * x + N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-9.0 * t), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -4.9999999999999998e199

   1. Initial program 84.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 84.0%

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

   6. Taylor expanded in y around inf

      \[\leadsto \left(-9 \cdot \left(t \cdot z\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 77.4%

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

   if -4.9999999999999998e199 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 1e30

   1. Initial program 99.2%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 90.2

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

   5. Applied rewrites 90.2%

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

   7. Applied rewrites 90.2%

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

   if 1e30 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 96.8%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 87.1%

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

   6. Taylor expanded in y around inf

      \[\leadsto \left(-9 \cdot \left(t \cdot z\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 75.2%

      \[\leadsto \left(\left(z \cdot t\right) \cdot -9\right) \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 75.3%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(9 \cdot y\right) \cdot z\right) \cdot t \leq -5 \cdot 10^{+199}:\\ \;\;\;\;\left(-9 \cdot \left(t \cdot z\right)\right) \cdot y\\ \mathbf{elif}\;\left(\left(9 \cdot y\right) \cdot z\right) \cdot t \leq 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(a \cdot 27\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-9 \cdot t\right) \cdot z\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(9 \cdot y\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+199}:\\ \;\;\;\;\left(-9 \cdot \left(t \cdot z\right)\right) \cdot y\\ \mathbf{elif}\;t\_1 \leq 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(2, x, a \cdot \left(27 \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-9 \cdot t\right) \cdot z\right) \cdot y\\ \end{array} \end{array} \]

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 y) z) t)))
   (if (<= t_1 -5e+199)
     (* (* -9.0 (* t z)) y)
     (if (<= t_1 1e+30) (fma 2.0 x (* a (* 27.0 b))) (* (* (* -9.0 t) z) y)))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((9.0 * y) * z) * t;
	double tmp;
	if (t_1 <= -5e+199) {
		tmp = (-9.0 * (t * z)) * y;
	} else if (t_1 <= 1e+30) {
		tmp = fma(2.0, x, (a * (27.0 * b)));
	} else {
		tmp = ((-9.0 * t) * z) * y;
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(9.0 * y) * z) * t)
	tmp = 0.0
	if (t_1 <= -5e+199)
		tmp = Float64(Float64(-9.0 * Float64(t * z)) * y);
	elseif (t_1 <= 1e+30)
		tmp = fma(2.0, x, Float64(a * Float64(27.0 * b)));
	else
		tmp = Float64(Float64(Float64(-9.0 * t) * z) * y);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(9.0 * y), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+199], N[(N[(-9.0 * N[(t * z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$1, 1e+30], N[(2.0 * x + N[(a * N[(27.0 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-9.0 * t), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -4.9999999999999998e199

   1. Initial program 84.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 84.0%

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

   6. Taylor expanded in y around inf

      \[\leadsto \left(-9 \cdot \left(t \cdot z\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 77.4%

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

   if -4.9999999999999998e199 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 1e30

   1. Initial program 99.2%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 90.2

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

   5. Applied rewrites 90.2%

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

   7. Applied rewrites 90.2%

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

   if 1e30 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 96.8%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 87.1%

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

   6. Taylor expanded in y around inf

      \[\leadsto \left(-9 \cdot \left(t \cdot z\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 75.2%

      \[\leadsto \left(\left(z \cdot t\right) \cdot -9\right) \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 75.3%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(9 \cdot y\right) \cdot z\right) \cdot t \leq -5 \cdot 10^{+199}:\\ \;\;\;\;\left(-9 \cdot \left(t \cdot z\right)\right) \cdot y\\ \mathbf{elif}\;\left(\left(9 \cdot y\right) \cdot z\right) \cdot t \leq 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(2, x, a \cdot \left(27 \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-9 \cdot t\right) \cdot z\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 93.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;\left(9 \cdot y\right) \cdot z \leq -5 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot z\right) \cdot -9, t, x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot b, 27, \mathsf{fma}\left(\left(y \cdot -9\right) \cdot t, z, x \cdot 2\right)\right)\\ \end{array} \end{array} \]

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 (<= (* (* 9.0 y) z) -5e+47)
   (fma (* (* y z) -9.0) t (* x 2.0))
   (fma (* a b) 27.0 (fma (* (* y -9.0) t) z (* x 2.0)))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((9.0 * y) * z) <= -5e+47) {
		tmp = fma(((y * z) * -9.0), t, (x * 2.0));
	} else {
		tmp = fma((a * b), 27.0, fma(((y * -9.0) * t), z, (x * 2.0)));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(9.0 * y) * z) <= -5e+47)
		tmp = fma(Float64(Float64(y * z) * -9.0), t, Float64(x * 2.0));
	else
		tmp = fma(Float64(a * b), 27.0, fma(Float64(Float64(y * -9.0) * t), z, Float64(x * 2.0)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(9.0 * y), $MachinePrecision] * z), $MachinePrecision], -5e+47], N[(N[(N[(y * z), $MachinePrecision] * -9.0), $MachinePrecision] * t + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * b), $MachinePrecision] * 27.0 + N[(N[(N[(y * -9.0), $MachinePrecision] * t), $MachinePrecision] * z + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y #s(literal 9 binary64)) z) < -5.00000000000000022e47

   1. Initial program 89.8%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 74.9

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

   5. Applied rewrites 74.9%

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

   6. Taylor expanded in a around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 82.7

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

   8. Applied rewrites 82.7%

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

   if -5.00000000000000022e47 < (*.f64 (*.f64 y #s(literal 9 binary64)) z)

   1. Initial program 98.3%

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

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 98.3

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l* N/A

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

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

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

      18. *-commutative N/A

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

      19. associate-*r* N/A

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

      20. lower-fma.f64 N/A

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

   4. Applied rewrites 93.8%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(9 \cdot y\right) \cdot z \leq -5 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot z\right) \cdot -9, t, x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot b, 27, \mathsf{fma}\left(\left(y \cdot -9\right) \cdot t, z, x \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 93.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;\left(9 \cdot y\right) \cdot z \leq -5 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot z\right) \cdot -9, t, x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(27 \cdot b, a, \mathsf{fma}\left(\left(y \cdot -9\right) \cdot t, z, x \cdot 2\right)\right)\\ \end{array} \end{array} \]

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 (<= (* (* 9.0 y) z) -5e+47)
   (fma (* (* y z) -9.0) t (* x 2.0))
   (fma (* 27.0 b) a (fma (* (* y -9.0) t) z (* x 2.0)))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((9.0 * y) * z) <= -5e+47) {
		tmp = fma(((y * z) * -9.0), t, (x * 2.0));
	} else {
		tmp = fma((27.0 * b), a, fma(((y * -9.0) * t), z, (x * 2.0)));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(9.0 * y) * z) <= -5e+47)
		tmp = fma(Float64(Float64(y * z) * -9.0), t, Float64(x * 2.0));
	else
		tmp = fma(Float64(27.0 * b), a, fma(Float64(Float64(y * -9.0) * t), z, Float64(x * 2.0)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(9.0 * y), $MachinePrecision] * z), $MachinePrecision], -5e+47], N[(N[(N[(y * z), $MachinePrecision] * -9.0), $MachinePrecision] * t + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(27.0 * b), $MachinePrecision] * a + N[(N[(N[(y * -9.0), $MachinePrecision] * t), $MachinePrecision] * z + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y #s(literal 9 binary64)) z) < -5.00000000000000022e47

   1. Initial program 89.8%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 74.9

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

   5. Applied rewrites 74.9%

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

   6. Taylor expanded in a around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 82.7

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

   8. Applied rewrites 82.7%

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

   if -5.00000000000000022e47 < (*.f64 (*.f64 y #s(literal 9 binary64)) z)

   1. Initial program 98.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 98.3

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

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

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

      17. *-commutative N/A

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

      18. associate-*r* N/A

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

      19. lower-fma.f64 N/A

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

   4. Applied rewrites 93.8%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(9 \cdot y\right) \cdot z \leq -5 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot z\right) \cdot -9, t, x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(27 \cdot b, a, \mathsf{fma}\left(\left(y \cdot -9\right) \cdot t, z, x \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 52.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(a \cdot 27\right) \cdot b\\ t_2 := \left(a \cdot b\right) \cdot 27\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-5}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+32}:\\ \;\;\;\;x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 b) 27.0)))
   (if (<= t_1 -1e-5) t_2 (if (<= t_1 5e+32) (* x 2.0) 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 * b) * 27.0;
	double tmp;
	if (t_1 <= -1e-5) {
		tmp = t_2;
	} else if (t_1 <= 5e+32) {
		tmp = x * 2.0;
	} 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.
real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (a * 27.0d0) * b
    t_2 = (a * b) * 27.0d0
    if (t_1 <= (-1d-5)) then
        tmp = t_2
    else if (t_1 <= 5d+32) then
        tmp = x * 2.0d0
    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 * b) * 27.0;
	double tmp;
	if (t_1 <= -1e-5) {
		tmp = t_2;
	} else if (t_1 <= 5e+32) {
		tmp = x * 2.0;
	} 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 * b) * 27.0
	tmp = 0
	if t_1 <= -1e-5:
		tmp = t_2
	elif t_1 <= 5e+32:
		tmp = x * 2.0
	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(a * b) * 27.0)
	tmp = 0.0
	if (t_1 <= -1e-5)
		tmp = t_2;
	elseif (t_1 <= 5e+32)
		tmp = Float64(x * 2.0);
	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 * b) * 27.0;
	tmp = 0.0;
	if (t_1 <= -1e-5)
		tmp = t_2;
	elseif (t_1 <= 5e+32)
		tmp = x * 2.0;
	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[(a * b), $MachinePrecision] * 27.0), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-5], t$95$2, If[LessEqual[t$95$1, 5e+32], N[(x * 2.0), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 a #s(literal 27 binary64)) b) < -1.00000000000000008e-5 or 4.9999999999999997e32 < (*.f64 (*.f64 a #s(literal 27 binary64)) b)

   1. Initial program 96.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 64.9

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

   5. Applied rewrites 64.9%

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

   7. Applied rewrites 65.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 54.2%

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

   if -1.00000000000000008e-5 < (*.f64 (*.f64 a #s(literal 27 binary64)) b) < 4.9999999999999997e32

   1. Initial program 95.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 95.4

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l* N/A

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

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

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

      18. *-commutative N/A

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

      19. associate-*r* N/A

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

      20. lower-fma.f64 N/A

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

   4. Applied rewrites 94.7%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 95.4

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 95.4

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

   6. Applied rewrites 95.4%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 49.6

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

   9. Applied rewrites 49.6%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a \cdot 27\right) \cdot b \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\left(a \cdot b\right) \cdot 27\\ \mathbf{elif}\;\left(a \cdot 27\right) \cdot b \leq 5 \cdot 10^{+32}:\\ \;\;\;\;x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot b\right) \cdot 27\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 97.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot b, 27, \mathsf{fma}\left(\left(y \cdot -9\right) \cdot t, z, x \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot b, 27, \mathsf{fma}\left(\left(-9 \cdot z\right) \cdot y, t, x \cdot 2\right)\right)\\ \end{array} \end{array} \]

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 -1.7e+49)
   (fma (* a b) 27.0 (fma (* (* y -9.0) t) z (* x 2.0)))
   (fma (* a b) 27.0 (fma (* (* -9.0 z) y) t (* x 2.0)))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.7e+49) {
		tmp = fma((a * b), 27.0, fma(((y * -9.0) * t), z, (x * 2.0)));
	} else {
		tmp = fma((a * b), 27.0, fma(((-9.0 * z) * y), t, (x * 2.0)));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.7e+49)
		tmp = fma(Float64(a * b), 27.0, fma(Float64(Float64(y * -9.0) * t), z, Float64(x * 2.0)));
	else
		tmp = fma(Float64(a * b), 27.0, fma(Float64(Float64(-9.0 * z) * y), t, Float64(x * 2.0)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.7e+49], N[(N[(a * b), $MachinePrecision] * 27.0 + N[(N[(N[(y * -9.0), $MachinePrecision] * t), $MachinePrecision] * z + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * b), $MachinePrecision] * 27.0 + N[(N[(N[(-9.0 * z), $MachinePrecision] * y), $MachinePrecision] * t + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.7e49

   1. Initial program 90.4%

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

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 90.4

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l* N/A

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

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

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

      18. *-commutative N/A

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

      19. associate-*r* N/A

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

      20. lower-fma.f64 N/A

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

   4. Applied rewrites 95.0%

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

   if -1.7e49 < z

   1. Initial program 97.9%

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

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 97.8

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l* N/A

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

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

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

      18. *-commutative N/A

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

      19. associate-*r* N/A

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

      20. lower-fma.f64 N/A

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

   4. Applied rewrites 93.5%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 97.8

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 97.8

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

   6. Applied rewrites 97.8%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot b, 27, \mathsf{fma}\left(\left(y \cdot -9\right) \cdot t, z, x \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot b, 27, \mathsf{fma}\left(\left(-9 \cdot z\right) \cdot y, t, x \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 30.2% accurate, 6.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ x \cdot 2 \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b) :precision binary64 (* x 2.0))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	return x * 2.0;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b)
    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
end function

Java

assert x < y && y < z && z < t && t < a && a < b;
public static double code(double x, double y, double z, double t, double a, double b) {
	return x * 2.0;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	return x * 2.0

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	return Float64(x * 2.0)
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp = code(x, y, z, t, a, b)
	tmp = x * 2.0;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := N[(x * 2.0), $MachinePrecision]

Derivation

1. Initial program 96.1%

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

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

   7. associate-*r* N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 96.1

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

   11. lift--.f64 N/A

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

   12. sub-neg N/A

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

   13. +-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. associate-*l* N/A

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

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

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

   18. *-commutative N/A

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

   19. associate-*r* N/A

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

   20. lower-fma.f64 N/A

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

4. Applied rewrites 93.9%

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

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 96.0

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 96.0

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

6. Applied rewrites 96.0%

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

7. Taylor expanded in x around inf

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

   1. lower-*.f64 31.2

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

9. Applied rewrites 31.2%

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

10. Final simplification 31.2%

    \[\leadsto x \cdot 2 \]
11. Add Preprocessing

Developer Target 1: 94.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024296 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, A"
  :precision binary64

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

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