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

Percentage Accurate: 95.3% → 98.0%

Time: 11.0s

Alternatives: 19

Speedup: 0.9×

• 27.7% 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: 98.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := z \cdot \left(9 \cdot y\right)\\ \mathbf{if}\;t\_1 \leq 10^{+160}:\\ \;\;\;\;\left(27 \cdot a\right) \cdot b - \left(t \cdot t\_1 - 2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-9 \cdot z, t \cdot y, \mathsf{fma}\left(a, b \cdot 27, x\right) + x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* z (* 9.0 y))))
   (if (<= t_1 1e+160)
     (- (* (* 27.0 a) b) (- (* t t_1) (* 2.0 x)))
     (fma (* -9.0 z) (* t y) (+ (fma a (* b 27.0) x) x)))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = z * (9.0 * y);
	double tmp;
	if (t_1 <= 1e+160) {
		tmp = ((27.0 * a) * b) - ((t * t_1) - (2.0 * x));
	} else {
		tmp = fma((-9.0 * z), (t * y), (fma(a, (b * 27.0), x) + x));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(z * Float64(9.0 * y))
	tmp = 0.0
	if (t_1 <= 1e+160)
		tmp = Float64(Float64(Float64(27.0 * a) * b) - Float64(Float64(t * t_1) - Float64(2.0 * x)));
	else
		tmp = fma(Float64(-9.0 * z), Float64(t * y), Float64(fma(a, Float64(b * 27.0), x) + x));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
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[(z * N[(9.0 * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e+160], N[(N[(N[(27.0 * a), $MachinePrecision] * b), $MachinePrecision] - N[(N[(t * t$95$1), $MachinePrecision] - N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-9.0 * z), $MachinePrecision] * N[(t * y), $MachinePrecision] + N[(N[(a * N[(b * 27.0), $MachinePrecision] + x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y #s(literal 9 binary64)) z) < 1.00000000000000001e160

   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

   if 1.00000000000000001e160 < (*.f64 (*.f64 y #s(literal 9 binary64)) 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. 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. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 99.8

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 99.8

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

      25. lift-*.f64 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-fma.f64 96.7

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 96.7

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

   6. Applied rewrites 96.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. associate-*l* N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. associate-+r+ N/A

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

      16. count-2-rev N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. associate-*r* N/A

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

      20. lower-fma.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

      23. count-2-rev N/A

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

   8. Applied rewrites 96.6%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(9 \cdot y\right) \leq 10^{+160}:\\ \;\;\;\;\left(27 \cdot a\right) \cdot b - \left(t \cdot \left(z \cdot \left(9 \cdot y\right)\right) - 2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-9 \cdot z, t \cdot y, \mathsf{fma}\left(a, b \cdot 27, x\right) + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b \cdot 27, a, \left(\left(z \cdot y\right) \cdot t\right) \cdot -9\right)\\ t_2 := t \cdot \left(z \cdot \left(9 \cdot y\right)\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(27 \cdot a\right) \cdot b\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+298}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot 27, a, \left(\left(-9 \cdot y\right) \cdot t\right) \cdot z\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
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 (* b 27.0) a (* (* (* z y) t) -9.0)))
        (t_2 (* t (* z (* 9.0 y)))))
   (if (<= t_2 -1e+20)
     t_1
     (if (<= t_2 5e+75)
       (fma 2.0 x (* (* 27.0 a) b))
       (if (<= t_2 1e+298) t_1 (fma (* b 27.0) a (* (* (* -9.0 y) t) z)))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((b * 27.0), a, (((z * y) * t) * -9.0));
	double t_2 = t * (z * (9.0 * y));
	double tmp;
	if (t_2 <= -1e+20) {
		tmp = t_1;
	} else if (t_2 <= 5e+75) {
		tmp = fma(2.0, x, ((27.0 * a) * b));
	} else if (t_2 <= 1e+298) {
		tmp = t_1;
	} else {
		tmp = fma((b * 27.0), a, (((-9.0 * y) * t) * z));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = fma(Float64(b * 27.0), a, Float64(Float64(Float64(z * y) * t) * -9.0))
	t_2 = Float64(t * Float64(z * Float64(9.0 * y)))
	tmp = 0.0
	if (t_2 <= -1e+20)
		tmp = t_1;
	elseif (t_2 <= 5e+75)
		tmp = fma(2.0, x, Float64(Float64(27.0 * a) * b));
	elseif (t_2 <= 1e+298)
		tmp = t_1;
	else
		tmp = fma(Float64(b * 27.0), a, Float64(Float64(Float64(-9.0 * y) * 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.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b * 27.0), $MachinePrecision] * a + N[(N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] * -9.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(z * N[(9.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+20], t$95$1, If[LessEqual[t$95$2, 5e+75], N[(2.0 * x + N[(N[(27.0 * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+298], t$95$1, N[(N[(b * 27.0), $MachinePrecision] * a + N[(N[(N[(-9.0 * y), $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) < -1e20 or 5.0000000000000002e75 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 9.9999999999999996e297

   1. Initial program 96.4%

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

   3. Taylor expanded in x around 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 84.8

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

   5. Applied rewrites 84.8%

      \[\leadsto \color{blue}{-9 \cdot \left(\left(z \cdot y\right) \cdot t\right)} + \left(a \cdot 27\right) \cdot b \]
   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. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 84.7

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 84.7

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

   7. Applied rewrites 78.5%

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

   9. Applied rewrites 78.4%

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

   10. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 84.7

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

   12. Applied rewrites 84.7%

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

   if -1e20 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 5.0000000000000002e75

   1. Initial program 98.4%

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

   3. Taylor expanded in 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 92.4

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

   5. Applied rewrites 92.4%

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

   7. Applied rewrites 92.4%

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

   if 9.9999999999999996e297 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 84.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} + \left(a \cdot 27\right) \cdot b \]
   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 82.1

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

   5. Applied rewrites 82.1%

      \[\leadsto \color{blue}{-9 \cdot \left(\left(z \cdot y\right) \cdot t\right)} + \left(a \cdot 27\right) \cdot b \]
   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. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 82.1

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 82.1

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

   7. Applied rewrites 91.3%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot \left(9 \cdot y\right)\right) \leq -1 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot 27, a, \left(\left(z \cdot y\right) \cdot t\right) \cdot -9\right)\\ \mathbf{elif}\;t \cdot \left(z \cdot \left(9 \cdot y\right)\right) \leq 5 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(27 \cdot a\right) \cdot b\right)\\ \mathbf{elif}\;t \cdot \left(z \cdot \left(9 \cdot y\right)\right) \leq 10^{+298}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot 27, a, \left(\left(z \cdot y\right) \cdot t\right) \cdot -9\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot 27, a, \left(\left(-9 \cdot y\right) \cdot t\right) \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b \cdot 27, a, \left(-9 \cdot t\right) \cdot \left(z \cdot y\right)\right)\\ t_2 := t \cdot \left(z \cdot \left(9 \cdot y\right)\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(27 \cdot a\right) \cdot b\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+298}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot 27, a, \left(\left(-9 \cdot y\right) \cdot t\right) \cdot z\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
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 (* b 27.0) a (* (* -9.0 t) (* z y))))
        (t_2 (* t (* z (* 9.0 y)))))
   (if (<= t_2 -1e+20)
     t_1
     (if (<= t_2 5e+75)
       (fma 2.0 x (* (* 27.0 a) b))
       (if (<= t_2 1e+298) t_1 (fma (* b 27.0) a (* (* (* -9.0 y) t) z)))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((b * 27.0), a, ((-9.0 * t) * (z * y)));
	double t_2 = t * (z * (9.0 * y));
	double tmp;
	if (t_2 <= -1e+20) {
		tmp = t_1;
	} else if (t_2 <= 5e+75) {
		tmp = fma(2.0, x, ((27.0 * a) * b));
	} else if (t_2 <= 1e+298) {
		tmp = t_1;
	} else {
		tmp = fma((b * 27.0), a, (((-9.0 * y) * t) * z));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = fma(Float64(b * 27.0), a, Float64(Float64(-9.0 * t) * Float64(z * y)))
	t_2 = Float64(t * Float64(z * Float64(9.0 * y)))
	tmp = 0.0
	if (t_2 <= -1e+20)
		tmp = t_1;
	elseif (t_2 <= 5e+75)
		tmp = fma(2.0, x, Float64(Float64(27.0 * a) * b));
	elseif (t_2 <= 1e+298)
		tmp = t_1;
	else
		tmp = fma(Float64(b * 27.0), a, Float64(Float64(Float64(-9.0 * y) * 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.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b * 27.0), $MachinePrecision] * a + N[(N[(-9.0 * t), $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(z * N[(9.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+20], t$95$1, If[LessEqual[t$95$2, 5e+75], N[(2.0 * x + N[(N[(27.0 * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+298], t$95$1, N[(N[(b * 27.0), $MachinePrecision] * a + N[(N[(N[(-9.0 * y), $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) < -1e20 or 5.0000000000000002e75 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 9.9999999999999996e297

   1. Initial program 96.4%

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

   3. Taylor expanded in x around 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 84.8

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

   5. Applied rewrites 84.8%

      \[\leadsto \color{blue}{-9 \cdot \left(\left(z \cdot y\right) \cdot t\right)} + \left(a \cdot 27\right) \cdot b \]
   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. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 84.7

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 84.7

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

   7. Applied rewrites 78.5%

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

   9. Applied rewrites 84.8%

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

   if -1e20 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 5.0000000000000002e75

   1. Initial program 98.4%

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

   3. Taylor expanded in 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 92.4

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

   5. Applied rewrites 92.4%

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

   7. Applied rewrites 92.4%

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

   if 9.9999999999999996e297 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 84.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} + \left(a \cdot 27\right) \cdot b \]
   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 82.1

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

   5. Applied rewrites 82.1%

      \[\leadsto \color{blue}{-9 \cdot \left(\left(z \cdot y\right) \cdot t\right)} + \left(a \cdot 27\right) \cdot b \]
   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. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 82.1

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 82.1

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

   7. Applied rewrites 91.3%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot \left(9 \cdot y\right)\right) \leq -1 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot 27, a, \left(-9 \cdot t\right) \cdot \left(z \cdot y\right)\right)\\ \mathbf{elif}\;t \cdot \left(z \cdot \left(9 \cdot y\right)\right) \leq 5 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(27 \cdot a\right) \cdot b\right)\\ \mathbf{elif}\;t \cdot \left(z \cdot \left(9 \cdot y\right)\right) \leq 10^{+298}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot 27, a, \left(-9 \cdot t\right) \cdot \left(z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot 27, a, \left(\left(-9 \cdot y\right) \cdot t\right) \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b \cdot 27, a, \left(-9 \cdot t\right) \cdot \left(z \cdot y\right)\right)\\ t_2 := t \cdot \left(z \cdot \left(9 \cdot y\right)\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(27 \cdot a\right) \cdot b\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+298}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot 27, a, \left(t \cdot y\right) \cdot \left(-9 \cdot z\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (* b 27.0) a (* (* -9.0 t) (* z y))))
        (t_2 (* t (* z (* 9.0 y)))))
   (if (<= t_2 -1e+20)
     t_1
     (if (<= t_2 5e+75)
       (fma 2.0 x (* (* 27.0 a) b))
       (if (<= t_2 1e+298) t_1 (fma (* b 27.0) a (* (* t y) (* -9.0 z))))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((b * 27.0), a, ((-9.0 * t) * (z * y)));
	double t_2 = t * (z * (9.0 * y));
	double tmp;
	if (t_2 <= -1e+20) {
		tmp = t_1;
	} else if (t_2 <= 5e+75) {
		tmp = fma(2.0, x, ((27.0 * a) * b));
	} else if (t_2 <= 1e+298) {
		tmp = t_1;
	} else {
		tmp = fma((b * 27.0), a, ((t * y) * (-9.0 * z)));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = fma(Float64(b * 27.0), a, Float64(Float64(-9.0 * t) * Float64(z * y)))
	t_2 = Float64(t * Float64(z * Float64(9.0 * y)))
	tmp = 0.0
	if (t_2 <= -1e+20)
		tmp = t_1;
	elseif (t_2 <= 5e+75)
		tmp = fma(2.0, x, Float64(Float64(27.0 * a) * b));
	elseif (t_2 <= 1e+298)
		tmp = t_1;
	else
		tmp = fma(Float64(b * 27.0), a, Float64(Float64(t * y) * Float64(-9.0 * z)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b * 27.0), $MachinePrecision] * a + N[(N[(-9.0 * t), $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(z * N[(9.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+20], t$95$1, If[LessEqual[t$95$2, 5e+75], N[(2.0 * x + N[(N[(27.0 * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+298], t$95$1, N[(N[(b * 27.0), $MachinePrecision] * a + N[(N[(t * y), $MachinePrecision] * N[(-9.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -1e20 or 5.0000000000000002e75 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 9.9999999999999996e297

   1. Initial program 96.4%

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

   3. Taylor expanded in x around 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 84.8

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

   5. Applied rewrites 84.8%

      \[\leadsto \color{blue}{-9 \cdot \left(\left(z \cdot y\right) \cdot t\right)} + \left(a \cdot 27\right) \cdot b \]
   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. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 84.7

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 84.7

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

   7. Applied rewrites 78.5%

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

   9. Applied rewrites 84.8%

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

   if -1e20 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 5.0000000000000002e75

   1. Initial program 98.4%

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

   3. Taylor expanded in 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 92.4

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

   5. Applied rewrites 92.4%

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

   7. Applied rewrites 92.4%

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

   if 9.9999999999999996e297 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 84.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} + \left(a \cdot 27\right) \cdot b \]
   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 82.1

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

   5. Applied rewrites 82.1%

      \[\leadsto \color{blue}{-9 \cdot \left(\left(z \cdot y\right) \cdot t\right)} + \left(a \cdot 27\right) \cdot b \]
   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. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 82.1

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 82.1

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

   7. Applied rewrites 91.3%

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

   9. Applied rewrites 91.2%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot \left(9 \cdot y\right)\right) \leq -1 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot 27, a, \left(-9 \cdot t\right) \cdot \left(z \cdot y\right)\right)\\ \mathbf{elif}\;t \cdot \left(z \cdot \left(9 \cdot y\right)\right) \leq 5 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(27 \cdot a\right) \cdot b\right)\\ \mathbf{elif}\;t \cdot \left(z \cdot \left(9 \cdot y\right)\right) \leq 10^{+298}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot 27, a, \left(-9 \cdot t\right) \cdot \left(z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot 27, a, \left(t \cdot y\right) \cdot \left(-9 \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(z \cdot y\right) \cdot t\\ t_2 := t \cdot \left(z \cdot \left(9 \cdot y\right)\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(-9, t\_1, \left(b \cdot a\right) \cdot 27\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(27 \cdot a\right) \cdot b\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+298}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, 27, t\_1 \cdot -9\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot 27, a, \left(t \cdot y\right) \cdot \left(-9 \cdot z\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* z y) t)) (t_2 (* t (* z (* 9.0 y)))))
   (if (<= t_2 -1e+20)
     (fma -9.0 t_1 (* (* b a) 27.0))
     (if (<= t_2 5e+75)
       (fma 2.0 x (* (* 27.0 a) b))
       (if (<= t_2 1e+298)
         (fma (* b a) 27.0 (* t_1 -9.0))
         (fma (* b 27.0) a (* (* t y) (* -9.0 z))))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (z * y) * t;
	double t_2 = t * (z * (9.0 * y));
	double tmp;
	if (t_2 <= -1e+20) {
		tmp = fma(-9.0, t_1, ((b * a) * 27.0));
	} else if (t_2 <= 5e+75) {
		tmp = fma(2.0, x, ((27.0 * a) * b));
	} else if (t_2 <= 1e+298) {
		tmp = fma((b * a), 27.0, (t_1 * -9.0));
	} else {
		tmp = fma((b * 27.0), a, ((t * y) * (-9.0 * z)));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(z * y) * t)
	t_2 = Float64(t * Float64(z * Float64(9.0 * y)))
	tmp = 0.0
	if (t_2 <= -1e+20)
		tmp = fma(-9.0, t_1, Float64(Float64(b * a) * 27.0));
	elseif (t_2 <= 5e+75)
		tmp = fma(2.0, x, Float64(Float64(27.0 * a) * b));
	elseif (t_2 <= 1e+298)
		tmp = fma(Float64(b * a), 27.0, Float64(t_1 * -9.0));
	else
		tmp = fma(Float64(b * 27.0), a, Float64(Float64(t * y) * Float64(-9.0 * z)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(z * N[(9.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+20], N[(-9.0 * t$95$1 + N[(N[(b * a), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+75], N[(2.0 * x + N[(N[(27.0 * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+298], N[(N[(b * a), $MachinePrecision] * 27.0 + N[(t$95$1 * -9.0), $MachinePrecision]), $MachinePrecision], N[(N[(b * 27.0), $MachinePrecision] * a + N[(N[(t * y), $MachinePrecision] * N[(-9.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -1e20

   1. Initial program 94.1%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   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-sub-sign-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 83.0

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

   5. Applied rewrites 83.0%

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

   if -1e20 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 5.0000000000000002e75

   1. Initial program 98.4%

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

   3. Taylor expanded in 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 92.4

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

   5. Applied rewrites 92.4%

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

   7. Applied rewrites 92.4%

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

   if 5.0000000000000002e75 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 9.9999999999999996e297

   1. Initial program 99.6%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   2. 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 31.1

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

   5. Applied rewrites 31.1%

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

   6. Taylor expanded in x around 0

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

      1. cancel-sub-sign-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. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 87.1

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

   8. Applied rewrites 87.1%

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

   if 9.9999999999999996e297 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 84.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} + \left(a \cdot 27\right) \cdot b \]
   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 82.1

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

   5. Applied rewrites 82.1%

      \[\leadsto \color{blue}{-9 \cdot \left(\left(z \cdot y\right) \cdot t\right)} + \left(a \cdot 27\right) \cdot b \]
   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. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 82.1

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 82.1

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

   7. Applied rewrites 91.3%

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

   9. Applied rewrites 91.2%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot \left(9 \cdot y\right)\right) \leq -1 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(b \cdot a\right) \cdot 27\right)\\ \mathbf{elif}\;t \cdot \left(z \cdot \left(9 \cdot y\right)\right) \leq 5 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(27 \cdot a\right) \cdot b\right)\\ \mathbf{elif}\;t \cdot \left(z \cdot \left(9 \cdot y\right)\right) \leq 10^{+298}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, 27, \left(\left(z \cdot y\right) \cdot t\right) \cdot -9\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot 27, a, \left(t \cdot y\right) \cdot \left(-9 \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t \cdot z, -9 \cdot y, \left(b \cdot 27\right) \cdot a + x\right)\\ t_2 := t \cdot \left(z \cdot \left(9 \cdot y\right)\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(27 \cdot a\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (* t z) (* -9.0 y) (+ (* (* b 27.0) a) x)))
        (t_2 (* t (* z (* 9.0 y)))))
   (if (<= t_2 -1e+20)
     t_1
     (if (<= t_2 5e+75) (fma 2.0 x (* (* 27.0 a) b)) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((t * z), (-9.0 * y), (((b * 27.0) * a) + x));
	double t_2 = t * (z * (9.0 * y));
	double tmp;
	if (t_2 <= -1e+20) {
		tmp = t_1;
	} else if (t_2 <= 5e+75) {
		tmp = fma(2.0, x, ((27.0 * a) * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = fma(Float64(t * z), Float64(-9.0 * y), Float64(Float64(Float64(b * 27.0) * a) + x))
	t_2 = Float64(t * Float64(z * Float64(9.0 * y)))
	tmp = 0.0
	if (t_2 <= -1e+20)
		tmp = t_1;
	elseif (t_2 <= 5e+75)
		tmp = fma(2.0, x, Float64(Float64(27.0 * a) * b));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] * N[(-9.0 * y), $MachinePrecision] + N[(N[(N[(b * 27.0), $MachinePrecision] * a), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(z * N[(9.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+20], t$95$1, If[LessEqual[t$95$2, 5e+75], N[(2.0 * x + N[(N[(27.0 * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -1e20 or 5.0000000000000002e75 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 93.3%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   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. 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. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 88.4

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 88.4

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

      25. lift-*.f64 N/A

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

   4. Applied rewrites 88.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-fma.f64 87.6

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 87.6

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

   6. Applied rewrites 87.6%

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

   7. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 80.9

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

   9. Applied rewrites 80.9%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 83.5%

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

   if -1e20 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 5.0000000000000002e75

   1. Initial program 98.4%

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

   3. Taylor expanded in 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 92.4

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

   5. Applied rewrites 92.4%

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

   7. Applied rewrites 92.4%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot \left(9 \cdot y\right)\right) \leq -1 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, -9 \cdot y, \left(b \cdot 27\right) \cdot a + x\right)\\ \mathbf{elif}\;t \cdot \left(z \cdot \left(9 \cdot y\right)\right) \leq 5 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(27 \cdot a\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, -9 \cdot y, \left(b \cdot 27\right) \cdot a + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := t \cdot \left(z \cdot \left(9 \cdot y\right)\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot 27, a, \left(\left(z \cdot y\right) \cdot t\right) \cdot -9\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(27 \cdot a\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, -9 \cdot y, \left(b \cdot a\right) \cdot 27\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* t (* z (* 9.0 y)))))
   (if (<= t_1 -1e+20)
     (fma (* b 27.0) a (* (* (* z y) t) -9.0))
     (if (<= t_1 5e+75)
       (fma 2.0 x (* (* 27.0 a) b))
       (fma (* t z) (* -9.0 y) (* (* b a) 27.0))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t * (z * (9.0 * y));
	double tmp;
	if (t_1 <= -1e+20) {
		tmp = fma((b * 27.0), a, (((z * y) * t) * -9.0));
	} else if (t_1 <= 5e+75) {
		tmp = fma(2.0, x, ((27.0 * a) * b));
	} else {
		tmp = fma((t * z), (-9.0 * y), ((b * a) * 27.0));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(t * Float64(z * Float64(9.0 * y)))
	tmp = 0.0
	if (t_1 <= -1e+20)
		tmp = fma(Float64(b * 27.0), a, Float64(Float64(Float64(z * y) * t) * -9.0));
	elseif (t_1 <= 5e+75)
		tmp = fma(2.0, x, Float64(Float64(27.0 * a) * b));
	else
		tmp = fma(Float64(t * z), Float64(-9.0 * y), Float64(Float64(b * a) * 27.0));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t * N[(z * N[(9.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+20], N[(N[(b * 27.0), $MachinePrecision] * a + N[(N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] * -9.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+75], N[(2.0 * x + N[(N[(27.0 * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(N[(t * z), $MachinePrecision] * N[(-9.0 * y), $MachinePrecision] + N[(N[(b * a), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -1e20

   1. Initial program 94.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} + \left(a \cdot 27\right) \cdot b \]
   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 83.0

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

   5. Applied rewrites 83.0%

      \[\leadsto \color{blue}{-9 \cdot \left(\left(z \cdot y\right) \cdot t\right)} + \left(a \cdot 27\right) \cdot b \]
   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. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 83.0

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 83.0

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

   7. Applied rewrites 81.2%

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

   9. Applied rewrites 81.2%

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

   10. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 83.0

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

   12. Applied rewrites 83.0%

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

   if -1e20 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 5.0000000000000002e75

   1. Initial program 98.4%

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

   3. Taylor expanded in 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 92.4

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

   5. Applied rewrites 92.4%

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

   7. Applied rewrites 92.4%

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

   if 5.0000000000000002e75 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 92.8%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   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. 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. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 87.2

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 87.2

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

      25. lift-*.f64 N/A

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

   4. Applied rewrites 87.2%

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

   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 82.8

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

   7. Applied rewrites 82.8%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot \left(9 \cdot y\right)\right) \leq -1 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot 27, a, \left(\left(z \cdot y\right) \cdot t\right) \cdot -9\right)\\ \mathbf{elif}\;t \cdot \left(z \cdot \left(9 \cdot y\right)\right) \leq 5 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(27 \cdot a\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, -9 \cdot y, \left(b \cdot a\right) \cdot 27\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(z \cdot y\right) \cdot t\\ t_2 := t \cdot \left(z \cdot \left(9 \cdot y\right)\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(-9, t\_1, \left(b \cdot a\right) \cdot 27\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(27 \cdot a\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, 27, t\_1 \cdot -9\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* z y) t)) (t_2 (* t (* z (* 9.0 y)))))
   (if (<= t_2 -1e+20)
     (fma -9.0 t_1 (* (* b a) 27.0))
     (if (<= t_2 5e+75)
       (fma 2.0 x (* (* 27.0 a) b))
       (fma (* b a) 27.0 (* t_1 -9.0))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (z * y) * t;
	double t_2 = t * (z * (9.0 * y));
	double tmp;
	if (t_2 <= -1e+20) {
		tmp = fma(-9.0, t_1, ((b * a) * 27.0));
	} else if (t_2 <= 5e+75) {
		tmp = fma(2.0, x, ((27.0 * a) * b));
	} else {
		tmp = fma((b * a), 27.0, (t_1 * -9.0));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(z * y) * t)
	t_2 = Float64(t * Float64(z * Float64(9.0 * y)))
	tmp = 0.0
	if (t_2 <= -1e+20)
		tmp = fma(-9.0, t_1, Float64(Float64(b * a) * 27.0));
	elseif (t_2 <= 5e+75)
		tmp = fma(2.0, x, Float64(Float64(27.0 * a) * b));
	else
		tmp = fma(Float64(b * a), 27.0, Float64(t_1 * -9.0));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(z * N[(9.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+20], N[(-9.0 * t$95$1 + N[(N[(b * a), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+75], N[(2.0 * x + N[(N[(27.0 * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(N[(b * a), $MachinePrecision] * 27.0 + N[(t$95$1 * -9.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -1e20

   1. Initial program 94.1%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   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-sub-sign-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 83.0

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

   5. Applied rewrites 83.0%

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

   if -1e20 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 5.0000000000000002e75

   1. Initial program 98.4%

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

   3. Taylor expanded in 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 92.4

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

   5. Applied rewrites 92.4%

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

   7. Applied rewrites 92.4%

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

   if 5.0000000000000002e75 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 92.8%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   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 24.0

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

   5. Applied rewrites 24.0%

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

   6. Taylor expanded in x around 0

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

      1. cancel-sub-sign-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. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 84.9

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

   8. Applied rewrites 84.9%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot \left(9 \cdot y\right)\right) \leq -1 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(b \cdot a\right) \cdot 27\right)\\ \mathbf{elif}\;t \cdot \left(z \cdot \left(9 \cdot y\right)\right) \leq 5 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(27 \cdot a\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, 27, \left(\left(z \cdot y\right) \cdot t\right) \cdot -9\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(b \cdot a\right) \cdot 27\right)\\ t_2 := t \cdot \left(z \cdot \left(9 \cdot y\right)\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(27 \cdot a\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma -9.0 (* (* z y) t) (* (* b a) 27.0)))
        (t_2 (* t (* z (* 9.0 y)))))
   (if (<= t_2 -1e+20)
     t_1
     (if (<= t_2 5e+75) (fma 2.0 x (* (* 27.0 a) b)) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(-9.0, ((z * y) * t), ((b * a) * 27.0));
	double t_2 = t * (z * (9.0 * y));
	double tmp;
	if (t_2 <= -1e+20) {
		tmp = t_1;
	} else if (t_2 <= 5e+75) {
		tmp = fma(2.0, x, ((27.0 * a) * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = fma(-9.0, Float64(Float64(z * y) * t), Float64(Float64(b * a) * 27.0))
	t_2 = Float64(t * Float64(z * Float64(9.0 * y)))
	tmp = 0.0
	if (t_2 <= -1e+20)
		tmp = t_1;
	elseif (t_2 <= 5e+75)
		tmp = fma(2.0, x, Float64(Float64(27.0 * a) * b));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(-9.0 * N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] + N[(N[(b * a), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(z * N[(9.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+20], t$95$1, If[LessEqual[t$95$2, 5e+75], N[(2.0 * x + N[(N[(27.0 * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -1e20 or 5.0000000000000002e75 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 93.3%

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

   3. Taylor expanded in x around 0

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

      1. cancel-sub-sign-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 84.1

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

   5. Applied rewrites 84.1%

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

   if -1e20 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 5.0000000000000002e75

   1. Initial program 98.4%

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

   3. Taylor expanded in 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 92.4

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

   5. Applied rewrites 92.4%

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

   7. Applied rewrites 92.4%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot \left(9 \cdot y\right)\right) \leq -1 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(b \cdot a\right) \cdot 27\right)\\ \mathbf{elif}\;t \cdot \left(z \cdot \left(9 \cdot y\right)\right) \leq 5 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(27 \cdot a\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(b \cdot a\right) \cdot 27\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 34.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;\left(27 \cdot a\right) \cdot b - \left(t \cdot \left(z \cdot \left(9 \cdot y\right)\right) - 2 \cdot x\right) \leq 2 \cdot 10^{+305}:\\ \;\;\;\;x + x\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (if (<= (- (* (* 27.0 a) b) (- (* t (* z (* 9.0 y))) (* 2.0 x))) 2e+305)
   (+ x x)
   (* 4.0 (* x x))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((((27.0 * a) * b) - ((t * (z * (9.0 * y))) - (2.0 * x))) <= 2e+305) {
		tmp = x + x;
	} else {
		tmp = 4.0 * (x * x);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
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 ((((27.0d0 * a) * b) - ((t * (z * (9.0d0 * y))) - (2.0d0 * x))) <= 2d+305) then
        tmp = x + x
    else
        tmp = 4.0d0 * (x * x)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b;
assert x < y && y < z && z < t && t < a && a < b;
public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((((27.0 * a) * b) - ((t * (z * (9.0 * y))) - (2.0 * x))) <= 2e+305) {
		tmp = x + x;
	} else {
		tmp = 4.0 * (x * x);
	}
	return tmp;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	tmp = 0
	if (((27.0 * a) * b) - ((t * (z * (9.0 * y))) - (2.0 * x))) <= 2e+305:
		tmp = x + x
	else:
		tmp = 4.0 * (x * x)
	return tmp

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(Float64(27.0 * a) * b) - Float64(Float64(t * Float64(z * Float64(9.0 * y))) - Float64(2.0 * x))) <= 2e+305)
		tmp = Float64(x + x);
	else
		tmp = Float64(4.0 * Float64(x * x));
	end
	return tmp
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((((27.0 * a) * b) - ((t * (z * (9.0 * y))) - (2.0 * x))) <= 2e+305)
		tmp = x + x;
	else
		tmp = 4.0 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
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[(N[(27.0 * a), $MachinePrecision] * b), $MachinePrecision] - N[(N[(t * N[(z * N[(9.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+305], N[(x + x), $MachinePrecision], N[(4.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)) (*.f64 (*.f64 a #s(literal 27 binary64)) b)) < 1.9999999999999999e305

   1. Initial program 97.6%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   2. 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. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 94.3

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 94.3

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

      25. lift-*.f64 N/A

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

   4. Applied rewrites 94.3%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 34.0

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

   7. Applied rewrites 34.0%

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

   9. Applied rewrites 34.0%

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

   if 1.9999999999999999e305 < (+.f64 (-.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)) (*.f64 (*.f64 a #s(literal 27 binary64)) b))

   1. Initial program 87.9%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   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. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 95.1

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 95.1

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

      25. lift-*.f64 N/A

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

   4. Applied rewrites 95.1%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 2.2

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

   7. Applied rewrites 2.2%

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

   9. Applied rewrites 2.2%

      \[\leadsto x + \color{blue}{x} \]
   10. Step-by-step derivation

   11. Applied rewrites 26.6%

       \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{4} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(27 \cdot a\right) \cdot b - \left(t \cdot \left(z \cdot \left(9 \cdot y\right)\right) - 2 \cdot x\right) \leq 2 \cdot 10^{+305}:\\ \;\;\;\;x + x\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;9 \cdot y \leq -2 \cdot 10^{-132}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, -9 \cdot y, \mathsf{fma}\left(b \cdot 27, a, x\right) + x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(-9 \cdot y\right) \cdot t, z, \left(27 \cdot a\right) \cdot b\right) + x\right) + x\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
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) -2e-132)
   (fma (* t z) (* -9.0 y) (+ (fma (* b 27.0) a x) x))
   (+ (+ (fma (* (* -9.0 y) t) z (* (* 27.0 a) b)) x) x)))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((9.0 * y) <= -2e-132) {
		tmp = fma((t * z), (-9.0 * y), (fma((b * 27.0), a, x) + x));
	} else {
		tmp = (fma(((-9.0 * y) * t), z, ((27.0 * a) * b)) + x) + x;
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(9.0 * y) <= -2e-132)
		tmp = fma(Float64(t * z), Float64(-9.0 * y), Float64(fma(Float64(b * 27.0), a, x) + x));
	else
		tmp = Float64(Float64(fma(Float64(Float64(-9.0 * y) * t), z, Float64(Float64(27.0 * a) * b)) + x) + x);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(9.0 * y), $MachinePrecision], -2e-132], N[(N[(t * z), $MachinePrecision] * N[(-9.0 * y), $MachinePrecision] + N[(N[(N[(b * 27.0), $MachinePrecision] * a + x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-9.0 * y), $MachinePrecision] * t), $MachinePrecision] * z + N[(N[(27.0 * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y #s(literal 9 binary64)) < -2e-132

   1. Initial program 94.3%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   2. Add Preprocessing
   3. 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. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 97.0

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 97.0

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

      25. lift-*.f64 N/A

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

   4. Applied rewrites 97.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-fma.f64 96.0

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 96.0

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

   6. Applied rewrites 96.0%

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

   if -2e-132 < (*.f64 y #s(literal 9 binary64))

   1. Initial program 97.2%

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

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. count-2-rev N/A

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

      8. associate-+l+ N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

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

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. lower-fma.f64 N/A

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

   4. Applied rewrites 98.0%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;9 \cdot y \leq -2 \cdot 10^{-132}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, -9 \cdot y, \mathsf{fma}\left(b \cdot 27, a, x\right) + x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(-9 \cdot y\right) \cdot t, z, \left(27 \cdot a\right) \cdot b\right) + x\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 65.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(27 \cdot a\right) \cdot b\\ \mathbf{if}\;t \cdot \left(z \cdot \left(9 \cdot y\right)\right) \leq -5 \cdot 10^{+219}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 4, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x, t\_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* 27.0 a) b)))
   (if (<= (* t (* z (* 9.0 y))) -5e+219)
     (fma (* x x) 4.0 t_1)
     (fma 2.0 x t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (27.0 * a) * b;
	double tmp;
	if ((t * (z * (9.0 * y))) <= -5e+219) {
		tmp = fma((x * x), 4.0, t_1);
	} else {
		tmp = fma(2.0, x, t_1);
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(27.0 * a) * b)
	tmp = 0.0
	if (Float64(t * Float64(z * Float64(9.0 * y))) <= -5e+219)
		tmp = fma(Float64(x * x), 4.0, t_1);
	else
		tmp = fma(2.0, x, t_1);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(27.0 * a), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[N[(t * N[(z * N[(9.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e+219], N[(N[(x * x), $MachinePrecision] * 4.0 + t$95$1), $MachinePrecision], N[(2.0 * x + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -5e219

   1. Initial program 91.3%

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

   3. Taylor expanded in y around 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 19.4

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

   5. Applied rewrites 19.4%

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

   7. Applied rewrites 19.5%

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

   9. Applied rewrites 43.2%

      \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{4}, \left(27 \cdot a\right) \cdot b\right) \]

   if -5e219 < (*.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 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 69.1

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

   5. Applied rewrites 69.1%

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

   7. Applied rewrites 69.1%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot \left(9 \cdot y\right)\right) \leq -5 \cdot 10^{+219}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 4, \left(27 \cdot a\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(27 \cdot a\right) \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 98.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;9 \cdot y \leq -5 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, -9 \cdot y, \mathsf{fma}\left(b \cdot 27, a, x\right) + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-9 \cdot z, t \cdot y, \mathsf{fma}\left(a, b \cdot 27, x\right) + x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* 9.0 y) -5e+39)
   (fma (* t z) (* -9.0 y) (+ (fma (* b 27.0) a x) x))
   (fma (* -9.0 z) (* t y) (+ (fma a (* b 27.0) x) x))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((9.0 * y) <= -5e+39) {
		tmp = fma((t * z), (-9.0 * y), (fma((b * 27.0), a, x) + x));
	} else {
		tmp = fma((-9.0 * z), (t * y), (fma(a, (b * 27.0), x) + x));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(9.0 * y) <= -5e+39)
		tmp = fma(Float64(t * z), Float64(-9.0 * y), Float64(fma(Float64(b * 27.0), a, x) + x));
	else
		tmp = fma(Float64(-9.0 * z), Float64(t * y), Float64(fma(a, Float64(b * 27.0), x) + x));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
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[(9.0 * y), $MachinePrecision], -5e+39], N[(N[(t * z), $MachinePrecision] * N[(-9.0 * y), $MachinePrecision] + N[(N[(N[(b * 27.0), $MachinePrecision] * a + x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[(-9.0 * z), $MachinePrecision] * N[(t * y), $MachinePrecision] + N[(N[(a * N[(b * 27.0), $MachinePrecision] + x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y #s(literal 9 binary64)) < -5.00000000000000015e39

   1. Initial program 91.4%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   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. 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. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 98.3

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 98.3

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

      25. lift-*.f64 N/A

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

   4. Applied rewrites 98.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-fma.f64 96.8

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 96.8

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

   6. Applied rewrites 96.8%

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

   if -5.00000000000000015e39 < (*.f64 y #s(literal 9 binary64))

   1. Initial program 97.7%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   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. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 93.0

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 93.0

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

      25. lift-*.f64 N/A

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

   4. Applied rewrites 93.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-fma.f64 93.0

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.0

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

   6. Applied rewrites 93.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. associate-*l* N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. associate-+r+ N/A

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

      16. count-2-rev N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. associate-*r* N/A

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

      20. lower-fma.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

      23. count-2-rev N/A

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

   8. Applied rewrites 98.3%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;9 \cdot y \leq -5 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, -9 \cdot y, \mathsf{fma}\left(b \cdot 27, a, x\right) + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-9 \cdot z, t \cdot y, \mathsf{fma}\left(a, b \cdot 27, x\right) + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 98.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 2.7 \cdot 10^{-31}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, -9 \cdot y, \mathsf{fma}\left(b \cdot 27, a, x\right) + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot 27, a, \mathsf{fma}\left(\left(-9 \cdot y\right) \cdot t, z, 2 \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (if (<= z 2.7e-31)
   (fma (* t z) (* -9.0 y) (+ (fma (* b 27.0) a x) x))
   (fma (* b 27.0) a (fma (* (* -9.0 y) t) z (* 2.0 x)))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= 2.7e-31) {
		tmp = fma((t * z), (-9.0 * y), (fma((b * 27.0), a, x) + x));
	} else {
		tmp = fma((b * 27.0), a, fma(((-9.0 * y) * t), z, (2.0 * x)));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= 2.7e-31)
		tmp = fma(Float64(t * z), Float64(-9.0 * y), Float64(fma(Float64(b * 27.0), a, x) + x));
	else
		tmp = fma(Float64(b * 27.0), a, fma(Float64(Float64(-9.0 * y) * t), z, Float64(2.0 * x)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, 2.7e-31], N[(N[(t * z), $MachinePrecision] * N[(-9.0 * y), $MachinePrecision] + N[(N[(N[(b * 27.0), $MachinePrecision] * a + x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[(b * 27.0), $MachinePrecision] * a + N[(N[(N[(-9.0 * y), $MachinePrecision] * t), $MachinePrecision] * z + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.70000000000000014e-31

   1. Initial program 97.2%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   2. Add Preprocessing
   3. 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. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 96.6

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 96.6

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

      25. lift-*.f64 N/A

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

   4. Applied rewrites 96.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-fma.f64 96.0

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 96.0

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

   6. Applied rewrites 96.0%

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

   if 2.70000000000000014e-31 < z

   1. Initial program 93.0%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   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 94.4

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

      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 98.4%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.7 \cdot 10^{-31}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, -9 \cdot y, \mathsf{fma}\left(b \cdot 27, a, x\right) + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot 27, a, \mathsf{fma}\left(\left(-9 \cdot y\right) \cdot t, z, 2 \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 65.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot \left(9 \cdot y\right)\right) \leq -1.56 \cdot 10^{+299}:\\ \;\;\;\;4 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(27 \cdot a\right) \cdot b\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* t (* z (* 9.0 y))) -1.56e+299)
   (* 4.0 (* x x))
   (fma 2.0 x (* (* 27.0 a) b))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t * (z * (9.0 * y))) <= -1.56e+299) {
		tmp = 4.0 * (x * x);
	} else {
		tmp = fma(2.0, x, ((27.0 * a) * b));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(t * Float64(z * Float64(9.0 * y))) <= -1.56e+299)
		tmp = Float64(4.0 * Float64(x * x));
	else
		tmp = fma(2.0, x, Float64(Float64(27.0 * a) * b));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
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[(t * N[(z * N[(9.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.56e+299], N[(4.0 * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(2.0 * x + N[(N[(27.0 * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -1.56e299

   1. Initial program 89.9%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   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. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 96.5

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 96.5

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

      25. lift-*.f64 N/A

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

   4. Applied rewrites 96.5%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 2.5

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

   7. Applied rewrites 2.5%

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

   9. Applied rewrites 2.5%

      \[\leadsto x + \color{blue}{x} \]
   10. Step-by-step derivation

   11. Applied rewrites 30.1%

       \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{4} \]

   if -1.56e299 < (*.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 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 68.9

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

   5. Applied rewrites 68.9%

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

   7. Applied rewrites 68.9%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot \left(9 \cdot y\right)\right) \leq -1.56 \cdot 10^{+299}:\\ \;\;\;\;4 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(27 \cdot a\right) \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 65.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot \left(9 \cdot y\right)\right) \leq -1.56 \cdot 10^{+299}:\\ \;\;\;\;4 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, 27, x\right) + x\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* t (* z (* 9.0 y))) -1.56e+299)
   (* 4.0 (* x x))
   (+ (fma (* b a) 27.0 x) x)))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t * (z * (9.0 * y))) <= -1.56e+299) {
		tmp = 4.0 * (x * x);
	} else {
		tmp = fma((b * a), 27.0, x) + x;
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(t * Float64(z * Float64(9.0 * y))) <= -1.56e+299)
		tmp = Float64(4.0 * Float64(x * x));
	else
		tmp = Float64(fma(Float64(b * a), 27.0, x) + x);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
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[(t * N[(z * N[(9.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.56e+299], N[(4.0 * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * a), $MachinePrecision] * 27.0 + x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -1.56e299

   1. Initial program 89.9%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   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. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 96.5

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 96.5

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

      25. lift-*.f64 N/A

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

   4. Applied rewrites 96.5%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 2.5

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

   7. Applied rewrites 2.5%

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

   9. Applied rewrites 2.5%

      \[\leadsto x + \color{blue}{x} \]
   10. Step-by-step derivation

   11. Applied rewrites 30.1%

       \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{4} \]

   if -1.56e299 < (*.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 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 68.9

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

   5. Applied rewrites 68.9%

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

   7. Applied rewrites 68.9%

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

   9. Applied rewrites 68.9%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot \left(9 \cdot y\right)\right) \leq -1.56 \cdot 10^{+299}:\\ \;\;\;\;4 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, 27, x\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 65.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot \left(9 \cdot y\right)\right) \leq -1.56 \cdot 10^{+299}:\\ \;\;\;\;4 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot 27, x\right) + x\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* t (* z (* 9.0 y))) -1.56e+299)
   (* 4.0 (* x x))
   (+ (fma a (* b 27.0) x) x)))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t * (z * (9.0 * y))) <= -1.56e+299) {
		tmp = 4.0 * (x * x);
	} else {
		tmp = fma(a, (b * 27.0), x) + x;
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(t * Float64(z * Float64(9.0 * y))) <= -1.56e+299)
		tmp = Float64(4.0 * Float64(x * x));
	else
		tmp = Float64(fma(a, Float64(b * 27.0), x) + x);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
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[(t * N[(z * N[(9.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.56e+299], N[(4.0 * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(b * 27.0), $MachinePrecision] + x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -1.56e299

   1. Initial program 89.9%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   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. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 96.5

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 96.5

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

      25. lift-*.f64 N/A

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

   4. Applied rewrites 96.5%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 2.5

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

   7. Applied rewrites 2.5%

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

   9. Applied rewrites 2.5%

      \[\leadsto x + \color{blue}{x} \]
   10. Step-by-step derivation

   11. Applied rewrites 30.1%

       \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{4} \]

   if -1.56e299 < (*.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 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 68.9

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

   5. Applied rewrites 68.9%

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

   7. Applied rewrites 68.9%

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

   9. Applied rewrites 68.9%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot \left(9 \cdot y\right)\right) \leq -1.56 \cdot 10^{+299}:\\ \;\;\;\;4 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot 27, x\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 93.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := N[(N[(-9.0 * z), $MachinePrecision] * N[(t * y), $MachinePrecision] + N[(N[(a * N[(b * 27.0), $MachinePrecision] + x), $MachinePrecision] + x), $MachinePrecision]), $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. 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. lower-fma.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

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

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

   17. lower-*.f64 N/A

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

   18. metadata-eval N/A

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. lower-fma.f64 94.4

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

   22. lift-*.f64 N/A

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

   23. *-commutative N/A

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

   24. lower-*.f64 94.4

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

   25. lift-*.f64 N/A

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

4. Applied rewrites 94.4%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lift-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. count-2-rev N/A

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

   10. associate-+r+ N/A

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

   11. lower-+.f64 N/A

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

   12. lower-fma.f64 94.0

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 94.0

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

6. Applied rewrites 94.0%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. associate-*l* N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-fma.f64 N/A

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

   14. +-commutative N/A

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

   15. associate-+r+ N/A

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

   16. count-2-rev N/A

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. associate-*r* N/A

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

   20. lower-fma.f64 N/A

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

   21. lower-*.f64 N/A

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

   22. lower-*.f64 N/A

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

   23. count-2-rev N/A

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

8. Applied rewrites 95.6%

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

Alternative 19: 31.1% accurate, 9.3× speedup

Math

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

FPCore

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

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	return x + x;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
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 + x
end function

Java

assert x < y && y < z && z < t && t < a && a < b;
assert x < y && y < z && z < t && t < a && a < b;
public static double code(double x, double y, double z, double t, double a, double b) {
	return x + x;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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. 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. lower-fma.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

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

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

   17. lower-*.f64 N/A

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

   18. metadata-eval N/A

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. lower-fma.f64 94.4

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

   22. lift-*.f64 N/A

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

   23. *-commutative N/A

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

   24. lower-*.f64 94.4

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

   25. lift-*.f64 N/A

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

4. Applied rewrites 94.4%

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

5. Taylor expanded in x around inf

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

   1. lower-*.f64 28.9

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

7. Applied rewrites 28.9%

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

9. Applied rewrites 28.9%

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

Developer Target 1: 95.0% 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 2024298 
(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)))