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

Percentage Accurate: 95.7% → 98.3%

Time: 10.8s

Alternatives: 10

Speedup: 1.0×

• 26.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 98.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 2.2 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(27 \cdot a, b, \mathsf{fma}\left(\left(-9 \cdot y\right) \cdot t, z, 2 \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(z \cdot y\right) \cdot t, -9, \mathsf{fma}\left(b \cdot a, 27, x\right) + x\right)\\ \end{array} \end{array} \]

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 (<= t 2.2e+27)
   (fma (* 27.0 a) b (fma (* (* -9.0 y) t) z (* 2.0 x)))
   (fma (* (* z y) t) -9.0 (+ (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 <= 2.2e+27) {
		tmp = fma((27.0 * a), b, fma(((-9.0 * y) * t), z, (2.0 * x)));
	} else {
		tmp = fma(((z * y) * t), -9.0, (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 (t <= 2.2e+27)
		tmp = fma(Float64(27.0 * a), b, fma(Float64(Float64(-9.0 * y) * t), z, Float64(2.0 * x)));
	else
		tmp = fma(Float64(Float64(z * y) * t), -9.0, 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[t, 2.2e+27], N[(N[(27.0 * a), $MachinePrecision] * b + N[(N[(N[(-9.0 * y), $MachinePrecision] * t), $MachinePrecision] * z + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] * -9.0 + N[(N[(N[(b * a), $MachinePrecision] * 27.0 + x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.1999999999999999e27

   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. +-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. lower-fma.f64 94.9

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 94.9

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

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

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. lower-fma.f64 N/A

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

   4. Applied rewrites 96.9%

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

   if 2.1999999999999999e27 < t

   1. Initial program 99.9%

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

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

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

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

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

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

      11. associate-*l* N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 89.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lower-*.f64 99.8

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

      18. lift-fma.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 99.8

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 99.8

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

      26. lower-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. count-2-rev N/A

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

      9. associate-+r+ N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 99.9

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

   8. Applied rewrites 99.9%

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

Alternative 2: 86.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+66} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+95}\right):\\ \;\;\;\;\mathsf{fma}\left(b \cdot 27, a, \left(\left(-9 \cdot y\right) \cdot z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(a \cdot 27\right) \cdot b\right)\\ \end{array} \end{array} \]

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
 (let* ((t_1 (* (* (* y 9.0) z) t)))
   (if (or (<= t_1 -2e+66) (not (<= t_1 2e+95)))
     (fma (* b 27.0) a (* (* (* -9.0 y) z) t))
     (fma 2.0 x (* (* a 27.0) 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 t_1 = ((y * 9.0) * z) * t;
	double tmp;
	if ((t_1 <= -2e+66) || !(t_1 <= 2e+95)) {
		tmp = fma((b * 27.0), a, (((-9.0 * y) * z) * t));
	} else {
		tmp = fma(2.0, x, ((a * 27.0) * 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)
	t_1 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if ((t_1 <= -2e+66) || !(t_1 <= 2e+95))
		tmp = fma(Float64(b * 27.0), a, Float64(Float64(Float64(-9.0 * y) * z) * t));
	else
		tmp = fma(2.0, x, Float64(Float64(a * 27.0) * 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_] := Block[{t$95$1 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+66], N[Not[LessEqual[t$95$1, 2e+95]], $MachinePrecision]], N[(N[(b * 27.0), $MachinePrecision] * a + N[(N[(N[(-9.0 * y), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 * x + N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -1.99999999999999989e66 or 2.00000000000000004e95 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 91.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 82.7

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

   5. Applied rewrites 82.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 83.6

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

   7. Applied rewrites 86.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 83.5

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 83.5

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

   9. Applied rewrites 83.5%

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

   if -1.99999999999999989e66 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 2.00000000000000004e95

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 88.4

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

   5. Applied rewrites 88.4%

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

   7. Applied rewrites 88.4%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \leq -2 \cdot 10^{+66} \lor \neg \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \leq 2 \cdot 10^{+95}\right):\\ \;\;\;\;\mathsf{fma}\left(b \cdot 27, a, \left(\left(-9 \cdot y\right) \cdot z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(a \cdot 27\right) \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+66} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+95}\right):\\ \;\;\;\;\mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(b \cdot a\right) \cdot 27\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(a \cdot 27\right) \cdot b\right)\\ \end{array} \end{array} \]

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
 (let* ((t_1 (* (* (* y 9.0) z) t)))
   (if (or (<= t_1 -2e+66) (not (<= t_1 2e+95)))
     (fma -9.0 (* (* z y) t) (* (* b a) 27.0))
     (fma 2.0 x (* (* a 27.0) 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 t_1 = ((y * 9.0) * z) * t;
	double tmp;
	if ((t_1 <= -2e+66) || !(t_1 <= 2e+95)) {
		tmp = fma(-9.0, ((z * y) * t), ((b * a) * 27.0));
	} else {
		tmp = fma(2.0, x, ((a * 27.0) * 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)
	t_1 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if ((t_1 <= -2e+66) || !(t_1 <= 2e+95))
		tmp = fma(-9.0, Float64(Float64(z * y) * t), Float64(Float64(b * a) * 27.0));
	else
		tmp = fma(2.0, x, Float64(Float64(a * 27.0) * 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_] := Block[{t$95$1 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+66], N[Not[LessEqual[t$95$1, 2e+95]], $MachinePrecision]], N[(-9.0 * N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] + N[(N[(b * a), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision], N[(2.0 * x + N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -1.99999999999999989e66 or 2.00000000000000004e95 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 91.7%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 82.7

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

   5. Applied rewrites 82.7%

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

   if -1.99999999999999989e66 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 2.00000000000000004e95

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 88.4

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

   5. Applied rewrites 88.4%

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

   7. Applied rewrites 88.4%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \leq -2 \cdot 10^{+66} \lor \neg \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \leq 2 \cdot 10^{+95}\right):\\ \;\;\;\;\mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(b \cdot a\right) \cdot 27\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(a \cdot 27\right) \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+66} \lor \neg \left(t\_1 \leq 10^{+231}\right):\\ \;\;\;\;\mathsf{fma}\left(-9, \left(t \cdot z\right) \cdot y, \left(b \cdot a\right) \cdot 27\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(a \cdot 27\right) \cdot b\right)\\ \end{array} \end{array} \]

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
 (let* ((t_1 (* (* (* y 9.0) z) t)))
   (if (or (<= t_1 -2e+66) (not (<= t_1 1e+231)))
     (fma -9.0 (* (* t z) y) (* (* b a) 27.0))
     (fma 2.0 x (* (* a 27.0) 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 t_1 = ((y * 9.0) * z) * t;
	double tmp;
	if ((t_1 <= -2e+66) || !(t_1 <= 1e+231)) {
		tmp = fma(-9.0, ((t * z) * y), ((b * a) * 27.0));
	} else {
		tmp = fma(2.0, x, ((a * 27.0) * 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)
	t_1 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if ((t_1 <= -2e+66) || !(t_1 <= 1e+231))
		tmp = fma(-9.0, Float64(Float64(t * z) * y), Float64(Float64(b * a) * 27.0));
	else
		tmp = fma(2.0, x, Float64(Float64(a * 27.0) * 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_] := Block[{t$95$1 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+66], N[Not[LessEqual[t$95$1, 1e+231]], $MachinePrecision]], N[(-9.0 * N[(N[(t * z), $MachinePrecision] * y), $MachinePrecision] + N[(N[(b * a), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision], N[(2.0 * x + N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -1.99999999999999989e66 or 1.0000000000000001e231 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 90.0%

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

   3. Taylor expanded in x around 0

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

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

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

      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 85.1

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

   5. Applied rewrites 85.1%

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

   7. Applied rewrites 86.3%

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

   if -1.99999999999999989e66 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 1.0000000000000001e231

   1. Initial program 99.2%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 85.1

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

   5. Applied rewrites 85.1%

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

   7. Applied rewrites 85.1%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \leq -2 \cdot 10^{+66} \lor \neg \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \leq 10^{+231}\right):\\ \;\;\;\;\mathsf{fma}\left(-9, \left(t \cdot z\right) \cdot y, \left(b \cdot a\right) \cdot 27\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(a \cdot 27\right) \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+66} \lor \neg \left(t\_1 \leq 10^{+231}\right):\\ \;\;\;\;\left(\left(t \cdot z\right) \cdot -9\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(a \cdot 27\right) \cdot b\right)\\ \end{array} \end{array} \]

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
 (let* ((t_1 (* (* (* y 9.0) z) t)))
   (if (or (<= t_1 -2e+66) (not (<= t_1 1e+231)))
     (* (* (* t z) -9.0) y)
     (fma 2.0 x (* (* a 27.0) 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 t_1 = ((y * 9.0) * z) * t;
	double tmp;
	if ((t_1 <= -2e+66) || !(t_1 <= 1e+231)) {
		tmp = ((t * z) * -9.0) * y;
	} else {
		tmp = fma(2.0, x, ((a * 27.0) * 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)
	t_1 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if ((t_1 <= -2e+66) || !(t_1 <= 1e+231))
		tmp = Float64(Float64(Float64(t * z) * -9.0) * y);
	else
		tmp = fma(2.0, x, Float64(Float64(a * 27.0) * 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_] := Block[{t$95$1 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+66], N[Not[LessEqual[t$95$1, 1e+231]], $MachinePrecision]], N[(N[(N[(t * z), $MachinePrecision] * -9.0), $MachinePrecision] * y), $MachinePrecision], N[(2.0 * x + N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -1.99999999999999989e66 or 1.0000000000000001e231 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 90.0%

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

   3. Taylor expanded in x around 0

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

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

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

      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 85.1

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 86.2%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 75.7%

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

   if -1.99999999999999989e66 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 1.0000000000000001e231

   1. Initial program 99.2%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 85.1

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

   5. Applied rewrites 85.1%

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

   7. Applied rewrites 85.1%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \leq -2 \cdot 10^{+66} \lor \neg \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \leq 10^{+231}\right):\\ \;\;\;\;\left(\left(t \cdot z\right) \cdot -9\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(a \cdot 27\right) \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+66} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+95}\right):\\ \;\;\;\;\left(\left(t \cdot z\right) \cdot -9\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + x\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* y 9.0) z) t)))
   (if (or (<= t_1 -2e+66) (not (<= t_1 2e+95)))
     (* (* (* t z) -9.0) y)
     (+ 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 = ((y * 9.0) * z) * t;
	double tmp;
	if ((t_1 <= -2e+66) || !(t_1 <= 2e+95)) {
		tmp = ((t * z) * -9.0) * y;
	} else {
		tmp = x + x;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((y * 9.0d0) * z) * t
    if ((t_1 <= (-2d+66)) .or. (.not. (t_1 <= 2d+95))) then
        tmp = ((t * z) * (-9.0d0)) * y
    else
        tmp = 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 t_1 = ((y * 9.0) * z) * t;
	double tmp;
	if ((t_1 <= -2e+66) || !(t_1 <= 2e+95)) {
		tmp = ((t * z) * -9.0) * y;
	} else {
		tmp = 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):
	t_1 = ((y * 9.0) * z) * t
	tmp = 0
	if (t_1 <= -2e+66) or not (t_1 <= 2e+95):
		tmp = ((t * z) * -9.0) * y
	else:
		tmp = 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(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if ((t_1 <= -2e+66) || !(t_1 <= 2e+95))
		tmp = Float64(Float64(Float64(t * z) * -9.0) * y);
	else
		tmp = 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)
	t_1 = ((y * 9.0) * z) * t;
	tmp = 0.0;
	if ((t_1 <= -2e+66) || ~((t_1 <= 2e+95)))
		tmp = ((t * z) * -9.0) * y;
	else
		tmp = x + x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+66], N[Not[LessEqual[t$95$1, 2e+95]], $MachinePrecision]], N[(N[(N[(t * z), $MachinePrecision] * -9.0), $MachinePrecision] * y), $MachinePrecision], N[(x + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -1.99999999999999989e66 or 2.00000000000000004e95 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 91.7%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 82.7

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

   5. Applied rewrites 82.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 78.7%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 66.7%

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

   if -1.99999999999999989e66 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 2.00000000000000004e95

   1. Initial program 99.1%

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

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

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

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

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

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

      11. associate-*l* N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lower-*.f64 99.2

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

      18. lift-fma.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 99.2

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 99.2

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

      26. lower-*.f64 99.2

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

   6. Applied rewrites 99.2%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 45.8

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

   9. Applied rewrites 45.8%

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

   11. Applied rewrites 45.8%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \leq -2 \cdot 10^{+66} \lor \neg \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \leq 2 \cdot 10^{+95}\right):\\ \;\;\;\;\left(\left(t \cdot z\right) \cdot -9\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.5% accurate, 0.8× speedup

Math

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

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* (* y 9.0) z) 1e+182)
   (fma (* (* z y) t) -9.0 (+ (fma (* b a) 27.0 x) x))
   (fma (* t z) (* -9.0 y) (fma (* b 27.0) a (+ 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 (((y * 9.0) * z) <= 1e+182) {
		tmp = fma(((z * y) * t), -9.0, (fma((b * a), 27.0, x) + x));
	} else {
		tmp = fma((t * z), (-9.0 * y), fma((b * 27.0), a, (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(y * 9.0) * z) <= 1e+182)
		tmp = fma(Float64(Float64(z * y) * t), -9.0, Float64(fma(Float64(b * a), 27.0, x) + x));
	else
		tmp = fma(Float64(t * z), Float64(-9.0 * y), fma(Float64(b * 27.0), a, Float64(x + x)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision], 1e+182], N[(N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] * -9.0 + N[(N[(N[(b * a), $MachinePrecision] * 27.0 + x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[(t * z), $MachinePrecision] * N[(-9.0 * y), $MachinePrecision] + N[(N[(b * 27.0), $MachinePrecision] * a + N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y #s(literal 9 binary64)) z) < 1.0000000000000001e182

   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. lift-*.f64 N/A

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

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

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

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

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

      11. associate-*l* N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 93.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lower-*.f64 97.2

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

      18. lift-fma.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 97.2

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 97.2

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

      26. lower-*.f64 97.2

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

   6. Applied rewrites 97.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. count-2-rev N/A

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

      9. associate-+r+ N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 97.2

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

   8. Applied rewrites 97.2%

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

   if 1.0000000000000001e182 < (*.f64 (*.f64 y #s(literal 9 binary64)) z)

   1. Initial program 86.0%

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

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

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

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 94.4%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 94.4

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

   6. Applied rewrites 94.4%

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

Alternative 8: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{-34}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, -9 \cdot y, \mathsf{fma}\left(b \cdot 27, a, x + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-9 \cdot y\right) \cdot t, z, \mathsf{fma}\left(27, a \cdot b, x\right) + x\right)\\ \end{array} \end{array} \]

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 (<= y -6.8e-34)
   (fma (* t z) (* -9.0 y) (fma (* b 27.0) a (+ x x)))
   (fma (* (* -9.0 y) t) z (+ (fma 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 (y <= -6.8e-34) {
		tmp = fma((t * z), (-9.0 * y), fma((b * 27.0), a, (x + x)));
	} else {
		tmp = fma(((-9.0 * y) * t), z, (fma(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 (y <= -6.8e-34)
		tmp = fma(Float64(t * z), Float64(-9.0 * y), fma(Float64(b * 27.0), a, Float64(x + x)));
	else
		tmp = fma(Float64(Float64(-9.0 * y) * t), z, Float64(fma(27.0, Float64(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[y, -6.8e-34], N[(N[(t * z), $MachinePrecision] * N[(-9.0 * y), $MachinePrecision] + N[(N[(b * 27.0), $MachinePrecision] * a + N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-9.0 * y), $MachinePrecision] * t), $MachinePrecision] * z + N[(N[(27.0 * N[(a * b), $MachinePrecision] + x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.8000000000000001e-34

   1. Initial program 95.9%

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

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

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

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 97.2%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 97.2

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

   6. Applied rewrites 97.2%

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

   if -6.8000000000000001e-34 < y

   1. Initial program 95.6%

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

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

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

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

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

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

      11. associate-*l* N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 92.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lower-*.f64 95.6

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

      18. lift-fma.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 95.6

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 95.6

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

      26. lower-*.f64 95.6

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

   6. Applied rewrites 95.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. count-2-rev N/A

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

      9. associate-+r+ N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 95.6

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

   8. Applied rewrites 95.6%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 96.7

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

      14. lift-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 96.7

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 96.7

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

   10. Applied rewrites 96.7%

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

Alternative 9: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 5.8 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, -9 \cdot y, \mathsf{fma}\left(b \cdot 27, a, x + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(b \cdot a\right) \cdot 27\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
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 5.8e+101)
   (fma (* t z) (* -9.0 y) (fma (* b 27.0) a (+ x x)))
   (fma -9.0 (* (* z y) t) (* (* 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 tmp;
	if (z <= 5.8e+101) {
		tmp = fma((t * z), (-9.0 * y), fma((b * 27.0), a, (x + x)));
	} else {
		tmp = fma(-9.0, ((z * y) * t), ((b * a) * 27.0));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
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 <= 5.8e+101)
		tmp = fma(Float64(t * z), Float64(-9.0 * y), fma(Float64(b * 27.0), a, Float64(x + x)));
	else
		tmp = fma(-9.0, Float64(Float64(z * y) * t), Float64(Float64(b * a) * 27.0));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
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, 5.8e+101], N[(N[(t * z), $MachinePrecision] * N[(-9.0 * y), $MachinePrecision] + N[(N[(b * 27.0), $MachinePrecision] * a + N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-9.0 * N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] + N[(N[(b * a), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 5.79999999999999974e101

   1. Initial program 95.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 96.1%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 96.1

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

   6. Applied rewrites 96.1%

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

   if 5.79999999999999974e101 < z

   1. Initial program 95.8%

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

   3. Taylor expanded in x around 0

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

      1. fp-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 64.2

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

   5. Applied rewrites 64.2%

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

Alternative 10: 30.8% 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 95.7%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

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

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

   5. +-commutative N/A

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

   6. associate-+l+ N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

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

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

   11. associate-*l* N/A

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

   12. +-commutative N/A

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

   13. lower-fma.f64 N/A

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

4. Applied rewrites 93.5%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. associate-*l* N/A

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

   15. *-commutative N/A

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

   16. lift-*.f64 N/A

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

   17. lower-*.f64 95.7

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

   18. lift-fma.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-fma.f64 95.7

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

   21. lift-*.f64 N/A

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

   22. *-commutative N/A

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

   23. lower-*.f64 95.7

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

   24. lift-*.f64 N/A

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

   25. *-commutative N/A

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

   26. lower-*.f64 95.7

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

6. Applied rewrites 95.7%

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

7. Taylor expanded in x around inf

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

   1. lower-*.f64 29.8

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

9. Applied rewrites 29.8%

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

11. Applied rewrites 29.8%

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

Developer Target 1: 95.2% 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 2024338 
(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)))