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

Percentage Accurate: 95.6% → 97.9%

Time: 19.8s

Alternatives: 19

Speedup: 0.9×

• 27.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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: 95.6% 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: 97.9% accurate, 0.7× speedup

Math

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

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* z (* 9.0 y))))
   (if (<= t_1 5e+94)
     (+ (* (* 27.0 a) b) (- (* 2.0 x) (* t t_1)))
     (fma (* (* -9.0 t) y) z (fma (* b a) 27.0 (* 2.0 x))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = z * (9.0 * y);
	double tmp;
	if (t_1 <= 5e+94) {
		tmp = ((27.0 * a) * b) + ((2.0 * x) - (t * t_1));
	} else {
		tmp = fma(((-9.0 * t) * y), z, fma((b * a), 27.0, (2.0 * x)));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(z * Float64(9.0 * y))
	tmp = 0.0
	if (t_1 <= 5e+94)
		tmp = Float64(Float64(Float64(27.0 * a) * b) + Float64(Float64(2.0 * x) - Float64(t * t_1)));
	else
		tmp = fma(Float64(Float64(-9.0 * t) * y), z, fma(Float64(b * a), 27.0, Float64(2.0 * x)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(9.0 * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+94], N[(N[(N[(27.0 * a), $MachinePrecision] * b), $MachinePrecision] + N[(N[(2.0 * x), $MachinePrecision] - N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-9.0 * t), $MachinePrecision] * y), $MachinePrecision] * z + N[(N[(b * a), $MachinePrecision] * 27.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y #s(literal 9 binary64)) z) < 5.0000000000000001e94

   1. Initial program 98.5%

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

   if 5.0000000000000001e94 < (*.f64 (*.f64 y #s(literal 9 binary64)) z)

   1. Initial program 82.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-*.f64 N/A

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

   4. Applied rewrites 84.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 95.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-neg.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 95.3

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

      14. lift-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. lower-fma.f64 95.3

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

      26. lower-*.f64 95.3

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

   6. Applied rewrites 95.3%

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

4. Final simplification 97.9%

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

Alternative 2: 54.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(27 \cdot a\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+169}:\\ \;\;\;\;\left(b \cdot 27\right) \cdot a\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-242}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \left(-9 \cdot t\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-199}:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+100}:\\ \;\;\;\;\left(\left(-9 \cdot y\right) \cdot z\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* 27.0 a) b)))
   (if (<= t_1 -2e+169)
     (* (* b 27.0) a)
     (if (<= t_1 -1e-242)
       (* (* z y) (* -9.0 t))
       (if (<= t_1 2e-199)
         (* 2.0 x)
         (if (<= t_1 2e+100) (* (* (* -9.0 y) z) t) t_1))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (27.0 * a) * b;
	double tmp;
	if (t_1 <= -2e+169) {
		tmp = (b * 27.0) * a;
	} else if (t_1 <= -1e-242) {
		tmp = (z * y) * (-9.0 * t);
	} else if (t_1 <= 2e-199) {
		tmp = 2.0 * x;
	} else if (t_1 <= 2e+100) {
		tmp = ((-9.0 * y) * z) * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (27.0d0 * a) * b
    if (t_1 <= (-2d+169)) then
        tmp = (b * 27.0d0) * a
    else if (t_1 <= (-1d-242)) then
        tmp = (z * y) * ((-9.0d0) * t)
    else if (t_1 <= 2d-199) then
        tmp = 2.0d0 * x
    else if (t_1 <= 2d+100) then
        tmp = (((-9.0d0) * y) * z) * t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b;
public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (27.0 * a) * b;
	double tmp;
	if (t_1 <= -2e+169) {
		tmp = (b * 27.0) * a;
	} else if (t_1 <= -1e-242) {
		tmp = (z * y) * (-9.0 * t);
	} else if (t_1 <= 2e-199) {
		tmp = 2.0 * x;
	} else if (t_1 <= 2e+100) {
		tmp = ((-9.0 * y) * z) * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	t_1 = (27.0 * a) * b
	tmp = 0
	if t_1 <= -2e+169:
		tmp = (b * 27.0) * a
	elif t_1 <= -1e-242:
		tmp = (z * y) * (-9.0 * t)
	elif t_1 <= 2e-199:
		tmp = 2.0 * x
	elif t_1 <= 2e+100:
		tmp = ((-9.0 * y) * z) * t
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(27.0 * a) * b)
	tmp = 0.0
	if (t_1 <= -2e+169)
		tmp = Float64(Float64(b * 27.0) * a);
	elseif (t_1 <= -1e-242)
		tmp = Float64(Float64(z * y) * Float64(-9.0 * t));
	elseif (t_1 <= 2e-199)
		tmp = Float64(2.0 * x);
	elseif (t_1 <= 2e+100)
		tmp = Float64(Float64(Float64(-9.0 * y) * z) * t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (27.0 * a) * b;
	tmp = 0.0;
	if (t_1 <= -2e+169)
		tmp = (b * 27.0) * a;
	elseif (t_1 <= -1e-242)
		tmp = (z * y) * (-9.0 * t);
	elseif (t_1 <= 2e-199)
		tmp = 2.0 * x;
	elseif (t_1 <= 2e+100)
		tmp = ((-9.0 * y) * z) * t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(27.0 * a), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+169], N[(N[(b * 27.0), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[t$95$1, -1e-242], N[(N[(z * y), $MachinePrecision] * N[(-9.0 * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-199], N[(2.0 * x), $MachinePrecision], If[LessEqual[t$95$1, 2e+100], N[(N[(N[(-9.0 * y), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 (*.f64 a #s(literal 27 binary64)) b) < -1.99999999999999987e169

   1. Initial program 92.9%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 87.6

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

   5. Applied rewrites 87.6%

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

   7. Applied rewrites 87.7%

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

   if -1.99999999999999987e169 < (*.f64 (*.f64 a #s(literal 27 binary64)) b) < -1e-242

   1. Initial program 98.6%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 52.2

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

   5. Applied rewrites 52.2%

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

   7. Applied rewrites 52.2%

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

   if -1e-242 < (*.f64 (*.f64 a #s(literal 27 binary64)) b) < 1.99999999999999996e-199

   1. Initial program 97.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 74.9

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

   5. Applied rewrites 74.9%

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

   if 1.99999999999999996e-199 < (*.f64 (*.f64 a #s(literal 27 binary64)) b) < 2.00000000000000003e100

   1. Initial program 95.3%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 54.4

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

   5. Applied rewrites 54.4%

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

   7. Applied rewrites 54.6%

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

   if 2.00000000000000003e100 < (*.f64 (*.f64 a #s(literal 27 binary64)) b)

   1. Initial program 92.1%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 74.5

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

   5. Applied rewrites 74.5%

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

   7. Applied rewrites 74.5%

      \[\leadsto \color{blue}{\left(27 \cdot a\right) \cdot b} \]
3. Recombined 5 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(27 \cdot a\right) \cdot b \leq -2 \cdot 10^{+169}:\\ \;\;\;\;\left(b \cdot 27\right) \cdot a\\ \mathbf{elif}\;\left(27 \cdot a\right) \cdot b \leq -1 \cdot 10^{-242}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \left(-9 \cdot t\right)\\ \mathbf{elif}\;\left(27 \cdot a\right) \cdot b \leq 2 \cdot 10^{-199}:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;\left(27 \cdot a\right) \cdot b \leq 2 \cdot 10^{+100}:\\ \;\;\;\;\left(\left(-9 \cdot y\right) \cdot z\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(27 \cdot a\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 54.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(27 \cdot a\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+169}:\\ \;\;\;\;\left(b \cdot 27\right) \cdot a\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-242}:\\ \;\;\;\;\left(\left(z \cdot y\right) \cdot t\right) \cdot -9\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-199}:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+100}:\\ \;\;\;\;\left(\left(-9 \cdot y\right) \cdot z\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* 27.0 a) b)))
   (if (<= t_1 -2e+169)
     (* (* b 27.0) a)
     (if (<= t_1 -1e-242)
       (* (* (* z y) t) -9.0)
       (if (<= t_1 2e-199)
         (* 2.0 x)
         (if (<= t_1 2e+100) (* (* (* -9.0 y) z) t) t_1))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (27.0 * a) * b;
	double tmp;
	if (t_1 <= -2e+169) {
		tmp = (b * 27.0) * a;
	} else if (t_1 <= -1e-242) {
		tmp = ((z * y) * t) * -9.0;
	} else if (t_1 <= 2e-199) {
		tmp = 2.0 * x;
	} else if (t_1 <= 2e+100) {
		tmp = ((-9.0 * y) * z) * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (27.0d0 * a) * b
    if (t_1 <= (-2d+169)) then
        tmp = (b * 27.0d0) * a
    else if (t_1 <= (-1d-242)) then
        tmp = ((z * y) * t) * (-9.0d0)
    else if (t_1 <= 2d-199) then
        tmp = 2.0d0 * x
    else if (t_1 <= 2d+100) then
        tmp = (((-9.0d0) * y) * z) * t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b;
public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (27.0 * a) * b;
	double tmp;
	if (t_1 <= -2e+169) {
		tmp = (b * 27.0) * a;
	} else if (t_1 <= -1e-242) {
		tmp = ((z * y) * t) * -9.0;
	} else if (t_1 <= 2e-199) {
		tmp = 2.0 * x;
	} else if (t_1 <= 2e+100) {
		tmp = ((-9.0 * y) * z) * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	t_1 = (27.0 * a) * b
	tmp = 0
	if t_1 <= -2e+169:
		tmp = (b * 27.0) * a
	elif t_1 <= -1e-242:
		tmp = ((z * y) * t) * -9.0
	elif t_1 <= 2e-199:
		tmp = 2.0 * x
	elif t_1 <= 2e+100:
		tmp = ((-9.0 * y) * z) * t
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(27.0 * a) * b)
	tmp = 0.0
	if (t_1 <= -2e+169)
		tmp = Float64(Float64(b * 27.0) * a);
	elseif (t_1 <= -1e-242)
		tmp = Float64(Float64(Float64(z * y) * t) * -9.0);
	elseif (t_1 <= 2e-199)
		tmp = Float64(2.0 * x);
	elseif (t_1 <= 2e+100)
		tmp = Float64(Float64(Float64(-9.0 * y) * z) * t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (27.0 * a) * b;
	tmp = 0.0;
	if (t_1 <= -2e+169)
		tmp = (b * 27.0) * a;
	elseif (t_1 <= -1e-242)
		tmp = ((z * y) * t) * -9.0;
	elseif (t_1 <= 2e-199)
		tmp = 2.0 * x;
	elseif (t_1 <= 2e+100)
		tmp = ((-9.0 * y) * z) * t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(27.0 * a), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+169], N[(N[(b * 27.0), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[t$95$1, -1e-242], N[(N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] * -9.0), $MachinePrecision], If[LessEqual[t$95$1, 2e-199], N[(2.0 * x), $MachinePrecision], If[LessEqual[t$95$1, 2e+100], N[(N[(N[(-9.0 * y), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 (*.f64 a #s(literal 27 binary64)) b) < -1.99999999999999987e169

   1. Initial program 92.9%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 87.6

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

   5. Applied rewrites 87.6%

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

   7. Applied rewrites 87.7%

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

   if -1.99999999999999987e169 < (*.f64 (*.f64 a #s(literal 27 binary64)) b) < -1e-242

   1. Initial program 98.6%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 52.2

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

   5. Applied rewrites 52.2%

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

   if -1e-242 < (*.f64 (*.f64 a #s(literal 27 binary64)) b) < 1.99999999999999996e-199

   1. Initial program 97.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 74.9

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

   5. Applied rewrites 74.9%

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

   if 1.99999999999999996e-199 < (*.f64 (*.f64 a #s(literal 27 binary64)) b) < 2.00000000000000003e100

   1. Initial program 95.3%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 54.4

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

   5. Applied rewrites 54.4%

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

   7. Applied rewrites 54.6%

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

   if 2.00000000000000003e100 < (*.f64 (*.f64 a #s(literal 27 binary64)) b)

   1. Initial program 92.1%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 74.5

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

   5. Applied rewrites 74.5%

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

   7. Applied rewrites 74.5%

      \[\leadsto \color{blue}{\left(27 \cdot a\right) \cdot b} \]
3. Recombined 5 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(27 \cdot a\right) \cdot b \leq -2 \cdot 10^{+169}:\\ \;\;\;\;\left(b \cdot 27\right) \cdot a\\ \mathbf{elif}\;\left(27 \cdot a\right) \cdot b \leq -1 \cdot 10^{-242}:\\ \;\;\;\;\left(\left(z \cdot y\right) \cdot t\right) \cdot -9\\ \mathbf{elif}\;\left(27 \cdot a\right) \cdot b \leq 2 \cdot 10^{-199}:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;\left(27 \cdot a\right) \cdot b \leq 2 \cdot 10^{+100}:\\ \;\;\;\;\left(\left(-9 \cdot y\right) \cdot z\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(27 \cdot a\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(z \cdot y\right) \cdot t\right) \cdot -9\\ t_2 := \left(27 \cdot a\right) \cdot b\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+169}:\\ \;\;\;\;\left(b \cdot 27\right) \cdot a\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-242}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-199}:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+100}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* z y) t) -9.0)) (t_2 (* (* 27.0 a) b)))
   (if (<= t_2 -2e+169)
     (* (* b 27.0) a)
     (if (<= t_2 -1e-242)
       t_1
       (if (<= t_2 2e-199) (* 2.0 x) (if (<= t_2 2e+100) t_1 t_2))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((z * y) * t) * -9.0;
	double t_2 = (27.0 * a) * b;
	double tmp;
	if (t_2 <= -2e+169) {
		tmp = (b * 27.0) * a;
	} else if (t_2 <= -1e-242) {
		tmp = t_1;
	} else if (t_2 <= 2e-199) {
		tmp = 2.0 * x;
	} else if (t_2 <= 2e+100) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((z * y) * t) * (-9.0d0)
    t_2 = (27.0d0 * a) * b
    if (t_2 <= (-2d+169)) then
        tmp = (b * 27.0d0) * a
    else if (t_2 <= (-1d-242)) then
        tmp = t_1
    else if (t_2 <= 2d-199) then
        tmp = 2.0d0 * x
    else if (t_2 <= 2d+100) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b;
public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((z * y) * t) * -9.0;
	double t_2 = (27.0 * a) * b;
	double tmp;
	if (t_2 <= -2e+169) {
		tmp = (b * 27.0) * a;
	} else if (t_2 <= -1e-242) {
		tmp = t_1;
	} else if (t_2 <= 2e-199) {
		tmp = 2.0 * x;
	} else if (t_2 <= 2e+100) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	t_1 = ((z * y) * t) * -9.0
	t_2 = (27.0 * a) * b
	tmp = 0
	if t_2 <= -2e+169:
		tmp = (b * 27.0) * a
	elif t_2 <= -1e-242:
		tmp = t_1
	elif t_2 <= 2e-199:
		tmp = 2.0 * x
	elif t_2 <= 2e+100:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(z * y) * t) * -9.0)
	t_2 = Float64(Float64(27.0 * a) * b)
	tmp = 0.0
	if (t_2 <= -2e+169)
		tmp = Float64(Float64(b * 27.0) * a);
	elseif (t_2 <= -1e-242)
		tmp = t_1;
	elseif (t_2 <= 2e-199)
		tmp = Float64(2.0 * x);
	elseif (t_2 <= 2e+100)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp_2 = code(x, y, z, t, a, b)
	t_1 = ((z * y) * t) * -9.0;
	t_2 = (27.0 * a) * b;
	tmp = 0.0;
	if (t_2 <= -2e+169)
		tmp = (b * 27.0) * a;
	elseif (t_2 <= -1e-242)
		tmp = t_1;
	elseif (t_2 <= 2e-199)
		tmp = 2.0 * x;
	elseif (t_2 <= 2e+100)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] * -9.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(27.0 * a), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+169], N[(N[(b * 27.0), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[t$95$2, -1e-242], t$95$1, If[LessEqual[t$95$2, 2e-199], N[(2.0 * x), $MachinePrecision], If[LessEqual[t$95$2, 2e+100], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 a #s(literal 27 binary64)) b) < -1.99999999999999987e169

   1. Initial program 92.9%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 87.6

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

   5. Applied rewrites 87.6%

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

   7. Applied rewrites 87.7%

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

   if -1.99999999999999987e169 < (*.f64 (*.f64 a #s(literal 27 binary64)) b) < -1e-242 or 1.99999999999999996e-199 < (*.f64 (*.f64 a #s(literal 27 binary64)) b) < 2.00000000000000003e100

   1. Initial program 97.1%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 53.2

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

   5. Applied rewrites 53.2%

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

   if -1e-242 < (*.f64 (*.f64 a #s(literal 27 binary64)) b) < 1.99999999999999996e-199

   1. Initial program 97.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 74.9

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

   5. Applied rewrites 74.9%

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

   if 2.00000000000000003e100 < (*.f64 (*.f64 a #s(literal 27 binary64)) b)

   1. Initial program 92.1%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 74.5

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

   5. Applied rewrites 74.5%

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

   7. Applied rewrites 74.5%

      \[\leadsto \color{blue}{\left(27 \cdot a\right) \cdot b} \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(27 \cdot a\right) \cdot b \leq -2 \cdot 10^{+169}:\\ \;\;\;\;\left(b \cdot 27\right) \cdot a\\ \mathbf{elif}\;\left(27 \cdot a\right) \cdot b \leq -1 \cdot 10^{-242}:\\ \;\;\;\;\left(\left(z \cdot y\right) \cdot t\right) \cdot -9\\ \mathbf{elif}\;\left(27 \cdot a\right) \cdot b \leq 2 \cdot 10^{-199}:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;\left(27 \cdot a\right) \cdot b \leq 2 \cdot 10^{+100}:\\ \;\;\;\;\left(\left(z \cdot y\right) \cdot t\right) \cdot -9\\ \mathbf{else}:\\ \;\;\;\;\left(27 \cdot a\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(t \cdot y\right) \cdot z\right) \cdot -9\\ t_2 := \left(27 \cdot a\right) \cdot b\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+105}:\\ \;\;\;\;\left(b \cdot 27\right) \cdot a\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-242}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-199}:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+100}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* t y) z) -9.0)) (t_2 (* (* 27.0 a) b)))
   (if (<= t_2 -5e+105)
     (* (* b 27.0) a)
     (if (<= t_2 -1e-242)
       t_1
       (if (<= t_2 2e-199) (* 2.0 x) (if (<= t_2 2e+100) t_1 t_2))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((t * y) * z) * -9.0;
	double t_2 = (27.0 * a) * b;
	double tmp;
	if (t_2 <= -5e+105) {
		tmp = (b * 27.0) * a;
	} else if (t_2 <= -1e-242) {
		tmp = t_1;
	} else if (t_2 <= 2e-199) {
		tmp = 2.0 * x;
	} else if (t_2 <= 2e+100) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((t * y) * z) * (-9.0d0)
    t_2 = (27.0d0 * a) * b
    if (t_2 <= (-5d+105)) then
        tmp = (b * 27.0d0) * a
    else if (t_2 <= (-1d-242)) then
        tmp = t_1
    else if (t_2 <= 2d-199) then
        tmp = 2.0d0 * x
    else if (t_2 <= 2d+100) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b;
public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((t * y) * z) * -9.0;
	double t_2 = (27.0 * a) * b;
	double tmp;
	if (t_2 <= -5e+105) {
		tmp = (b * 27.0) * a;
	} else if (t_2 <= -1e-242) {
		tmp = t_1;
	} else if (t_2 <= 2e-199) {
		tmp = 2.0 * x;
	} else if (t_2 <= 2e+100) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	t_1 = ((t * y) * z) * -9.0
	t_2 = (27.0 * a) * b
	tmp = 0
	if t_2 <= -5e+105:
		tmp = (b * 27.0) * a
	elif t_2 <= -1e-242:
		tmp = t_1
	elif t_2 <= 2e-199:
		tmp = 2.0 * x
	elif t_2 <= 2e+100:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(t * y) * z) * -9.0)
	t_2 = Float64(Float64(27.0 * a) * b)
	tmp = 0.0
	if (t_2 <= -5e+105)
		tmp = Float64(Float64(b * 27.0) * a);
	elseif (t_2 <= -1e-242)
		tmp = t_1;
	elseif (t_2 <= 2e-199)
		tmp = Float64(2.0 * x);
	elseif (t_2 <= 2e+100)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp_2 = code(x, y, z, t, a, b)
	t_1 = ((t * y) * z) * -9.0;
	t_2 = (27.0 * a) * b;
	tmp = 0.0;
	if (t_2 <= -5e+105)
		tmp = (b * 27.0) * a;
	elseif (t_2 <= -1e-242)
		tmp = t_1;
	elseif (t_2 <= 2e-199)
		tmp = 2.0 * x;
	elseif (t_2 <= 2e+100)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(t * y), $MachinePrecision] * z), $MachinePrecision] * -9.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(27.0 * a), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+105], N[(N[(b * 27.0), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[t$95$2, -1e-242], t$95$1, If[LessEqual[t$95$2, 2e-199], N[(2.0 * x), $MachinePrecision], If[LessEqual[t$95$2, 2e+100], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 a #s(literal 27 binary64)) b) < -5.00000000000000046e105

   1. Initial program 94.6%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 77.2

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

   5. Applied rewrites 77.2%

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

   7. Applied rewrites 77.3%

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

   if -5.00000000000000046e105 < (*.f64 (*.f64 a #s(literal 27 binary64)) b) < -1e-242 or 1.99999999999999996e-199 < (*.f64 (*.f64 a #s(literal 27 binary64)) b) < 2.00000000000000003e100

   1. Initial program 96.7%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 53.4

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

   5. Applied rewrites 53.4%

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

   7. Applied rewrites 52.7%

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

   if -1e-242 < (*.f64 (*.f64 a #s(literal 27 binary64)) b) < 1.99999999999999996e-199

   1. Initial program 97.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 74.9

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

   5. Applied rewrites 74.9%

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

   if 2.00000000000000003e100 < (*.f64 (*.f64 a #s(literal 27 binary64)) b)

   1. Initial program 92.1%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 74.5

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

   5. Applied rewrites 74.5%

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

   7. Applied rewrites 74.5%

      \[\leadsto \color{blue}{\left(27 \cdot a\right) \cdot b} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(27 \cdot a\right) \cdot b \leq -5 \cdot 10^{+105}:\\ \;\;\;\;\left(b \cdot 27\right) \cdot a\\ \mathbf{elif}\;\left(27 \cdot a\right) \cdot b \leq -1 \cdot 10^{-242}:\\ \;\;\;\;\left(\left(t \cdot y\right) \cdot z\right) \cdot -9\\ \mathbf{elif}\;\left(27 \cdot a\right) \cdot b \leq 2 \cdot 10^{-199}:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;\left(27 \cdot a\right) \cdot b \leq 2 \cdot 10^{+100}:\\ \;\;\;\;\left(\left(t \cdot y\right) \cdot z\right) \cdot -9\\ \mathbf{else}:\\ \;\;\;\;\left(27 \cdot a\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\left(t \cdot z\right) \cdot -9, y, \left(b \cdot a\right) \cdot 27\right)\\ t_2 := t \cdot \left(z \cdot \left(9 \cdot y\right)\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot 27, a, 2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (* (* t z) -9.0) y (* (* b a) 27.0)))
        (t_2 (* t (* z (* 9.0 y)))))
   (if (<= t_2 -1e-19)
     t_1
     (if (<= t_2 2e-42) (fma (* b 27.0) a (* 2.0 x)) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(((t * z) * -9.0), y, ((b * a) * 27.0));
	double t_2 = t * (z * (9.0 * y));
	double tmp;
	if (t_2 <= -1e-19) {
		tmp = t_1;
	} else if (t_2 <= 2e-42) {
		tmp = fma((b * 27.0), a, (2.0 * x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = fma(Float64(Float64(t * z) * -9.0), y, Float64(Float64(b * a) * 27.0))
	t_2 = Float64(t * Float64(z * Float64(9.0 * y)))
	tmp = 0.0
	if (t_2 <= -1e-19)
		tmp = t_1;
	elseif (t_2 <= 2e-42)
		tmp = fma(Float64(b * 27.0), a, Float64(2.0 * x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(t * z), $MachinePrecision] * -9.0), $MachinePrecision] * y + N[(N[(b * a), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(z * N[(9.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-19], t$95$1, If[LessEqual[t$95$2, 2e-42], N[(N[(b * 27.0), $MachinePrecision] * a + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -9.9999999999999998e-20 or 2.00000000000000008e-42 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 92.3%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 77.5

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

   5. Applied rewrites 77.5%

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

   if -9.9999999999999998e-20 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 2.00000000000000008e-42

   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. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 99.0

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

   5. Applied rewrites 99.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.1

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 99.1

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

   7. Applied rewrites 99.1%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot \left(9 \cdot y\right)\right) \leq -1 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(\left(t \cdot z\right) \cdot -9, y, \left(b \cdot a\right) \cdot 27\right)\\ \mathbf{elif}\;t \cdot \left(z \cdot \left(9 \cdot y\right)\right) \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot 27, a, 2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(t \cdot z\right) \cdot -9, y, \left(b \cdot a\right) \cdot 27\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := t \cdot \left(z \cdot \left(9 \cdot y\right)\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(\left(-t\right) \cdot 9, z \cdot y, 2 \cdot x\right)\\ \mathbf{elif}\;t\_1 \leq 1.5 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot 27, a, 2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-9 \cdot t\right) \cdot y, z, 2 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* t (* z (* 9.0 y)))))
   (if (<= t_1 -2e+87)
     (fma (* (- t) 9.0) (* z y) (* 2.0 x))
     (if (<= t_1 1.5e+41)
       (fma (* b 27.0) a (* 2.0 x))
       (fma (* (* -9.0 t) y) z (* 2.0 x))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t * (z * (9.0 * y));
	double tmp;
	if (t_1 <= -2e+87) {
		tmp = fma((-t * 9.0), (z * y), (2.0 * x));
	} else if (t_1 <= 1.5e+41) {
		tmp = fma((b * 27.0), a, (2.0 * x));
	} else {
		tmp = fma(((-9.0 * t) * y), z, (2.0 * x));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(t * Float64(z * Float64(9.0 * y)))
	tmp = 0.0
	if (t_1 <= -2e+87)
		tmp = fma(Float64(Float64(-t) * 9.0), Float64(z * y), Float64(2.0 * x));
	elseif (t_1 <= 1.5e+41)
		tmp = fma(Float64(b * 27.0), a, Float64(2.0 * x));
	else
		tmp = fma(Float64(Float64(-9.0 * t) * y), z, Float64(2.0 * x));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t * N[(z * N[(9.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+87], N[(N[((-t) * 9.0), $MachinePrecision] * N[(z * y), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.5e+41], N[(N[(b * 27.0), $MachinePrecision] * a + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-9.0 * t), $MachinePrecision] * y), $MachinePrecision] * z + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -1.9999999999999999e87

   1. Initial program 88.0%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-*.f64 N/A

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

   4. Applied rewrites 87.9%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 78.1

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

   7. Applied rewrites 78.1%

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

   if -1.9999999999999999e87 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 1.4999999999999999e41

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 91.5

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

   5. Applied rewrites 91.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 91.6

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 91.6

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

   7. Applied rewrites 91.6%

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

   if 1.4999999999999999e41 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 92.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-*.f64 N/A

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

   4. Applied rewrites 92.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 87.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-neg.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 87.1

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

      14. lift-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. lower-fma.f64 87.1

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

      26. lower-*.f64 87.1

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

   6. Applied rewrites 87.1%

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

   7. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 76.4

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

   9. Applied rewrites 76.4%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot \left(9 \cdot y\right)\right) \leq -2 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(\left(-t\right) \cdot 9, z \cdot y, 2 \cdot x\right)\\ \mathbf{elif}\;t \cdot \left(z \cdot \left(9 \cdot y\right)\right) \leq 1.5 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot 27, a, 2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-9 \cdot t\right) \cdot y, z, 2 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := t \cdot \left(z \cdot \left(9 \cdot y\right)\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(x, 2, \left(\left(z \cdot y\right) \cdot t\right) \cdot -9\right)\\ \mathbf{elif}\;t\_1 \leq 1.5 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot 27, a, 2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-9 \cdot t\right) \cdot y, z, 2 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* t (* z (* 9.0 y)))))
   (if (<= t_1 -2e+87)
     (fma x 2.0 (* (* (* z y) t) -9.0))
     (if (<= t_1 1.5e+41)
       (fma (* b 27.0) a (* 2.0 x))
       (fma (* (* -9.0 t) y) z (* 2.0 x))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t * (z * (9.0 * y));
	double tmp;
	if (t_1 <= -2e+87) {
		tmp = fma(x, 2.0, (((z * y) * t) * -9.0));
	} else if (t_1 <= 1.5e+41) {
		tmp = fma((b * 27.0), a, (2.0 * x));
	} else {
		tmp = fma(((-9.0 * t) * y), z, (2.0 * x));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(t * Float64(z * Float64(9.0 * y)))
	tmp = 0.0
	if (t_1 <= -2e+87)
		tmp = fma(x, 2.0, Float64(Float64(Float64(z * y) * t) * -9.0));
	elseif (t_1 <= 1.5e+41)
		tmp = fma(Float64(b * 27.0), a, Float64(2.0 * x));
	else
		tmp = fma(Float64(Float64(-9.0 * t) * y), z, Float64(2.0 * x));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t * N[(z * N[(9.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+87], N[(x * 2.0 + N[(N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] * -9.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.5e+41], N[(N[(b * 27.0), $MachinePrecision] * a + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-9.0 * t), $MachinePrecision] * y), $MachinePrecision] * z + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -1.9999999999999999e87

   1. Initial program 88.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 7.5

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

   5. Applied rewrites 7.5%

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

   6. Taylor expanded in b around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 78.0

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

   8. Applied rewrites 78.0%

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

   if -1.9999999999999999e87 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 1.4999999999999999e41

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 91.5

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

   5. Applied rewrites 91.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 91.6

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 91.6

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

   7. Applied rewrites 91.6%

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

   if 1.4999999999999999e41 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 92.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-*.f64 N/A

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

   4. Applied rewrites 92.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 87.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-neg.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 87.1

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

      14. lift-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. lower-fma.f64 87.1

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

      26. lower-*.f64 87.1

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

   6. Applied rewrites 87.1%

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

   7. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 76.4

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

   9. Applied rewrites 76.4%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot \left(9 \cdot y\right)\right) \leq -2 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(x, 2, \left(\left(z \cdot y\right) \cdot t\right) \cdot -9\right)\\ \mathbf{elif}\;t \cdot \left(z \cdot \left(9 \cdot y\right)\right) \leq 1.5 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot 27, a, 2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-9 \cdot t\right) \cdot y, z, 2 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, 2, \left(\left(z \cdot y\right) \cdot t\right) \cdot -9\right)\\ t_2 := t \cdot \left(z \cdot \left(9 \cdot y\right)\right)\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+87}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 1.5 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot 27, a, 2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma x 2.0 (* (* (* z y) t) -9.0))) (t_2 (* t (* z (* 9.0 y)))))
   (if (<= t_2 -2e+87)
     t_1
     (if (<= t_2 1.5e+41) (fma (* b 27.0) a (* 2.0 x)) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(x, 2.0, (((z * y) * t) * -9.0));
	double t_2 = t * (z * (9.0 * y));
	double tmp;
	if (t_2 <= -2e+87) {
		tmp = t_1;
	} else if (t_2 <= 1.5e+41) {
		tmp = fma((b * 27.0), a, (2.0 * x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = fma(x, 2.0, Float64(Float64(Float64(z * y) * t) * -9.0))
	t_2 = Float64(t * Float64(z * Float64(9.0 * y)))
	tmp = 0.0
	if (t_2 <= -2e+87)
		tmp = t_1;
	elseif (t_2 <= 1.5e+41)
		tmp = fma(Float64(b * 27.0), a, Float64(2.0 * x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * 2.0 + N[(N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] * -9.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(z * N[(9.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+87], t$95$1, If[LessEqual[t$95$2, 1.5e+41], N[(N[(b * 27.0), $MachinePrecision] * a + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -1.9999999999999999e87 or 1.4999999999999999e41 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 90.2%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 7.9

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

   5. Applied rewrites 7.9%

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

   6. Taylor expanded in b around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 80.7

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

   8. Applied rewrites 80.7%

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

   if -1.9999999999999999e87 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 1.4999999999999999e41

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 91.5

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

   5. Applied rewrites 91.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 91.6

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 91.6

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

   7. Applied rewrites 91.6%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot \left(9 \cdot y\right)\right) \leq -2 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(x, 2, \left(\left(z \cdot y\right) \cdot t\right) \cdot -9\right)\\ \mathbf{elif}\;t \cdot \left(z \cdot \left(9 \cdot y\right)\right) \leq 1.5 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot 27, a, 2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 2, \left(\left(z \cdot y\right) \cdot t\right) \cdot -9\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := t \cdot \left(z \cdot \left(9 \cdot y\right)\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+114}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \left(-9 \cdot t\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot 27, a, 2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t \cdot y\right) \cdot z\right) \cdot -9\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* t (* z (* 9.0 y)))))
   (if (<= t_1 -2e+114)
     (* (* z y) (* -9.0 t))
     (if (<= t_1 5e+151)
       (fma (* b 27.0) a (* 2.0 x))
       (* (* (* t y) z) -9.0)))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t * (z * (9.0 * y));
	double tmp;
	if (t_1 <= -2e+114) {
		tmp = (z * y) * (-9.0 * t);
	} else if (t_1 <= 5e+151) {
		tmp = fma((b * 27.0), a, (2.0 * x));
	} else {
		tmp = ((t * y) * z) * -9.0;
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(t * Float64(z * Float64(9.0 * y)))
	tmp = 0.0
	if (t_1 <= -2e+114)
		tmp = Float64(Float64(z * y) * Float64(-9.0 * t));
	elseif (t_1 <= 5e+151)
		tmp = fma(Float64(b * 27.0), a, Float64(2.0 * x));
	else
		tmp = Float64(Float64(Float64(t * y) * z) * -9.0);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t * N[(z * N[(9.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+114], N[(N[(z * y), $MachinePrecision] * N[(-9.0 * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+151], N[(N[(b * 27.0), $MachinePrecision] * a + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t * y), $MachinePrecision] * z), $MachinePrecision] * -9.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -2e114

   1. Initial program 87.1%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 76.3

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

   5. Applied rewrites 76.3%

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

   7. Applied rewrites 76.4%

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

   if -2e114 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 5.0000000000000002e151

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 87.0

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

   5. Applied rewrites 87.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 87.1

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 87.1

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

   7. Applied rewrites 87.1%

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

   if 5.0000000000000002e151 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 90.2%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 84.9

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

   5. Applied rewrites 84.9%

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

   7. Applied rewrites 79.8%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot \left(9 \cdot y\right)\right) \leq -2 \cdot 10^{+114}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \left(-9 \cdot t\right)\\ \mathbf{elif}\;t \cdot \left(z \cdot \left(9 \cdot y\right)\right) \leq 5 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot 27, a, 2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t \cdot y\right) \cdot z\right) \cdot -9\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 82.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := t \cdot \left(z \cdot \left(9 \cdot y\right)\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+114}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \left(-9 \cdot t\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, 27, 2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t \cdot y\right) \cdot z\right) \cdot -9\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* t (* z (* 9.0 y)))))
   (if (<= t_1 -2e+114)
     (* (* z y) (* -9.0 t))
     (if (<= t_1 5e+151)
       (fma (* b a) 27.0 (* 2.0 x))
       (* (* (* t y) z) -9.0)))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t * (z * (9.0 * y));
	double tmp;
	if (t_1 <= -2e+114) {
		tmp = (z * y) * (-9.0 * t);
	} else if (t_1 <= 5e+151) {
		tmp = fma((b * a), 27.0, (2.0 * x));
	} else {
		tmp = ((t * y) * z) * -9.0;
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(t * Float64(z * Float64(9.0 * y)))
	tmp = 0.0
	if (t_1 <= -2e+114)
		tmp = Float64(Float64(z * y) * Float64(-9.0 * t));
	elseif (t_1 <= 5e+151)
		tmp = fma(Float64(b * a), 27.0, Float64(2.0 * x));
	else
		tmp = Float64(Float64(Float64(t * y) * z) * -9.0);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t * N[(z * N[(9.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+114], N[(N[(z * y), $MachinePrecision] * N[(-9.0 * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+151], N[(N[(b * a), $MachinePrecision] * 27.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t * y), $MachinePrecision] * z), $MachinePrecision] * -9.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -2e114

   1. Initial program 87.1%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 76.3

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

   5. Applied rewrites 76.3%

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

   7. Applied rewrites 76.4%

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

   if -2e114 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 5.0000000000000002e151

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 87.0

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

   5. Applied rewrites 87.0%

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

   if 5.0000000000000002e151 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 90.2%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 84.9

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

   5. Applied rewrites 84.9%

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

   7. Applied rewrites 79.8%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot \left(9 \cdot y\right)\right) \leq -2 \cdot 10^{+114}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \left(-9 \cdot t\right)\\ \mathbf{elif}\;t \cdot \left(z \cdot \left(9 \cdot y\right)\right) \leq 5 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, 27, 2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t \cdot y\right) \cdot z\right) \cdot -9\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot \left(9 \cdot y\right) \leq 2 \cdot 10^{+282}:\\ \;\;\;\;\mathsf{fma}\left(\left(z \cdot y\right) \cdot -9, t, \mathsf{fma}\left(b \cdot 27, a, 2 \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-9 \cdot t\right) \cdot y, z, 2 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* z (* 9.0 y)) 2e+282)
   (fma (* (* z y) -9.0) t (fma (* b 27.0) a (* 2.0 x)))
   (fma (* (* -9.0 t) y) z (* 2.0 x))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z * (9.0 * y)) <= 2e+282) {
		tmp = fma(((z * y) * -9.0), t, fma((b * 27.0), a, (2.0 * x)));
	} else {
		tmp = fma(((-9.0 * t) * y), z, (2.0 * x));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(z * Float64(9.0 * y)) <= 2e+282)
		tmp = fma(Float64(Float64(z * y) * -9.0), t, fma(Float64(b * 27.0), a, Float64(2.0 * x)));
	else
		tmp = fma(Float64(Float64(-9.0 * t) * y), z, Float64(2.0 * x));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(z * N[(9.0 * y), $MachinePrecision]), $MachinePrecision], 2e+282], N[(N[(N[(z * y), $MachinePrecision] * -9.0), $MachinePrecision] * t + N[(N[(b * 27.0), $MachinePrecision] * a + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-9.0 * t), $MachinePrecision] * y), $MachinePrecision] * z + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y #s(literal 9 binary64)) z) < 2.00000000000000007e282

   1. Initial program 97.8%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

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

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

   4. Applied rewrites 97.4%

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

   if 2.00000000000000007e282 < (*.f64 (*.f64 y #s(literal 9 binary64)) z)

   1. Initial program 64.8%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-*.f64 N/A

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

   4. Applied rewrites 71.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 99.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-neg.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 99.9

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

      14. lift-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. lower-fma.f64 99.9

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

      26. lower-*.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 87.4

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

   9. Applied rewrites 87.4%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(9 \cdot y\right) \leq 2 \cdot 10^{+282}:\\ \;\;\;\;\mathsf{fma}\left(\left(z \cdot y\right) \cdot -9, t, \mathsf{fma}\left(b \cdot 27, a, 2 \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-9 \cdot t\right) \cdot y, z, 2 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 51.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(27 \cdot a\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-39}:\\ \;\;\;\;\left(b \cdot 27\right) \cdot a\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-92}:\\ \;\;\;\;2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot a\right) \cdot 27\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* 27.0 a) b)))
   (if (<= t_1 -1e-39)
     (* (* b 27.0) a)
     (if (<= t_1 2e-92) (* 2.0 x) (* (* b a) 27.0)))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (27.0 * a) * b;
	double tmp;
	if (t_1 <= -1e-39) {
		tmp = (b * 27.0) * a;
	} else if (t_1 <= 2e-92) {
		tmp = 2.0 * x;
	} else {
		tmp = (b * a) * 27.0;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (27.0d0 * a) * b
    if (t_1 <= (-1d-39)) then
        tmp = (b * 27.0d0) * a
    else if (t_1 <= 2d-92) then
        tmp = 2.0d0 * x
    else
        tmp = (b * a) * 27.0d0
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b;
public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (27.0 * a) * b;
	double tmp;
	if (t_1 <= -1e-39) {
		tmp = (b * 27.0) * a;
	} else if (t_1 <= 2e-92) {
		tmp = 2.0 * x;
	} else {
		tmp = (b * a) * 27.0;
	}
	return tmp;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	t_1 = (27.0 * a) * b
	tmp = 0
	if t_1 <= -1e-39:
		tmp = (b * 27.0) * a
	elif t_1 <= 2e-92:
		tmp = 2.0 * x
	else:
		tmp = (b * a) * 27.0
	return tmp

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(27.0 * a) * b)
	tmp = 0.0
	if (t_1 <= -1e-39)
		tmp = Float64(Float64(b * 27.0) * a);
	elseif (t_1 <= 2e-92)
		tmp = Float64(2.0 * x);
	else
		tmp = Float64(Float64(b * a) * 27.0);
	end
	return tmp
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (27.0 * a) * b;
	tmp = 0.0;
	if (t_1 <= -1e-39)
		tmp = (b * 27.0) * a;
	elseif (t_1 <= 2e-92)
		tmp = 2.0 * x;
	else
		tmp = (b * a) * 27.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(27.0 * a), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-39], N[(N[(b * 27.0), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[t$95$1, 2e-92], N[(2.0 * x), $MachinePrecision], N[(N[(b * a), $MachinePrecision] * 27.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 a #s(literal 27 binary64)) b) < -9.99999999999999929e-40

   1. Initial program 96.4%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 61.0

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

   5. Applied rewrites 61.0%

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

   7. Applied rewrites 61.1%

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

   if -9.99999999999999929e-40 < (*.f64 (*.f64 a #s(literal 27 binary64)) b) < 1.99999999999999998e-92

   1. Initial program 96.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 55.3

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

   5. Applied rewrites 55.3%

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

   if 1.99999999999999998e-92 < (*.f64 (*.f64 a #s(literal 27 binary64)) b)

   1. Initial program 93.8%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 50.2

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

   5. Applied rewrites 50.2%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(27 \cdot a\right) \cdot b \leq -1 \cdot 10^{-39}:\\ \;\;\;\;\left(b \cdot 27\right) \cdot a\\ \mathbf{elif}\;\left(27 \cdot a\right) \cdot b \leq 2 \cdot 10^{-92}:\\ \;\;\;\;2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot a\right) \cdot 27\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 51.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(27 \cdot a\right) \cdot b\\ t_2 := \left(b \cdot 27\right) \cdot a\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-39}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-92}:\\ \;\;\;\;2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* 27.0 a) b)) (t_2 (* (* b 27.0) a)))
   (if (<= t_1 -1e-39) t_2 (if (<= t_1 2e-92) (* 2.0 x) t_2))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (27.0 * a) * b;
	double t_2 = (b * 27.0) * a;
	double tmp;
	if (t_1 <= -1e-39) {
		tmp = t_2;
	} else if (t_1 <= 2e-92) {
		tmp = 2.0 * x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (27.0d0 * a) * b
    t_2 = (b * 27.0d0) * a
    if (t_1 <= (-1d-39)) then
        tmp = t_2
    else if (t_1 <= 2d-92) then
        tmp = 2.0d0 * x
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b;
public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (27.0 * a) * b;
	double t_2 = (b * 27.0) * a;
	double tmp;
	if (t_1 <= -1e-39) {
		tmp = t_2;
	} else if (t_1 <= 2e-92) {
		tmp = 2.0 * x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	t_1 = (27.0 * a) * b
	t_2 = (b * 27.0) * a
	tmp = 0
	if t_1 <= -1e-39:
		tmp = t_2
	elif t_1 <= 2e-92:
		tmp = 2.0 * x
	else:
		tmp = t_2
	return tmp

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(27.0 * a) * b)
	t_2 = Float64(Float64(b * 27.0) * a)
	tmp = 0.0
	if (t_1 <= -1e-39)
		tmp = t_2;
	elseif (t_1 <= 2e-92)
		tmp = Float64(2.0 * x);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (27.0 * a) * b;
	t_2 = (b * 27.0) * a;
	tmp = 0.0;
	if (t_1 <= -1e-39)
		tmp = t_2;
	elseif (t_1 <= 2e-92)
		tmp = 2.0 * x;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(27.0 * a), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b * 27.0), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-39], t$95$2, If[LessEqual[t$95$1, 2e-92], N[(2.0 * x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 a #s(literal 27 binary64)) b) < -9.99999999999999929e-40 or 1.99999999999999998e-92 < (*.f64 (*.f64 a #s(literal 27 binary64)) b)

   1. Initial program 95.2%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 56.0

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

   5. Applied rewrites 56.0%

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

   7. Applied rewrites 56.0%

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

   if -9.99999999999999929e-40 < (*.f64 (*.f64 a #s(literal 27 binary64)) b) < 1.99999999999999998e-92

   1. Initial program 96.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 55.3

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

   5. Applied rewrites 55.3%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(27 \cdot a\right) \cdot b \leq -1 \cdot 10^{-39}:\\ \;\;\;\;\left(b \cdot 27\right) \cdot a\\ \mathbf{elif}\;\left(27 \cdot a\right) \cdot b \leq 2 \cdot 10^{-92}:\\ \;\;\;\;2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot 27\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 51.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(27 \cdot a\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-92}:\\ \;\;\;\;2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* 27.0 a) b)))
   (if (<= t_1 -1e-39) t_1 (if (<= t_1 2e-92) (* 2.0 x) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (27.0 * a) * b;
	double tmp;
	if (t_1 <= -1e-39) {
		tmp = t_1;
	} else if (t_1 <= 2e-92) {
		tmp = 2.0 * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (27.0d0 * a) * b
    if (t_1 <= (-1d-39)) then
        tmp = t_1
    else if (t_1 <= 2d-92) then
        tmp = 2.0d0 * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b;
public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (27.0 * a) * b;
	double tmp;
	if (t_1 <= -1e-39) {
		tmp = t_1;
	} else if (t_1 <= 2e-92) {
		tmp = 2.0 * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	t_1 = (27.0 * a) * b
	tmp = 0
	if t_1 <= -1e-39:
		tmp = t_1
	elif t_1 <= 2e-92:
		tmp = 2.0 * x
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(27.0 * a) * b)
	tmp = 0.0
	if (t_1 <= -1e-39)
		tmp = t_1;
	elseif (t_1 <= 2e-92)
		tmp = Float64(2.0 * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (27.0 * a) * b;
	tmp = 0.0;
	if (t_1 <= -1e-39)
		tmp = t_1;
	elseif (t_1 <= 2e-92)
		tmp = 2.0 * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(27.0 * a), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-39], t$95$1, If[LessEqual[t$95$1, 2e-92], N[(2.0 * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 a #s(literal 27 binary64)) b) < -9.99999999999999929e-40 or 1.99999999999999998e-92 < (*.f64 (*.f64 a #s(literal 27 binary64)) b)

   1. Initial program 95.2%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 56.0

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

   5. Applied rewrites 56.0%

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

   7. Applied rewrites 56.0%

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

   if -9.99999999999999929e-40 < (*.f64 (*.f64 a #s(literal 27 binary64)) b) < 1.99999999999999998e-92

   1. Initial program 96.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 55.3

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

   5. Applied rewrites 55.3%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(27 \cdot a\right) \cdot b \leq -1 \cdot 10^{-39}:\\ \;\;\;\;\left(27 \cdot a\right) \cdot b\\ \mathbf{elif}\;\left(27 \cdot a\right) \cdot b \leq 2 \cdot 10^{-92}:\\ \;\;\;\;2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(27 \cdot a\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 98.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(\left(-9 \cdot t\right) \cdot y, z, \mathsf{fma}\left(b \cdot a, 27, 2 \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-t\right) \cdot 9, z \cdot y, \mathsf{fma}\left(b \cdot 27, a, 2 \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -2e-51)
   (fma (* (* -9.0 t) y) z (fma (* b a) 27.0 (* 2.0 x)))
   (fma (* (- t) 9.0) (* z y) (fma (* b 27.0) a (* 2.0 x)))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -2e-51) {
		tmp = fma(((-9.0 * t) * y), z, fma((b * a), 27.0, (2.0 * x)));
	} else {
		tmp = fma((-t * 9.0), (z * y), fma((b * 27.0), a, (2.0 * x)));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -2e-51)
		tmp = fma(Float64(Float64(-9.0 * t) * y), z, fma(Float64(b * a), 27.0, Float64(2.0 * x)));
	else
		tmp = fma(Float64(Float64(-t) * 9.0), Float64(z * y), fma(Float64(b * 27.0), a, Float64(2.0 * x)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -2e-51], N[(N[(N[(-9.0 * t), $MachinePrecision] * y), $MachinePrecision] * z + N[(N[(b * a), $MachinePrecision] * 27.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-t) * 9.0), $MachinePrecision] * N[(z * y), $MachinePrecision] + N[(N[(b * 27.0), $MachinePrecision] * a + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2e-51

   1. Initial program 90.5%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-*.f64 N/A

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

   4. Applied rewrites 91.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 97.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-neg.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 97.1

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

      14. lift-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. lower-fma.f64 97.1

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

      26. lower-*.f64 97.1

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

   6. Applied rewrites 97.1%

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

   if -2e-51 < z

   1. Initial program 97.8%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-*.f64 N/A

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

   4. Applied rewrites 97.3%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(\left(-9 \cdot t\right) \cdot y, z, \mathsf{fma}\left(b \cdot a, 27, 2 \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-t\right) \cdot 9, z \cdot y, \mathsf{fma}\left(b \cdot 27, a, 2 \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 97.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{-157}:\\ \;\;\;\;\mathsf{fma}\left(\left(-9 \cdot t\right) \cdot y, z, \mathsf{fma}\left(b \cdot a, 27, 2 \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(z \cdot y\right) \cdot -9, t, \mathsf{fma}\left(b \cdot 27, a, 2 \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -4.3e-157)
   (fma (* (* -9.0 t) y) z (fma (* b a) 27.0 (* 2.0 x)))
   (fma (* (* z y) -9.0) t (fma (* b 27.0) a (* 2.0 x)))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -4.3e-157) {
		tmp = fma(((-9.0 * t) * y), z, fma((b * a), 27.0, (2.0 * x)));
	} else {
		tmp = fma(((z * y) * -9.0), t, fma((b * 27.0), a, (2.0 * x)));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -4.3e-157)
		tmp = fma(Float64(Float64(-9.0 * t) * y), z, fma(Float64(b * a), 27.0, Float64(2.0 * x)));
	else
		tmp = fma(Float64(Float64(z * y) * -9.0), t, fma(Float64(b * 27.0), a, Float64(2.0 * x)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -4.3e-157], N[(N[(N[(-9.0 * t), $MachinePrecision] * y), $MachinePrecision] * z + N[(N[(b * a), $MachinePrecision] * 27.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z * y), $MachinePrecision] * -9.0), $MachinePrecision] * t + N[(N[(b * 27.0), $MachinePrecision] * a + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.2999999999999998e-157

   1. Initial program 92.8%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-*.f64 N/A

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

   4. Applied rewrites 93.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 97.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-neg.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 97.8

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

      14. lift-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. lower-fma.f64 97.8

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

      26. lower-*.f64 97.8

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

   6. Applied rewrites 97.8%

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

   if -4.2999999999999998e-157 < z

   1. Initial program 97.5%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

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

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

   4. Applied rewrites 96.9%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{-157}:\\ \;\;\;\;\mathsf{fma}\left(\left(-9 \cdot t\right) \cdot y, z, \mathsf{fma}\left(b \cdot a, 27, 2 \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(z \cdot y\right) \cdot -9, t, \mathsf{fma}\left(b \cdot 27, a, 2 \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 93.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-neg N/A

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

   4. +-commutative N/A

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

   5. associate-+l+ N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

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

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

   13. +-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. metadata-eval N/A

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

   16. *-commutative N/A

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

   17. associate-*r* N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 N/A

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

   20. lower-*.f64 N/A

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

4. Applied rewrites 95.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-9, \left(t \cdot y\right) \cdot z, \mathsf{fma}\left(b \cdot 27, a, 2 \cdot x\right)\right)} \]
5. Add Preprocessing

Alternative 19: 30.4% accurate, 6.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ 2 \cdot x \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b) :precision binary64 (* 2.0 x))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	return 2.0 * x;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
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 = 2.0d0 * x
end function

Java

assert x < y && y < z && z < t && t < a && a < b;
public static double code(double x, double y, double z, double t, double a, double b) {
	return 2.0 * x;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	return 2.0 * x

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	return Float64(2.0 * x)
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp = code(x, y, z, t, a, b)
	tmp = 2.0 * x;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := N[(2.0 * x), $MachinePrecision]

Derivation

1. Initial program 95.8%

   \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{2 \cdot x} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{x \cdot 2} \]

   2. lower-*.f64 27.2

      \[\leadsto \color{blue}{x \cdot 2} \]

5. Applied rewrites 27.2%

   \[\leadsto \color{blue}{x \cdot 2} \]

6. Final simplification 27.2%

   \[\leadsto 2 \cdot x \]
7. 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 2024268 
(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)))