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

Percentage Accurate: 95.4% → 98.3%

Time: 20.0s

Alternatives: 11

Speedup: 0.9×

• 28.1% 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: 95.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(27 \cdot b, a, x \cdot 2\right)\\ \mathbf{if}\;9 \cdot y \leq -1 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(9, \left(t \cdot \left(-z\right)\right) \cdot y, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot y, -9 \cdot z, t\_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (* 27.0 b) a (* x 2.0))))
   (if (<= (* 9.0 y) -1e-33)
     (fma 9.0 (* (* t (- z)) y) t_1)
     (fma (* t y) (* -9.0 z) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((27.0 * b), a, (x * 2.0));
	double tmp;
	if ((9.0 * y) <= -1e-33) {
		tmp = fma(9.0, ((t * -z) * y), t_1);
	} else {
		tmp = fma((t * y), (-9.0 * z), t_1);
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = fma(Float64(27.0 * b), a, Float64(x * 2.0))
	tmp = 0.0
	if (Float64(9.0 * y) <= -1e-33)
		tmp = fma(9.0, Float64(Float64(t * Float64(-z)) * y), t_1);
	else
		tmp = fma(Float64(t * y), Float64(-9.0 * z), t_1);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(27.0 * b), $MachinePrecision] * a + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(9.0 * y), $MachinePrecision], -1e-33], N[(9.0 * N[(N[(t * (-z)), $MachinePrecision] * y), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(t * y), $MachinePrecision] * N[(-9.0 * z), $MachinePrecision] + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y #s(literal 9 binary64)) < -1.0000000000000001e-33

   1. Initial program 95.4%

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

      1. lift-+.f64 N/A

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

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

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 N/A

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

   4. Applied rewrites 98.3%

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

   if -1.0000000000000001e-33 < (*.f64 y #s(literal 9 binary64))

   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. *-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. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

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

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

   4. Applied rewrites 96.8%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;9 \cdot y \leq -1 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(9, \left(t \cdot \left(-z\right)\right) \cdot y, \mathsf{fma}\left(27 \cdot b, a, x \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot y, -9 \cdot z, \mathsf{fma}\left(27 \cdot b, a, x \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(z \cdot \left(9 \cdot y\right)\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(27 \cdot b, a, \left(\left(z \cdot y\right) \cdot t\right) \cdot -9\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(a \cdot 27\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, -9 \cdot t, \left(a \cdot b\right) \cdot 27\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
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)) t)))
   (if (<= t_1 -4e+33)
     (fma (* 27.0 b) a (* (* (* z y) t) -9.0))
     (if (<= t_1 1e+26)
       (fma 2.0 x (* (* a 27.0) b))
       (fma (* z y) (* -9.0 t) (* (* a b) 27.0))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (z * (9.0 * y)) * t;
	double tmp;
	if (t_1 <= -4e+33) {
		tmp = fma((27.0 * b), a, (((z * y) * t) * -9.0));
	} else if (t_1 <= 1e+26) {
		tmp = fma(2.0, x, ((a * 27.0) * b));
	} else {
		tmp = fma((z * y), (-9.0 * t), ((a * b) * 27.0));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(z * Float64(9.0 * y)) * t)
	tmp = 0.0
	if (t_1 <= -4e+33)
		tmp = fma(Float64(27.0 * b), a, Float64(Float64(Float64(z * y) * t) * -9.0));
	elseif (t_1 <= 1e+26)
		tmp = fma(2.0, x, Float64(Float64(a * 27.0) * b));
	else
		tmp = fma(Float64(z * y), Float64(-9.0 * t), Float64(Float64(a * b) * 27.0));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(z * N[(9.0 * y), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+33], N[(N[(27.0 * b), $MachinePrecision] * a + N[(N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] * -9.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+26], N[(2.0 * x + N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(N[(z * y), $MachinePrecision] * N[(-9.0 * t), $MachinePrecision] + N[(N[(a * b), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -3.9999999999999998e33

   1. Initial program 91.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 88.9

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

   5. Applied rewrites 88.9%

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

   7. Applied rewrites 88.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 90.7

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 90.7

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

   9. Applied rewrites 90.8%

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

   if -3.9999999999999998e33 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 1.00000000000000005e26

   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 y around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 96.5

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

   5. Applied rewrites 96.5%

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

   7. Applied rewrites 96.6%

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

   if 1.00000000000000005e26 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 90.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 32.1

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

   5. Applied rewrites 32.1%

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

   6. Taylor expanded in x around 0

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

      1. cancel-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. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 79.6

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

   8. Applied rewrites 79.6%

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

   10. Applied rewrites 78.2%

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

   12. Applied rewrites 82.7%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot \left(9 \cdot y\right)\right) \cdot t \leq -4 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(27 \cdot b, a, \left(\left(z \cdot y\right) \cdot t\right) \cdot -9\right)\\ \mathbf{elif}\;\left(z \cdot \left(9 \cdot y\right)\right) \cdot t \leq 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(a \cdot 27\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, -9 \cdot t, \left(a \cdot b\right) \cdot 27\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(a \cdot b\right) \cdot 27\\ t_2 := \left(z \cdot \left(9 \cdot y\right)\right) \cdot t\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, t\_1\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(a \cdot 27\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, -9 \cdot t, t\_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* a b) 27.0)) (t_2 (* (* z (* 9.0 y)) t)))
   (if (<= t_2 -4e+33)
     (fma -9.0 (* (* z y) t) t_1)
     (if (<= t_2 1e+26)
       (fma 2.0 x (* (* a 27.0) b))
       (fma (* z y) (* -9.0 t) t_1)))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * b) * 27.0;
	double t_2 = (z * (9.0 * y)) * t;
	double tmp;
	if (t_2 <= -4e+33) {
		tmp = fma(-9.0, ((z * y) * t), t_1);
	} else if (t_2 <= 1e+26) {
		tmp = fma(2.0, x, ((a * 27.0) * b));
	} else {
		tmp = fma((z * y), (-9.0 * t), t_1);
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a * b) * 27.0)
	t_2 = Float64(Float64(z * Float64(9.0 * y)) * t)
	tmp = 0.0
	if (t_2 <= -4e+33)
		tmp = fma(-9.0, Float64(Float64(z * y) * t), t_1);
	elseif (t_2 <= 1e+26)
		tmp = fma(2.0, x, Float64(Float64(a * 27.0) * b));
	else
		tmp = fma(Float64(z * y), Float64(-9.0 * t), t_1);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] * 27.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * N[(9.0 * y), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+33], N[(-9.0 * N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 1e+26], N[(2.0 * x + N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(N[(z * y), $MachinePrecision] * N[(-9.0 * t), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -3.9999999999999998e33

   1. Initial program 91.1%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 89.0

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

   5. Applied rewrites 89.0%

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

   if -3.9999999999999998e33 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 1.00000000000000005e26

   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 y around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 96.5

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

   5. Applied rewrites 96.5%

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

   7. Applied rewrites 96.6%

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

   if 1.00000000000000005e26 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 90.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 32.1

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

   5. Applied rewrites 32.1%

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

   6. Taylor expanded in x around 0

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

      1. cancel-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. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 79.6

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

   8. Applied rewrites 79.6%

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

   10. Applied rewrites 78.2%

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

   12. Applied rewrites 82.7%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot \left(9 \cdot y\right)\right) \cdot t \leq -4 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(a \cdot b\right) \cdot 27\right)\\ \mathbf{elif}\;\left(z \cdot \left(9 \cdot y\right)\right) \cdot t \leq 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(a \cdot 27\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, -9 \cdot t, \left(a \cdot b\right) \cdot 27\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(a \cdot b\right) \cdot 27\right)\\ t_2 := \left(z \cdot \left(9 \cdot y\right)\right) \cdot t\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(a \cdot 27\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma -9.0 (* (* z y) t) (* (* a b) 27.0)))
        (t_2 (* (* z (* 9.0 y)) t)))
   (if (<= t_2 -4e+33)
     t_1
     (if (<= t_2 1e+26) (fma 2.0 x (* (* a 27.0) b)) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(-9.0, ((z * y) * t), ((a * b) * 27.0));
	double t_2 = (z * (9.0 * y)) * t;
	double tmp;
	if (t_2 <= -4e+33) {
		tmp = t_1;
	} else if (t_2 <= 1e+26) {
		tmp = fma(2.0, x, ((a * 27.0) * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = fma(-9.0, Float64(Float64(z * y) * t), Float64(Float64(a * b) * 27.0))
	t_2 = Float64(Float64(z * Float64(9.0 * y)) * t)
	tmp = 0.0
	if (t_2 <= -4e+33)
		tmp = t_1;
	elseif (t_2 <= 1e+26)
		tmp = fma(2.0, x, Float64(Float64(a * 27.0) * b));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(-9.0 * N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] + N[(N[(a * b), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * N[(9.0 * y), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+33], t$95$1, If[LessEqual[t$95$2, 1e+26], N[(2.0 * x + N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -3.9999999999999998e33 or 1.00000000000000005e26 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 90.9%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 83.9

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

   5. Applied rewrites 83.9%

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

   if -3.9999999999999998e33 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 1.00000000000000005e26

   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 y around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 96.5

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

   5. Applied rewrites 96.5%

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

   7. Applied rewrites 96.6%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot \left(9 \cdot y\right)\right) \cdot t \leq -4 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(a \cdot b\right) \cdot 27\right)\\ \mathbf{elif}\;\left(z \cdot \left(9 \cdot y\right)\right) \cdot t \leq 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(a \cdot 27\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(a \cdot b\right) \cdot 27\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(a \cdot 27\right) \cdot b\\ t_2 := \left(a \cdot b\right) \cdot 27\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+97}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+46}:\\ \;\;\;\;x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* a 27.0) b)) (t_2 (* (* a b) 27.0)))
   (if (<= t_1 -5e+97) t_2 (if (<= t_1 5e+46) (* x 2.0) t_2))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * 27.0) * b;
	double t_2 = (a * b) * 27.0;
	double tmp;
	if (t_1 <= -5e+97) {
		tmp = t_2;
	} else if (t_1 <= 5e+46) {
		tmp = x * 2.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (a * 27.0d0) * b
    t_2 = (a * b) * 27.0d0
    if (t_1 <= (-5d+97)) then
        tmp = t_2
    else if (t_1 <= 5d+46) then
        tmp = x * 2.0d0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b;
assert x < y && y < z && z < t && t < a && a < b;
public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * 27.0) * b;
	double t_2 = (a * b) * 27.0;
	double tmp;
	if (t_1 <= -5e+97) {
		tmp = t_2;
	} else if (t_1 <= 5e+46) {
		tmp = x * 2.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	t_1 = (a * 27.0) * b
	t_2 = (a * b) * 27.0
	tmp = 0
	if t_1 <= -5e+97:
		tmp = t_2
	elif t_1 <= 5e+46:
		tmp = x * 2.0
	else:
		tmp = t_2
	return tmp

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a * 27.0) * b)
	t_2 = Float64(Float64(a * b) * 27.0)
	tmp = 0.0
	if (t_1 <= -5e+97)
		tmp = t_2;
	elseif (t_1 <= 5e+46)
		tmp = Float64(x * 2.0);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a * 27.0) * b;
	t_2 = (a * b) * 27.0;
	tmp = 0.0;
	if (t_1 <= -5e+97)
		tmp = t_2;
	elseif (t_1 <= 5e+46)
		tmp = x * 2.0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a * b), $MachinePrecision] * 27.0), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+97], t$95$2, If[LessEqual[t$95$1, 5e+46], N[(x * 2.0), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 a #s(literal 27 binary64)) b) < -4.99999999999999999e97 or 5.0000000000000002e46 < (*.f64 (*.f64 a #s(literal 27 binary64)) b)

   1. Initial program 92.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 78.3

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

   5. Applied rewrites 78.3%

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

   7. Applied rewrites 78.4%

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

   9. Applied rewrites 78.4%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 66.2%

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

   if -4.99999999999999999e97 < (*.f64 (*.f64 a #s(literal 27 binary64)) b) < 5.0000000000000002e46

   1. Initial program 97.9%

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

      1. lift-+.f64 N/A

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

      2. 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. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

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

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

   4. Applied rewrites 96.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 98.0

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

      12. lift-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 98.0

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 98.0

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 98.0

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

   6. Applied rewrites 98.0%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 49.0

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

   9. Applied rewrites 49.0%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a \cdot 27\right) \cdot b \leq -5 \cdot 10^{+97}:\\ \;\;\;\;\left(a \cdot b\right) \cdot 27\\ \mathbf{elif}\;\left(a \cdot 27\right) \cdot b \leq 5 \cdot 10^{+46}:\\ \;\;\;\;x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot b\right) \cdot 27\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(27 \cdot b, a, x \cdot 2\right)\\ \mathbf{if}\;z \leq 5 \cdot 10^{-50}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(-9 \cdot z\right) \cdot t, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot y, -9 \cdot z, t\_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (* 27.0 b) a (* x 2.0))))
   (if (<= z 5e-50)
     (fma y (* (* -9.0 z) t) t_1)
     (fma (* t y) (* -9.0 z) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((27.0 * b), a, (x * 2.0));
	double tmp;
	if (z <= 5e-50) {
		tmp = fma(y, ((-9.0 * z) * t), t_1);
	} else {
		tmp = fma((t * y), (-9.0 * z), t_1);
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = fma(Float64(27.0 * b), a, Float64(x * 2.0))
	tmp = 0.0
	if (z <= 5e-50)
		tmp = fma(y, Float64(Float64(-9.0 * z) * t), t_1);
	else
		tmp = fma(Float64(t * y), Float64(-9.0 * z), t_1);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(27.0 * b), $MachinePrecision] * a + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 5e-50], N[(y * N[(N[(-9.0 * z), $MachinePrecision] * t), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(t * y), $MachinePrecision] * N[(-9.0 * z), $MachinePrecision] + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 4.99999999999999968e-50

   1. Initial program 97.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. 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. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

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

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

      12. associate-*l* N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

   4. Applied rewrites 96.7%

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

   if 4.99999999999999968e-50 < z

   1. Initial program 90.8%

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

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

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

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

   4. Applied rewrites 97.2%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{-50}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(-9 \cdot z\right) \cdot t, \mathsf{fma}\left(27 \cdot b, a, x \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot y, -9 \cdot z, \mathsf{fma}\left(27 \cdot b, a, x \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 4.6 \cdot 10^{-49}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(-9 \cdot z\right) \cdot t, \mathsf{fma}\left(27 \cdot b, a, x \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(27 \cdot b, a, \mathsf{fma}\left(\left(-9 \cdot y\right) \cdot t, z, x \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
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 4.6e-49)
   (fma y (* (* -9.0 z) t) (fma (* 27.0 b) a (* x 2.0)))
   (fma (* 27.0 b) a (fma (* (* -9.0 y) t) z (* x 2.0)))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= 4.6e-49) {
		tmp = fma(y, ((-9.0 * z) * t), fma((27.0 * b), a, (x * 2.0)));
	} else {
		tmp = fma((27.0 * b), a, fma(((-9.0 * y) * t), z, (x * 2.0)));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= 4.6e-49)
		tmp = fma(y, Float64(Float64(-9.0 * z) * t), fma(Float64(27.0 * b), a, Float64(x * 2.0)));
	else
		tmp = fma(Float64(27.0 * b), a, fma(Float64(Float64(-9.0 * y) * t), z, Float64(x * 2.0)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
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, 4.6e-49], N[(y * N[(N[(-9.0 * z), $MachinePrecision] * t), $MachinePrecision] + N[(N[(27.0 * b), $MachinePrecision] * a + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(27.0 * b), $MachinePrecision] * a + N[(N[(N[(-9.0 * y), $MachinePrecision] * t), $MachinePrecision] * z + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.5999999999999998e-49

   1. Initial program 97.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. 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. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

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

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

      12. associate-*l* N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

   4. Applied rewrites 96.7%

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

   if 4.5999999999999998e-49 < z

   1. Initial program 90.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 93.4

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

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

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

      17. *-commutative N/A

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

      18. associate-*r* N/A

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

      19. lower-fma.f64 N/A

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

   4. Applied rewrites 97.1%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.6 \cdot 10^{-49}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(-9 \cdot z\right) \cdot t, \mathsf{fma}\left(27 \cdot b, a, x \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(27 \cdot b, a, \mathsf{fma}\left(\left(-9 \cdot y\right) \cdot t, z, x \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 2.65 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(-9 \cdot z\right) \cdot t, \mathsf{fma}\left(27 \cdot b, a, x \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot y, -9 \cdot z, \left(a \cdot b\right) \cdot 27\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (if (<= z 2.65e+57)
   (fma y (* (* -9.0 z) t) (fma (* 27.0 b) a (* x 2.0)))
   (fma (* t y) (* -9.0 z) (* (* a b) 27.0))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= 2.65e+57) {
		tmp = fma(y, ((-9.0 * z) * t), fma((27.0 * b), a, (x * 2.0)));
	} else {
		tmp = fma((t * y), (-9.0 * z), ((a * b) * 27.0));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= 2.65e+57)
		tmp = fma(y, Float64(Float64(-9.0 * z) * t), fma(Float64(27.0 * b), a, Float64(x * 2.0)));
	else
		tmp = fma(Float64(t * y), Float64(-9.0 * z), Float64(Float64(a * b) * 27.0));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, 2.65e+57], N[(y * N[(N[(-9.0 * z), $MachinePrecision] * t), $MachinePrecision] + N[(N[(27.0 * b), $MachinePrecision] * a + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t * y), $MachinePrecision] * N[(-9.0 * z), $MachinePrecision] + N[(N[(a * b), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.64999999999999993e57

   1. Initial program 97.4%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. 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. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

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

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

      12. associate-*l* N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

   4. Applied rewrites 96.5%

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

   if 2.64999999999999993e57 < z

   1. Initial program 89.2%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. *-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. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

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

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 82.8

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

   7. Applied rewrites 82.8%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.65 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(-9 \cdot z\right) \cdot t, \mathsf{fma}\left(27 \cdot b, a, x \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot y, -9 \cdot z, \left(a \cdot b\right) \cdot 27\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \mathsf{fma}\left(a, 27 \cdot b, x \cdot 2\right) \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b) :precision binary64 (fma a (* 27.0 b) (* x 2.0)))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	return fma(a, (27.0 * b), (x * 2.0));
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	return fma(a, Float64(27.0 * b), Float64(x * 2.0))
end

Wolfram

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

Derivation

1. Initial program 95.7%

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

3. Taylor expanded in y around 0

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

   1. lower-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 65.4

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

5. Applied rewrites 65.4%

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

7. Applied rewrites 65.4%

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

Alternative 10: 64.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \mathsf{fma}\left(2, x, \left(a \cdot 27\right) \cdot b\right) \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b) :precision binary64 (fma 2.0 x (* (* a 27.0) b)))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	return fma(2.0, x, ((a * 27.0) * b));
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	return fma(2.0, x, Float64(Float64(a * 27.0) * b))
end

Wolfram

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

Derivation

1. Initial program 95.7%

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

3. Taylor expanded in y around 0

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

   1. lower-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 65.4

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

5. Applied rewrites 65.4%

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

7. Applied rewrites 65.5%

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

Alternative 11: 30.5% accurate, 6.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ x \cdot 2 \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b) :precision binary64 (* x 2.0))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	return x * 2.0;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x * 2.0d0
end function

Java

assert x < y && y < z && z < t && t < a && a < b;
assert x < y && y < z && z < t && t < a && a < b;
public static double code(double x, double y, double z, double t, double a, double b) {
	return x * 2.0;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	return x * 2.0

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	return Float64(x * 2.0)
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp = code(x, y, z, t, a, b)
	tmp = x * 2.0;
end

Wolfram

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

Derivation

1. Initial program 95.7%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. 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. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*l* N/A

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

   11. associate-*r* N/A

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

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

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

   13. +-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-*.f64 N/A

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

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

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

   17. lower-*.f64 N/A

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

   18. metadata-eval N/A

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

4. Applied rewrites 95.7%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. lower-fma.f64 95.7

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

   12. lift-fma.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 95.7

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 95.7

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 95.7

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

6. Applied rewrites 95.7%

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

7. Taylor expanded in x around inf

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

   1. lower-*.f64 34.8

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

9. Applied rewrites 34.8%

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

10. Final simplification 34.8%

    \[\leadsto x \cdot 2 \]
11. Add Preprocessing

Developer Target 1: 94.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y < 7.590524218811189 \cdot 10^{-161}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - 9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (< y 7.590524218811189e-161)
   (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* a (* 27.0 b)))
   (+ (- (* x 2.0) (* 9.0 (* y (* t z)))) (* (* a 27.0) b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y < 7.590524218811189e-161) {
		tmp = ((x * 2.0) - (((y * 9.0) * z) * t)) + (a * (27.0 * b));
	} else {
		tmp = ((x * 2.0) - (9.0 * (y * (t * z)))) + ((a * 27.0) * b);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y < 7.590524218811189d-161) then
        tmp = ((x * 2.0d0) - (((y * 9.0d0) * z) * t)) + (a * (27.0d0 * b))
    else
        tmp = ((x * 2.0d0) - (9.0d0 * (y * (t * z)))) + ((a * 27.0d0) * b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y < 7.590524218811189e-161) {
		tmp = ((x * 2.0) - (((y * 9.0) * z) * t)) + (a * (27.0 * b));
	} else {
		tmp = ((x * 2.0) - (9.0 * (y * (t * z)))) + ((a * 27.0) * b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y < 7.590524218811189e-161:
		tmp = ((x * 2.0) - (((y * 9.0) * z) * t)) + (a * (27.0 * b))
	else:
		tmp = ((x * 2.0) - (9.0 * (y * (t * z)))) + ((a * 27.0) * b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y < 7.590524218811189e-161)
		tmp = Float64(Float64(Float64(x * 2.0) - Float64(Float64(Float64(y * 9.0) * z) * t)) + Float64(a * Float64(27.0 * b)));
	else
		tmp = Float64(Float64(Float64(x * 2.0) - Float64(9.0 * Float64(y * Float64(t * z)))) + Float64(Float64(a * 27.0) * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y < 7.590524218811189e-161)
		tmp = ((x * 2.0) - (((y * 9.0) * z) * t)) + (a * (27.0 * b));
	else
		tmp = ((x * 2.0) - (9.0 * (y * (t * z)))) + ((a * 27.0) * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Less[y, 7.590524218811189e-161], N[(N[(N[(x * 2.0), $MachinePrecision] - N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(27.0 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * 2.0), $MachinePrecision] - N[(9.0 * N[(y * N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024294 
(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)))