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

Percentage Accurate: 95.6% → 98.4%

Time: 11.0s

Alternatives: 19

Speedup: 0.9×

• 26.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b \cdot 27, a, 2 \cdot x\right)\\ \mathbf{if}\;y \cdot 9 \leq -5 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, -9 \cdot y, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(t \cdot y\right) \cdot 9, -z, t\_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (* b 27.0) a (* 2.0 x))))
   (if (<= (* y 9.0) -5e+54)
     (fma (* t z) (* -9.0 y) t_1)
     (fma (* (* t y) 9.0) (- z) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((b * 27.0), a, (2.0 * x));
	double tmp;
	if ((y * 9.0) <= -5e+54) {
		tmp = fma((t * z), (-9.0 * 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])
function code(x, y, z, t, a, b)
	t_1 = fma(Float64(b * 27.0), a, Float64(2.0 * x))
	tmp = 0.0
	if (Float64(y * 9.0) <= -5e+54)
		tmp = fma(Float64(t * z), Float64(-9.0 * y), t_1);
	else
		tmp = fma(Float64(Float64(t * y) * 9.0), Float64(-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.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b * 27.0), $MachinePrecision] * a + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y * 9.0), $MachinePrecision], -5e+54], N[(N[(t * z), $MachinePrecision] * N[(-9.0 * y), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(N[(t * y), $MachinePrecision] * 9.0), $MachinePrecision] * (-z) + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y #s(literal 9 binary64)) < -5.00000000000000005e54

   1. Initial program 85.4%

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

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

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

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   if -5.00000000000000005e54 < (*.f64 y #s(literal 9 binary64))

   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. lift-*.f64 N/A

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

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

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. associate-*r* N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-neg.f64 N/A

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

   4. Applied rewrites 94.7%

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

Alternative 2: 85.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+303}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, 27, \left(\left(-9 \cdot y\right) \cdot t\right) \cdot z\right)\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+90}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, t \cdot -9, \left(27 \cdot a\right) \cdot b\right)\\ \mathbf{elif}\;t\_1 \leq 0.74:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(b \cdot a\right) \cdot 27\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(t \cdot \left(-9 \cdot y\right), z, x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* y 9.0) z) t)))
   (if (<= t_1 -4e+303)
     (fma (* b a) 27.0 (* (* (* -9.0 y) t) z))
     (if (<= t_1 -1e+90)
       (fma (* y z) (* t -9.0) (* (* 27.0 a) b))
       (if (<= t_1 0.74)
         (fma 2.0 x (* (* b a) 27.0))
         (+ x (fma (* t (* -9.0 y)) z x)))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((y * 9.0) * z) * t;
	double tmp;
	if (t_1 <= -4e+303) {
		tmp = fma((b * a), 27.0, (((-9.0 * y) * t) * z));
	} else if (t_1 <= -1e+90) {
		tmp = fma((y * z), (t * -9.0), ((27.0 * a) * b));
	} else if (t_1 <= 0.74) {
		tmp = fma(2.0, x, ((b * a) * 27.0));
	} else {
		tmp = x + fma((t * (-9.0 * y)), z, x);
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if (t_1 <= -4e+303)
		tmp = fma(Float64(b * a), 27.0, Float64(Float64(Float64(-9.0 * y) * t) * z));
	elseif (t_1 <= -1e+90)
		tmp = fma(Float64(y * z), Float64(t * -9.0), Float64(Float64(27.0 * a) * b));
	elseif (t_1 <= 0.74)
		tmp = fma(2.0, x, Float64(Float64(b * a) * 27.0));
	else
		tmp = Float64(x + fma(Float64(t * Float64(-9.0 * y)), z, x));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+303], N[(N[(b * a), $MachinePrecision] * 27.0 + N[(N[(N[(-9.0 * y), $MachinePrecision] * t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1e+90], N[(N[(y * z), $MachinePrecision] * N[(t * -9.0), $MachinePrecision] + N[(N[(27.0 * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.74], N[(2.0 * x + N[(N[(b * a), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(t * N[(-9.0 * y), $MachinePrecision]), $MachinePrecision] * z + x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -4e303

   1. Initial program 70.1%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 70.1

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

   5. Applied rewrites 70.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 70.1

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

   7. Applied rewrites 99.9%

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

   if -4e303 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -9.99999999999999966e89

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 90.9

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

   5. Applied rewrites 90.9%

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

   7. Applied rewrites 90.8%

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

   if -9.99999999999999966e89 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 0.73999999999999999

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 93.4

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

   5. Applied rewrites 93.4%

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

   if 0.73999999999999999 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 90.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. count-2-rev N/A

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

      7. associate--l+ N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval 92.2

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 92.2

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 92.2

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

      25. lift-*.f64 N/A

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

      26. *-commutative N/A

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

      27. lower-*.f64 92.2

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

   4. Applied rewrites 92.2%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 89.0

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

   7. Applied rewrites 89.0%

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

   8. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 82.8

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

   10. Applied rewrites 82.8%

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

   12. Applied rewrites 82.8%

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

Alternative 3: 85.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+90} \lor \neg \left(t\_1 \leq 0.74\right):\\ \;\;\;\;x + \mathsf{fma}\left(y, t \cdot \left(-9 \cdot z\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(b \cdot a\right) \cdot 27\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* y 9.0) z) t)))
   (if (or (<= t_1 -1e+90) (not (<= t_1 0.74)))
     (+ x (fma y (* t (* -9.0 z)) x))
     (fma 2.0 x (* (* b a) 27.0)))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((y * 9.0) * z) * t;
	double tmp;
	if ((t_1 <= -1e+90) || !(t_1 <= 0.74)) {
		tmp = x + fma(y, (t * (-9.0 * z)), x);
	} else {
		tmp = fma(2.0, x, ((b * a) * 27.0));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if ((t_1 <= -1e+90) || !(t_1 <= 0.74))
		tmp = Float64(x + fma(y, Float64(t * Float64(-9.0 * z)), x));
	else
		tmp = fma(2.0, x, Float64(Float64(b * a) * 27.0));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e+90], N[Not[LessEqual[t$95$1, 0.74]], $MachinePrecision]], N[(x + N[(y * N[(t * N[(-9.0 * z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(2.0 * x + N[(N[(b * a), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -9.99999999999999966e89 or 0.73999999999999999 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 88.3%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. count-2-rev N/A

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

      7. associate--l+ N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval 89.2

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 89.2

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 89.2

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

      25. lift-*.f64 N/A

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

      26. *-commutative N/A

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

      27. lower-*.f64 89.2

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

   4. Applied rewrites 89.2%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 89.2

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

   7. Applied rewrites 89.2%

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

   8. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 81.1

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

   10. Applied rewrites 81.1%

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

   12. Applied rewrites 80.3%

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

   if -9.99999999999999966e89 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 0.73999999999999999

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 93.4

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

   5. Applied rewrites 93.4%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \leq -1 \cdot 10^{+90} \lor \neg \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \leq 0.74\right):\\ \;\;\;\;x + \mathsf{fma}\left(y, t \cdot \left(-9 \cdot z\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(b \cdot a\right) \cdot 27\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+90}:\\ \;\;\;\;\left(t \cdot \left(-9 \cdot z\right)\right) \cdot y + \left(a \cdot 27\right) \cdot b\\ \mathbf{elif}\;t\_1 \leq 0.74:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(b \cdot a\right) \cdot 27\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(t \cdot \left(-9 \cdot y\right), z, x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* y 9.0) z) t)))
   (if (<= t_1 -1e+90)
     (+ (* (* t (* -9.0 z)) y) (* (* a 27.0) b))
     (if (<= t_1 0.74)
       (fma 2.0 x (* (* b a) 27.0))
       (+ x (fma (* t (* -9.0 y)) z x))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((y * 9.0) * z) * t;
	double tmp;
	if (t_1 <= -1e+90) {
		tmp = ((t * (-9.0 * z)) * y) + ((a * 27.0) * b);
	} else if (t_1 <= 0.74) {
		tmp = fma(2.0, x, ((b * a) * 27.0));
	} else {
		tmp = x + fma((t * (-9.0 * y)), z, x);
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if (t_1 <= -1e+90)
		tmp = Float64(Float64(Float64(t * Float64(-9.0 * z)) * y) + Float64(Float64(a * 27.0) * b));
	elseif (t_1 <= 0.74)
		tmp = fma(2.0, x, Float64(Float64(b * a) * 27.0));
	else
		tmp = Float64(x + fma(Float64(t * Float64(-9.0 * y)), z, x));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+90], N[(N[(N[(t * N[(-9.0 * z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] + N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.74], N[(2.0 * x + N[(N[(b * a), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(t * N[(-9.0 * y), $MachinePrecision]), $MachinePrecision] * z + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -9.99999999999999966e89

   1. Initial program 85.1%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 80.8

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

   5. Applied rewrites 80.8%

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

   7. Applied rewrites 85.0%

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

   if -9.99999999999999966e89 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 0.73999999999999999

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 93.4

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

   5. Applied rewrites 93.4%

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

   if 0.73999999999999999 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 90.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. count-2-rev N/A

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

      7. associate--l+ N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval 92.2

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 92.2

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 92.2

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

      25. lift-*.f64 N/A

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

      26. *-commutative N/A

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

      27. lower-*.f64 92.2

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

   4. Applied rewrites 92.2%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 89.0

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

   7. Applied rewrites 89.0%

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

   8. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 82.8

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

   10. Applied rewrites 82.8%

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

   12. Applied rewrites 82.8%

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

Alternative 5: 85.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+90}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, t \cdot -9, \left(27 \cdot a\right) \cdot b\right)\\ \mathbf{elif}\;t\_1 \leq 0.74:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(b \cdot a\right) \cdot 27\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(t \cdot \left(-9 \cdot y\right), z, x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* y 9.0) z) t)))
   (if (<= t_1 -1e+90)
     (fma (* y z) (* t -9.0) (* (* 27.0 a) b))
     (if (<= t_1 0.74)
       (fma 2.0 x (* (* b a) 27.0))
       (+ x (fma (* t (* -9.0 y)) z x))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((y * 9.0) * z) * t;
	double tmp;
	if (t_1 <= -1e+90) {
		tmp = fma((y * z), (t * -9.0), ((27.0 * a) * b));
	} else if (t_1 <= 0.74) {
		tmp = fma(2.0, x, ((b * a) * 27.0));
	} else {
		tmp = x + fma((t * (-9.0 * y)), z, x);
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if (t_1 <= -1e+90)
		tmp = fma(Float64(y * z), Float64(t * -9.0), Float64(Float64(27.0 * a) * b));
	elseif (t_1 <= 0.74)
		tmp = fma(2.0, x, Float64(Float64(b * a) * 27.0));
	else
		tmp = Float64(x + fma(Float64(t * Float64(-9.0 * y)), z, x));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+90], N[(N[(y * z), $MachinePrecision] * N[(t * -9.0), $MachinePrecision] + N[(N[(27.0 * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.74], N[(2.0 * x + N[(N[(b * a), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(t * N[(-9.0 * y), $MachinePrecision]), $MachinePrecision] * z + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -9.99999999999999966e89

   1. Initial program 85.1%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   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. fp-cancel-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 80.7

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

   5. Applied rewrites 80.7%

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

   7. Applied rewrites 80.6%

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

   if -9.99999999999999966e89 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 0.73999999999999999

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 93.4

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

   5. Applied rewrites 93.4%

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

   if 0.73999999999999999 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 90.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. count-2-rev N/A

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

      7. associate--l+ N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval 92.2

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 92.2

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 92.2

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

      25. lift-*.f64 N/A

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

      26. *-commutative N/A

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

      27. lower-*.f64 92.2

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

   4. Applied rewrites 92.2%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 89.0

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

   7. Applied rewrites 89.0%

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

   8. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 82.8

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

   10. Applied rewrites 82.8%

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

   12. Applied rewrites 82.8%

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

Alternative 6: 85.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(b \cdot a\right) \cdot 27\\ t_2 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+90}:\\ \;\;\;\;\mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, t\_1\right)\\ \mathbf{elif}\;t\_2 \leq 0.74:\\ \;\;\;\;\mathsf{fma}\left(2, x, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(t \cdot \left(-9 \cdot y\right), z, x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* b a) 27.0)) (t_2 (* (* (* y 9.0) z) t)))
   (if (<= t_2 -1e+90)
     (fma -9.0 (* (* z y) t) t_1)
     (if (<= t_2 0.74) (fma 2.0 x t_1) (+ x (fma (* t (* -9.0 y)) z x))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (b * a) * 27.0;
	double t_2 = ((y * 9.0) * z) * t;
	double tmp;
	if (t_2 <= -1e+90) {
		tmp = fma(-9.0, ((z * y) * t), t_1);
	} else if (t_2 <= 0.74) {
		tmp = fma(2.0, x, t_1);
	} else {
		tmp = x + fma((t * (-9.0 * y)), z, x);
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(b * a) * 27.0)
	t_2 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if (t_2 <= -1e+90)
		tmp = fma(-9.0, Float64(Float64(z * y) * t), t_1);
	elseif (t_2 <= 0.74)
		tmp = fma(2.0, x, t_1);
	else
		tmp = Float64(x + fma(Float64(t * Float64(-9.0 * y)), z, x));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b * a), $MachinePrecision] * 27.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+90], N[(-9.0 * N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 0.74], N[(2.0 * x + t$95$1), $MachinePrecision], N[(x + N[(N[(t * N[(-9.0 * y), $MachinePrecision]), $MachinePrecision] * z + x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -9.99999999999999966e89

   1. Initial program 85.1%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   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. fp-cancel-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 80.7

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

   5. Applied rewrites 80.7%

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

   if -9.99999999999999966e89 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 0.73999999999999999

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 93.4

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

   5. Applied rewrites 93.4%

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

   if 0.73999999999999999 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 90.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. count-2-rev N/A

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

      7. associate--l+ N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval 92.2

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 92.2

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 92.2

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

      25. lift-*.f64 N/A

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

      26. *-commutative N/A

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

      27. lower-*.f64 92.2

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

   4. Applied rewrites 92.2%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 89.0

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

   7. Applied rewrites 89.0%

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

   8. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 82.8

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

   10. Applied rewrites 82.8%

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

   12. Applied rewrites 82.8%

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

Alternative 7: 84.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+90}:\\ \;\;\;\;x + \mathsf{fma}\left(\left(z \cdot t\right) \cdot -9, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 0.74:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(b \cdot a\right) \cdot 27\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(t \cdot \left(-9 \cdot y\right), z, x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* y 9.0) z) t)))
   (if (<= t_1 -1e+90)
     (+ x (fma (* (* z t) -9.0) y x))
     (if (<= t_1 0.74)
       (fma 2.0 x (* (* b a) 27.0))
       (+ x (fma (* t (* -9.0 y)) z x))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((y * 9.0) * z) * t;
	double tmp;
	if (t_1 <= -1e+90) {
		tmp = x + fma(((z * t) * -9.0), y, x);
	} else if (t_1 <= 0.74) {
		tmp = fma(2.0, x, ((b * a) * 27.0));
	} else {
		tmp = x + fma((t * (-9.0 * y)), z, x);
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if (t_1 <= -1e+90)
		tmp = Float64(x + fma(Float64(Float64(z * t) * -9.0), y, x));
	elseif (t_1 <= 0.74)
		tmp = fma(2.0, x, Float64(Float64(b * a) * 27.0));
	else
		tmp = Float64(x + fma(Float64(t * Float64(-9.0 * y)), z, x));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+90], N[(x + N[(N[(N[(z * t), $MachinePrecision] * -9.0), $MachinePrecision] * y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.74], N[(2.0 * x + N[(N[(b * a), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(t * N[(-9.0 * y), $MachinePrecision]), $MachinePrecision] * z + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -9.99999999999999966e89

   1. Initial program 85.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. count-2-rev N/A

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

      7. associate--l+ N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval 85.1

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 85.1

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 85.1

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

      25. lift-*.f64 N/A

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

      26. *-commutative N/A

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

      27. lower-*.f64 85.1

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

   4. Applied rewrites 85.1%

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

   5. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 74.7

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

   7. Applied rewrites 74.7%

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

   if -9.99999999999999966e89 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 0.73999999999999999

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 93.4

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

   5. Applied rewrites 93.4%

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

   if 0.73999999999999999 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 90.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. count-2-rev N/A

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

      7. associate--l+ N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval 92.2

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 92.2

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 92.2

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

      25. lift-*.f64 N/A

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

      26. *-commutative N/A

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

      27. lower-*.f64 92.2

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

   4. Applied rewrites 92.2%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 89.0

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

   7. Applied rewrites 89.0%

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

   8. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 82.8

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

   10. Applied rewrites 82.8%

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

   12. Applied rewrites 82.8%

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

Alternative 8: 84.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+90}:\\ \;\;\;\;x + \mathsf{fma}\left(y, t \cdot \left(-9 \cdot z\right), x\right)\\ \mathbf{elif}\;t\_1 \leq 0.74:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(b \cdot a\right) \cdot 27\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(t \cdot \left(-9 \cdot y\right), z, x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* y 9.0) z) t)))
   (if (<= t_1 -1e+90)
     (+ x (fma y (* t (* -9.0 z)) x))
     (if (<= t_1 0.74)
       (fma 2.0 x (* (* b a) 27.0))
       (+ x (fma (* t (* -9.0 y)) z x))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((y * 9.0) * z) * t;
	double tmp;
	if (t_1 <= -1e+90) {
		tmp = x + fma(y, (t * (-9.0 * z)), x);
	} else if (t_1 <= 0.74) {
		tmp = fma(2.0, x, ((b * a) * 27.0));
	} else {
		tmp = x + fma((t * (-9.0 * y)), z, x);
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if (t_1 <= -1e+90)
		tmp = Float64(x + fma(y, Float64(t * Float64(-9.0 * z)), x));
	elseif (t_1 <= 0.74)
		tmp = fma(2.0, x, Float64(Float64(b * a) * 27.0));
	else
		tmp = Float64(x + fma(Float64(t * Float64(-9.0 * y)), z, x));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+90], N[(x + N[(y * N[(t * N[(-9.0 * z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.74], N[(2.0 * x + N[(N[(b * a), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(t * N[(-9.0 * y), $MachinePrecision]), $MachinePrecision] * z + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -9.99999999999999966e89

   1. Initial program 85.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. count-2-rev N/A

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

      7. associate--l+ N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval 85.1

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 85.1

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 85.1

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

      25. lift-*.f64 N/A

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

      26. *-commutative N/A

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

      27. lower-*.f64 85.1

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

   4. Applied rewrites 85.1%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 89.4

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

   7. Applied rewrites 89.4%

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

   8. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 78.8

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

   10. Applied rewrites 78.8%

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

   12. Applied rewrites 74.6%

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

   if -9.99999999999999966e89 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 0.73999999999999999

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 93.4

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

   5. Applied rewrites 93.4%

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

   if 0.73999999999999999 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 90.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. count-2-rev N/A

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

      7. associate--l+ N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval 92.2

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 92.2

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 92.2

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

      25. lift-*.f64 N/A

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

      26. *-commutative N/A

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

      27. lower-*.f64 92.2

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

   4. Applied rewrites 92.2%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 89.0

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

   7. Applied rewrites 89.0%

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

   8. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 82.8

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

   10. Applied rewrites 82.8%

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

   12. Applied rewrites 82.8%

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

Alternative 9: 84.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+90}:\\ \;\;\;\;x + \mathsf{fma}\left(y, t \cdot \left(-9 \cdot z\right), x\right)\\ \mathbf{elif}\;t\_1 \leq 0.74:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(b \cdot a\right) \cdot 27\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(-9 \cdot z, y \cdot t, x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* y 9.0) z) t)))
   (if (<= t_1 -1e+90)
     (+ x (fma y (* t (* -9.0 z)) x))
     (if (<= t_1 0.74)
       (fma 2.0 x (* (* b a) 27.0))
       (+ x (fma (* -9.0 z) (* y t) x))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((y * 9.0) * z) * t;
	double tmp;
	if (t_1 <= -1e+90) {
		tmp = x + fma(y, (t * (-9.0 * z)), x);
	} else if (t_1 <= 0.74) {
		tmp = fma(2.0, x, ((b * a) * 27.0));
	} else {
		tmp = x + fma((-9.0 * z), (y * t), x);
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if (t_1 <= -1e+90)
		tmp = Float64(x + fma(y, Float64(t * Float64(-9.0 * z)), x));
	elseif (t_1 <= 0.74)
		tmp = fma(2.0, x, Float64(Float64(b * a) * 27.0));
	else
		tmp = Float64(x + fma(Float64(-9.0 * z), Float64(y * t), x));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+90], N[(x + N[(y * N[(t * N[(-9.0 * z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.74], N[(2.0 * x + N[(N[(b * a), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(-9.0 * z), $MachinePrecision] * N[(y * t), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -9.99999999999999966e89

   1. Initial program 85.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. count-2-rev N/A

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

      7. associate--l+ N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval 85.1

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 85.1

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 85.1

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

      25. lift-*.f64 N/A

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

      26. *-commutative N/A

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

      27. lower-*.f64 85.1

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

   4. Applied rewrites 85.1%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 89.4

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

   7. Applied rewrites 89.4%

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

   8. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 78.8

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

   10. Applied rewrites 78.8%

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

   12. Applied rewrites 74.6%

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

   if -9.99999999999999966e89 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 0.73999999999999999

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 93.4

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

   5. Applied rewrites 93.4%

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

   if 0.73999999999999999 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 90.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. count-2-rev N/A

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

      7. associate--l+ N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval 92.2

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

      19. lift-*.f64 N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, \color{blue}{\left(\left(y \cdot 9\right) \cdot z\right) \cdot t}\right)\right) \]

      20. *-commutative N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, \color{blue}{t \cdot \left(\left(y \cdot 9\right) \cdot z\right)}\right)\right) \]

      21. lower-*.f64 92.2

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, \color{blue}{t \cdot \left(\left(y \cdot 9\right) \cdot z\right)}\right)\right) \]

      22. lift-*.f64 N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \color{blue}{\left(\left(y \cdot 9\right) \cdot z\right)}\right)\right) \]

      23. *-commutative N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \color{blue}{\left(z \cdot \left(y \cdot 9\right)\right)}\right)\right) \]

      24. lower-*.f64 92.2

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \color{blue}{\left(z \cdot \left(y \cdot 9\right)\right)}\right)\right) \]

      25. lift-*.f64 N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \left(z \cdot \color{blue}{\left(y \cdot 9\right)}\right)\right)\right) \]

      26. *-commutative N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \left(z \cdot \color{blue}{\left(9 \cdot y\right)}\right)\right)\right) \]

      27. lower-*.f64 92.2

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \left(z \cdot \color{blue}{\left(9 \cdot y\right)}\right)\right)\right) \]

   4. Applied rewrites 92.2%

      \[\leadsto \color{blue}{x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \left(z \cdot \left(9 \cdot y\right)\right)\right)\right)} \]

   5. Taylor expanded in y around inf

      \[\leadsto x + \left(x - \color{blue}{y \cdot \left(-27 \cdot \frac{a \cdot b}{y} + 9 \cdot \left(t \cdot z\right)\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x + \left(x - \color{blue}{\left(-27 \cdot \frac{a \cdot b}{y} + 9 \cdot \left(t \cdot z\right)\right) \cdot y}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto x + \left(x - \color{blue}{\left(-27 \cdot \frac{a \cdot b}{y} + 9 \cdot \left(t \cdot z\right)\right) \cdot y}\right) \]

      3. +-commutative N/A

         \[\leadsto x + \left(x - \color{blue}{\left(9 \cdot \left(t \cdot z\right) + -27 \cdot \frac{a \cdot b}{y}\right)} \cdot y\right) \]

      4. *-commutative N/A

         \[\leadsto x + \left(x - \left(\color{blue}{\left(t \cdot z\right) \cdot 9} + -27 \cdot \frac{a \cdot b}{y}\right) \cdot y\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto x + \left(x - \color{blue}{\mathsf{fma}\left(t \cdot z, 9, -27 \cdot \frac{a \cdot b}{y}\right)} \cdot y\right) \]

      6. *-commutative N/A

         \[\leadsto x + \left(x - \mathsf{fma}\left(\color{blue}{z \cdot t}, 9, -27 \cdot \frac{a \cdot b}{y}\right) \cdot y\right) \]

      7. lower-*.f64 N/A

         \[\leadsto x + \left(x - \mathsf{fma}\left(\color{blue}{z \cdot t}, 9, -27 \cdot \frac{a \cdot b}{y}\right) \cdot y\right) \]

      8. *-commutative N/A

         \[\leadsto x + \left(x - \mathsf{fma}\left(z \cdot t, 9, \color{blue}{\frac{a \cdot b}{y} \cdot -27}\right) \cdot y\right) \]

      9. lower-*.f64 N/A

         \[\leadsto x + \left(x - \mathsf{fma}\left(z \cdot t, 9, \color{blue}{\frac{a \cdot b}{y} \cdot -27}\right) \cdot y\right) \]

      10. lower-/.f64 N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(z \cdot t, 9, \color{blue}{\frac{a \cdot b}{y}} \cdot -27\right) \cdot y\right) \]

      11. *-commutative N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(z \cdot t, 9, \frac{\color{blue}{b \cdot a}}{y} \cdot -27\right) \cdot y\right) \]

      12. lower-*.f64 89.0

          \[\leadsto x + \left(x - \mathsf{fma}\left(z \cdot t, 9, \frac{\color{blue}{b \cdot a}}{y} \cdot -27\right) \cdot y\right) \]

   7. Applied rewrites 89.0%

      \[\leadsto x + \left(x - \color{blue}{\mathsf{fma}\left(z \cdot t, 9, \frac{b \cdot a}{y} \cdot -27\right) \cdot y}\right) \]

   8. Taylor expanded in a around 0

      \[\leadsto x + \color{blue}{\left(x - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right)} \]
   9. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto x + \left(x - \color{blue}{\left(\mathsf{neg}\left(-9\right)\right)} \cdot \left(t \cdot \left(y \cdot z\right)\right)\right) \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto x + \color{blue}{\left(x + -9 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto x + \color{blue}{\left(-9 \cdot \left(t \cdot \left(y \cdot z\right)\right) + x\right)} \]

      4. associate-*r* N/A

         \[\leadsto x + \left(-9 \cdot \color{blue}{\left(\left(t \cdot y\right) \cdot z\right)} + x\right) \]

      5. associate-*r* N/A

         \[\leadsto x + \left(\color{blue}{\left(-9 \cdot \left(t \cdot y\right)\right) \cdot z} + x\right) \]

      6. *-commutative N/A

         \[\leadsto x + \left(\color{blue}{z \cdot \left(-9 \cdot \left(t \cdot y\right)\right)} + x\right) \]

      7. associate-*r* N/A

         \[\leadsto x + \left(\color{blue}{\left(z \cdot -9\right) \cdot \left(t \cdot y\right)} + x\right) \]

      8. *-commutative N/A

         \[\leadsto x + \left(\color{blue}{\left(-9 \cdot z\right)} \cdot \left(t \cdot y\right) + x\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto x + \color{blue}{\mathsf{fma}\left(-9 \cdot z, t \cdot y, x\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto x + \mathsf{fma}\left(\color{blue}{-9 \cdot z}, t \cdot y, x\right) \]

      11. *-commutative N/A

          \[\leadsto x + \mathsf{fma}\left(-9 \cdot z, \color{blue}{y \cdot t}, x\right) \]

      12. lower-*.f64 82.8

          \[\leadsto x + \mathsf{fma}\left(-9 \cdot z, \color{blue}{y \cdot t}, x\right) \]

   10. Applied rewrites 82.8%

       \[\leadsto x + \color{blue}{\mathsf{fma}\left(-9 \cdot z, y \cdot t, x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 82.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+90} \lor \neg \left(t\_1 \leq 10^{+82}\right):\\ \;\;\;\;-9 \cdot \left(\left(y \cdot z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(b \cdot a\right) \cdot 27\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* y 9.0) z) t)))
   (if (or (<= t_1 -1e+90) (not (<= t_1 1e+82)))
     (* -9.0 (* (* y z) t))
     (fma 2.0 x (* (* b a) 27.0)))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((y * 9.0) * z) * t;
	double tmp;
	if ((t_1 <= -1e+90) || !(t_1 <= 1e+82)) {
		tmp = -9.0 * ((y * z) * t);
	} else {
		tmp = fma(2.0, x, ((b * a) * 27.0));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if ((t_1 <= -1e+90) || !(t_1 <= 1e+82))
		tmp = Float64(-9.0 * Float64(Float64(y * z) * t));
	else
		tmp = fma(2.0, x, Float64(Float64(b * a) * 27.0));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e+90], N[Not[LessEqual[t$95$1, 1e+82]], $MachinePrecision]], N[(-9.0 * N[(N[(y * z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 * x + N[(N[(b * a), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -9.99999999999999966e89 or 9.9999999999999996e81 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 86.8%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   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. associate-+l- N/A

         \[\leadsto \color{blue}{x \cdot 2 - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot 2} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{2 \cdot x} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      6. count-2-rev N/A

         \[\leadsto \color{blue}{\left(x + x\right)} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      7. associate--l+ N/A

         \[\leadsto \color{blue}{x + \left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)\right)} \]

      8. lower-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto x + \color{blue}{\left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto x + \left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \color{blue}{\left(a \cdot 27\right) \cdot b}\right)\right) \]

      11. fp-cancel-sub-sign-inv N/A

          \[\leadsto x + \left(x - \color{blue}{\left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t + \left(\mathsf{neg}\left(a \cdot 27\right)\right) \cdot b\right)}\right) \]

      12. +-commutative N/A

          \[\leadsto x + \left(x - \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot 27\right)\right) \cdot b + \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)}\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto x + \left(x - \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a \cdot 27\right), b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{a \cdot 27}\right), b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) \]

      15. *-commutative N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{27 \cdot a}\right), b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot a}, b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) \]

      17. lower-*.f64 N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot a}, b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) \]

      18. metadata-eval 87.9

          \[\leadsto x + \left(x - \mathsf{fma}\left(\color{blue}{-27} \cdot a, b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) \]

      19. lift-*.f64 N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, \color{blue}{\left(\left(y \cdot 9\right) \cdot z\right) \cdot t}\right)\right) \]

      20. *-commutative N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, \color{blue}{t \cdot \left(\left(y \cdot 9\right) \cdot z\right)}\right)\right) \]

      21. lower-*.f64 87.9

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, \color{blue}{t \cdot \left(\left(y \cdot 9\right) \cdot z\right)}\right)\right) \]

      22. lift-*.f64 N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \color{blue}{\left(\left(y \cdot 9\right) \cdot z\right)}\right)\right) \]

      23. *-commutative N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \color{blue}{\left(z \cdot \left(y \cdot 9\right)\right)}\right)\right) \]

      24. lower-*.f64 87.9

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \color{blue}{\left(z \cdot \left(y \cdot 9\right)\right)}\right)\right) \]

      25. lift-*.f64 N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \left(z \cdot \color{blue}{\left(y \cdot 9\right)}\right)\right)\right) \]

      26. *-commutative N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \left(z \cdot \color{blue}{\left(9 \cdot y\right)}\right)\right)\right) \]

      27. lower-*.f64 87.9

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \left(z \cdot \color{blue}{\left(9 \cdot y\right)}\right)\right)\right) \]

   4. Applied rewrites 87.9%

      \[\leadsto \color{blue}{x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \left(z \cdot \left(9 \cdot y\right)\right)\right)\right)} \]

   5. Taylor expanded in y around inf

      \[\leadsto x + \left(x - \color{blue}{y \cdot \left(-27 \cdot \frac{a \cdot b}{y} + 9 \cdot \left(t \cdot z\right)\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x + \left(x - \color{blue}{\left(-27 \cdot \frac{a \cdot b}{y} + 9 \cdot \left(t \cdot z\right)\right) \cdot y}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto x + \left(x - \color{blue}{\left(-27 \cdot \frac{a \cdot b}{y} + 9 \cdot \left(t \cdot z\right)\right) \cdot y}\right) \]

      3. +-commutative N/A

         \[\leadsto x + \left(x - \color{blue}{\left(9 \cdot \left(t \cdot z\right) + -27 \cdot \frac{a \cdot b}{y}\right)} \cdot y\right) \]

      4. *-commutative N/A

         \[\leadsto x + \left(x - \left(\color{blue}{\left(t \cdot z\right) \cdot 9} + -27 \cdot \frac{a \cdot b}{y}\right) \cdot y\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto x + \left(x - \color{blue}{\mathsf{fma}\left(t \cdot z, 9, -27 \cdot \frac{a \cdot b}{y}\right)} \cdot y\right) \]

      6. *-commutative N/A

         \[\leadsto x + \left(x - \mathsf{fma}\left(\color{blue}{z \cdot t}, 9, -27 \cdot \frac{a \cdot b}{y}\right) \cdot y\right) \]

      7. lower-*.f64 N/A

         \[\leadsto x + \left(x - \mathsf{fma}\left(\color{blue}{z \cdot t}, 9, -27 \cdot \frac{a \cdot b}{y}\right) \cdot y\right) \]

      8. *-commutative N/A

         \[\leadsto x + \left(x - \mathsf{fma}\left(z \cdot t, 9, \color{blue}{\frac{a \cdot b}{y} \cdot -27}\right) \cdot y\right) \]

      9. lower-*.f64 N/A

         \[\leadsto x + \left(x - \mathsf{fma}\left(z \cdot t, 9, \color{blue}{\frac{a \cdot b}{y} \cdot -27}\right) \cdot y\right) \]

      10. lower-/.f64 N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(z \cdot t, 9, \color{blue}{\frac{a \cdot b}{y}} \cdot -27\right) \cdot y\right) \]

      11. *-commutative N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(z \cdot t, 9, \frac{\color{blue}{b \cdot a}}{y} \cdot -27\right) \cdot y\right) \]

      12. lower-*.f64 88.9

          \[\leadsto x + \left(x - \mathsf{fma}\left(z \cdot t, 9, \frac{\color{blue}{b \cdot a}}{y} \cdot -27\right) \cdot y\right) \]

   7. Applied rewrites 88.9%

      \[\leadsto x + \left(x - \color{blue}{\mathsf{fma}\left(z \cdot t, 9, \frac{b \cdot a}{y} \cdot -27\right) \cdot y}\right) \]

   8. Taylor expanded in y around inf

      \[\leadsto \color{blue}{-9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{-9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto -9 \cdot \color{blue}{\left(\left(y \cdot z\right) \cdot t\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto -9 \cdot \color{blue}{\left(\left(y \cdot z\right) \cdot t\right)} \]

      4. lower-*.f64 76.6

         \[\leadsto -9 \cdot \left(\color{blue}{\left(y \cdot z\right)} \cdot t\right) \]

   10. Applied rewrites 76.6%

       \[\leadsto \color{blue}{-9 \cdot \left(\left(y \cdot z\right) \cdot t\right)} \]

   if -9.99999999999999966e89 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 9.9999999999999996e81

   1. Initial program 99.8%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{2 \cdot x + 27 \cdot \left(a \cdot b\right)} \]
   4. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(2, x, \color{blue}{\left(a \cdot b\right) \cdot 27}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x, \color{blue}{\left(a \cdot b\right) \cdot 27}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(2, x, \color{blue}{\left(b \cdot a\right)} \cdot 27\right) \]

      5. lower-*.f64 90.3

         \[\leadsto \mathsf{fma}\left(2, x, \color{blue}{\left(b \cdot a\right)} \cdot 27\right) \]

   5. Applied rewrites 90.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, \left(b \cdot a\right) \cdot 27\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \leq -1 \cdot 10^{+90} \lor \neg \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \leq 10^{+82}\right):\\ \;\;\;\;-9 \cdot \left(\left(y \cdot z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(b \cdot a\right) \cdot 27\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 82.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+90} \lor \neg \left(t\_1 \leq 10^{+82}\right):\\ \;\;\;\;-9 \cdot \left(\left(y \cdot z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(b \cdot a, 27, x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* y 9.0) z) t)))
   (if (or (<= t_1 -1e+90) (not (<= t_1 1e+82)))
     (* -9.0 (* (* y z) t))
     (+ x (fma (* b a) 27.0 x)))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((y * 9.0) * z) * t;
	double tmp;
	if ((t_1 <= -1e+90) || !(t_1 <= 1e+82)) {
		tmp = -9.0 * ((y * z) * t);
	} else {
		tmp = x + fma((b * a), 27.0, x);
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if ((t_1 <= -1e+90) || !(t_1 <= 1e+82))
		tmp = Float64(-9.0 * Float64(Float64(y * z) * t));
	else
		tmp = Float64(x + fma(Float64(b * a), 27.0, x));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e+90], N[Not[LessEqual[t$95$1, 1e+82]], $MachinePrecision]], N[(-9.0 * N[(N[(y * z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(b * a), $MachinePrecision] * 27.0 + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -9.99999999999999966e89 or 9.9999999999999996e81 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 86.8%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   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. associate-+l- N/A

         \[\leadsto \color{blue}{x \cdot 2 - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot 2} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{2 \cdot x} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      6. count-2-rev N/A

         \[\leadsto \color{blue}{\left(x + x\right)} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      7. associate--l+ N/A

         \[\leadsto \color{blue}{x + \left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)\right)} \]

      8. lower-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto x + \color{blue}{\left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto x + \left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \color{blue}{\left(a \cdot 27\right) \cdot b}\right)\right) \]

      11. fp-cancel-sub-sign-inv N/A

          \[\leadsto x + \left(x - \color{blue}{\left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t + \left(\mathsf{neg}\left(a \cdot 27\right)\right) \cdot b\right)}\right) \]

      12. +-commutative N/A

          \[\leadsto x + \left(x - \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot 27\right)\right) \cdot b + \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)}\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto x + \left(x - \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a \cdot 27\right), b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{a \cdot 27}\right), b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) \]

      15. *-commutative N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{27 \cdot a}\right), b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot a}, b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) \]

      17. lower-*.f64 N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot a}, b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) \]

      18. metadata-eval 87.9

          \[\leadsto x + \left(x - \mathsf{fma}\left(\color{blue}{-27} \cdot a, b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) \]

      19. lift-*.f64 N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, \color{blue}{\left(\left(y \cdot 9\right) \cdot z\right) \cdot t}\right)\right) \]

      20. *-commutative N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, \color{blue}{t \cdot \left(\left(y \cdot 9\right) \cdot z\right)}\right)\right) \]

      21. lower-*.f64 87.9

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, \color{blue}{t \cdot \left(\left(y \cdot 9\right) \cdot z\right)}\right)\right) \]

      22. lift-*.f64 N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \color{blue}{\left(\left(y \cdot 9\right) \cdot z\right)}\right)\right) \]

      23. *-commutative N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \color{blue}{\left(z \cdot \left(y \cdot 9\right)\right)}\right)\right) \]

      24. lower-*.f64 87.9

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \color{blue}{\left(z \cdot \left(y \cdot 9\right)\right)}\right)\right) \]

      25. lift-*.f64 N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \left(z \cdot \color{blue}{\left(y \cdot 9\right)}\right)\right)\right) \]

      26. *-commutative N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \left(z \cdot \color{blue}{\left(9 \cdot y\right)}\right)\right)\right) \]

      27. lower-*.f64 87.9

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \left(z \cdot \color{blue}{\left(9 \cdot y\right)}\right)\right)\right) \]

   4. Applied rewrites 87.9%

      \[\leadsto \color{blue}{x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \left(z \cdot \left(9 \cdot y\right)\right)\right)\right)} \]

   5. Taylor expanded in y around inf

      \[\leadsto x + \left(x - \color{blue}{y \cdot \left(-27 \cdot \frac{a \cdot b}{y} + 9 \cdot \left(t \cdot z\right)\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x + \left(x - \color{blue}{\left(-27 \cdot \frac{a \cdot b}{y} + 9 \cdot \left(t \cdot z\right)\right) \cdot y}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto x + \left(x - \color{blue}{\left(-27 \cdot \frac{a \cdot b}{y} + 9 \cdot \left(t \cdot z\right)\right) \cdot y}\right) \]

      3. +-commutative N/A

         \[\leadsto x + \left(x - \color{blue}{\left(9 \cdot \left(t \cdot z\right) + -27 \cdot \frac{a \cdot b}{y}\right)} \cdot y\right) \]

      4. *-commutative N/A

         \[\leadsto x + \left(x - \left(\color{blue}{\left(t \cdot z\right) \cdot 9} + -27 \cdot \frac{a \cdot b}{y}\right) \cdot y\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto x + \left(x - \color{blue}{\mathsf{fma}\left(t \cdot z, 9, -27 \cdot \frac{a \cdot b}{y}\right)} \cdot y\right) \]

      6. *-commutative N/A

         \[\leadsto x + \left(x - \mathsf{fma}\left(\color{blue}{z \cdot t}, 9, -27 \cdot \frac{a \cdot b}{y}\right) \cdot y\right) \]

      7. lower-*.f64 N/A

         \[\leadsto x + \left(x - \mathsf{fma}\left(\color{blue}{z \cdot t}, 9, -27 \cdot \frac{a \cdot b}{y}\right) \cdot y\right) \]

      8. *-commutative N/A

         \[\leadsto x + \left(x - \mathsf{fma}\left(z \cdot t, 9, \color{blue}{\frac{a \cdot b}{y} \cdot -27}\right) \cdot y\right) \]

      9. lower-*.f64 N/A

         \[\leadsto x + \left(x - \mathsf{fma}\left(z \cdot t, 9, \color{blue}{\frac{a \cdot b}{y} \cdot -27}\right) \cdot y\right) \]

      10. lower-/.f64 N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(z \cdot t, 9, \color{blue}{\frac{a \cdot b}{y}} \cdot -27\right) \cdot y\right) \]

      11. *-commutative N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(z \cdot t, 9, \frac{\color{blue}{b \cdot a}}{y} \cdot -27\right) \cdot y\right) \]

      12. lower-*.f64 88.9

          \[\leadsto x + \left(x - \mathsf{fma}\left(z \cdot t, 9, \frac{\color{blue}{b \cdot a}}{y} \cdot -27\right) \cdot y\right) \]

   7. Applied rewrites 88.9%

      \[\leadsto x + \left(x - \color{blue}{\mathsf{fma}\left(z \cdot t, 9, \frac{b \cdot a}{y} \cdot -27\right) \cdot y}\right) \]

   8. Taylor expanded in y around inf

      \[\leadsto \color{blue}{-9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{-9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto -9 \cdot \color{blue}{\left(\left(y \cdot z\right) \cdot t\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto -9 \cdot \color{blue}{\left(\left(y \cdot z\right) \cdot t\right)} \]

      4. lower-*.f64 76.6

         \[\leadsto -9 \cdot \left(\color{blue}{\left(y \cdot z\right)} \cdot t\right) \]

   10. Applied rewrites 76.6%

       \[\leadsto \color{blue}{-9 \cdot \left(\left(y \cdot z\right) \cdot t\right)} \]

   if -9.99999999999999966e89 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 9.9999999999999996e81

   1. Initial program 99.8%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b} \]

      2. 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. associate-+l- N/A

         \[\leadsto \color{blue}{x \cdot 2 - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot 2} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{2 \cdot x} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      6. count-2-rev N/A

         \[\leadsto \color{blue}{\left(x + x\right)} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      7. associate--l+ N/A

         \[\leadsto \color{blue}{x + \left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)\right)} \]

      8. lower-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto x + \color{blue}{\left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto x + \left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \color{blue}{\left(a \cdot 27\right) \cdot b}\right)\right) \]

      11. fp-cancel-sub-sign-inv N/A

          \[\leadsto x + \left(x - \color{blue}{\left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t + \left(\mathsf{neg}\left(a \cdot 27\right)\right) \cdot b\right)}\right) \]

      12. +-commutative N/A

          \[\leadsto x + \left(x - \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot 27\right)\right) \cdot b + \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)}\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto x + \left(x - \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a \cdot 27\right), b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{a \cdot 27}\right), b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) \]

      15. *-commutative N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{27 \cdot a}\right), b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot a}, b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) \]

      17. lower-*.f64 N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot a}, b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) \]

      18. metadata-eval 99.8

          \[\leadsto x + \left(x - \mathsf{fma}\left(\color{blue}{-27} \cdot a, b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) \]

      19. lift-*.f64 N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, \color{blue}{\left(\left(y \cdot 9\right) \cdot z\right) \cdot t}\right)\right) \]

      20. *-commutative N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, \color{blue}{t \cdot \left(\left(y \cdot 9\right) \cdot z\right)}\right)\right) \]

      21. lower-*.f64 99.8

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, \color{blue}{t \cdot \left(\left(y \cdot 9\right) \cdot z\right)}\right)\right) \]

      22. lift-*.f64 N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \color{blue}{\left(\left(y \cdot 9\right) \cdot z\right)}\right)\right) \]

      23. *-commutative N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \color{blue}{\left(z \cdot \left(y \cdot 9\right)\right)}\right)\right) \]

      24. lower-*.f64 99.8

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \color{blue}{\left(z \cdot \left(y \cdot 9\right)\right)}\right)\right) \]

      25. lift-*.f64 N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \left(z \cdot \color{blue}{\left(y \cdot 9\right)}\right)\right)\right) \]

      26. *-commutative N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \left(z \cdot \color{blue}{\left(9 \cdot y\right)}\right)\right)\right) \]

      27. lower-*.f64 99.8

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \left(z \cdot \color{blue}{\left(9 \cdot y\right)}\right)\right)\right) \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \left(z \cdot \left(9 \cdot y\right)\right)\right)\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto x + \color{blue}{\left(x - -27 \cdot \left(a \cdot b\right)\right)} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto x + \left(x - \color{blue}{\left(\mathsf{neg}\left(27\right)\right)} \cdot \left(a \cdot b\right)\right) \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto x + \color{blue}{\left(x + 27 \cdot \left(a \cdot b\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto x + \color{blue}{\left(27 \cdot \left(a \cdot b\right) + x\right)} \]

      4. *-commutative N/A

         \[\leadsto x + \left(\color{blue}{\left(a \cdot b\right) \cdot 27} + x\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto x + \color{blue}{\mathsf{fma}\left(a \cdot b, 27, x\right)} \]

      6. *-commutative N/A

         \[\leadsto x + \mathsf{fma}\left(\color{blue}{b \cdot a}, 27, x\right) \]

      7. lower-*.f64 90.3

         \[\leadsto x + \mathsf{fma}\left(\color{blue}{b \cdot a}, 27, x\right) \]

   7. Applied rewrites 90.3%

      \[\leadsto x + \color{blue}{\mathsf{fma}\left(b \cdot a, 27, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \leq -1 \cdot 10^{+90} \lor \neg \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \leq 10^{+82}\right):\\ \;\;\;\;-9 \cdot \left(\left(y \cdot z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(b \cdot a, 27, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;\left(y \cdot 9\right) \cdot z \leq 5 \cdot 10^{+283}:\\ \;\;\;\;x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \left(z \cdot \left(9 \cdot y\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, 27, \left(\left(-9 \cdot y\right) \cdot t\right) \cdot z\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* (* y 9.0) z) 5e+283)
   (+ x (- x (fma (* -27.0 a) b (* t (* z (* 9.0 y))))))
   (fma (* b a) 27.0 (* (* (* -9.0 y) t) z))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((y * 9.0) * z) <= 5e+283) {
		tmp = x + (x - fma((-27.0 * a), b, (t * (z * (9.0 * y)))));
	} else {
		tmp = fma((b * a), 27.0, (((-9.0 * y) * t) * z));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(y * 9.0) * z) <= 5e+283)
		tmp = Float64(x + Float64(x - fma(Float64(-27.0 * a), b, Float64(t * Float64(z * Float64(9.0 * y))))));
	else
		tmp = fma(Float64(b * a), 27.0, Float64(Float64(Float64(-9.0 * y) * t) * z));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision], 5e+283], N[(x + N[(x - N[(N[(-27.0 * a), $MachinePrecision] * b + N[(t * N[(z * N[(9.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * a), $MachinePrecision] * 27.0 + N[(N[(N[(-9.0 * y), $MachinePrecision] * t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y #s(literal 9 binary64)) z) < 5.0000000000000004e283

   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. associate-+l- N/A

         \[\leadsto \color{blue}{x \cdot 2 - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot 2} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{2 \cdot x} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      6. count-2-rev N/A

         \[\leadsto \color{blue}{\left(x + x\right)} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      7. associate--l+ N/A

         \[\leadsto \color{blue}{x + \left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)\right)} \]

      8. lower-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto x + \color{blue}{\left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto x + \left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \color{blue}{\left(a \cdot 27\right) \cdot b}\right)\right) \]

      11. fp-cancel-sub-sign-inv N/A

          \[\leadsto x + \left(x - \color{blue}{\left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t + \left(\mathsf{neg}\left(a \cdot 27\right)\right) \cdot b\right)}\right) \]

      12. +-commutative N/A

          \[\leadsto x + \left(x - \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot 27\right)\right) \cdot b + \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)}\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto x + \left(x - \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a \cdot 27\right), b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{a \cdot 27}\right), b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) \]

      15. *-commutative N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{27 \cdot a}\right), b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot a}, b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) \]

      17. lower-*.f64 N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot a}, b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) \]

      18. metadata-eval 97.8

          \[\leadsto x + \left(x - \mathsf{fma}\left(\color{blue}{-27} \cdot a, b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) \]

      19. lift-*.f64 N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, \color{blue}{\left(\left(y \cdot 9\right) \cdot z\right) \cdot t}\right)\right) \]

      20. *-commutative N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, \color{blue}{t \cdot \left(\left(y \cdot 9\right) \cdot z\right)}\right)\right) \]

      21. lower-*.f64 97.8

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, \color{blue}{t \cdot \left(\left(y \cdot 9\right) \cdot z\right)}\right)\right) \]

      22. lift-*.f64 N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \color{blue}{\left(\left(y \cdot 9\right) \cdot z\right)}\right)\right) \]

      23. *-commutative N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \color{blue}{\left(z \cdot \left(y \cdot 9\right)\right)}\right)\right) \]

      24. lower-*.f64 97.8

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \color{blue}{\left(z \cdot \left(y \cdot 9\right)\right)}\right)\right) \]

      25. lift-*.f64 N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \left(z \cdot \color{blue}{\left(y \cdot 9\right)}\right)\right)\right) \]

      26. *-commutative N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \left(z \cdot \color{blue}{\left(9 \cdot y\right)}\right)\right)\right) \]

      27. lower-*.f64 97.8

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \left(z \cdot \color{blue}{\left(9 \cdot y\right)}\right)\right)\right) \]

   4. Applied rewrites 97.8%

      \[\leadsto \color{blue}{x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \left(z \cdot \left(9 \cdot y\right)\right)\right)\right)} \]

   if 5.0000000000000004e283 < (*.f64 (*.f64 y #s(literal 9 binary64)) z)

   1. Initial program 46.0%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} + \left(a \cdot 27\right) \cdot b \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto -9 \cdot \color{blue}{\left(\left(y \cdot z\right) \cdot t\right)} + \left(a \cdot 27\right) \cdot b \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-9 \cdot \left(y \cdot z\right)\right) \cdot t} + \left(a \cdot 27\right) \cdot b \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-9 \cdot \left(y \cdot z\right)\right) \cdot t} + \left(a \cdot 27\right) \cdot b \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-9 \cdot \left(y \cdot z\right)\right)} \cdot t + \left(a \cdot 27\right) \cdot b \]

      5. *-commutative N/A

         \[\leadsto \left(-9 \cdot \color{blue}{\left(z \cdot y\right)}\right) \cdot t + \left(a \cdot 27\right) \cdot b \]

      6. lower-*.f64 38.2

         \[\leadsto \left(-9 \cdot \color{blue}{\left(z \cdot y\right)}\right) \cdot t + \left(a \cdot 27\right) \cdot b \]

   5. Applied rewrites 38.2%

      \[\leadsto \color{blue}{\left(-9 \cdot \left(z \cdot y\right)\right) \cdot t} + \left(a \cdot 27\right) \cdot b \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-9 \cdot \left(z \cdot y\right)\right) \cdot t + \left(a \cdot 27\right) \cdot b} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot 27\right) \cdot b + \left(-9 \cdot \left(z \cdot y\right)\right) \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(a \cdot 27\right) \cdot b} + \left(-9 \cdot \left(z \cdot y\right)\right) \cdot t \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{b \cdot \left(a \cdot 27\right)} + \left(-9 \cdot \left(z \cdot y\right)\right) \cdot t \]

      5. lift-*.f64 N/A

         \[\leadsto b \cdot \color{blue}{\left(a \cdot 27\right)} + \left(-9 \cdot \left(z \cdot y\right)\right) \cdot t \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left(b \cdot a\right) \cdot 27} + \left(-9 \cdot \left(z \cdot y\right)\right) \cdot t \]

      7. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(b \cdot a\right)} \cdot 27 + \left(-9 \cdot \left(z \cdot y\right)\right) \cdot t \]

      8. lower-fma.f64 38.2

         \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot a, 27, \left(-9 \cdot \left(z \cdot y\right)\right) \cdot t\right)} \]

   7. Applied rewrites 92.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot a, 27, \left(\left(-9 \cdot y\right) \cdot t\right) \cdot z\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 98.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;y \cdot 9 \leq -5 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, -9 \cdot y, \mathsf{fma}\left(b \cdot 27, a, 2 \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot 27, a, \mathsf{fma}\left(\left(-9 \cdot y\right) \cdot t, z, 2 \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* y 9.0) -5e+54)
   (fma (* t z) (* -9.0 y) (fma (* b 27.0) a (* 2.0 x)))
   (fma (* b 27.0) a (fma (* (* -9.0 y) t) z (* 2.0 x)))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y * 9.0) <= -5e+54) {
		tmp = fma((t * z), (-9.0 * y), fma((b * 27.0), a, (2.0 * x)));
	} else {
		tmp = fma((b * 27.0), a, fma(((-9.0 * y) * t), z, (2.0 * x)));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(y * 9.0) <= -5e+54)
		tmp = fma(Float64(t * z), Float64(-9.0 * y), fma(Float64(b * 27.0), a, Float64(2.0 * x)));
	else
		tmp = fma(Float64(b * 27.0), a, fma(Float64(Float64(-9.0 * y) * t), z, Float64(2.0 * x)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(y * 9.0), $MachinePrecision], -5e+54], N[(N[(t * z), $MachinePrecision] * N[(-9.0 * y), $MachinePrecision] + N[(N[(b * 27.0), $MachinePrecision] * a + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * 27.0), $MachinePrecision] * a + N[(N[(N[(-9.0 * y), $MachinePrecision] * t), $MachinePrecision] * z + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y #s(literal 9 binary64)) < -5.00000000000000005e54

   1. Initial program 85.4%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   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. lift-*.f64 N/A

         \[\leadsto \left(x \cdot 2 - \color{blue}{\left(\left(y \cdot 9\right) \cdot z\right) \cdot t}\right) + \left(a \cdot 27\right) \cdot b \]

      4. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(x \cdot 2 + \left(\mathsf{neg}\left(\left(y \cdot 9\right) \cdot z\right)\right) \cdot t\right)} + \left(a \cdot 27\right) \cdot b \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\left(y \cdot 9\right) \cdot z\right)\right) \cdot t + x \cdot 2\right)} + \left(a \cdot 27\right) \cdot b \]

      6. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y \cdot 9\right) \cdot z\right)\right) \cdot t + \left(x \cdot 2 + \left(a \cdot 27\right) \cdot b\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y \cdot 9\right) \cdot z}\right)\right) \cdot t + \left(x \cdot 2 + \left(a \cdot 27\right) \cdot b\right) \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(y \cdot 9\right)\right) \cdot z\right)} \cdot t + \left(x \cdot 2 + \left(a \cdot 27\right) \cdot b\right) \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y \cdot 9\right)\right) \cdot \left(z \cdot t\right)} + \left(x \cdot 2 + \left(a \cdot 27\right) \cdot b\right) \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(z \cdot t\right) \cdot \left(\mathsf{neg}\left(y \cdot 9\right)\right)} + \left(x \cdot 2 + \left(a \cdot 27\right) \cdot b\right) \]

      11. +-commutative N/A

          \[\leadsto \left(z \cdot t\right) \cdot \left(\mathsf{neg}\left(y \cdot 9\right)\right) + \color{blue}{\left(\left(a \cdot 27\right) \cdot b + x \cdot 2\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot t, \mathsf{neg}\left(y \cdot 9\right), \left(a \cdot 27\right) \cdot b + x \cdot 2\right)} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot z, -9 \cdot y, \mathsf{fma}\left(b \cdot 27, a, 2 \cdot x\right)\right)} \]

   if -5.00000000000000005e54 < (*.f64 y #s(literal 9 binary64))

   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. +-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot 27\right) \cdot b + \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(a \cdot 27\right) \cdot b} + \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(a \cdot 27\right)} \cdot b + \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{a \cdot \left(27 \cdot b\right)} + \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(27 \cdot b\right) \cdot a} + \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(27 \cdot b, a, x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)} \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot 27}, a, x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) \]

      9. lower-*.f64 98.4

         \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot 27}, a, x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) \]

      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. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot 27, a, x \cdot 2 - \color{blue}{\left(\left(y \cdot 9\right) \cdot z\right) \cdot t}\right) \]

      12. fp-cancel-sub-sign-inv N/A

          \[\leadsto \mathsf{fma}\left(b \cdot 27, a, \color{blue}{x \cdot 2 + \left(\mathsf{neg}\left(\left(y \cdot 9\right) \cdot z\right)\right) \cdot t}\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(b \cdot 27, a, \color{blue}{\left(\mathsf{neg}\left(\left(y \cdot 9\right) \cdot z\right)\right) \cdot t + x \cdot 2}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot 27, a, \left(\mathsf{neg}\left(\color{blue}{\left(y \cdot 9\right) \cdot z}\right)\right) \cdot t + x \cdot 2\right) \]

      15. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(b \cdot 27, a, \color{blue}{\left(\left(\mathsf{neg}\left(y \cdot 9\right)\right) \cdot z\right)} \cdot t + x \cdot 2\right) \]

      16. associate-*r* 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 95.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot 27, a, \mathsf{fma}\left(\left(-9 \cdot y\right) \cdot t, z, 2 \cdot x\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 54.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(a \cdot 27\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+63} \lor \neg \left(t\_1 \leq 10^{+44}\right):\\ \;\;\;\;\left(b \cdot a\right) \cdot 27\\ \mathbf{else}:\\ \;\;\;\;x + x\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* a 27.0) b)))
   (if (or (<= t_1 -1e+63) (not (<= t_1 1e+44))) (* (* b a) 27.0) (+ x x))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * 27.0) * b;
	double tmp;
	if ((t_1 <= -1e+63) || !(t_1 <= 1e+44)) {
		tmp = (b * a) * 27.0;
	} else {
		tmp = x + x;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a * 27.0d0) * b
    if ((t_1 <= (-1d+63)) .or. (.not. (t_1 <= 1d+44))) then
        tmp = (b * a) * 27.0d0
    else
        tmp = x + x
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b;
public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * 27.0) * b;
	double tmp;
	if ((t_1 <= -1e+63) || !(t_1 <= 1e+44)) {
		tmp = (b * a) * 27.0;
	} else {
		tmp = x + x;
	}
	return tmp;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	t_1 = (a * 27.0) * b
	tmp = 0
	if (t_1 <= -1e+63) or not (t_1 <= 1e+44):
		tmp = (b * a) * 27.0
	else:
		tmp = x + x
	return tmp

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a * 27.0) * b)
	tmp = 0.0
	if ((t_1 <= -1e+63) || !(t_1 <= 1e+44))
		tmp = Float64(Float64(b * a) * 27.0);
	else
		tmp = Float64(x + x);
	end
	return tmp
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a * 27.0) * b;
	tmp = 0.0;
	if ((t_1 <= -1e+63) || ~((t_1 <= 1e+44)))
		tmp = (b * a) * 27.0;
	else
		tmp = x + x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e+63], N[Not[LessEqual[t$95$1, 1e+44]], $MachinePrecision]], N[(N[(b * a), $MachinePrecision] * 27.0), $MachinePrecision], N[(x + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 a #s(literal 27 binary64)) b) < -1.00000000000000006e63 or 1.0000000000000001e44 < (*.f64 (*.f64 a #s(literal 27 binary64)) b)

   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. 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 74.7

         \[\leadsto \mathsf{fma}\left(2, x, \color{blue}{\left(b \cdot a\right)} \cdot 27\right) \]

   5. Applied rewrites 74.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, \left(b \cdot a\right) \cdot 27\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto 27 \cdot \color{blue}{\left(a \cdot b\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 69.3%

      \[\leadsto \left(b \cdot a\right) \cdot \color{blue}{27} \]

   if -1.00000000000000006e63 < (*.f64 (*.f64 a #s(literal 27 binary64)) b) < 1.0000000000000001e44

   1. Initial program 94.7%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   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. associate-+l- N/A

         \[\leadsto \color{blue}{x \cdot 2 - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot 2} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{2 \cdot x} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      6. count-2-rev N/A

         \[\leadsto \color{blue}{\left(x + x\right)} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      7. associate--l+ N/A

         \[\leadsto \color{blue}{x + \left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)\right)} \]

      8. lower-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto x + \color{blue}{\left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto x + \left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \color{blue}{\left(a \cdot 27\right) \cdot b}\right)\right) \]

      11. fp-cancel-sub-sign-inv N/A

          \[\leadsto x + \left(x - \color{blue}{\left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t + \left(\mathsf{neg}\left(a \cdot 27\right)\right) \cdot b\right)}\right) \]

      12. +-commutative N/A

          \[\leadsto x + \left(x - \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot 27\right)\right) \cdot b + \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)}\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto x + \left(x - \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a \cdot 27\right), b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{a \cdot 27}\right), b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) \]

      15. *-commutative N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{27 \cdot a}\right), b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot a}, b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) \]

      17. lower-*.f64 N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot a}, b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) \]

      18. metadata-eval 94.7

          \[\leadsto x + \left(x - \mathsf{fma}\left(\color{blue}{-27} \cdot a, b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) \]

      19. lift-*.f64 N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, \color{blue}{\left(\left(y \cdot 9\right) \cdot z\right) \cdot t}\right)\right) \]

      20. *-commutative N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, \color{blue}{t \cdot \left(\left(y \cdot 9\right) \cdot z\right)}\right)\right) \]

      21. lower-*.f64 94.7

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, \color{blue}{t \cdot \left(\left(y \cdot 9\right) \cdot z\right)}\right)\right) \]

      22. lift-*.f64 N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \color{blue}{\left(\left(y \cdot 9\right) \cdot z\right)}\right)\right) \]

      23. *-commutative N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \color{blue}{\left(z \cdot \left(y \cdot 9\right)\right)}\right)\right) \]

      24. lower-*.f64 94.7

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \color{blue}{\left(z \cdot \left(y \cdot 9\right)\right)}\right)\right) \]

      25. lift-*.f64 N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \left(z \cdot \color{blue}{\left(y \cdot 9\right)}\right)\right)\right) \]

      26. *-commutative N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \left(z \cdot \color{blue}{\left(9 \cdot y\right)}\right)\right)\right) \]

      27. lower-*.f64 94.7

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \left(z \cdot \color{blue}{\left(9 \cdot y\right)}\right)\right)\right) \]

   4. Applied rewrites 94.7%

      \[\leadsto \color{blue}{x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \left(z \cdot \left(9 \cdot y\right)\right)\right)\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot x} \]
   6. Step-by-step derivation

      1. lower-*.f64 46.5

         \[\leadsto \color{blue}{2 \cdot x} \]

   7. Applied rewrites 46.5%

      \[\leadsto \color{blue}{2 \cdot x} \]
   8. Step-by-step derivation

   9. Applied rewrites 46.5%

      \[\leadsto x + \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a \cdot 27\right) \cdot b \leq -1 \cdot 10^{+63} \lor \neg \left(\left(a \cdot 27\right) \cdot b \leq 10^{+44}\right):\\ \;\;\;\;\left(b \cdot a\right) \cdot 27\\ \mathbf{else}:\\ \;\;\;\;x + x\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 4 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(-9 \cdot z\right) \cdot t, \mathsf{fma}\left(b \cdot 27, a, 2 \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \left(z \cdot \left(9 \cdot y\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (if (<= z 4e-16)
   (fma y (* (* -9.0 z) t) (fma (* b 27.0) a (* 2.0 x)))
   (+ x (- x (fma (* -27.0 a) b (* t (* z (* 9.0 y))))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= 4e-16) {
		tmp = fma(y, ((-9.0 * z) * t), fma((b * 27.0), a, (2.0 * x)));
	} else {
		tmp = x + (x - fma((-27.0 * a), b, (t * (z * (9.0 * y)))));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= 4e-16)
		tmp = fma(y, Float64(Float64(-9.0 * z) * t), fma(Float64(b * 27.0), a, Float64(2.0 * x)));
	else
		tmp = Float64(x + Float64(x - fma(Float64(-27.0 * a), b, Float64(t * Float64(z * Float64(9.0 * y))))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, 4e-16], N[(y * N[(N[(-9.0 * z), $MachinePrecision] * t), $MachinePrecision] + N[(N[(b * 27.0), $MachinePrecision] * a + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(x - N[(N[(-27.0 * a), $MachinePrecision] * b + N[(t * N[(z * N[(9.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.9999999999999999e-16

   1. Initial program 95.7%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)} + \left(a \cdot 27\right) \cdot b \]

      3. lift-*.f64 N/A

         \[\leadsto \left(x \cdot 2 - \color{blue}{\left(\left(y \cdot 9\right) \cdot z\right) \cdot t}\right) + \left(a \cdot 27\right) \cdot b \]

      4. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(x \cdot 2 + \left(\mathsf{neg}\left(\left(y \cdot 9\right) \cdot z\right)\right) \cdot t\right)} + \left(a \cdot 27\right) \cdot b \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\left(y \cdot 9\right) \cdot z\right)\right) \cdot t + x \cdot 2\right)} + \left(a \cdot 27\right) \cdot b \]

      6. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y \cdot 9\right) \cdot z\right)\right) \cdot t + \left(x \cdot 2 + \left(a \cdot 27\right) \cdot b\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y \cdot 9\right) \cdot z}\right)\right) \cdot t + \left(x \cdot 2 + \left(a \cdot 27\right) \cdot b\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y \cdot 9\right)} \cdot z\right)\right) \cdot t + \left(x \cdot 2 + \left(a \cdot 27\right) \cdot b\right) \]

      9. associate-*l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{y \cdot \left(9 \cdot z\right)}\right)\right) \cdot t + \left(x \cdot 2 + \left(a \cdot 27\right) \cdot b\right) \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{\left(y \cdot \left(\mathsf{neg}\left(9 \cdot z\right)\right)\right)} \cdot t + \left(x \cdot 2 + \left(a \cdot 27\right) \cdot b\right) \]

      11. associate-*l* N/A

          \[\leadsto \color{blue}{y \cdot \left(\left(\mathsf{neg}\left(9 \cdot z\right)\right) \cdot t\right)} + \left(x \cdot 2 + \left(a \cdot 27\right) \cdot b\right) \]

      12. +-commutative N/A

          \[\leadsto y \cdot \left(\left(\mathsf{neg}\left(9 \cdot z\right)\right) \cdot t\right) + \color{blue}{\left(\left(a \cdot 27\right) \cdot b + x \cdot 2\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, \left(\mathsf{neg}\left(9 \cdot z\right)\right) \cdot t, \left(a \cdot 27\right) \cdot b + x \cdot 2\right)} \]

   4. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \left(-9 \cdot z\right) \cdot t, \mathsf{fma}\left(b \cdot 27, a, 2 \cdot x\right)\right)} \]

   if 3.9999999999999999e-16 < z

   1. Initial program 92.9%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   2. Add Preprocessing
   3. 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. associate-+l- N/A

         \[\leadsto \color{blue}{x \cdot 2 - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot 2} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{2 \cdot x} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      6. count-2-rev N/A

         \[\leadsto \color{blue}{\left(x + x\right)} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

      7. associate--l+ N/A

         \[\leadsto \color{blue}{x + \left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)\right)} \]

      8. lower-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto x + \color{blue}{\left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto x + \left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \color{blue}{\left(a \cdot 27\right) \cdot b}\right)\right) \]

      11. fp-cancel-sub-sign-inv N/A

          \[\leadsto x + \left(x - \color{blue}{\left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t + \left(\mathsf{neg}\left(a \cdot 27\right)\right) \cdot b\right)}\right) \]

      12. +-commutative N/A

          \[\leadsto x + \left(x - \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot 27\right)\right) \cdot b + \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)}\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto x + \left(x - \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a \cdot 27\right), b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{a \cdot 27}\right), b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) \]

      15. *-commutative N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{27 \cdot a}\right), b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot a}, b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) \]

      17. lower-*.f64 N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot a}, b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) \]

      18. metadata-eval 93.0

          \[\leadsto x + \left(x - \mathsf{fma}\left(\color{blue}{-27} \cdot a, b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) \]

      19. lift-*.f64 N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, \color{blue}{\left(\left(y \cdot 9\right) \cdot z\right) \cdot t}\right)\right) \]

      20. *-commutative N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, \color{blue}{t \cdot \left(\left(y \cdot 9\right) \cdot z\right)}\right)\right) \]

      21. lower-*.f64 93.0

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, \color{blue}{t \cdot \left(\left(y \cdot 9\right) \cdot z\right)}\right)\right) \]

      22. lift-*.f64 N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \color{blue}{\left(\left(y \cdot 9\right) \cdot z\right)}\right)\right) \]

      23. *-commutative N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \color{blue}{\left(z \cdot \left(y \cdot 9\right)\right)}\right)\right) \]

      24. lower-*.f64 93.0

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \color{blue}{\left(z \cdot \left(y \cdot 9\right)\right)}\right)\right) \]

      25. lift-*.f64 N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \left(z \cdot \color{blue}{\left(y \cdot 9\right)}\right)\right)\right) \]

      26. *-commutative N/A

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \left(z \cdot \color{blue}{\left(9 \cdot y\right)}\right)\right)\right) \]

      27. lower-*.f64 93.0

          \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \left(z \cdot \color{blue}{\left(9 \cdot y\right)}\right)\right)\right) \]

   4. Applied rewrites 93.0%

      \[\leadsto \color{blue}{x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \left(z \cdot \left(9 \cdot y\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 65.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ x + \mathsf{fma}\left(b \cdot a, 27, x\right) \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b) :precision binary64 (+ x (fma (* b a) 27.0 x)))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	return x + fma((b * a), 27.0, x);
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	return Float64(x + fma(Float64(b * a), 27.0, x))
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := N[(x + N[(N[(b * a), $MachinePrecision] * 27.0 + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.0%

   \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b} \]

   2. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)} + \left(a \cdot 27\right) \cdot b \]

   3. associate-+l- N/A

      \[\leadsto \color{blue}{x \cdot 2 - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot 2} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{2 \cdot x} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

   6. count-2-rev N/A

      \[\leadsto \color{blue}{\left(x + x\right)} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

   7. associate--l+ N/A

      \[\leadsto \color{blue}{x + \left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)\right)} \]

   8. lower-+.f64 N/A

      \[\leadsto \color{blue}{x + \left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)\right)} \]

   9. lower--.f64 N/A

      \[\leadsto x + \color{blue}{\left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)\right)} \]

   10. lift-*.f64 N/A

       \[\leadsto x + \left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \color{blue}{\left(a \cdot 27\right) \cdot b}\right)\right) \]

   11. fp-cancel-sub-sign-inv N/A

       \[\leadsto x + \left(x - \color{blue}{\left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t + \left(\mathsf{neg}\left(a \cdot 27\right)\right) \cdot b\right)}\right) \]

   12. +-commutative N/A

       \[\leadsto x + \left(x - \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot 27\right)\right) \cdot b + \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)}\right) \]

   13. lower-fma.f64 N/A

       \[\leadsto x + \left(x - \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a \cdot 27\right), b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)}\right) \]

   14. lift-*.f64 N/A

       \[\leadsto x + \left(x - \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{a \cdot 27}\right), b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) \]

   15. *-commutative N/A

       \[\leadsto x + \left(x - \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{27 \cdot a}\right), b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) \]

   16. distribute-lft-neg-in N/A

       \[\leadsto x + \left(x - \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot a}, b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) \]

   17. lower-*.f64 N/A

       \[\leadsto x + \left(x - \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot a}, b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) \]

   18. metadata-eval 95.4

       \[\leadsto x + \left(x - \mathsf{fma}\left(\color{blue}{-27} \cdot a, b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) \]

   19. lift-*.f64 N/A

       \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, \color{blue}{\left(\left(y \cdot 9\right) \cdot z\right) \cdot t}\right)\right) \]

   20. *-commutative N/A

       \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, \color{blue}{t \cdot \left(\left(y \cdot 9\right) \cdot z\right)}\right)\right) \]

   21. lower-*.f64 95.4

       \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, \color{blue}{t \cdot \left(\left(y \cdot 9\right) \cdot z\right)}\right)\right) \]

   22. lift-*.f64 N/A

       \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \color{blue}{\left(\left(y \cdot 9\right) \cdot z\right)}\right)\right) \]

   23. *-commutative N/A

       \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \color{blue}{\left(z \cdot \left(y \cdot 9\right)\right)}\right)\right) \]

   24. lower-*.f64 95.4

       \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \color{blue}{\left(z \cdot \left(y \cdot 9\right)\right)}\right)\right) \]

   25. lift-*.f64 N/A

       \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \left(z \cdot \color{blue}{\left(y \cdot 9\right)}\right)\right)\right) \]

   26. *-commutative N/A

       \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \left(z \cdot \color{blue}{\left(9 \cdot y\right)}\right)\right)\right) \]

   27. lower-*.f64 95.4

       \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \left(z \cdot \color{blue}{\left(9 \cdot y\right)}\right)\right)\right) \]

4. Applied rewrites 95.4%

   \[\leadsto \color{blue}{x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \left(z \cdot \left(9 \cdot y\right)\right)\right)\right)} \]

5. Taylor expanded in y around 0

   \[\leadsto x + \color{blue}{\left(x - -27 \cdot \left(a \cdot b\right)\right)} \]
6. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto x + \left(x - \color{blue}{\left(\mathsf{neg}\left(27\right)\right)} \cdot \left(a \cdot b\right)\right) \]

   2. fp-cancel-sign-sub-inv N/A

      \[\leadsto x + \color{blue}{\left(x + 27 \cdot \left(a \cdot b\right)\right)} \]

   3. +-commutative N/A

      \[\leadsto x + \color{blue}{\left(27 \cdot \left(a \cdot b\right) + x\right)} \]

   4. *-commutative N/A

      \[\leadsto x + \left(\color{blue}{\left(a \cdot b\right) \cdot 27} + x\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto x + \color{blue}{\mathsf{fma}\left(a \cdot b, 27, x\right)} \]

   6. *-commutative N/A

      \[\leadsto x + \mathsf{fma}\left(\color{blue}{b \cdot a}, 27, x\right) \]

   7. lower-*.f64 63.3

      \[\leadsto x + \mathsf{fma}\left(\color{blue}{b \cdot a}, 27, x\right) \]

7. Applied rewrites 63.3%

   \[\leadsto x + \color{blue}{\mathsf{fma}\left(b \cdot a, 27, x\right)} \]
8. Add Preprocessing

Alternative 17: 65.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \mathsf{fma}\left(27 \cdot a, b, x + x\right) \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b) :precision binary64 (fma (* 27.0 a) b (+ x x)))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	return fma((27.0 * a), b, (x + x));
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	return fma(Float64(27.0 * a), b, Float64(x + x))
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := N[(N[(27.0 * a), $MachinePrecision] * b + N[(x + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.0%

   \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
2. Add Preprocessing

3. Taylor expanded in y around 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 63.3

      \[\leadsto \mathsf{fma}\left(2, x, \color{blue}{\left(b \cdot a\right)} \cdot 27\right) \]

5. Applied rewrites 63.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, \left(b \cdot a\right) \cdot 27\right)} \]
6. Step-by-step derivation

7. Applied rewrites 63.3%

   \[\leadsto \mathsf{fma}\left(27 \cdot a, \color{blue}{b}, 2 \cdot x\right) \]
8. Step-by-step derivation

9. Applied rewrites 63.3%

   \[\leadsto \mathsf{fma}\left(27 \cdot a, b, x + x\right) \]
10. Add Preprocessing

Alternative 18: 65.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \mathsf{fma}\left(27 \cdot b, a, x\right) + x \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b) :precision binary64 (+ (fma (* 27.0 b) a x) x))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	return fma((27.0 * b), a, x) + x;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	return Float64(fma(Float64(27.0 * b), a, x) + x)
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(27.0 * b), $MachinePrecision] * a + x), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 95.0%

   \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
2. Add Preprocessing

3. Taylor expanded in y around 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 63.3

      \[\leadsto \mathsf{fma}\left(2, x, \color{blue}{\left(b \cdot a\right)} \cdot 27\right) \]

5. Applied rewrites 63.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, \left(b \cdot a\right) \cdot 27\right)} \]
6. Step-by-step derivation

7. Applied rewrites 63.3%

   \[\leadsto \mathsf{fma}\left(27 \cdot b, a, x\right) + \color{blue}{x} \]
8. Add Preprocessing

Alternative 19: 31.6% accurate, 9.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ x + x \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b) :precision binary64 (+ x x))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	return x + x;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x + x
end function

Java

assert x < y && y < z && z < t && t < a && a < b;
public static double code(double x, double y, double z, double t, double a, double b) {
	return x + x;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	return x + x

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	return Float64(x + x)
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp = code(x, y, z, t, a, b)
	tmp = x + x;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := N[(x + x), $MachinePrecision]

Derivation

1. Initial program 95.0%

   \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b} \]

   2. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)} + \left(a \cdot 27\right) \cdot b \]

   3. associate-+l- N/A

      \[\leadsto \color{blue}{x \cdot 2 - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot 2} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{2 \cdot x} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

   6. count-2-rev N/A

      \[\leadsto \color{blue}{\left(x + x\right)} - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right) \]

   7. associate--l+ N/A

      \[\leadsto \color{blue}{x + \left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)\right)} \]

   8. lower-+.f64 N/A

      \[\leadsto \color{blue}{x + \left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)\right)} \]

   9. lower--.f64 N/A

      \[\leadsto x + \color{blue}{\left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)\right)} \]

   10. lift-*.f64 N/A

       \[\leadsto x + \left(x - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \color{blue}{\left(a \cdot 27\right) \cdot b}\right)\right) \]

   11. fp-cancel-sub-sign-inv N/A

       \[\leadsto x + \left(x - \color{blue}{\left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t + \left(\mathsf{neg}\left(a \cdot 27\right)\right) \cdot b\right)}\right) \]

   12. +-commutative N/A

       \[\leadsto x + \left(x - \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot 27\right)\right) \cdot b + \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)}\right) \]

   13. lower-fma.f64 N/A

       \[\leadsto x + \left(x - \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a \cdot 27\right), b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)}\right) \]

   14. lift-*.f64 N/A

       \[\leadsto x + \left(x - \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{a \cdot 27}\right), b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) \]

   15. *-commutative N/A

       \[\leadsto x + \left(x - \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{27 \cdot a}\right), b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) \]

   16. distribute-lft-neg-in N/A

       \[\leadsto x + \left(x - \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot a}, b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) \]

   17. lower-*.f64 N/A

       \[\leadsto x + \left(x - \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot a}, b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) \]

   18. metadata-eval 95.4

       \[\leadsto x + \left(x - \mathsf{fma}\left(\color{blue}{-27} \cdot a, b, \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) \]

   19. lift-*.f64 N/A

       \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, \color{blue}{\left(\left(y \cdot 9\right) \cdot z\right) \cdot t}\right)\right) \]

   20. *-commutative N/A

       \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, \color{blue}{t \cdot \left(\left(y \cdot 9\right) \cdot z\right)}\right)\right) \]

   21. lower-*.f64 95.4

       \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, \color{blue}{t \cdot \left(\left(y \cdot 9\right) \cdot z\right)}\right)\right) \]

   22. lift-*.f64 N/A

       \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \color{blue}{\left(\left(y \cdot 9\right) \cdot z\right)}\right)\right) \]

   23. *-commutative N/A

       \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \color{blue}{\left(z \cdot \left(y \cdot 9\right)\right)}\right)\right) \]

   24. lower-*.f64 95.4

       \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \color{blue}{\left(z \cdot \left(y \cdot 9\right)\right)}\right)\right) \]

   25. lift-*.f64 N/A

       \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \left(z \cdot \color{blue}{\left(y \cdot 9\right)}\right)\right)\right) \]

   26. *-commutative N/A

       \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \left(z \cdot \color{blue}{\left(9 \cdot y\right)}\right)\right)\right) \]

   27. lower-*.f64 95.4

       \[\leadsto x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \left(z \cdot \color{blue}{\left(9 \cdot y\right)}\right)\right)\right) \]

4. Applied rewrites 95.4%

   \[\leadsto \color{blue}{x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \left(z \cdot \left(9 \cdot y\right)\right)\right)\right)} \]

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{2 \cdot x} \]
6. Step-by-step derivation

   1. lower-*.f64 33.1

      \[\leadsto \color{blue}{2 \cdot x} \]

7. Applied rewrites 33.1%

   \[\leadsto \color{blue}{2 \cdot x} \]
8. Step-by-step derivation

9. Applied rewrites 33.1%

   \[\leadsto x + \color{blue}{x} \]
10. Add Preprocessing

Developer Target 1: 95.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y < 7.590524218811189 \cdot 10^{-161}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - 9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (< y 7.590524218811189e-161)
   (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* a (* 27.0 b)))
   (+ (- (* x 2.0) (* 9.0 (* y (* t z)))) (* (* a 27.0) b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y < 7.590524218811189e-161) {
		tmp = ((x * 2.0) - (((y * 9.0) * z) * t)) + (a * (27.0 * b));
	} else {
		tmp = ((x * 2.0) - (9.0 * (y * (t * z)))) + ((a * 27.0) * b);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y < 7.590524218811189d-161) then
        tmp = ((x * 2.0d0) - (((y * 9.0d0) * z) * t)) + (a * (27.0d0 * b))
    else
        tmp = ((x * 2.0d0) - (9.0d0 * (y * (t * z)))) + ((a * 27.0d0) * b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y < 7.590524218811189e-161) {
		tmp = ((x * 2.0) - (((y * 9.0) * z) * t)) + (a * (27.0 * b));
	} else {
		tmp = ((x * 2.0) - (9.0 * (y * (t * z)))) + ((a * 27.0) * b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y < 7.590524218811189e-161:
		tmp = ((x * 2.0) - (((y * 9.0) * z) * t)) + (a * (27.0 * b))
	else:
		tmp = ((x * 2.0) - (9.0 * (y * (t * z)))) + ((a * 27.0) * b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y < 7.590524218811189e-161)
		tmp = Float64(Float64(Float64(x * 2.0) - Float64(Float64(Float64(y * 9.0) * z) * t)) + Float64(a * Float64(27.0 * b)));
	else
		tmp = Float64(Float64(Float64(x * 2.0) - Float64(9.0 * Float64(y * Float64(t * z)))) + Float64(Float64(a * 27.0) * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y < 7.590524218811189e-161)
		tmp = ((x * 2.0) - (((y * 9.0) * z) * t)) + (a * (27.0 * b));
	else
		tmp = ((x * 2.0) - (9.0 * (y * (t * z)))) + ((a * 27.0) * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Less[y, 7.590524218811189e-161], N[(N[(N[(x * 2.0), $MachinePrecision] - N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(27.0 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * 2.0), $MachinePrecision] - N[(9.0 * N[(y * N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024326 
(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)))