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

Percentage Accurate: 95.5% → 98.2%

Time: 27.2s

Alternatives: 21

Speedup: 0.9×

• 27.6% 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: 95.5% 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: 98.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;y \cdot 9 \leq -1 \cdot 10^{+41}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(27, \frac{a \cdot b}{y}, \mathsf{fma}\left(t, z \cdot -9, 2 \cdot \frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot t, z \cdot -9, \mathsf{fma}\left(a, 27 \cdot b, 2 \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* y 9.0) -1e+41)
   (* y (fma 27.0 (/ (* a b) y) (fma t (* z -9.0) (* 2.0 (/ x y)))))
   (fma (* y t) (* z -9.0) (fma a (* 27.0 b) (* 2.0 x)))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y * 9.0) <= -1e+41) {
		tmp = y * fma(27.0, ((a * b) / y), fma(t, (z * -9.0), (2.0 * (x / y))));
	} else {
		tmp = fma((y * t), (z * -9.0), fma(a, (27.0 * b), (2.0 * x)));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(y * 9.0) <= -1e+41)
		tmp = Float64(y * fma(27.0, Float64(Float64(a * b) / y), fma(t, Float64(z * -9.0), Float64(2.0 * Float64(x / y)))));
	else
		tmp = fma(Float64(y * t), Float64(z * -9.0), fma(a, Float64(27.0 * b), Float64(2.0 * x)));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 y #s(literal 9 binary64)) < -1.00000000000000001e41

   1. Initial program 80.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

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

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. neg-sub0 N/A

          \[\leadsto y \cdot \mathsf{fma}\left(27, \frac{a \cdot b}{y}, \color{blue}{0 - \left(-2 \cdot \frac{x}{y} - -9 \cdot \left(t \cdot z\right)\right)}\right) \]

      14. associate--r- N/A

          \[\leadsto y \cdot \mathsf{fma}\left(27, \frac{a \cdot b}{y}, \color{blue}{\left(0 - -2 \cdot \frac{x}{y}\right) + -9 \cdot \left(t \cdot z\right)}\right) \]

   5. Applied rewrites 99.8%

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

   if -1.00000000000000001e41 < (*.f64 y #s(literal 9 binary64))

   1. Initial program 96.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. associate-*r* N/A

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

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

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

      17. +-commutative N/A

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

   4. Applied rewrites 95.9%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot 9 \leq -1 \cdot 10^{+41}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(27, \frac{a \cdot b}{y}, \mathsf{fma}\left(t, z \cdot -9, 2 \cdot \frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot t, z \cdot -9, \mathsf{fma}\left(a, 27 \cdot b, 2 \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 57.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := 2 \cdot x - t \cdot \left(\left(y \cdot 9\right) \cdot z\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+96}:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;t\_1 \leq 1.4 \cdot 10^{+181}:\\ \;\;\;\;a \cdot \left(27 \cdot b\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+254}:\\ \;\;\;\;t \cdot \left(-9 \cdot \left(y \cdot z\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+297}:\\ \;\;\;\;2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(z \cdot -9\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (* 2.0 x) (* t (* (* y 9.0) z)))))
   (if (<= t_1 (- INFINITY))
     (* -9.0 (* y (* t z)))
     (if (<= t_1 -5e+96)
       (* 2.0 x)
       (if (<= t_1 1.4e+181)
         (* a (* 27.0 b))
         (if (<= t_1 2e+254)
           (* t (* -9.0 (* y z)))
           (if (<= t_1 2e+297) (* 2.0 x) (* y (* t (* z -9.0))))))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (2.0 * x) - (t * ((y * 9.0) * z));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -9.0 * (y * (t * z));
	} else if (t_1 <= -5e+96) {
		tmp = 2.0 * x;
	} else if (t_1 <= 1.4e+181) {
		tmp = a * (27.0 * b);
	} else if (t_1 <= 2e+254) {
		tmp = t * (-9.0 * (y * z));
	} else if (t_1 <= 2e+297) {
		tmp = 2.0 * x;
	} else {
		tmp = y * (t * (z * -9.0));
	}
	return tmp;
}

Java

assert x < y && y < z && z < t && t < a && a < b;
assert x < y && y < z && z < t && t < a && a < b;
public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (2.0 * x) - (t * ((y * 9.0) * z));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -9.0 * (y * (t * z));
	} else if (t_1 <= -5e+96) {
		tmp = 2.0 * x;
	} else if (t_1 <= 1.4e+181) {
		tmp = a * (27.0 * b);
	} else if (t_1 <= 2e+254) {
		tmp = t * (-9.0 * (y * z));
	} else if (t_1 <= 2e+297) {
		tmp = 2.0 * x;
	} else {
		tmp = y * (t * (z * -9.0));
	}
	return tmp;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	t_1 = (2.0 * x) - (t * ((y * 9.0) * z))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -9.0 * (y * (t * z))
	elif t_1 <= -5e+96:
		tmp = 2.0 * x
	elif t_1 <= 1.4e+181:
		tmp = a * (27.0 * b)
	elif t_1 <= 2e+254:
		tmp = t * (-9.0 * (y * z))
	elif t_1 <= 2e+297:
		tmp = 2.0 * x
	else:
		tmp = y * (t * (z * -9.0))
	return tmp

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(2.0 * x) - Float64(t * Float64(Float64(y * 9.0) * z)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-9.0 * Float64(y * Float64(t * z)));
	elseif (t_1 <= -5e+96)
		tmp = Float64(2.0 * x);
	elseif (t_1 <= 1.4e+181)
		tmp = Float64(a * Float64(27.0 * b));
	elseif (t_1 <= 2e+254)
		tmp = Float64(t * Float64(-9.0 * Float64(y * z)));
	elseif (t_1 <= 2e+297)
		tmp = Float64(2.0 * x);
	else
		tmp = Float64(y * Float64(t * Float64(z * -9.0)));
	end
	return tmp
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (2.0 * x) - (t * ((y * 9.0) * z));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -9.0 * (y * (t * z));
	elseif (t_1 <= -5e+96)
		tmp = 2.0 * x;
	elseif (t_1 <= 1.4e+181)
		tmp = a * (27.0 * b);
	elseif (t_1 <= 2e+254)
		tmp = t * (-9.0 * (y * z));
	elseif (t_1 <= 2e+297)
		tmp = 2.0 * x;
	else
		tmp = y * (t * (z * -9.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
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[(2.0 * x), $MachinePrecision] - N[(t * N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(-9.0 * N[(y * N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e+96], N[(2.0 * x), $MachinePrecision], If[LessEqual[t$95$1, 1.4e+181], N[(a * N[(27.0 * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+254], N[(t * N[(-9.0 * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+297], N[(2.0 * x), $MachinePrecision], N[(y * N[(t * N[(z * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (-.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)) < -inf.0

   1. Initial program 70.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

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

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

      15. +-commutative N/A

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

      16. 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 99.9%

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

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   6. 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. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 80.0

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

   7. Applied rewrites 80.0%

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

   if -inf.0 < (-.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)) < -5.0000000000000004e96 or 1.9999999999999999e254 < (-.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)) < 2e297

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 58.1

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

   5. Applied rewrites 58.1%

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

   if -5.0000000000000004e96 < (-.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)) < 1.39999999999999992e181

   1. Initial program 99.7%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.8

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

   5. Applied rewrites 56.8%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 56.9

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

   7. Applied rewrites 56.9%

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

   if 1.39999999999999992e181 < (-.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)) < 1.9999999999999999e254

   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 y around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 66.0

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

   5. Applied rewrites 66.0%

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

   if 2e297 < (-.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t))

   1. Initial program 71.6%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 74.7

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

   5. Applied rewrites 74.7%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 85.7

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

   7. Applied rewrites 85.7%

      \[\leadsto \color{blue}{\left(t \cdot \left(z \cdot -9\right)\right) \cdot y} \]
3. Recombined 5 regimes into one program.

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot x - t \cdot \left(\left(y \cdot 9\right) \cdot z\right) \leq -\infty:\\ \;\;\;\;-9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\\ \mathbf{elif}\;2 \cdot x - t \cdot \left(\left(y \cdot 9\right) \cdot z\right) \leq -5 \cdot 10^{+96}:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;2 \cdot x - t \cdot \left(\left(y \cdot 9\right) \cdot z\right) \leq 1.4 \cdot 10^{+181}:\\ \;\;\;\;a \cdot \left(27 \cdot b\right)\\ \mathbf{elif}\;2 \cdot x - t \cdot \left(\left(y \cdot 9\right) \cdot z\right) \leq 2 \cdot 10^{+254}:\\ \;\;\;\;t \cdot \left(-9 \cdot \left(y \cdot z\right)\right)\\ \mathbf{elif}\;2 \cdot x - t \cdot \left(\left(y \cdot 9\right) \cdot z\right) \leq 2 \cdot 10^{+297}:\\ \;\;\;\;2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(z \cdot -9\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := -9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\\ t_2 := 2 \cdot x - t \cdot \left(\left(y \cdot 9\right) \cdot z\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{+96}:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;t\_2 \leq 1.4 \cdot 10^{+181}:\\ \;\;\;\;a \cdot \left(27 \cdot b\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+254}:\\ \;\;\;\;t \cdot \left(-9 \cdot \left(y \cdot z\right)\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+300}:\\ \;\;\;\;2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
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 (* -9.0 (* y (* t z)))) (t_2 (- (* 2.0 x) (* t (* (* y 9.0) z)))))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 -5e+96)
       (* 2.0 x)
       (if (<= t_2 1.4e+181)
         (* a (* 27.0 b))
         (if (<= t_2 2e+254)
           (* t (* -9.0 (* y z)))
           (if (<= t_2 5e+300) (* 2.0 x) t_1)))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = -9.0 * (y * (t * z));
	double t_2 = (2.0 * x) - (t * ((y * 9.0) * z));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= -5e+96) {
		tmp = 2.0 * x;
	} else if (t_2 <= 1.4e+181) {
		tmp = a * (27.0 * b);
	} else if (t_2 <= 2e+254) {
		tmp = t * (-9.0 * (y * z));
	} else if (t_2 <= 5e+300) {
		tmp = 2.0 * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

assert x < y && y < z && z < t && t < a && a < b;
assert x < y && y < z && z < t && t < a && a < b;
public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = -9.0 * (y * (t * z));
	double t_2 = (2.0 * x) - (t * ((y * 9.0) * z));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= -5e+96) {
		tmp = 2.0 * x;
	} else if (t_2 <= 1.4e+181) {
		tmp = a * (27.0 * b);
	} else if (t_2 <= 2e+254) {
		tmp = t * (-9.0 * (y * z));
	} else if (t_2 <= 5e+300) {
		tmp = 2.0 * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	t_1 = -9.0 * (y * (t * z))
	t_2 = (2.0 * x) - (t * ((y * 9.0) * z))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= -5e+96:
		tmp = 2.0 * x
	elif t_2 <= 1.4e+181:
		tmp = a * (27.0 * b)
	elif t_2 <= 2e+254:
		tmp = t * (-9.0 * (y * z))
	elif t_2 <= 5e+300:
		tmp = 2.0 * x
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(-9.0 * Float64(y * Float64(t * z)))
	t_2 = Float64(Float64(2.0 * x) - Float64(t * Float64(Float64(y * 9.0) * z)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= -5e+96)
		tmp = Float64(2.0 * x);
	elseif (t_2 <= 1.4e+181)
		tmp = Float64(a * Float64(27.0 * b));
	elseif (t_2 <= 2e+254)
		tmp = Float64(t * Float64(-9.0 * Float64(y * z)));
	elseif (t_2 <= 5e+300)
		tmp = Float64(2.0 * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
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 = -9.0 * (y * (t * z));
	t_2 = (2.0 * x) - (t * ((y * 9.0) * z));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= -5e+96)
		tmp = 2.0 * x;
	elseif (t_2 <= 1.4e+181)
		tmp = a * (27.0 * b);
	elseif (t_2 <= 2e+254)
		tmp = t * (-9.0 * (y * z));
	elseif (t_2 <= 5e+300)
		tmp = 2.0 * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(-9.0 * N[(y * N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * x), $MachinePrecision] - N[(t * N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, -5e+96], N[(2.0 * x), $MachinePrecision], If[LessEqual[t$95$2, 1.4e+181], N[(a * N[(27.0 * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+254], N[(t * N[(-9.0 * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+300], N[(2.0 * x), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)) < -inf.0 or 5.00000000000000026e300 < (-.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t))

   1. Initial program 70.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

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

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

      15. +-commutative N/A

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

      16. 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 95.0%

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

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   6. 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. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 84.3

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

   7. Applied rewrites 84.3%

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

   if -inf.0 < (-.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)) < -5.0000000000000004e96 or 1.9999999999999999e254 < (-.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)) < 5.00000000000000026e300

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 58.7

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

   5. Applied rewrites 58.7%

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

   if -5.0000000000000004e96 < (-.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)) < 1.39999999999999992e181

   1. Initial program 99.7%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.8

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

   5. Applied rewrites 56.8%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 56.9

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

   7. Applied rewrites 56.9%

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

   if 1.39999999999999992e181 < (-.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)) < 1.9999999999999999e254

   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 y around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 66.0

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

   5. Applied rewrites 66.0%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot x - t \cdot \left(\left(y \cdot 9\right) \cdot z\right) \leq -\infty:\\ \;\;\;\;-9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\\ \mathbf{elif}\;2 \cdot x - t \cdot \left(\left(y \cdot 9\right) \cdot z\right) \leq -5 \cdot 10^{+96}:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;2 \cdot x - t \cdot \left(\left(y \cdot 9\right) \cdot z\right) \leq 1.4 \cdot 10^{+181}:\\ \;\;\;\;a \cdot \left(27 \cdot b\right)\\ \mathbf{elif}\;2 \cdot x - t \cdot \left(\left(y \cdot 9\right) \cdot z\right) \leq 2 \cdot 10^{+254}:\\ \;\;\;\;t \cdot \left(-9 \cdot \left(y \cdot z\right)\right)\\ \mathbf{elif}\;2 \cdot x - t \cdot \left(\left(y \cdot 9\right) \cdot z\right) \leq 5 \cdot 10^{+300}:\\ \;\;\;\;2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;-9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := -9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\\ t_2 := 2 \cdot x - t \cdot \left(\left(y \cdot 9\right) \cdot z\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{+96}:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+87}:\\ \;\;\;\;a \cdot \left(27 \cdot b\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+189}:\\ \;\;\;\;2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
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 (* -9.0 (* y (* t z)))) (t_2 (- (* 2.0 x) (* t (* (* y 9.0) z)))))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 -5e+96)
       (* 2.0 x)
       (if (<= t_2 2e+87)
         (* a (* 27.0 b))
         (if (<= t_2 1e+189) (* 2.0 x) t_1))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = -9.0 * (y * (t * z));
	double t_2 = (2.0 * x) - (t * ((y * 9.0) * z));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= -5e+96) {
		tmp = 2.0 * x;
	} else if (t_2 <= 2e+87) {
		tmp = a * (27.0 * b);
	} else if (t_2 <= 1e+189) {
		tmp = 2.0 * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

assert x < y && y < z && z < t && t < a && a < b;
assert x < y && y < z && z < t && t < a && a < b;
public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = -9.0 * (y * (t * z));
	double t_2 = (2.0 * x) - (t * ((y * 9.0) * z));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= -5e+96) {
		tmp = 2.0 * x;
	} else if (t_2 <= 2e+87) {
		tmp = a * (27.0 * b);
	} else if (t_2 <= 1e+189) {
		tmp = 2.0 * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	t_1 = -9.0 * (y * (t * z))
	t_2 = (2.0 * x) - (t * ((y * 9.0) * z))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= -5e+96:
		tmp = 2.0 * x
	elif t_2 <= 2e+87:
		tmp = a * (27.0 * b)
	elif t_2 <= 1e+189:
		tmp = 2.0 * x
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(-9.0 * Float64(y * Float64(t * z)))
	t_2 = Float64(Float64(2.0 * x) - Float64(t * Float64(Float64(y * 9.0) * z)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= -5e+96)
		tmp = Float64(2.0 * x);
	elseif (t_2 <= 2e+87)
		tmp = Float64(a * Float64(27.0 * b));
	elseif (t_2 <= 1e+189)
		tmp = Float64(2.0 * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
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 = -9.0 * (y * (t * z));
	t_2 = (2.0 * x) - (t * ((y * 9.0) * z));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= -5e+96)
		tmp = 2.0 * x;
	elseif (t_2 <= 2e+87)
		tmp = a * (27.0 * b);
	elseif (t_2 <= 1e+189)
		tmp = 2.0 * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(-9.0 * N[(y * N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * x), $MachinePrecision] - N[(t * N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, -5e+96], N[(2.0 * x), $MachinePrecision], If[LessEqual[t$95$2, 2e+87], N[(a * N[(27.0 * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+189], N[(2.0 * x), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)) < -inf.0 or 1e189 < (-.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t))

   1. Initial program 80.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

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

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

      15. +-commutative N/A

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

      16. 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 95.6%

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

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   6. 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. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 70.3

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

   7. Applied rewrites 70.3%

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

   if -inf.0 < (-.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)) < -5.0000000000000004e96 or 1.9999999999999999e87 < (-.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)) < 1e189

   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 x around inf

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

      1. lower-*.f64 52.6

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

   5. Applied rewrites 52.6%

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

   if -5.0000000000000004e96 < (-.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)) < 1.9999999999999999e87

   1. Initial program 99.7%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 63.9

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

   5. Applied rewrites 63.9%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 64.0

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

   7. Applied rewrites 64.0%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot x - t \cdot \left(\left(y \cdot 9\right) \cdot z\right) \leq -\infty:\\ \;\;\;\;-9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\\ \mathbf{elif}\;2 \cdot x - t \cdot \left(\left(y \cdot 9\right) \cdot z\right) \leq -5 \cdot 10^{+96}:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;2 \cdot x - t \cdot \left(\left(y \cdot 9\right) \cdot z\right) \leq 2 \cdot 10^{+87}:\\ \;\;\;\;a \cdot \left(27 \cdot b\right)\\ \mathbf{elif}\;2 \cdot x - t \cdot \left(\left(y \cdot 9\right) \cdot z\right) \leq 10^{+189}:\\ \;\;\;\;2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;-9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, -9 \cdot \left(y \cdot z\right), 2 \cdot x\right)\\ t_2 := t \cdot \left(\left(y \cdot 9\right) \cdot z\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+248}:\\ \;\;\;\;z \cdot \left(-9 \cdot \left(y \cdot t\right)\right)\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{+98}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(27 \cdot b, a, 2 \cdot x\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+212}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;-9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma t (* -9.0 (* y z)) (* 2.0 x))) (t_2 (* t (* (* y 9.0) z))))
   (if (<= t_2 -1e+248)
     (* z (* -9.0 (* y t)))
     (if (<= t_2 -5e+98)
       t_1
       (if (<= t_2 2e+71)
         (fma (* 27.0 b) a (* 2.0 x))
         (if (<= t_2 1e+212) t_1 (* -9.0 (* y (* t z)))))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(t, (-9.0 * (y * z)), (2.0 * x));
	double t_2 = t * ((y * 9.0) * z);
	double tmp;
	if (t_2 <= -1e+248) {
		tmp = z * (-9.0 * (y * t));
	} else if (t_2 <= -5e+98) {
		tmp = t_1;
	} else if (t_2 <= 2e+71) {
		tmp = fma((27.0 * b), a, (2.0 * x));
	} else if (t_2 <= 1e+212) {
		tmp = t_1;
	} else {
		tmp = -9.0 * (y * (t * z));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = fma(t, Float64(-9.0 * Float64(y * z)), Float64(2.0 * x))
	t_2 = Float64(t * Float64(Float64(y * 9.0) * z))
	tmp = 0.0
	if (t_2 <= -1e+248)
		tmp = Float64(z * Float64(-9.0 * Float64(y * t)));
	elseif (t_2 <= -5e+98)
		tmp = t_1;
	elseif (t_2 <= 2e+71)
		tmp = fma(Float64(27.0 * b), a, Float64(2.0 * x));
	elseif (t_2 <= 1e+212)
		tmp = t_1;
	else
		tmp = Float64(-9.0 * Float64(y * Float64(t * z)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t * N[(-9.0 * N[(y * z), $MachinePrecision]), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+248], N[(z * N[(-9.0 * N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -5e+98], t$95$1, If[LessEqual[t$95$2, 2e+71], N[(N[(27.0 * b), $MachinePrecision] * a + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+212], t$95$1, N[(-9.0 * N[(y * N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -1.00000000000000005e248

   1. Initial program 75.2%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 70.7

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

   5. Applied rewrites 70.7%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 82.5

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

   7. Applied rewrites 82.5%

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

   if -1.00000000000000005e248 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -4.9999999999999998e98 or 2.0000000000000001e71 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 9.9999999999999991e211

   1. Initial program 99.6%

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

   3. Taylor expanded in a around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 87.3

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

   5. Applied rewrites 87.3%

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

   if -4.9999999999999998e98 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 2.0000000000000001e71

   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. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 89.6

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

   5. Applied rewrites 89.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 89.6

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

   7. Applied rewrites 89.6%

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

   if 9.9999999999999991e211 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 73.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

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

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

   4. Applied rewrites 97.2%

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

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   6. 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. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 76.8

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

   7. Applied rewrites 76.8%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(\left(y \cdot 9\right) \cdot z\right) \leq -1 \cdot 10^{+248}:\\ \;\;\;\;z \cdot \left(-9 \cdot \left(y \cdot t\right)\right)\\ \mathbf{elif}\;t \cdot \left(\left(y \cdot 9\right) \cdot z\right) \leq -5 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(t, -9 \cdot \left(y \cdot z\right), 2 \cdot x\right)\\ \mathbf{elif}\;t \cdot \left(\left(y \cdot 9\right) \cdot z\right) \leq 2 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(27 \cdot b, a, 2 \cdot x\right)\\ \mathbf{elif}\;t \cdot \left(\left(y \cdot 9\right) \cdot z\right) \leq 10^{+212}:\\ \;\;\;\;\mathsf{fma}\left(t, -9 \cdot \left(y \cdot z\right), 2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z \cdot \left(t \cdot -9\right), y, a \cdot \left(27 \cdot b\right)\right)\\ t_2 := t \cdot \left(\left(y \cdot 9\right) \cdot z\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+248}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(t, -9 \cdot \left(y \cdot z\right), 2 \cdot x\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(27 \cdot b, a, 2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
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 (* z (* t -9.0)) y (* a (* 27.0 b))))
        (t_2 (* t (* (* y 9.0) z))))
   (if (<= t_2 -1e+248)
     t_1
     (if (<= t_2 -5e+98)
       (fma t (* -9.0 (* y z)) (* 2.0 x))
       (if (<= t_2 2e+82) (fma (* 27.0 b) a (* 2.0 x)) t_1)))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((z * (t * -9.0)), y, (a * (27.0 * b)));
	double t_2 = t * ((y * 9.0) * z);
	double tmp;
	if (t_2 <= -1e+248) {
		tmp = t_1;
	} else if (t_2 <= -5e+98) {
		tmp = fma(t, (-9.0 * (y * z)), (2.0 * x));
	} else if (t_2 <= 2e+82) {
		tmp = fma((27.0 * b), a, (2.0 * x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = fma(Float64(z * Float64(t * -9.0)), y, Float64(a * Float64(27.0 * b)))
	t_2 = Float64(t * Float64(Float64(y * 9.0) * z))
	tmp = 0.0
	if (t_2 <= -1e+248)
		tmp = t_1;
	elseif (t_2 <= -5e+98)
		tmp = fma(t, Float64(-9.0 * Float64(y * z)), Float64(2.0 * x));
	elseif (t_2 <= 2e+82)
		tmp = fma(Float64(27.0 * b), a, Float64(2.0 * x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(z * N[(t * -9.0), $MachinePrecision]), $MachinePrecision] * y + N[(a * N[(27.0 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+248], t$95$1, If[LessEqual[t$95$2, -5e+98], N[(t * N[(-9.0 * N[(y * z), $MachinePrecision]), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+82], N[(N[(27.0 * b), $MachinePrecision] * a + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -1.00000000000000005e248 or 1.9999999999999999e82 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 79.8%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 80.8

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

   5. Applied rewrites 80.8%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. associate-*r* N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 86.2

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

   7. Applied rewrites 86.2%

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

   if -1.00000000000000005e248 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -4.9999999999999998e98

   1. Initial program 99.5%

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

   3. Taylor expanded in a around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 95.1

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

   5. Applied rewrites 95.1%

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

   if -4.9999999999999998e98 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 1.9999999999999999e82

   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. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 89.0

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

   5. Applied rewrites 89.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 89.1

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

   7. Applied rewrites 89.1%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(\left(y \cdot 9\right) \cdot z\right) \leq -1 \cdot 10^{+248}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(t \cdot -9\right), y, a \cdot \left(27 \cdot b\right)\right)\\ \mathbf{elif}\;t \cdot \left(\left(y \cdot 9\right) \cdot z\right) \leq -5 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(t, -9 \cdot \left(y \cdot z\right), 2 \cdot x\right)\\ \mathbf{elif}\;t \cdot \left(\left(y \cdot 9\right) \cdot z\right) \leq 2 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(27 \cdot b, a, 2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(t \cdot -9\right), y, a \cdot \left(27 \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := t \cdot \left(\left(y \cdot 9\right) \cdot z\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-70}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot t, z \cdot -9, 2 \cdot x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(27 \cdot b, a, 2 \cdot x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+212}:\\ \;\;\;\;\mathsf{fma}\left(t, -9 \cdot \left(y \cdot z\right), 2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* t (* (* y 9.0) z))))
   (if (<= t_1 -1e-70)
     (fma (* y t) (* z -9.0) (* 2.0 x))
     (if (<= t_1 2e+71)
       (fma (* 27.0 b) a (* 2.0 x))
       (if (<= t_1 1e+212)
         (fma t (* -9.0 (* y z)) (* 2.0 x))
         (* -9.0 (* y (* t z))))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t * ((y * 9.0) * z);
	double tmp;
	if (t_1 <= -1e-70) {
		tmp = fma((y * t), (z * -9.0), (2.0 * x));
	} else if (t_1 <= 2e+71) {
		tmp = fma((27.0 * b), a, (2.0 * x));
	} else if (t_1 <= 1e+212) {
		tmp = fma(t, (-9.0 * (y * z)), (2.0 * x));
	} else {
		tmp = -9.0 * (y * (t * z));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(t * Float64(Float64(y * 9.0) * z))
	tmp = 0.0
	if (t_1 <= -1e-70)
		tmp = fma(Float64(y * t), Float64(z * -9.0), Float64(2.0 * x));
	elseif (t_1 <= 2e+71)
		tmp = fma(Float64(27.0 * b), a, Float64(2.0 * x));
	elseif (t_1 <= 1e+212)
		tmp = fma(t, Float64(-9.0 * Float64(y * z)), Float64(2.0 * x));
	else
		tmp = Float64(-9.0 * Float64(y * Float64(t * z)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t * N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-70], N[(N[(y * t), $MachinePrecision] * N[(z * -9.0), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+71], N[(N[(27.0 * b), $MachinePrecision] * a + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+212], N[(t * N[(-9.0 * N[(y * z), $MachinePrecision]), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], N[(-9.0 * N[(y * N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -9.99999999999999996e-71

   1. Initial program 87.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. associate-*r* N/A

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

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

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

      17. +-commutative N/A

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

   4. Applied rewrites 88.8%

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

   5. Taylor expanded in a around 0

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

      1. lower-*.f64 78.1

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

   7. Applied rewrites 78.1%

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

   if -9.99999999999999996e-71 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 2.0000000000000001e71

   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. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 94.4

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

   5. Applied rewrites 94.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 94.4

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

   7. Applied rewrites 94.4%

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

   if 2.0000000000000001e71 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 9.9999999999999991e211

   1. Initial program 99.7%

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

   3. Taylor expanded in a around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 79.1

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

   5. Applied rewrites 79.1%

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

   if 9.9999999999999991e211 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 73.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

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

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

   4. Applied rewrites 97.2%

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

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   6. 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. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 76.8

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

   7. Applied rewrites 76.8%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(\left(y \cdot 9\right) \cdot z\right) \leq -1 \cdot 10^{-70}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot t, z \cdot -9, 2 \cdot x\right)\\ \mathbf{elif}\;t \cdot \left(\left(y \cdot 9\right) \cdot z\right) \leq 2 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(27 \cdot b, a, 2 \cdot x\right)\\ \mathbf{elif}\;t \cdot \left(\left(y \cdot 9\right) \cdot z\right) \leq 10^{+212}:\\ \;\;\;\;\mathsf{fma}\left(t, -9 \cdot \left(y \cdot z\right), 2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := t \cdot \left(\left(y \cdot 9\right) \cdot z\right)\\ \mathbf{if}\;t\_1 \leq -0.1:\\ \;\;\;\;z \cdot \left(-9 \cdot \left(y \cdot t\right)\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-308}:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+71}:\\ \;\;\;\;b \cdot \left(27 \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -9\right) \cdot \left(t \cdot z\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* t (* (* y 9.0) z))))
   (if (<= t_1 -0.1)
     (* z (* -9.0 (* y t)))
     (if (<= t_1 5e-308)
       (* 2.0 x)
       (if (<= t_1 2e+71) (* b (* 27.0 a)) (* (* y -9.0) (* t z)))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t * ((y * 9.0) * z);
	double tmp;
	if (t_1 <= -0.1) {
		tmp = z * (-9.0 * (y * t));
	} else if (t_1 <= 5e-308) {
		tmp = 2.0 * x;
	} else if (t_1 <= 2e+71) {
		tmp = b * (27.0 * a);
	} else {
		tmp = (y * -9.0) * (t * z);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * ((y * 9.0d0) * z)
    if (t_1 <= (-0.1d0)) then
        tmp = z * ((-9.0d0) * (y * t))
    else if (t_1 <= 5d-308) then
        tmp = 2.0d0 * x
    else if (t_1 <= 2d+71) then
        tmp = b * (27.0d0 * a)
    else
        tmp = (y * (-9.0d0)) * (t * z)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b;
assert x < y && y < z && z < t && t < a && a < b;
public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t * ((y * 9.0) * z);
	double tmp;
	if (t_1 <= -0.1) {
		tmp = z * (-9.0 * (y * t));
	} else if (t_1 <= 5e-308) {
		tmp = 2.0 * x;
	} else if (t_1 <= 2e+71) {
		tmp = b * (27.0 * a);
	} else {
		tmp = (y * -9.0) * (t * z);
	}
	return tmp;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	t_1 = t * ((y * 9.0) * z)
	tmp = 0
	if t_1 <= -0.1:
		tmp = z * (-9.0 * (y * t))
	elif t_1 <= 5e-308:
		tmp = 2.0 * x
	elif t_1 <= 2e+71:
		tmp = b * (27.0 * a)
	else:
		tmp = (y * -9.0) * (t * z)
	return tmp

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(t * Float64(Float64(y * 9.0) * z))
	tmp = 0.0
	if (t_1 <= -0.1)
		tmp = Float64(z * Float64(-9.0 * Float64(y * t)));
	elseif (t_1 <= 5e-308)
		tmp = Float64(2.0 * x);
	elseif (t_1 <= 2e+71)
		tmp = Float64(b * Float64(27.0 * a));
	else
		tmp = Float64(Float64(y * -9.0) * Float64(t * z));
	end
	return tmp
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp_2 = code(x, y, z, t, a, b)
	t_1 = t * ((y * 9.0) * z);
	tmp = 0.0;
	if (t_1 <= -0.1)
		tmp = z * (-9.0 * (y * t));
	elseif (t_1 <= 5e-308)
		tmp = 2.0 * x;
	elseif (t_1 <= 2e+71)
		tmp = b * (27.0 * a);
	else
		tmp = (y * -9.0) * (t * z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t * N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.1], N[(z * N[(-9.0 * N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-308], N[(2.0 * x), $MachinePrecision], If[LessEqual[t$95$1, 2e+71], N[(b * N[(27.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(y * -9.0), $MachinePrecision] * N[(t * z), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -0.10000000000000001

   1. Initial program 86.2%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 66.5

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

   5. Applied rewrites 66.5%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 69.0

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

   7. Applied rewrites 69.0%

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

   if -0.10000000000000001 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 4.99999999999999955e-308

   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 x around inf

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

      1. lower-*.f64 50.7

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

   5. Applied rewrites 50.7%

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

   if 4.99999999999999955e-308 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 2.0000000000000001e71

   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 a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 58.7

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

   5. Applied rewrites 58.7%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 58.8

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 58.8

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

   7. Applied rewrites 58.8%

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

   if 2.0000000000000001e71 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 83.8%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 68.3

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

   5. Applied rewrites 68.3%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 65.0

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 65.0

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

   7. Applied rewrites 65.0%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(\left(y \cdot 9\right) \cdot z\right) \leq -0.1:\\ \;\;\;\;z \cdot \left(-9 \cdot \left(y \cdot t\right)\right)\\ \mathbf{elif}\;t \cdot \left(\left(y \cdot 9\right) \cdot z\right) \leq 5 \cdot 10^{-308}:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;t \cdot \left(\left(y \cdot 9\right) \cdot z\right) \leq 2 \cdot 10^{+71}:\\ \;\;\;\;b \cdot \left(27 \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -9\right) \cdot \left(t \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := t \cdot \left(\left(y \cdot 9\right) \cdot z\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-70}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot t, z \cdot -9, 2 \cdot x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(27 \cdot b, a, 2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, y \cdot -9, 2 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* t (* (* y 9.0) z))))
   (if (<= t_1 -1e-70)
     (fma (* y t) (* z -9.0) (* 2.0 x))
     (if (<= t_1 2e+71)
       (fma (* 27.0 b) a (* 2.0 x))
       (fma (* t z) (* y -9.0) (* 2.0 x))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t * ((y * 9.0) * z);
	double tmp;
	if (t_1 <= -1e-70) {
		tmp = fma((y * t), (z * -9.0), (2.0 * x));
	} else if (t_1 <= 2e+71) {
		tmp = fma((27.0 * b), a, (2.0 * x));
	} else {
		tmp = fma((t * z), (y * -9.0), (2.0 * x));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(t * Float64(Float64(y * 9.0) * z))
	tmp = 0.0
	if (t_1 <= -1e-70)
		tmp = fma(Float64(y * t), Float64(z * -9.0), Float64(2.0 * x));
	elseif (t_1 <= 2e+71)
		tmp = fma(Float64(27.0 * b), a, Float64(2.0 * x));
	else
		tmp = fma(Float64(t * z), Float64(y * -9.0), Float64(2.0 * x));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t * N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-70], N[(N[(y * t), $MachinePrecision] * N[(z * -9.0), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+71], N[(N[(27.0 * b), $MachinePrecision] * a + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], N[(N[(t * z), $MachinePrecision] * N[(y * -9.0), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -9.99999999999999996e-71

   1. Initial program 87.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. associate-*r* N/A

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

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

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

      17. +-commutative N/A

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

   4. Applied rewrites 88.8%

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

   5. Taylor expanded in a around 0

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

      1. lower-*.f64 78.1

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

   7. Applied rewrites 78.1%

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

   if -9.99999999999999996e-71 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 2.0000000000000001e71

   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. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 94.4

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

   5. Applied rewrites 94.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 94.4

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

   7. Applied rewrites 94.4%

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

   if 2.0000000000000001e71 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 83.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

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

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

      15. +-commutative N/A

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

      16. 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 91.0%

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

   5. Taylor expanded in a around 0

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

      1. lower-*.f64 73.9

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

   7. Applied rewrites 73.9%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(\left(y \cdot 9\right) \cdot z\right) \leq -1 \cdot 10^{-70}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot t, z \cdot -9, 2 \cdot x\right)\\ \mathbf{elif}\;t \cdot \left(\left(y \cdot 9\right) \cdot z\right) \leq 2 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(27 \cdot b, a, 2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, y \cdot -9, 2 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 83.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := b \cdot \left(27 \cdot a\right)\\ t_2 := \mathsf{fma}\left(t, -9 \cdot \left(y \cdot z\right), 27 \cdot \left(a \cdot b\right)\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+30}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-52}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, y \cdot -9, 2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (* 27.0 a))) (t_2 (fma t (* -9.0 (* y z)) (* 27.0 (* a b)))))
   (if (<= t_1 -1e+30)
     t_2
     (if (<= t_1 2e-52) (fma (* t z) (* y -9.0) (* 2.0 x)) t_2))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (27.0 * a);
	double t_2 = fma(t, (-9.0 * (y * z)), (27.0 * (a * b)));
	double tmp;
	if (t_1 <= -1e+30) {
		tmp = t_2;
	} else if (t_1 <= 2e-52) {
		tmp = fma((t * z), (y * -9.0), (2.0 * x));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(27.0 * a))
	t_2 = fma(t, Float64(-9.0 * Float64(y * z)), Float64(27.0 * Float64(a * b)))
	tmp = 0.0
	if (t_1 <= -1e+30)
		tmp = t_2;
	elseif (t_1 <= 2e-52)
		tmp = fma(Float64(t * z), Float64(y * -9.0), Float64(2.0 * x));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(27.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(-9.0 * N[(y * z), $MachinePrecision]), $MachinePrecision] + N[(27.0 * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+30], t$95$2, If[LessEqual[t$95$1, 2e-52], N[(N[(t * z), $MachinePrecision] * N[(y * -9.0), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 a #s(literal 27 binary64)) b) < -1e30 or 2e-52 < (*.f64 (*.f64 a #s(literal 27 binary64)) b)

   1. Initial program 92.8%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 83.1

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

   5. Applied rewrites 83.1%

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

   if -1e30 < (*.f64 (*.f64 a #s(literal 27 binary64)) b) < 2e-52

   1. Initial program 92.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

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

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

   4. Applied rewrites 94.0%

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

   5. Taylor expanded in a around 0

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

      1. lower-*.f64 90.8

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

   7. Applied rewrites 90.8%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(27 \cdot a\right) \leq -1 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(t, -9 \cdot \left(y \cdot z\right), 27 \cdot \left(a \cdot b\right)\right)\\ \mathbf{elif}\;b \cdot \left(27 \cdot a\right) \leq 2 \cdot 10^{-52}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, y \cdot -9, 2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, -9 \cdot \left(y \cdot z\right), 27 \cdot \left(a \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := t \cdot \left(\left(y \cdot 9\right) \cdot z\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+101}:\\ \;\;\;\;z \cdot \left(-9 \cdot \left(y \cdot t\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(27 \cdot b, a, 2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -9\right) \cdot \left(t \cdot z\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* t (* (* y 9.0) z))))
   (if (<= t_1 -2e+101)
     (* z (* -9.0 (* y t)))
     (if (<= t_1 2e+82) (fma (* 27.0 b) a (* 2.0 x)) (* (* y -9.0) (* t z))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t * ((y * 9.0) * z);
	double tmp;
	if (t_1 <= -2e+101) {
		tmp = z * (-9.0 * (y * t));
	} else if (t_1 <= 2e+82) {
		tmp = fma((27.0 * b), a, (2.0 * x));
	} else {
		tmp = (y * -9.0) * (t * z);
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(t * Float64(Float64(y * 9.0) * z))
	tmp = 0.0
	if (t_1 <= -2e+101)
		tmp = Float64(z * Float64(-9.0 * Float64(y * t)));
	elseif (t_1 <= 2e+82)
		tmp = fma(Float64(27.0 * b), a, Float64(2.0 * x));
	else
		tmp = Float64(Float64(y * -9.0) * Float64(t * z));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t * N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+101], N[(z * N[(-9.0 * N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+82], N[(N[(27.0 * b), $MachinePrecision] * a + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], N[(N[(y * -9.0), $MachinePrecision] * N[(t * z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -2e101

   1. Initial program 83.7%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 70.1

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

   5. Applied rewrites 70.1%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 73.1

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

   7. Applied rewrites 73.1%

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

   if -2e101 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 1.9999999999999999e82

   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. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 88.9

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

   5. Applied rewrites 88.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 88.9

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

   7. Applied rewrites 88.9%

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

   if 1.9999999999999999e82 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 83.2%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 69.1

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

   5. Applied rewrites 69.1%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 65.6

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 65.6

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

   7. Applied rewrites 65.6%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(\left(y \cdot 9\right) \cdot z\right) \leq -2 \cdot 10^{+101}:\\ \;\;\;\;z \cdot \left(-9 \cdot \left(y \cdot t\right)\right)\\ \mathbf{elif}\;t \cdot \left(\left(y \cdot 9\right) \cdot z\right) \leq 2 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(27 \cdot b, a, 2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -9\right) \cdot \left(t \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := t \cdot \left(\left(y \cdot 9\right) \cdot z\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+101}:\\ \;\;\;\;z \cdot \left(-9 \cdot \left(y \cdot t\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(27, a \cdot b, 2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -9\right) \cdot \left(t \cdot z\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* t (* (* y 9.0) z))))
   (if (<= t_1 -2e+101)
     (* z (* -9.0 (* y t)))
     (if (<= t_1 2e+82) (fma 27.0 (* a b) (* 2.0 x)) (* (* y -9.0) (* t z))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t * ((y * 9.0) * z);
	double tmp;
	if (t_1 <= -2e+101) {
		tmp = z * (-9.0 * (y * t));
	} else if (t_1 <= 2e+82) {
		tmp = fma(27.0, (a * b), (2.0 * x));
	} else {
		tmp = (y * -9.0) * (t * z);
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(t * Float64(Float64(y * 9.0) * z))
	tmp = 0.0
	if (t_1 <= -2e+101)
		tmp = Float64(z * Float64(-9.0 * Float64(y * t)));
	elseif (t_1 <= 2e+82)
		tmp = fma(27.0, Float64(a * b), Float64(2.0 * x));
	else
		tmp = Float64(Float64(y * -9.0) * Float64(t * z));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t * N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+101], N[(z * N[(-9.0 * N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+82], N[(27.0 * N[(a * b), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], N[(N[(y * -9.0), $MachinePrecision] * N[(t * z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -2e101

   1. Initial program 83.7%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 70.1

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

   5. Applied rewrites 70.1%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 73.1

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

   7. Applied rewrites 73.1%

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

   if -2e101 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 1.9999999999999999e82

   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. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 88.9

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

   5. Applied rewrites 88.9%

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

   if 1.9999999999999999e82 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 83.2%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 69.1

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

   5. Applied rewrites 69.1%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 65.6

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 65.6

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

   7. Applied rewrites 65.6%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(\left(y \cdot 9\right) \cdot z\right) \leq -2 \cdot 10^{+101}:\\ \;\;\;\;z \cdot \left(-9 \cdot \left(y \cdot t\right)\right)\\ \mathbf{elif}\;t \cdot \left(\left(y \cdot 9\right) \cdot z\right) \leq 2 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(27, a \cdot b, 2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -9\right) \cdot \left(t \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 8.5 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, y \cdot -9, \mathsf{fma}\left(a, 27 \cdot b, 2 \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(t, y \cdot -9, \frac{\mathsf{fma}\left(27, a \cdot b, 2 \cdot x\right)}{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (if (<= z 8.5e+19)
   (fma (* t z) (* y -9.0) (fma a (* 27.0 b) (* 2.0 x)))
   (* z (fma t (* y -9.0) (/ (fma 27.0 (* a b) (* 2.0 x)) z)))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= 8.5e+19) {
		tmp = fma((t * z), (y * -9.0), fma(a, (27.0 * b), (2.0 * x)));
	} else {
		tmp = z * fma(t, (y * -9.0), (fma(27.0, (a * b), (2.0 * x)) / z));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= 8.5e+19)
		tmp = fma(Float64(t * z), Float64(y * -9.0), fma(a, Float64(27.0 * b), Float64(2.0 * x)));
	else
		tmp = Float64(z * fma(t, Float64(y * -9.0), Float64(fma(27.0, Float64(a * b), Float64(2.0 * x)) / z)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, 8.5e+19], N[(N[(t * z), $MachinePrecision] * N[(y * -9.0), $MachinePrecision] + N[(a * N[(27.0 * b), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(t * N[(y * -9.0), $MachinePrecision] + N[(N[(27.0 * N[(a * b), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 8.5e19

   1. Initial program 94.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

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

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

   4. Applied rewrites 96.8%

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

   if 8.5e19 < z

   1. Initial program 87.5%

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

   3. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(z \cdot \left(-1 \cdot \frac{2 \cdot x + 27 \cdot \left(a \cdot b\right)}{z} - -9 \cdot \left(t \cdot y\right)\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. neg-sub0 N/A

         \[\leadsto z \cdot \color{blue}{\left(0 - \left(-1 \cdot \frac{2 \cdot x + 27 \cdot \left(a \cdot b\right)}{z} - -9 \cdot \left(t \cdot y\right)\right)\right)} \]

      5. associate--r- N/A

         \[\leadsto z \cdot \color{blue}{\left(\left(0 - -1 \cdot \frac{2 \cdot x + 27 \cdot \left(a \cdot b\right)}{z}\right) + -9 \cdot \left(t \cdot y\right)\right)} \]

      6. neg-sub0 N/A

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

      7. mul-1-neg N/A

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

      8. remove-double-neg N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

   5. Applied rewrites 98.2%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 8.5 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, y \cdot -9, \mathsf{fma}\left(a, 27 \cdot b, 2 \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(t, y \cdot -9, \frac{\mathsf{fma}\left(27, a \cdot b, 2 \cdot x\right)}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 51.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := b \cdot \left(27 \cdot a\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+31}:\\ \;\;\;\;a \cdot \left(27 \cdot b\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-51}:\\ \;\;\;\;2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
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 (* 27.0 a))))
   (if (<= t_1 -2e+31) (* a (* 27.0 b)) (if (<= t_1 4e-51) (* 2.0 x) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (27.0 * a);
	double tmp;
	if (t_1 <= -2e+31) {
		tmp = a * (27.0 * b);
	} else if (t_1 <= 4e-51) {
		tmp = 2.0 * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
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 = b * (27.0d0 * a)
    if (t_1 <= (-2d+31)) then
        tmp = a * (27.0d0 * b)
    else if (t_1 <= 4d-51) then
        tmp = 2.0d0 * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b;
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 = b * (27.0 * a);
	double tmp;
	if (t_1 <= -2e+31) {
		tmp = a * (27.0 * b);
	} else if (t_1 <= 4e-51) {
		tmp = 2.0 * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	t_1 = b * (27.0 * a)
	tmp = 0
	if t_1 <= -2e+31:
		tmp = a * (27.0 * b)
	elif t_1 <= 4e-51:
		tmp = 2.0 * x
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(27.0 * a))
	tmp = 0.0
	if (t_1 <= -2e+31)
		tmp = Float64(a * Float64(27.0 * b));
	elseif (t_1 <= 4e-51)
		tmp = Float64(2.0 * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
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 = b * (27.0 * a);
	tmp = 0.0;
	if (t_1 <= -2e+31)
		tmp = a * (27.0 * b);
	elseif (t_1 <= 4e-51)
		tmp = 2.0 * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
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[(b * N[(27.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+31], N[(a * N[(27.0 * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e-51], N[(2.0 * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 a #s(literal 27 binary64)) b) < -1.9999999999999999e31

   1. Initial program 90.8%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 67.1

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

   5. Applied rewrites 67.1%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 67.2

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

   7. Applied rewrites 67.2%

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

   if -1.9999999999999999e31 < (*.f64 (*.f64 a #s(literal 27 binary64)) b) < 4e-51

   1. Initial program 92.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 45.5

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

   5. Applied rewrites 45.5%

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

   if 4e-51 < (*.f64 (*.f64 a #s(literal 27 binary64)) b)

   1. Initial program 94.3%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 53.6

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

   5. Applied rewrites 53.6%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 53.6

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 53.6

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

   7. Applied rewrites 53.6%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(27 \cdot a\right) \leq -2 \cdot 10^{+31}:\\ \;\;\;\;a \cdot \left(27 \cdot b\right)\\ \mathbf{elif}\;b \cdot \left(27 \cdot a\right) \leq 4 \cdot 10^{-51}:\\ \;\;\;\;2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(27 \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 51.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := b \cdot \left(27 \cdot a\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+31}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-51}:\\ \;\;\;\;2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
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 (* 27.0 a))))
   (if (<= t_1 -2e+31) t_1 (if (<= t_1 4e-51) (* 2.0 x) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (27.0 * a);
	double tmp;
	if (t_1 <= -2e+31) {
		tmp = t_1;
	} else if (t_1 <= 4e-51) {
		tmp = 2.0 * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
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 = b * (27.0d0 * a)
    if (t_1 <= (-2d+31)) then
        tmp = t_1
    else if (t_1 <= 4d-51) then
        tmp = 2.0d0 * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b;
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 = b * (27.0 * a);
	double tmp;
	if (t_1 <= -2e+31) {
		tmp = t_1;
	} else if (t_1 <= 4e-51) {
		tmp = 2.0 * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	t_1 = b * (27.0 * a)
	tmp = 0
	if t_1 <= -2e+31:
		tmp = t_1
	elif t_1 <= 4e-51:
		tmp = 2.0 * x
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(27.0 * a))
	tmp = 0.0
	if (t_1 <= -2e+31)
		tmp = t_1;
	elseif (t_1 <= 4e-51)
		tmp = Float64(2.0 * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
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 = b * (27.0 * a);
	tmp = 0.0;
	if (t_1 <= -2e+31)
		tmp = t_1;
	elseif (t_1 <= 4e-51)
		tmp = 2.0 * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
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[(b * N[(27.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+31], t$95$1, If[LessEqual[t$95$1, 4e-51], N[(2.0 * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 a #s(literal 27 binary64)) b) < -1.9999999999999999e31 or 4e-51 < (*.f64 (*.f64 a #s(literal 27 binary64)) b)

   1. Initial program 92.6%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 60.0

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

   5. Applied rewrites 60.0%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 60.1

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 60.1

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

   7. Applied rewrites 60.1%

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

   if -1.9999999999999999e31 < (*.f64 (*.f64 a #s(literal 27 binary64)) b) < 4e-51

   1. Initial program 92.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 45.5

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

   5. Applied rewrites 45.5%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(27 \cdot a\right) \leq -2 \cdot 10^{+31}:\\ \;\;\;\;b \cdot \left(27 \cdot a\right)\\ \mathbf{elif}\;b \cdot \left(27 \cdot a\right) \leq 4 \cdot 10^{-51}:\\ \;\;\;\;2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(27 \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 51.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := b \cdot \left(27 \cdot a\right)\\ t_2 := 27 \cdot \left(a \cdot b\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+31}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-51}:\\ \;\;\;\;2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
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 (* 27.0 a))) (t_2 (* 27.0 (* a b))))
   (if (<= t_1 -2e+31) t_2 (if (<= t_1 4e-51) (* 2.0 x) t_2))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (27.0 * a);
	double t_2 = 27.0 * (a * b);
	double tmp;
	if (t_1 <= -2e+31) {
		tmp = t_2;
	} else if (t_1 <= 4e-51) {
		tmp = 2.0 * x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = b * (27.0d0 * a)
    t_2 = 27.0d0 * (a * b)
    if (t_1 <= (-2d+31)) then
        tmp = t_2
    else if (t_1 <= 4d-51) then
        tmp = 2.0d0 * x
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b;
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 = b * (27.0 * a);
	double t_2 = 27.0 * (a * b);
	double tmp;
	if (t_1 <= -2e+31) {
		tmp = t_2;
	} else if (t_1 <= 4e-51) {
		tmp = 2.0 * x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	t_1 = b * (27.0 * a)
	t_2 = 27.0 * (a * b)
	tmp = 0
	if t_1 <= -2e+31:
		tmp = t_2
	elif t_1 <= 4e-51:
		tmp = 2.0 * x
	else:
		tmp = t_2
	return tmp

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(27.0 * a))
	t_2 = Float64(27.0 * Float64(a * b))
	tmp = 0.0
	if (t_1 <= -2e+31)
		tmp = t_2;
	elseif (t_1 <= 4e-51)
		tmp = Float64(2.0 * x);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
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 = b * (27.0 * a);
	t_2 = 27.0 * (a * b);
	tmp = 0.0;
	if (t_1 <= -2e+31)
		tmp = t_2;
	elseif (t_1 <= 4e-51)
		tmp = 2.0 * x;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
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[(b * N[(27.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(27.0 * N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+31], t$95$2, If[LessEqual[t$95$1, 4e-51], N[(2.0 * x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 a #s(literal 27 binary64)) b) < -1.9999999999999999e31 or 4e-51 < (*.f64 (*.f64 a #s(literal 27 binary64)) b)

   1. Initial program 92.6%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 60.0

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

   5. Applied rewrites 60.0%

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

   if -1.9999999999999999e31 < (*.f64 (*.f64 a #s(literal 27 binary64)) b) < 4e-51

   1. Initial program 92.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 45.5

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

   5. Applied rewrites 45.5%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(27 \cdot a\right) \leq -2 \cdot 10^{+31}:\\ \;\;\;\;27 \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;b \cdot \left(27 \cdot a\right) \leq 4 \cdot 10^{-51}:\\ \;\;\;\;2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;27 \cdot \left(a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 98.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, 27 \cdot b, 2 \cdot x\right)\\ \mathbf{if}\;z \leq 1.2 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, y \cdot -9, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot t, z \cdot -9, t\_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma a (* 27.0 b) (* 2.0 x))))
   (if (<= z 1.2e+18)
     (fma (* t z) (* y -9.0) t_1)
     (fma (* y t) (* z -9.0) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(a, (27.0 * b), (2.0 * x));
	double tmp;
	if (z <= 1.2e+18) {
		tmp = fma((t * z), (y * -9.0), t_1);
	} else {
		tmp = fma((y * t), (z * -9.0), t_1);
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = fma(a, Float64(27.0 * b), Float64(2.0 * x))
	tmp = 0.0
	if (z <= 1.2e+18)
		tmp = fma(Float64(t * z), Float64(y * -9.0), t_1);
	else
		tmp = fma(Float64(y * t), Float64(z * -9.0), t_1);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a * N[(27.0 * b), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 1.2e+18], N[(N[(t * z), $MachinePrecision] * N[(y * -9.0), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(y * t), $MachinePrecision] * N[(z * -9.0), $MachinePrecision] + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 1.2e18

   1. Initial program 94.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

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

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

   4. Applied rewrites 96.8%

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

   if 1.2e18 < z

   1. Initial program 87.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. associate-*r* N/A

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

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

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

      17. +-commutative N/A

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

   4. Applied rewrites 96.6%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.2 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, y \cdot -9, \mathsf{fma}\left(a, 27 \cdot b, 2 \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot t, z \cdot -9, \mathsf{fma}\left(a, 27 \cdot b, 2 \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, 27 \cdot b, 2 \cdot x\right)\\ \mathbf{if}\;z \leq 1.2 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(y, t \cdot \left(z \cdot -9\right), t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot t, z \cdot -9, t\_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma a (* 27.0 b) (* 2.0 x))))
   (if (<= z 1.2e+18)
     (fma y (* t (* z -9.0)) t_1)
     (fma (* y t) (* z -9.0) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(a, (27.0 * b), (2.0 * x));
	double tmp;
	if (z <= 1.2e+18) {
		tmp = fma(y, (t * (z * -9.0)), t_1);
	} else {
		tmp = fma((y * t), (z * -9.0), t_1);
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = fma(a, Float64(27.0 * b), Float64(2.0 * x))
	tmp = 0.0
	if (z <= 1.2e+18)
		tmp = fma(y, Float64(t * Float64(z * -9.0)), t_1);
	else
		tmp = fma(Float64(y * t), Float64(z * -9.0), t_1);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a * N[(27.0 * b), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 1.2e+18], N[(y * N[(t * N[(z * -9.0), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(y * t), $MachinePrecision] * N[(z * -9.0), $MachinePrecision] + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 1.2e18

   1. Initial program 94.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

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

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

      16. associate-*l* N/A

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

      17. +-commutative N/A

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

   4. Applied rewrites 96.8%

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

   if 1.2e18 < z

   1. Initial program 87.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. lift-*.f64 N/A

         \[\leadsto \left(x \cdot 2 - \color{blue}{\left(\left(y \cdot 9\right) \cdot z\right) \cdot t}\right) + \left(a \cdot 27\right) \cdot b \]

      5. lift-*.f64 N/A

         \[\leadsto \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \color{blue}{\left(a \cdot 27\right)} \cdot b \]

      6. lift-*.f64 N/A

         \[\leadsto \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \color{blue}{\left(a \cdot 27\right) \cdot b} \]

      7. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot 2 + \left(\mathsf{neg}\left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right)\right)} + \left(a \cdot 27\right) \cdot b \]

      8. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) + x \cdot 2\right)} + \left(a \cdot 27\right) \cdot b \]

      9. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) + \left(x \cdot 2 + \left(a \cdot 27\right) \cdot b\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\left(y \cdot 9\right) \cdot z\right) \cdot t}\right)\right) + \left(x \cdot 2 + \left(a \cdot 27\right) \cdot b\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{t \cdot \left(\left(y \cdot 9\right) \cdot z\right)}\right)\right) + \left(x \cdot 2 + \left(a \cdot 27\right) \cdot b\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(t \cdot \color{blue}{\left(\left(y \cdot 9\right) \cdot z\right)}\right)\right) + \left(x \cdot 2 + \left(a \cdot 27\right) \cdot b\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(t \cdot \left(\color{blue}{\left(y \cdot 9\right)} \cdot z\right)\right)\right) + \left(x \cdot 2 + \left(a \cdot 27\right) \cdot b\right) \]

      14. associate-*l* N/A

          \[\leadsto \left(\mathsf{neg}\left(t \cdot \color{blue}{\left(y \cdot \left(9 \cdot z\right)\right)}\right)\right) + \left(x \cdot 2 + \left(a \cdot 27\right) \cdot b\right) \]

      15. associate-*r* N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(t \cdot y\right) \cdot \left(9 \cdot z\right)}\right)\right) + \left(x \cdot 2 + \left(a \cdot 27\right) \cdot b\right) \]

      16. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{\left(t \cdot y\right) \cdot \left(\mathsf{neg}\left(9 \cdot z\right)\right)} + \left(x \cdot 2 + \left(a \cdot 27\right) \cdot b\right) \]

      17. +-commutative N/A

          \[\leadsto \left(t \cdot y\right) \cdot \left(\mathsf{neg}\left(9 \cdot z\right)\right) + \color{blue}{\left(\left(a \cdot 27\right) \cdot b + x \cdot 2\right)} \]

   4. Applied rewrites 96.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot t, z \cdot -9, \mathsf{fma}\left(a, 27 \cdot b, x \cdot 2\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.2 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(y, t \cdot \left(z \cdot -9\right), \mathsf{fma}\left(a, 27 \cdot b, 2 \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot t, z \cdot -9, \mathsf{fma}\left(a, 27 \cdot b, 2 \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 97.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 6.3 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(y, t \cdot \left(z \cdot -9\right), \mathsf{fma}\left(a, 27 \cdot b, 2 \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, -9 \cdot \left(y \cdot z\right), 27 \cdot \left(a \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (if (<= z 6.3e+49)
   (fma y (* t (* z -9.0)) (fma a (* 27.0 b) (* 2.0 x)))
   (fma t (* -9.0 (* y z)) (* 27.0 (* a b)))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= 6.3e+49) {
		tmp = fma(y, (t * (z * -9.0)), fma(a, (27.0 * b), (2.0 * x)));
	} else {
		tmp = fma(t, (-9.0 * (y * z)), (27.0 * (a * b)));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= 6.3e+49)
		tmp = fma(y, Float64(t * Float64(z * -9.0)), fma(a, Float64(27.0 * b), Float64(2.0 * x)));
	else
		tmp = fma(t, Float64(-9.0 * Float64(y * z)), Float64(27.0 * Float64(a * b)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, 6.3e+49], N[(y * N[(t * N[(z * -9.0), $MachinePrecision]), $MachinePrecision] + N[(a * N[(27.0 * b), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(-9.0 * N[(y * z), $MachinePrecision]), $MachinePrecision] + N[(27.0 * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 6.30000000000000007e49

   1. Initial program 94.4%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot 2} - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

      2. lift-*.f64 N/A

         \[\leadsto \left(x \cdot 2 - \left(\color{blue}{\left(y \cdot 9\right)} \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

      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. lift-*.f64 N/A

         \[\leadsto \left(x \cdot 2 - \color{blue}{\left(\left(y \cdot 9\right) \cdot z\right) \cdot t}\right) + \left(a \cdot 27\right) \cdot b \]

      5. lift-*.f64 N/A

         \[\leadsto \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \color{blue}{\left(a \cdot 27\right)} \cdot b \]

      6. lift-*.f64 N/A

         \[\leadsto \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \color{blue}{\left(a \cdot 27\right) \cdot b} \]

      7. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot 2 + \left(\mathsf{neg}\left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right)\right)} + \left(a \cdot 27\right) \cdot b \]

      8. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) + x \cdot 2\right)} + \left(a \cdot 27\right) \cdot b \]

      9. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) + \left(x \cdot 2 + \left(a \cdot 27\right) \cdot b\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\left(y \cdot 9\right) \cdot z\right) \cdot t}\right)\right) + \left(x \cdot 2 + \left(a \cdot 27\right) \cdot b\right) \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y \cdot 9\right) \cdot z\right)\right) \cdot t} + \left(x \cdot 2 + \left(a \cdot 27\right) \cdot b\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y \cdot 9\right) \cdot z}\right)\right) \cdot t + \left(x \cdot 2 + \left(a \cdot 27\right) \cdot b\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y \cdot 9\right)} \cdot z\right)\right) \cdot t + \left(x \cdot 2 + \left(a \cdot 27\right) \cdot b\right) \]

      14. associate-*l* N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{y \cdot \left(9 \cdot z\right)}\right)\right) \cdot t + \left(x \cdot 2 + \left(a \cdot 27\right) \cdot b\right) \]

      15. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{\left(y \cdot \left(\mathsf{neg}\left(9 \cdot z\right)\right)\right)} \cdot t + \left(x \cdot 2 + \left(a \cdot 27\right) \cdot b\right) \]

      16. associate-*l* N/A

          \[\leadsto \color{blue}{y \cdot \left(\left(\mathsf{neg}\left(9 \cdot z\right)\right) \cdot t\right)} + \left(x \cdot 2 + \left(a \cdot 27\right) \cdot b\right) \]

      17. +-commutative N/A

          \[\leadsto y \cdot \left(\left(\mathsf{neg}\left(9 \cdot z\right)\right) \cdot t\right) + \color{blue}{\left(\left(a \cdot 27\right) \cdot b + x \cdot 2\right)} \]

   4. Applied rewrites 96.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \left(z \cdot -9\right) \cdot t, \mathsf{fma}\left(a, 27 \cdot b, x \cdot 2\right)\right)} \]

   if 6.30000000000000007e49 < z

   1. Initial program 86.7%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{27 \cdot \left(a \cdot b\right) - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{27 \cdot \left(a \cdot b\right) + \left(\mathsf{neg}\left(9\right)\right) \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto 27 \cdot \left(a \cdot b\right) + \color{blue}{-9} \cdot \left(t \cdot \left(y \cdot z\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{-9 \cdot \left(t \cdot \left(y \cdot z\right)\right) + 27 \cdot \left(a \cdot b\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot \left(y \cdot z\right)\right) \cdot -9} + 27 \cdot \left(a \cdot b\right) \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{t \cdot \left(\left(y \cdot z\right) \cdot -9\right)} + 27 \cdot \left(a \cdot b\right) \]

      6. *-commutative N/A

         \[\leadsto t \cdot \color{blue}{\left(-9 \cdot \left(y \cdot z\right)\right)} + 27 \cdot \left(a \cdot b\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -9 \cdot \left(y \cdot z\right), 27 \cdot \left(a \cdot b\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\left(y \cdot z\right) \cdot -9}, 27 \cdot \left(a \cdot b\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\left(y \cdot z\right) \cdot -9}, 27 \cdot \left(a \cdot b\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{\left(y \cdot z\right)} \cdot -9, 27 \cdot \left(a \cdot b\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \left(y \cdot z\right) \cdot -9, \color{blue}{27 \cdot \left(a \cdot b\right)}\right) \]

      12. lower-*.f64 70.3

          \[\leadsto \mathsf{fma}\left(t, \left(y \cdot z\right) \cdot -9, 27 \cdot \color{blue}{\left(a \cdot b\right)}\right) \]

   5. Applied rewrites 70.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \left(y \cdot z\right) \cdot -9, 27 \cdot \left(a \cdot b\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6.3 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(y, t \cdot \left(z \cdot -9\right), \mathsf{fma}\left(a, 27 \cdot b, 2 \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, -9 \cdot \left(y \cdot z\right), 27 \cdot \left(a \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 97.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 6.3 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(-9, y \cdot \left(t \cdot z\right), \mathsf{fma}\left(a, 27 \cdot b, 2 \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, -9 \cdot \left(y \cdot z\right), 27 \cdot \left(a \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (if (<= z 6.3e+49)
   (fma -9.0 (* y (* t z)) (fma a (* 27.0 b) (* 2.0 x)))
   (fma t (* -9.0 (* y z)) (* 27.0 (* a b)))))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= 6.3e+49) {
		tmp = fma(-9.0, (y * (t * z)), fma(a, (27.0 * b), (2.0 * x)));
	} else {
		tmp = fma(t, (-9.0 * (y * z)), (27.0 * (a * b)));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= 6.3e+49)
		tmp = fma(-9.0, Float64(y * Float64(t * z)), fma(a, Float64(27.0 * b), Float64(2.0 * x)));
	else
		tmp = fma(t, Float64(-9.0 * Float64(y * z)), Float64(27.0 * Float64(a * b)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, 6.3e+49], N[(-9.0 * N[(y * N[(t * z), $MachinePrecision]), $MachinePrecision] + N[(a * N[(27.0 * b), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(-9.0 * N[(y * z), $MachinePrecision]), $MachinePrecision] + N[(27.0 * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 6.30000000000000007e49

   1. Initial program 94.4%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot 2} - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

      2. lift-*.f64 N/A

         \[\leadsto \left(x \cdot 2 - \left(\color{blue}{\left(y \cdot 9\right)} \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

      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. lift-*.f64 N/A

         \[\leadsto \left(x \cdot 2 - \color{blue}{\left(\left(y \cdot 9\right) \cdot z\right) \cdot t}\right) + \left(a \cdot 27\right) \cdot b \]

      5. lift-*.f64 N/A

         \[\leadsto \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \color{blue}{\left(a \cdot 27\right)} \cdot b \]

      6. lift-*.f64 N/A

         \[\leadsto \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \color{blue}{\left(a \cdot 27\right) \cdot b} \]

      7. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot 2 + \left(\mathsf{neg}\left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right)\right)} + \left(a \cdot 27\right) \cdot b \]

      8. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) + x \cdot 2\right)} + \left(a \cdot 27\right) \cdot b \]

      9. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right) + \left(x \cdot 2 + \left(a \cdot 27\right) \cdot b\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\left(y \cdot 9\right) \cdot z\right) \cdot t}\right)\right) + \left(x \cdot 2 + \left(a \cdot 27\right) \cdot b\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\left(y \cdot 9\right) \cdot z\right)} \cdot t\right)\right) + \left(x \cdot 2 + \left(a \cdot 27\right) \cdot b\right) \]

      12. associate-*l* N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y \cdot 9\right) \cdot \left(z \cdot t\right)}\right)\right) + \left(x \cdot 2 + \left(a \cdot 27\right) \cdot b\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y \cdot 9\right)} \cdot \left(z \cdot t\right)\right)\right) + \left(x \cdot 2 + \left(a \cdot 27\right) \cdot b\right) \]

      14. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(9 \cdot y\right)} \cdot \left(z \cdot t\right)\right)\right) + \left(x \cdot 2 + \left(a \cdot 27\right) \cdot b\right) \]

      15. associate-*l* N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{9 \cdot \left(y \cdot \left(z \cdot t\right)\right)}\right)\right) + \left(x \cdot 2 + \left(a \cdot 27\right) \cdot b\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(9\right)\right) \cdot \left(y \cdot \left(z \cdot t\right)\right)} + \left(x \cdot 2 + \left(a \cdot 27\right) \cdot b\right) \]

      17. +-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(9\right)\right) \cdot \left(y \cdot \left(z \cdot t\right)\right) + \color{blue}{\left(\left(a \cdot 27\right) \cdot b + x \cdot 2\right)} \]

   4. Applied rewrites 96.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-9, y \cdot \left(z \cdot t\right), \mathsf{fma}\left(a, 27 \cdot b, x \cdot 2\right)\right)} \]

   if 6.30000000000000007e49 < z

   1. Initial program 86.7%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{27 \cdot \left(a \cdot b\right) - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{27 \cdot \left(a \cdot b\right) + \left(\mathsf{neg}\left(9\right)\right) \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto 27 \cdot \left(a \cdot b\right) + \color{blue}{-9} \cdot \left(t \cdot \left(y \cdot z\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{-9 \cdot \left(t \cdot \left(y \cdot z\right)\right) + 27 \cdot \left(a \cdot b\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot \left(y \cdot z\right)\right) \cdot -9} + 27 \cdot \left(a \cdot b\right) \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{t \cdot \left(\left(y \cdot z\right) \cdot -9\right)} + 27 \cdot \left(a \cdot b\right) \]

      6. *-commutative N/A

         \[\leadsto t \cdot \color{blue}{\left(-9 \cdot \left(y \cdot z\right)\right)} + 27 \cdot \left(a \cdot b\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -9 \cdot \left(y \cdot z\right), 27 \cdot \left(a \cdot b\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\left(y \cdot z\right) \cdot -9}, 27 \cdot \left(a \cdot b\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\left(y \cdot z\right) \cdot -9}, 27 \cdot \left(a \cdot b\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{\left(y \cdot z\right)} \cdot -9, 27 \cdot \left(a \cdot b\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \left(y \cdot z\right) \cdot -9, \color{blue}{27 \cdot \left(a \cdot b\right)}\right) \]

      12. lower-*.f64 70.3

          \[\leadsto \mathsf{fma}\left(t, \left(y \cdot z\right) \cdot -9, 27 \cdot \color{blue}{\left(a \cdot b\right)}\right) \]

   5. Applied rewrites 70.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \left(y \cdot z\right) \cdot -9, 27 \cdot \left(a \cdot b\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6.3 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(-9, y \cdot \left(t \cdot z\right), \mathsf{fma}\left(a, 27 \cdot b, 2 \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, -9 \cdot \left(y \cdot z\right), 27 \cdot \left(a \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 30.6% accurate, 6.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\\\ [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ 2 \cdot x \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b) :precision binary64 (* 2.0 x))

C

assert(x < y && y < z && z < t && t < a && a < b);
assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	return 2.0 * x;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 2.0d0 * x
end function

Java

assert x < y && y < z && z < t && t < a && a < b;
assert x < y && y < z && z < t && t < a && a < b;
public static double code(double x, double y, double z, double t, double a, double b) {
	return 2.0 * x;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	return 2.0 * x

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	return Float64(2.0 * x)
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp = code(x, y, z, t, a, b)
	tmp = 2.0 * x;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := N[(2.0 * x), $MachinePrecision]

Derivation

1. Initial program 92.7%

   \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{2 \cdot x} \]
4. Step-by-step derivation

   1. lower-*.f64 27.9

      \[\leadsto \color{blue}{2 \cdot x} \]

5. Applied rewrites 27.9%

   \[\leadsto \color{blue}{2 \cdot x} \]
6. 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 2024216 
(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)))