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

Percentage Accurate: 95.9% → 98.3%

Time: 14.9s

Alternatives: 10

Speedup: 0.9×

• 27.0% 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: 95.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(27 \cdot b, a, x \cdot 2\right)\\ \mathbf{if}\;t \leq 10^{+202}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot t, y \cdot -9, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot z\right) \cdot -9, t, t\_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (* 27.0 b) a (* x 2.0))))
   (if (<= t 1e+202)
     (fma (* z t) (* y -9.0) t_1)
     (fma (* (* y z) -9.0) t t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((27.0 * b), a, (x * 2.0));
	double tmp;
	if (t <= 1e+202) {
		tmp = fma((z * t), (y * -9.0), t_1);
	} else {
		tmp = fma(((y * z) * -9.0), t, t_1);
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = fma(Float64(27.0 * b), a, Float64(x * 2.0))
	tmp = 0.0
	if (t <= 1e+202)
		tmp = fma(Float64(z * t), Float64(y * -9.0), t_1);
	else
		tmp = fma(Float64(Float64(y * z) * -9.0), t, t_1);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(27.0 * b), $MachinePrecision] * a + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 1e+202], N[(N[(z * t), $MachinePrecision] * N[(y * -9.0), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(N[(y * z), $MachinePrecision] * -9.0), $MachinePrecision] * t + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 9.999999999999999e201

   1. Initial program 96.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

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

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

      18. lower-*.f64 N/A

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

      19. metadata-eval N/A

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

   4. Applied rewrites 97.5%

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

   if 9.999999999999999e201 < t

   1. Initial program 95.0%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

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

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

   4. Applied rewrites 99.8%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 10^{+202}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot t, y \cdot -9, \mathsf{fma}\left(27 \cdot b, a, x \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot z\right) \cdot -9, t, \mathsf{fma}\left(27 \cdot b, a, x \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(9 \cdot y\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(\left(-9 \cdot t\right) \cdot z, y, x \cdot 2\right)\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(-9, \left(y \cdot z\right) \cdot t, \left(a \cdot b\right) \cdot 27\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(a \cdot 27\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-9 \cdot \left(z \cdot t\right), y, x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* 9.0 y) z) t)))
   (if (<= t_1 -2e+71)
     (fma (* (* -9.0 t) z) y (* x 2.0))
     (if (<= t_1 -5e-33)
       (fma -9.0 (* (* y z) t) (* (* a b) 27.0))
       (if (<= t_1 5e+16)
         (fma 2.0 x (* (* a 27.0) b))
         (fma (* -9.0 (* z t)) y (* x 2.0)))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((9.0 * y) * z) * t;
	double tmp;
	if (t_1 <= -2e+71) {
		tmp = fma(((-9.0 * t) * z), y, (x * 2.0));
	} else if (t_1 <= -5e-33) {
		tmp = fma(-9.0, ((y * z) * t), ((a * b) * 27.0));
	} else if (t_1 <= 5e+16) {
		tmp = fma(2.0, x, ((a * 27.0) * b));
	} else {
		tmp = fma((-9.0 * (z * t)), y, (x * 2.0));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(9.0 * y) * z) * t)
	tmp = 0.0
	if (t_1 <= -2e+71)
		tmp = fma(Float64(Float64(-9.0 * t) * z), y, Float64(x * 2.0));
	elseif (t_1 <= -5e-33)
		tmp = fma(-9.0, Float64(Float64(y * z) * t), Float64(Float64(a * b) * 27.0));
	elseif (t_1 <= 5e+16)
		tmp = fma(2.0, x, Float64(Float64(a * 27.0) * b));
	else
		tmp = fma(Float64(-9.0 * Float64(z * t)), y, Float64(x * 2.0));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(9.0 * y), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+71], N[(N[(N[(-9.0 * t), $MachinePrecision] * z), $MachinePrecision] * y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e-33], N[(-9.0 * N[(N[(y * z), $MachinePrecision] * t), $MachinePrecision] + N[(N[(a * b), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+16], N[(2.0 * x + N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(N[(-9.0 * N[(z * t), $MachinePrecision]), $MachinePrecision] * y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 90.5%

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

   3. 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. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 86.9%

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

   6. Taylor expanded in a around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 84.6

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

   8. Applied rewrites 84.6%

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

   10. Applied rewrites 84.7%

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

   if -2.0000000000000001e71 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -5.00000000000000028e-33

   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 x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 86.7

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

   5. Applied rewrites 86.7%

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

   if -5.00000000000000028e-33 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 5e16

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 95.5

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

   5. Applied rewrites 95.5%

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

   7. Applied rewrites 95.5%

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

   if 5e16 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 92.3%

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

   3. Taylor expanded in 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. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 84.1%

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

   6. Taylor expanded in a around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 80.5

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

   8. Applied rewrites 80.5%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(9 \cdot y\right) \cdot z\right) \cdot t \leq -2 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(\left(-9 \cdot t\right) \cdot z, y, x \cdot 2\right)\\ \mathbf{elif}\;\left(\left(9 \cdot y\right) \cdot z\right) \cdot t \leq -5 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(-9, \left(y \cdot z\right) \cdot t, \left(a \cdot b\right) \cdot 27\right)\\ \mathbf{elif}\;\left(\left(9 \cdot y\right) \cdot z\right) \cdot t \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(a \cdot 27\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-9 \cdot \left(z \cdot t\right), y, x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(9 \cdot y\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot z\right) \cdot -9, t, \left(a \cdot b\right) \cdot 27\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(a \cdot 27\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-9 \cdot \left(z \cdot t\right), y, x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* 9.0 y) z) t)))
   (if (<= t_1 -5e-33)
     (fma (* (* y z) -9.0) t (* (* a b) 27.0))
     (if (<= t_1 5e+16)
       (fma 2.0 x (* (* a 27.0) b))
       (fma (* -9.0 (* z t)) y (* x 2.0))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((9.0 * y) * z) * t;
	double tmp;
	if (t_1 <= -5e-33) {
		tmp = fma(((y * z) * -9.0), t, ((a * b) * 27.0));
	} else if (t_1 <= 5e+16) {
		tmp = fma(2.0, x, ((a * 27.0) * b));
	} else {
		tmp = fma((-9.0 * (z * t)), y, (x * 2.0));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(9.0 * y) * z) * t)
	tmp = 0.0
	if (t_1 <= -5e-33)
		tmp = fma(Float64(Float64(y * z) * -9.0), t, Float64(Float64(a * b) * 27.0));
	elseif (t_1 <= 5e+16)
		tmp = fma(2.0, x, Float64(Float64(a * 27.0) * b));
	else
		tmp = fma(Float64(-9.0 * Float64(z * t)), y, Float64(x * 2.0));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(9.0 * y), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-33], N[(N[(N[(y * z), $MachinePrecision] * -9.0), $MachinePrecision] * t + N[(N[(a * b), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+16], N[(2.0 * x + N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(N[(-9.0 * N[(z * t), $MachinePrecision]), $MachinePrecision] * y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -5.00000000000000028e-33

   1. Initial program 92.5%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

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

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

   4. Applied rewrites 92.5%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 73.6

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

   7. Applied rewrites 73.6%

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

   if -5.00000000000000028e-33 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 5e16

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 95.5

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

   5. Applied rewrites 95.5%

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

   7. Applied rewrites 95.5%

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

   if 5e16 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 92.3%

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

   3. Taylor expanded in 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. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 84.1%

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

   6. Taylor expanded in a around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 80.5

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

   8. Applied rewrites 80.5%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(9 \cdot y\right) \cdot z\right) \cdot t \leq -5 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot z\right) \cdot -9, t, \left(a \cdot b\right) \cdot 27\right)\\ \mathbf{elif}\;\left(\left(9 \cdot y\right) \cdot z\right) \cdot t \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(a \cdot 27\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-9 \cdot \left(z \cdot t\right), y, x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(9 \cdot y\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(\left(-9 \cdot t\right) \cdot z, y, x \cdot 2\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(a \cdot 27\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-9 \cdot \left(z \cdot t\right), y, x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* 9.0 y) z) t)))
   (if (<= t_1 -5e+62)
     (fma (* (* -9.0 t) z) y (* x 2.0))
     (if (<= t_1 5e+16)
       (fma 2.0 x (* (* a 27.0) b))
       (fma (* -9.0 (* z t)) y (* x 2.0))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((9.0 * y) * z) * t;
	double tmp;
	if (t_1 <= -5e+62) {
		tmp = fma(((-9.0 * t) * z), y, (x * 2.0));
	} else if (t_1 <= 5e+16) {
		tmp = fma(2.0, x, ((a * 27.0) * b));
	} else {
		tmp = fma((-9.0 * (z * t)), y, (x * 2.0));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(9.0 * y) * z) * t)
	tmp = 0.0
	if (t_1 <= -5e+62)
		tmp = fma(Float64(Float64(-9.0 * t) * z), y, Float64(x * 2.0));
	elseif (t_1 <= 5e+16)
		tmp = fma(2.0, x, Float64(Float64(a * 27.0) * b));
	else
		tmp = fma(Float64(-9.0 * Float64(z * t)), y, Float64(x * 2.0));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(9.0 * y), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+62], N[(N[(N[(-9.0 * t), $MachinePrecision] * z), $MachinePrecision] * y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+16], N[(2.0 * x + N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(N[(-9.0 * N[(z * t), $MachinePrecision]), $MachinePrecision] * y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -5.00000000000000029e62

   1. Initial program 90.6%

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

   3. 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. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 87.2%

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

   6. Taylor expanded in a around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 84.9

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

   8. Applied rewrites 84.9%

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

   10. Applied rewrites 84.9%

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

   if -5.00000000000000029e62 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 5e16

   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 y around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 91.0

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

   5. Applied rewrites 91.0%

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

   7. Applied rewrites 91.0%

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

   if 5e16 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 92.3%

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

   3. Taylor expanded in 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. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 84.1%

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

   6. Taylor expanded in a around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 80.5

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

   8. Applied rewrites 80.5%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(9 \cdot y\right) \cdot z\right) \cdot t \leq -5 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(\left(-9 \cdot t\right) \cdot z, y, x \cdot 2\right)\\ \mathbf{elif}\;\left(\left(9 \cdot y\right) \cdot z\right) \cdot t \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(a \cdot 27\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-9 \cdot \left(z \cdot t\right), y, x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\left(-9 \cdot t\right) \cdot z, y, x \cdot 2\right)\\ t_2 := \left(\left(9 \cdot y\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+62}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(a \cdot 27\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (* (* -9.0 t) z) y (* x 2.0))) (t_2 (* (* (* 9.0 y) z) t)))
   (if (<= t_2 -5e+62)
     t_1
     (if (<= t_2 5e+16) (fma 2.0 x (* (* a 27.0) b)) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(((-9.0 * t) * z), y, (x * 2.0));
	double t_2 = ((9.0 * y) * z) * t;
	double tmp;
	if (t_2 <= -5e+62) {
		tmp = t_1;
	} else if (t_2 <= 5e+16) {
		tmp = fma(2.0, x, ((a * 27.0) * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = fma(Float64(Float64(-9.0 * t) * z), y, Float64(x * 2.0))
	t_2 = Float64(Float64(Float64(9.0 * y) * z) * t)
	tmp = 0.0
	if (t_2 <= -5e+62)
		tmp = t_1;
	elseif (t_2 <= 5e+16)
		tmp = fma(2.0, x, Float64(Float64(a * 27.0) * b));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(-9.0 * t), $MachinePrecision] * z), $MachinePrecision] * y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(9.0 * y), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+62], t$95$1, If[LessEqual[t$95$2, 5e+16], N[(2.0 * x + N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -5.00000000000000029e62 or 5e16 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 91.6%

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

   3. Taylor expanded in 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. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 85.5%

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

   6. Taylor expanded in a around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 82.4

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

   8. Applied rewrites 82.4%

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

   10. Applied rewrites 82.5%

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

   if -5.00000000000000029e62 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 5e16

   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 y around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 91.0

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

   5. Applied rewrites 91.0%

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

   7. Applied rewrites 91.0%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(9 \cdot y\right) \cdot z\right) \cdot t \leq -5 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(\left(-9 \cdot t\right) \cdot z, y, x \cdot 2\right)\\ \mathbf{elif}\;\left(\left(9 \cdot y\right) \cdot z\right) \cdot t \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(a \cdot 27\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-9 \cdot t\right) \cdot z, y, x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(y \cdot -9\right) \cdot \left(z \cdot t\right)\\ t_2 := \left(\left(9 \cdot y\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+259}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+129}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(a \cdot 27\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* y -9.0) (* z t))) (t_2 (* (* (* 9.0 y) z) t)))
   (if (<= t_2 -1e+259)
     t_1
     (if (<= t_2 1e+129) (fma 2.0 x (* (* a 27.0) b)) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * -9.0) * (z * t);
	double t_2 = ((9.0 * y) * z) * t;
	double tmp;
	if (t_2 <= -1e+259) {
		tmp = t_1;
	} else if (t_2 <= 1e+129) {
		tmp = fma(2.0, x, ((a * 27.0) * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * -9.0) * Float64(z * t))
	t_2 = Float64(Float64(Float64(9.0 * y) * z) * t)
	tmp = 0.0
	if (t_2 <= -1e+259)
		tmp = t_1;
	elseif (t_2 <= 1e+129)
		tmp = fma(2.0, x, Float64(Float64(a * 27.0) * b));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * -9.0), $MachinePrecision] * N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(9.0 * y), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+259], t$95$1, If[LessEqual[t$95$2, 1e+129], N[(2.0 * x + N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -9.999999999999999e258 or 1e129 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 86.8%

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

   3. 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. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 92.2%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 82.0%

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

   10. Applied rewrites 80.9%

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

   if -9.999999999999999e258 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 1e129

   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 y around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 83.4

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

   5. Applied rewrites 83.4%

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

   7. Applied rewrites 83.4%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(9 \cdot y\right) \cdot z\right) \cdot t \leq -1 \cdot 10^{+259}:\\ \;\;\;\;\left(y \cdot -9\right) \cdot \left(z \cdot t\right)\\ \mathbf{elif}\;\left(\left(9 \cdot y\right) \cdot z\right) \cdot t \leq 10^{+129}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(a \cdot 27\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -9\right) \cdot \left(z \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(9 \cdot y\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-33}:\\ \;\;\;\;\left(\left(y \cdot -9\right) \cdot z\right) \cdot t\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\left(a \cdot 27\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -9\right) \cdot \left(z \cdot t\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* 9.0 y) z) t)))
   (if (<= t_1 -5e-33)
     (* (* (* y -9.0) z) t)
     (if (<= t_1 5e+16) (* (* a 27.0) b) (* (* y -9.0) (* z t))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((9.0 * y) * z) * t;
	double tmp;
	if (t_1 <= -5e-33) {
		tmp = ((y * -9.0) * z) * t;
	} else if (t_1 <= 5e+16) {
		tmp = (a * 27.0) * b;
	} else {
		tmp = (y * -9.0) * (z * t);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((9.0d0 * y) * z) * t
    if (t_1 <= (-5d-33)) then
        tmp = ((y * (-9.0d0)) * z) * t
    else if (t_1 <= 5d+16) then
        tmp = (a * 27.0d0) * b
    else
        tmp = (y * (-9.0d0)) * (z * t)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b;
public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((9.0 * y) * z) * t;
	double tmp;
	if (t_1 <= -5e-33) {
		tmp = ((y * -9.0) * z) * t;
	} else if (t_1 <= 5e+16) {
		tmp = (a * 27.0) * b;
	} else {
		tmp = (y * -9.0) * (z * t);
	}
	return tmp;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	t_1 = ((9.0 * y) * z) * t
	tmp = 0
	if t_1 <= -5e-33:
		tmp = ((y * -9.0) * z) * t
	elif t_1 <= 5e+16:
		tmp = (a * 27.0) * b
	else:
		tmp = (y * -9.0) * (z * t)
	return tmp

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(9.0 * y) * z) * t)
	tmp = 0.0
	if (t_1 <= -5e-33)
		tmp = Float64(Float64(Float64(y * -9.0) * z) * t);
	elseif (t_1 <= 5e+16)
		tmp = Float64(Float64(a * 27.0) * b);
	else
		tmp = Float64(Float64(y * -9.0) * Float64(z * t));
	end
	return tmp
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp_2 = code(x, y, z, t, a, b)
	t_1 = ((9.0 * y) * z) * t;
	tmp = 0.0;
	if (t_1 <= -5e-33)
		tmp = ((y * -9.0) * z) * t;
	elseif (t_1 <= 5e+16)
		tmp = (a * 27.0) * b;
	else
		tmp = (y * -9.0) * (z * t);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(9.0 * y), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-33], N[(N[(N[(y * -9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 5e+16], N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision], N[(N[(y * -9.0), $MachinePrecision] * N[(z * t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -5.00000000000000028e-33

   1. Initial program 92.5%

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

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

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

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 60.5%

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

   10. Applied rewrites 60.4%

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

   if -5.00000000000000028e-33 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 5e16

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 95.5

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

   5. Applied rewrites 95.5%

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

   7. Applied rewrites 95.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 52.2%

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

   12. Applied rewrites 52.2%

       \[\leadsto \left(27 \cdot a\right) \cdot b \]

   if 5e16 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 92.3%

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

   3. Taylor expanded in 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. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 84.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 71.4%

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

   10. Applied rewrites 67.2%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(9 \cdot y\right) \cdot z\right) \cdot t \leq -5 \cdot 10^{-33}:\\ \;\;\;\;\left(\left(y \cdot -9\right) \cdot z\right) \cdot t\\ \mathbf{elif}\;\left(\left(9 \cdot y\right) \cdot z\right) \cdot t \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\left(a \cdot 27\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -9\right) \cdot \left(z \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot -9\right) \cdot z\right) \cdot t\\ t_2 := \left(\left(9 \cdot y\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\left(a \cdot 27\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* y -9.0) z) t)) (t_2 (* (* (* 9.0 y) z) t)))
   (if (<= t_2 -5e-33) t_1 (if (<= t_2 5e+16) (* (* a 27.0) b) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((y * -9.0) * z) * t;
	double t_2 = ((9.0 * y) * z) * t;
	double tmp;
	if (t_2 <= -5e-33) {
		tmp = t_1;
	} else if (t_2 <= 5e+16) {
		tmp = (a * 27.0) * b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((y * (-9.0d0)) * z) * t
    t_2 = ((9.0d0 * y) * z) * t
    if (t_2 <= (-5d-33)) then
        tmp = t_1
    else if (t_2 <= 5d+16) then
        tmp = (a * 27.0d0) * b
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b;
public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((y * -9.0) * z) * t;
	double t_2 = ((9.0 * y) * z) * t;
	double tmp;
	if (t_2 <= -5e-33) {
		tmp = t_1;
	} else if (t_2 <= 5e+16) {
		tmp = (a * 27.0) * b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	t_1 = ((y * -9.0) * z) * t
	t_2 = ((9.0 * y) * z) * t
	tmp = 0
	if t_2 <= -5e-33:
		tmp = t_1
	elif t_2 <= 5e+16:
		tmp = (a * 27.0) * b
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(y * -9.0) * z) * t)
	t_2 = Float64(Float64(Float64(9.0 * y) * z) * t)
	tmp = 0.0
	if (t_2 <= -5e-33)
		tmp = t_1;
	elseif (t_2 <= 5e+16)
		tmp = Float64(Float64(a * 27.0) * b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp_2 = code(x, y, z, t, a, b)
	t_1 = ((y * -9.0) * z) * t;
	t_2 = ((9.0 * y) * z) * t;
	tmp = 0.0;
	if (t_2 <= -5e-33)
		tmp = t_1;
	elseif (t_2 <= 5e+16)
		tmp = (a * 27.0) * b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y * -9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(9.0 * y), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-33], t$95$1, If[LessEqual[t$95$2, 5e+16], N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -5.00000000000000028e-33 or 5e16 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 92.4%

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

   3. Taylor expanded in 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. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 65.9%

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

   10. Applied rewrites 65.9%

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

   if -5.00000000000000028e-33 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 5e16

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 95.5

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

   5. Applied rewrites 95.5%

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

   7. Applied rewrites 95.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 52.2%

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

   12. Applied rewrites 52.2%

       \[\leadsto \left(27 \cdot a\right) \cdot b \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(9 \cdot y\right) \cdot z\right) \cdot t \leq -5 \cdot 10^{-33}:\\ \;\;\;\;\left(\left(y \cdot -9\right) \cdot z\right) \cdot t\\ \mathbf{elif}\;\left(\left(9 \cdot y\right) \cdot z\right) \cdot t \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\left(a \cdot 27\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot -9\right) \cdot z\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;\left(9 \cdot y\right) \cdot z \leq 10^{+259}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot z\right) \cdot -9, t, \mathsf{fma}\left(27 \cdot b, a, x \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-9 \cdot \left(z \cdot t\right), y, x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* (* 9.0 y) z) 1e+259)
   (fma (* (* y z) -9.0) t (fma (* 27.0 b) a (* x 2.0)))
   (fma (* -9.0 (* z t)) y (* x 2.0))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((9.0 * y) * z) <= 1e+259) {
		tmp = fma(((y * z) * -9.0), t, fma((27.0 * b), a, (x * 2.0)));
	} else {
		tmp = fma((-9.0 * (z * t)), y, (x * 2.0));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(9.0 * y) * z) <= 1e+259)
		tmp = fma(Float64(Float64(y * z) * -9.0), t, fma(Float64(27.0 * b), a, Float64(x * 2.0)));
	else
		tmp = fma(Float64(-9.0 * Float64(z * t)), y, Float64(x * 2.0));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(9.0 * y), $MachinePrecision] * z), $MachinePrecision], 1e+259], N[(N[(N[(y * z), $MachinePrecision] * -9.0), $MachinePrecision] * t + N[(N[(27.0 * b), $MachinePrecision] * a + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-9.0 * N[(z * t), $MachinePrecision]), $MachinePrecision] * y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y #s(literal 9 binary64)) z) < 9.999999999999999e258

   1. Initial program 98.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

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

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

   4. Applied rewrites 98.5%

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

   if 9.999999999999999e258 < (*.f64 (*.f64 y #s(literal 9 binary64)) z)

   1. Initial program 71.1%

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

   3. Taylor expanded in 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. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in a around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 90.0

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

   8. Applied rewrites 90.0%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(9 \cdot y\right) \cdot z \leq 10^{+259}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot z\right) \cdot -9, t, \mathsf{fma}\left(27 \cdot b, a, x \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-9 \cdot \left(z \cdot t\right), y, x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 34.9% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (a * 27.0d0) * b
end function

Java

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

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	return (a * 27.0) * b

Julia

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

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp = code(x, y, z, t, a, b)
	tmp = (a * 27.0) * b;
end

Wolfram

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

Derivation

1. Initial program 96.0%

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

3. Taylor expanded in y around 0

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

   1. lower-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 65.2

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

5. Applied rewrites 65.2%

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

7. Applied rewrites 65.2%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 35.5%

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

12. Applied rewrites 35.5%

    \[\leadsto \left(27 \cdot a\right) \cdot b \]

13. Final simplification 35.5%

    \[\leadsto \left(a \cdot 27\right) \cdot b \]
14. Add Preprocessing

Developer Target 1: 95.4% 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 2024332 
(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)))