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

Percentage Accurate: 95.8% → 98.8%

Time: 11.9s

Alternatives: 16

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: 95.8% 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: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 2.1 \cdot 10^{-116}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot y, -9 \cdot z, \mathsf{fma}\left(a \cdot 27, b, x\right) + x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(x - \mathsf{fma}\left(-27 \cdot a, b, t \cdot \left(z \cdot \left(9 \cdot y\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (if (<= t 2.1e-116)
   (fma (* t y) (* -9.0 z) (+ (fma (* a 27.0) b x) x))
   (+ x (- x (fma (* -27.0 a) b (* t (* z (* 9.0 y))))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 2.1e-116) {
		tmp = fma((t * y), (-9.0 * z), (fma((a * 27.0), b, x) + x));
	} else {
		tmp = x + (x - fma((-27.0 * a), b, (t * (z * (9.0 * y)))));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= 2.1e-116)
		tmp = fma(Float64(t * y), Float64(-9.0 * z), Float64(fma(Float64(a * 27.0), b, x) + x));
	else
		tmp = Float64(x + Float64(x - fma(Float64(-27.0 * a), b, Float64(t * Float64(z * Float64(9.0 * y))))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, 2.1e-116], N[(N[(t * y), $MachinePrecision] * N[(-9.0 * z), $MachinePrecision] + N[(N[(N[(a * 27.0), $MachinePrecision] * b + x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(x + N[(x - N[(N[(-27.0 * a), $MachinePrecision] * b + N[(t * N[(z * N[(9.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.0999999999999999e-116

   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. lift-*.f64 N/A

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

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

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

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

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

      12. associate-*r* N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 96.2%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 96.2

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

   6. Applied rewrites 96.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+r+ N/A

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

      4. lower-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 96.8

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 96.8

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

   8. Applied rewrites 96.8%

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

   if 2.0999999999999999e-116 < t

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. count-2-rev N/A

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

      7. associate--l+ N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval 99.8

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 99.8

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 99.8

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

      25. lift-*.f64 N/A

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

      26. *-commutative N/A

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

      27. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

Alternative 2: 82.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+82} \lor \neg \left(t\_1 \leq 4 \cdot 10^{-110}\right):\\ \;\;\;\;x + \mathsf{fma}\left(-9, \left(y \cdot t\right) \cdot z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(27 \cdot b, a, x\right) + x\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* y 9.0) z) t)))
   (if (or (<= t_1 -1e+82) (not (<= t_1 4e-110)))
     (+ x (fma -9.0 (* (* y t) z) x))
     (+ (fma (* 27.0 b) a x) x))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((y * 9.0) * z) * t;
	double tmp;
	if ((t_1 <= -1e+82) || !(t_1 <= 4e-110)) {
		tmp = x + fma(-9.0, ((y * t) * z), x);
	} else {
		tmp = fma((27.0 * b), a, x) + x;
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if ((t_1 <= -1e+82) || !(t_1 <= 4e-110))
		tmp = Float64(x + fma(-9.0, Float64(Float64(y * t) * z), x));
	else
		tmp = Float64(fma(Float64(27.0 * b), a, x) + x);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e+82], N[Not[LessEqual[t$95$1, 4e-110]], $MachinePrecision]], N[(x + N[(-9.0 * N[(N[(y * t), $MachinePrecision] * z), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(27.0 * b), $MachinePrecision] * a + x), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -9.9999999999999996e81 or 4.0000000000000002e-110 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 95.4%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. count-2-rev N/A

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

      7. associate--l+ N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval 96.9

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 96.9

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 96.9

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

      25. lift-*.f64 N/A

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

      26. *-commutative N/A

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

      27. lower-*.f64 96.9

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

   4. Applied rewrites 96.9%

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

   5. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 85.4

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

   7. Applied rewrites 85.4%

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

   9. Applied rewrites 80.1%

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

   if -9.9999999999999996e81 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 4.0000000000000002e-110

   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 94.9

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

   5. Applied rewrites 94.9%

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

   7. Applied rewrites 94.9%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \leq -1 \cdot 10^{+82} \lor \neg \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \leq 4 \cdot 10^{-110}\right):\\ \;\;\;\;x + \mathsf{fma}\left(-9, \left(y \cdot t\right) \cdot z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(27 \cdot b, a, x\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(b \cdot a\right) \cdot 27\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-110}:\\ \;\;\;\;\mathsf{fma}\left(27 \cdot b, a, x\right) + x\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(-9, \left(y \cdot t\right) \cdot z, x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* y 9.0) z) t)))
   (if (<= t_1 -5e+33)
     (fma -9.0 (* (* z y) t) (* (* b a) 27.0))
     (if (<= t_1 4e-110)
       (+ (fma (* 27.0 b) a x) x)
       (+ x (fma -9.0 (* (* y t) z) x))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((y * 9.0) * z) * t;
	double tmp;
	if (t_1 <= -5e+33) {
		tmp = fma(-9.0, ((z * y) * t), ((b * a) * 27.0));
	} else if (t_1 <= 4e-110) {
		tmp = fma((27.0 * b), a, x) + x;
	} else {
		tmp = x + fma(-9.0, ((y * t) * z), x);
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if (t_1 <= -5e+33)
		tmp = fma(-9.0, Float64(Float64(z * y) * t), Float64(Float64(b * a) * 27.0));
	elseif (t_1 <= 4e-110)
		tmp = Float64(fma(Float64(27.0 * b), a, x) + x);
	else
		tmp = Float64(x + fma(-9.0, Float64(Float64(y * t) * z), x));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+33], N[(-9.0 * N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] + N[(N[(b * a), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e-110], N[(N[(N[(27.0 * b), $MachinePrecision] * a + x), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(-9.0 * N[(N[(y * t), $MachinePrecision] * z), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -4.99999999999999973e33

   1. Initial program 96.9%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 84.8

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

   5. Applied rewrites 84.8%

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

   if -4.99999999999999973e33 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 4.0000000000000002e-110

   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 96.5

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

   5. Applied rewrites 96.5%

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

   7. Applied rewrites 96.5%

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

   if 4.0000000000000002e-110 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 94.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. count-2-rev N/A

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

      7. associate--l+ N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval 97.2

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 97.2

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 97.2

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

      25. lift-*.f64 N/A

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

      26. *-commutative N/A

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

      27. lower-*.f64 97.2

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

   4. Applied rewrites 97.2%

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

   5. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 82.0

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

   7. Applied rewrites 82.0%

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

   9. Applied rewrites 78.6%

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

Alternative 4: 83.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+82}:\\ \;\;\;\;x + \mathsf{fma}\left(-9, \left(y \cdot z\right) \cdot t, x\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-110}:\\ \;\;\;\;\mathsf{fma}\left(27 \cdot b, a, x\right) + x\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(-9, \left(y \cdot t\right) \cdot z, x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* y 9.0) z) t)))
   (if (<= t_1 -1e+82)
     (+ x (fma -9.0 (* (* y z) t) x))
     (if (<= t_1 4e-110)
       (+ (fma (* 27.0 b) a x) x)
       (+ x (fma -9.0 (* (* y t) z) x))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((y * 9.0) * z) * t;
	double tmp;
	if (t_1 <= -1e+82) {
		tmp = x + fma(-9.0, ((y * z) * t), x);
	} else if (t_1 <= 4e-110) {
		tmp = fma((27.0 * b), a, x) + x;
	} else {
		tmp = x + fma(-9.0, ((y * t) * z), x);
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if (t_1 <= -1e+82)
		tmp = Float64(x + fma(-9.0, Float64(Float64(y * z) * t), x));
	elseif (t_1 <= 4e-110)
		tmp = Float64(fma(Float64(27.0 * b), a, x) + x);
	else
		tmp = Float64(x + fma(-9.0, Float64(Float64(y * t) * z), x));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+82], N[(x + N[(-9.0 * N[(N[(y * z), $MachinePrecision] * t), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e-110], N[(N[(N[(27.0 * b), $MachinePrecision] * a + x), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(-9.0 * N[(N[(y * t), $MachinePrecision] * z), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -9.9999999999999996e81

   1. Initial program 96.5%

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

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. count-2-rev N/A

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

      7. associate--l+ N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval 96.5

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 96.5

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 96.5

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

      25. lift-*.f64 N/A

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

      26. *-commutative N/A

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

      27. lower-*.f64 96.5

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

   4. Applied rewrites 96.5%

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

   5. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 90.0

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

   7. Applied rewrites 90.0%

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

   if -9.9999999999999996e81 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 4.0000000000000002e-110

   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 94.9

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

   5. Applied rewrites 94.9%

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

   7. Applied rewrites 94.9%

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

   if 4.0000000000000002e-110 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 94.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. count-2-rev N/A

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

      7. associate--l+ N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval 97.2

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 97.2

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 97.2

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

      25. lift-*.f64 N/A

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

      26. *-commutative N/A

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

      27. lower-*.f64 97.2

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

   4. Applied rewrites 97.2%

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

   5. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 82.0

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

   7. Applied rewrites 82.0%

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

   9. Applied rewrites 78.6%

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

Alternative 5: 83.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+116} \lor \neg \left(t\_1 \leq 4 \cdot 10^{+123}\right):\\ \;\;\;\;x + \left(\left(z \cdot y\right) \cdot t\right) \cdot -9\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(b \cdot a\right) \cdot 27\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* y 9.0) z) t)))
   (if (or (<= t_1 -2e+116) (not (<= t_1 4e+123)))
     (+ x (* (* (* z y) t) -9.0))
     (fma 2.0 x (* (* b a) 27.0)))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((y * 9.0) * z) * t;
	double tmp;
	if ((t_1 <= -2e+116) || !(t_1 <= 4e+123)) {
		tmp = x + (((z * y) * t) * -9.0);
	} else {
		tmp = fma(2.0, x, ((b * a) * 27.0));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if ((t_1 <= -2e+116) || !(t_1 <= 4e+123))
		tmp = Float64(x + Float64(Float64(Float64(z * y) * t) * -9.0));
	else
		tmp = fma(2.0, x, Float64(Float64(b * a) * 27.0));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+116], N[Not[LessEqual[t$95$1, 4e+123]], $MachinePrecision]], N[(x + N[(N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] * -9.0), $MachinePrecision]), $MachinePrecision], N[(2.0 * x + N[(N[(b * a), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -2.00000000000000003e116 or 3.99999999999999991e123 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 93.9%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. count-2-rev N/A

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

      7. associate--l+ N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval 95.9

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 95.9

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 95.9

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

      25. lift-*.f64 N/A

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

      26. *-commutative N/A

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

      27. lower-*.f64 95.9

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

   4. Applied rewrites 95.9%

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

   5. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 90.2

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

   7. Applied rewrites 90.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 83.6%

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

   if -2.00000000000000003e116 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 3.99999999999999991e123

   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 86.5

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

   5. Applied rewrites 86.5%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \leq -2 \cdot 10^{+116} \lor \neg \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \leq 4 \cdot 10^{+123}\right):\\ \;\;\;\;x + \left(\left(z \cdot y\right) \cdot t\right) \cdot -9\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(b \cdot a\right) \cdot 27\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+145} \lor \neg \left(t\_1 \leq 4 \cdot 10^{+123}\right):\\ \;\;\;\;\left(\left(z \cdot y\right) \cdot t\right) \cdot -9\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(b \cdot a\right) \cdot 27\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* y 9.0) z) t)))
   (if (or (<= t_1 -5e+145) (not (<= t_1 4e+123)))
     (* (* (* z y) t) -9.0)
     (fma 2.0 x (* (* b a) 27.0)))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((y * 9.0) * z) * t;
	double tmp;
	if ((t_1 <= -5e+145) || !(t_1 <= 4e+123)) {
		tmp = ((z * y) * t) * -9.0;
	} else {
		tmp = fma(2.0, x, ((b * a) * 27.0));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if ((t_1 <= -5e+145) || !(t_1 <= 4e+123))
		tmp = Float64(Float64(Float64(z * y) * t) * -9.0);
	else
		tmp = fma(2.0, x, Float64(Float64(b * a) * 27.0));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e+145], N[Not[LessEqual[t$95$1, 4e+123]], $MachinePrecision]], N[(N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] * -9.0), $MachinePrecision], N[(2.0 * x + N[(N[(b * a), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -4.99999999999999967e145 or 3.99999999999999991e123 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 93.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

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

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

      12. associate-*r* N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 89.9%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 89.9

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

   6. Applied rewrites 89.9%

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

   7. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 83.7

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

   9. Applied rewrites 83.7%

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

   if -4.99999999999999967e145 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 3.99999999999999991e123

   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 85.6

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

   5. Applied rewrites 85.6%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \leq -5 \cdot 10^{+145} \lor \neg \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \leq 4 \cdot 10^{+123}\right):\\ \;\;\;\;\left(\left(z \cdot y\right) \cdot t\right) \cdot -9\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(b \cdot a\right) \cdot 27\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(y \cdot 9\right) \cdot z\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-42}:\\ \;\;\;\;\left(\left(-9 \cdot z\right) \cdot y\right) \cdot t + \left(a \cdot 27\right) \cdot b\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(27 \cdot b, a, x\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(27 \cdot a, b, \left(\left(-9 \cdot y\right) \cdot t\right) \cdot z\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* y 9.0) z)))
   (if (<= t_1 -4e-42)
     (+ (* (* (* -9.0 z) y) t) (* (* a 27.0) b))
     (if (<= t_1 2e-23)
       (+ (fma (* 27.0 b) a x) x)
       (fma (* 27.0 a) b (* (* (* -9.0 y) t) z))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * 9.0) * z;
	double tmp;
	if (t_1 <= -4e-42) {
		tmp = (((-9.0 * z) * y) * t) + ((a * 27.0) * b);
	} else if (t_1 <= 2e-23) {
		tmp = fma((27.0 * b), a, x) + x;
	} else {
		tmp = fma((27.0 * a), b, (((-9.0 * y) * t) * z));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * 9.0) * z)
	tmp = 0.0
	if (t_1 <= -4e-42)
		tmp = Float64(Float64(Float64(Float64(-9.0 * z) * y) * t) + Float64(Float64(a * 27.0) * b));
	elseif (t_1 <= 2e-23)
		tmp = Float64(fma(Float64(27.0 * b), a, x) + x);
	else
		tmp = fma(Float64(27.0 * a), b, Float64(Float64(Float64(-9.0 * y) * t) * z));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-42], N[(N[(N[(N[(-9.0 * z), $MachinePrecision] * y), $MachinePrecision] * t), $MachinePrecision] + N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-23], N[(N[(N[(27.0 * b), $MachinePrecision] * a + x), $MachinePrecision] + x), $MachinePrecision], N[(N[(27.0 * a), $MachinePrecision] * b + N[(N[(N[(-9.0 * y), $MachinePrecision] * t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y #s(literal 9 binary64)) z) < -4.00000000000000015e-42

   1. Initial program 97.2%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 74.5

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

   5. Applied rewrites 74.5%

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

   7. Applied rewrites 74.5%

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

   if -4.00000000000000015e-42 < (*.f64 (*.f64 y #s(literal 9 binary64)) z) < 1.99999999999999992e-23

   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 89.9

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

   5. Applied rewrites 89.9%

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

   7. Applied rewrites 89.9%

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

   if 1.99999999999999992e-23 < (*.f64 (*.f64 y #s(literal 9 binary64)) z)

   1. Initial program 94.2%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 82.9

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

   5. Applied rewrites 82.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 82.9

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 82.9

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

   7. Applied rewrites 80.4%

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

Alternative 8: 83.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(y \cdot 9\right) \cdot z\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-42}:\\ \;\;\;\;\left(-9 \cdot \left(z \cdot y\right)\right) \cdot t + \left(a \cdot 27\right) \cdot b\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(27 \cdot b, a, x\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(27 \cdot a, b, \left(\left(-9 \cdot y\right) \cdot t\right) \cdot z\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* y 9.0) z)))
   (if (<= t_1 -4e-42)
     (+ (* (* -9.0 (* z y)) t) (* (* a 27.0) b))
     (if (<= t_1 2e-23)
       (+ (fma (* 27.0 b) a x) x)
       (fma (* 27.0 a) b (* (* (* -9.0 y) t) z))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * 9.0) * z;
	double tmp;
	if (t_1 <= -4e-42) {
		tmp = ((-9.0 * (z * y)) * t) + ((a * 27.0) * b);
	} else if (t_1 <= 2e-23) {
		tmp = fma((27.0 * b), a, x) + x;
	} else {
		tmp = fma((27.0 * a), b, (((-9.0 * y) * t) * z));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * 9.0) * z)
	tmp = 0.0
	if (t_1 <= -4e-42)
		tmp = Float64(Float64(Float64(-9.0 * Float64(z * y)) * t) + Float64(Float64(a * 27.0) * b));
	elseif (t_1 <= 2e-23)
		tmp = Float64(fma(Float64(27.0 * b), a, x) + x);
	else
		tmp = fma(Float64(27.0 * a), b, Float64(Float64(Float64(-9.0 * y) * t) * z));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-42], N[(N[(N[(-9.0 * N[(z * y), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] + N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-23], N[(N[(N[(27.0 * b), $MachinePrecision] * a + x), $MachinePrecision] + x), $MachinePrecision], N[(N[(27.0 * a), $MachinePrecision] * b + N[(N[(N[(-9.0 * y), $MachinePrecision] * t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y #s(literal 9 binary64)) z) < -4.00000000000000015e-42

   1. Initial program 97.2%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 74.5

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

   5. Applied rewrites 74.5%

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

   if -4.00000000000000015e-42 < (*.f64 (*.f64 y #s(literal 9 binary64)) z) < 1.99999999999999992e-23

   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 89.9

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

   5. Applied rewrites 89.9%

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

   7. Applied rewrites 89.9%

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

   if 1.99999999999999992e-23 < (*.f64 (*.f64 y #s(literal 9 binary64)) z)

   1. Initial program 94.2%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 82.9

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

   5. Applied rewrites 82.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 82.9

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 82.9

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

   7. Applied rewrites 80.4%

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

Alternative 9: 83.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(y \cdot 9\right) \cdot z\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-42}:\\ \;\;\;\;\mathsf{fma}\left(-9, \left(z \cdot y\right) \cdot t, \left(b \cdot a\right) \cdot 27\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(27 \cdot b, a, x\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(27 \cdot a, b, \left(\left(-9 \cdot y\right) \cdot t\right) \cdot z\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* y 9.0) z)))
   (if (<= t_1 -4e-42)
     (fma -9.0 (* (* z y) t) (* (* b a) 27.0))
     (if (<= t_1 2e-23)
       (+ (fma (* 27.0 b) a x) x)
       (fma (* 27.0 a) b (* (* (* -9.0 y) t) z))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * 9.0) * z;
	double tmp;
	if (t_1 <= -4e-42) {
		tmp = fma(-9.0, ((z * y) * t), ((b * a) * 27.0));
	} else if (t_1 <= 2e-23) {
		tmp = fma((27.0 * b), a, x) + x;
	} else {
		tmp = fma((27.0 * a), b, (((-9.0 * y) * t) * z));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * 9.0) * z)
	tmp = 0.0
	if (t_1 <= -4e-42)
		tmp = fma(-9.0, Float64(Float64(z * y) * t), Float64(Float64(b * a) * 27.0));
	elseif (t_1 <= 2e-23)
		tmp = Float64(fma(Float64(27.0 * b), a, x) + x);
	else
		tmp = fma(Float64(27.0 * a), b, Float64(Float64(Float64(-9.0 * y) * t) * z));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-42], N[(-9.0 * N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] + N[(N[(b * a), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-23], N[(N[(N[(27.0 * b), $MachinePrecision] * a + x), $MachinePrecision] + x), $MachinePrecision], N[(N[(27.0 * a), $MachinePrecision] * b + N[(N[(N[(-9.0 * y), $MachinePrecision] * t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y #s(literal 9 binary64)) z) < -4.00000000000000015e-42

   1. Initial program 97.2%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 74.5

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

   5. Applied rewrites 74.5%

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

   if -4.00000000000000015e-42 < (*.f64 (*.f64 y #s(literal 9 binary64)) z) < 1.99999999999999992e-23

   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 89.9

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

   5. Applied rewrites 89.9%

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

   7. Applied rewrites 89.9%

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

   if 1.99999999999999992e-23 < (*.f64 (*.f64 y #s(literal 9 binary64)) z)

   1. Initial program 94.2%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 82.9

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

   5. Applied rewrites 82.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 82.9

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 82.9

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

   7. Applied rewrites 80.4%

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

Alternative 10: 82.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+145} \lor \neg \left(t\_1 \leq 4 \cdot 10^{+123}\right):\\ \;\;\;\;\left(\left(z \cdot y\right) \cdot t\right) \cdot -9\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(27 \cdot a, b, x + x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* y 9.0) z) t)))
   (if (or (<= t_1 -5e+145) (not (<= t_1 4e+123)))
     (* (* (* z y) t) -9.0)
     (fma (* 27.0 a) b (+ x x)))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((y * 9.0) * z) * t;
	double tmp;
	if ((t_1 <= -5e+145) || !(t_1 <= 4e+123)) {
		tmp = ((z * y) * t) * -9.0;
	} else {
		tmp = fma((27.0 * a), b, (x + x));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(y * 9.0) * z) * t)
	tmp = 0.0
	if ((t_1 <= -5e+145) || !(t_1 <= 4e+123))
		tmp = Float64(Float64(Float64(z * y) * t) * -9.0);
	else
		tmp = fma(Float64(27.0 * a), b, Float64(x + x));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e+145], N[Not[LessEqual[t$95$1, 4e+123]], $MachinePrecision]], N[(N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] * -9.0), $MachinePrecision], N[(N[(27.0 * a), $MachinePrecision] * b + N[(x + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -4.99999999999999967e145 or 3.99999999999999991e123 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 93.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

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

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

      12. associate-*r* N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 89.9%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 89.9

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

   6. Applied rewrites 89.9%

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

   7. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 83.7

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

   9. Applied rewrites 83.7%

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

   if -4.99999999999999967e145 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 3.99999999999999991e123

   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 85.6

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

   5. Applied rewrites 85.6%

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

   7. Applied rewrites 85.6%

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

   9. Applied rewrites 85.6%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \leq -5 \cdot 10^{+145} \lor \neg \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \leq 4 \cdot 10^{+123}\right):\\ \;\;\;\;\left(\left(z \cdot y\right) \cdot t\right) \cdot -9\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(27 \cdot a, b, x + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 92.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	return fma(Float64(t * y), Float64(-9.0 * z), Float64(fma(Float64(a * 27.0), b, x) + x))
end

Wolfram

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

Derivation

1. Initial program 97.5%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

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

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

   5. +-commutative N/A

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

   6. associate-+l+ N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*l* N/A

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

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

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

   12. associate-*r* N/A

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

   13. +-commutative N/A

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

   14. lower-fma.f64 N/A

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

4. Applied rewrites 94.3%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 94.3

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

6. Applied rewrites 94.3%

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

   1. lift-fma.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-+r+ N/A

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

   4. lower-+.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 94.6

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 94.6

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

8. Applied rewrites 94.6%

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

Alternative 12: 47.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+26} \lor \neg \left(x \leq 1.9 \cdot 10^{+76}\right):\\ \;\;\;\;2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot 27\right) \cdot b\\ \end{array} \end{array} \]

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 (or (<= x -7.2e+26) (not (<= x 1.9e+76))) (* 2.0 x) (* (* 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) {
	double tmp;
	if ((x <= -7.2e+26) || !(x <= 1.9e+76)) {
		tmp = 2.0 * x;
	} else {
		tmp = (a * 27.0) * b;
	}
	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) :: tmp
    if ((x <= (-7.2d+26)) .or. (.not. (x <= 1.9d+76))) then
        tmp = 2.0d0 * x
    else
        tmp = (a * 27.0d0) * b
    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 tmp;
	if ((x <= -7.2e+26) || !(x <= 1.9e+76)) {
		tmp = 2.0 * x;
	} else {
		tmp = (a * 27.0) * b;
	}
	return tmp;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	tmp = 0
	if (x <= -7.2e+26) or not (x <= 1.9e+76):
		tmp = 2.0 * x
	else:
		tmp = (a * 27.0) * b
	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 ((x <= -7.2e+26) || !(x <= 1.9e+76))
		tmp = Float64(2.0 * x);
	else
		tmp = Float64(Float64(a * 27.0) * b);
	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)
	tmp = 0.0;
	if ((x <= -7.2e+26) || ~((x <= 1.9e+76)))
		tmp = 2.0 * x;
	else
		tmp = (a * 27.0) * b;
	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_] := If[Or[LessEqual[x, -7.2e+26], N[Not[LessEqual[x, 1.9e+76]], $MachinePrecision]], N[(2.0 * x), $MachinePrecision], N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.20000000000000048e26 or 1.90000000000000012e76 < x

   1. Initial program 95.5%

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

   3. Taylor expanded in y around 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 77.5

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

   5. Applied rewrites 77.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 77.5%

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

   9. Taylor expanded in x around inf

      \[\leadsto 2 \cdot x \]
   10. Step-by-step derivation

   11. Applied rewrites 58.8%

       \[\leadsto 2 \cdot x \]

   if -7.20000000000000048e26 < x < 1.90000000000000012e76

   1. Initial program 99.0%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   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 49.1

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

   5. Applied rewrites 49.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 43.0%

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

   10. Applied rewrites 43.0%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+26} \lor \neg \left(x \leq 1.9 \cdot 10^{+76}\right):\\ \;\;\;\;2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot 27\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 47.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+26} \lor \neg \left(x \leq 1.9 \cdot 10^{+76}\right):\\ \;\;\;\;2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(27 \cdot b\right) \cdot a\\ \end{array} \end{array} \]

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 (or (<= x -7.2e+26) (not (<= x 1.9e+76))) (* 2.0 x) (* (* 27.0 b) a)))

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 ((x <= -7.2e+26) || !(x <= 1.9e+76)) {
		tmp = 2.0 * x;
	} else {
		tmp = (27.0 * b) * a;
	}
	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) :: tmp
    if ((x <= (-7.2d+26)) .or. (.not. (x <= 1.9d+76))) then
        tmp = 2.0d0 * x
    else
        tmp = (27.0d0 * b) * a
    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 tmp;
	if ((x <= -7.2e+26) || !(x <= 1.9e+76)) {
		tmp = 2.0 * x;
	} else {
		tmp = (27.0 * b) * a;
	}
	return tmp;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	tmp = 0
	if (x <= -7.2e+26) or not (x <= 1.9e+76):
		tmp = 2.0 * x
	else:
		tmp = (27.0 * b) * a
	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 ((x <= -7.2e+26) || !(x <= 1.9e+76))
		tmp = Float64(2.0 * x);
	else
		tmp = Float64(Float64(27.0 * b) * a);
	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)
	tmp = 0.0;
	if ((x <= -7.2e+26) || ~((x <= 1.9e+76)))
		tmp = 2.0 * x;
	else
		tmp = (27.0 * b) * a;
	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_] := If[Or[LessEqual[x, -7.2e+26], N[Not[LessEqual[x, 1.9e+76]], $MachinePrecision]], N[(2.0 * x), $MachinePrecision], N[(N[(27.0 * b), $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.20000000000000048e26 or 1.90000000000000012e76 < x

   1. Initial program 95.5%

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

   3. Taylor expanded in y around 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 77.5

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

   5. Applied rewrites 77.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 77.5%

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

   9. Taylor expanded in x around inf

      \[\leadsto 2 \cdot x \]
   10. Step-by-step derivation

   11. Applied rewrites 58.8%

       \[\leadsto 2 \cdot x \]

   if -7.20000000000000048e26 < x < 1.90000000000000012e76

   1. Initial program 99.0%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
   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 49.1

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

   5. Applied rewrites 49.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 43.0%

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

   10. Applied rewrites 42.4%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+26} \lor \neg \left(x \leq 1.9 \cdot 10^{+76}\right):\\ \;\;\;\;2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(27 \cdot b\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 63.5% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.5%

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

3. 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 61.2

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

5. Applied rewrites 61.2%

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

7. Applied rewrites 61.2%

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

9. Applied rewrites 61.2%

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

Alternative 15: 63.5% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.5%

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

3. 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 61.2

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

5. Applied rewrites 61.2%

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

7. Applied rewrites 60.9%

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

Alternative 16: 30.3% accurate, 6.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.5%

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

3. 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 61.2

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

5. Applied rewrites 61.2%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 58.8%

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

9. Taylor expanded in x around inf

   \[\leadsto 2 \cdot x \]
10. Step-by-step derivation

11. Applied rewrites 29.6%

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

Developer Target 1: 95.3% 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 2024337 
(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)))