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

Percentage Accurate: 85.4% → 91.8%

Time: 31.7s

Alternatives: 23

Speedup: 0.5×

• 51.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \end{array} \]

FPCore

(FPCore (x y z t a b c i j k)
 :precision binary64
 (-
  (-
   (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c))
   (* (* x 4.0) i))
  (* (* j 27.0) k)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	return (((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - ((j * 27.0) * k);
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j, k)
    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), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    code = (((((((x * 18.0d0) * y) * z) * t) - ((a * 4.0d0) * t)) + (b * c)) - ((x * 4.0d0) * i)) - ((j * 27.0d0) * k)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	return (((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - ((j * 27.0) * k);
}

Python

def code(x, y, z, t, a, b, c, i, j, k):
	return (((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - ((j * 27.0) * k)

Julia

function code(x, y, z, t, a, b, c, i, j, k)
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * 18.0) * y) * z) * t) - Float64(Float64(a * 4.0) * t)) + Float64(b * c)) - Float64(Float64(x * 4.0) * i)) - Float64(Float64(j * 27.0) * k))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j, k)
	tmp = (((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - ((j * 27.0) * k);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := N[(N[(N[(N[(N[(N[(N[(N[(x * 18.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision] - N[(N[(x * 4.0), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision] - N[(N[(j * 27.0), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]

Initial Program: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \end{array} \]

FPCore

(FPCore (x y z t a b c i j k)
 :precision binary64
 (-
  (-
   (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c))
   (* (* x 4.0) i))
  (* (* j 27.0) k)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	return (((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - ((j * 27.0) * k);
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j, k)
    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), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    code = (((((((x * 18.0d0) * y) * z) * t) - ((a * 4.0d0) * t)) + (b * c)) - ((x * 4.0d0) * i)) - ((j * 27.0d0) * k)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	return (((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - ((j * 27.0) * k);
}

Python

def code(x, y, z, t, a, b, c, i, j, k):
	return (((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - ((j * 27.0) * k)

Julia

function code(x, y, z, t, a, b, c, i, j, k)
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * 18.0) * y) * z) * t) - Float64(Float64(a * 4.0) * t)) + Float64(b * c)) - Float64(Float64(x * 4.0) * i)) - Float64(Float64(j * 27.0) * k))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j, k)
	tmp = (((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - ((j * 27.0) * k);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := N[(N[(N[(N[(N[(N[(N[(N[(x * 18.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision] - N[(N[(x * 4.0), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision] - N[(N[(j * 27.0), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]

Alternative 1: 91.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\\\ [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \left(18 \cdot x\right) \cdot y\\ t_2 := \left(\left(c \cdot b + \left(t \cdot \left(z \cdot t\_1\right) - \left(4 \cdot a\right) \cdot t\right)\right) - i \cdot \left(4 \cdot x\right)\right) - k \cdot \left(27 \cdot j\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-4, i, \left(\left(z \cdot y\right) \cdot t\right) \cdot 18\right), x, \mathsf{fma}\left(-27 \cdot k, j, \left(a \cdot t\right) \cdot -4\right)\right)}{b} + c\right) \cdot b\\ \mathbf{elif}\;t\_2 \leq 8 \cdot 10^{+305}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot j, -27, \mathsf{fma}\left(i \cdot x, -4, \mathsf{fma}\left(\mathsf{fma}\left(z, t\_1, -4 \cdot a\right), t, c \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(\mathsf{fma}\left(\left(t \cdot 18\right) \cdot z, y, -4 \cdot i\right), x, c \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (* (* 18.0 x) y))
        (t_2
         (-
          (- (+ (* c b) (- (* t (* z t_1)) (* (* 4.0 a) t))) (* i (* 4.0 x)))
          (* k (* 27.0 j)))))
   (if (<= t_2 (- INFINITY))
     (*
      (+
       (/
        (fma
         (fma -4.0 i (* (* (* z y) t) 18.0))
         x
         (fma (* -27.0 k) j (* (* a t) -4.0)))
        b)
       c)
      b)
     (if (<= t_2 8e+305)
       (fma
        (* k j)
        -27.0
        (fma (* i x) -4.0 (fma (fma z t_1 (* -4.0 a)) t (* c b))))
       (fma
        (* -27.0 k)
        j
        (fma (fma (* (* t 18.0) z) y (* -4.0 i)) x (* c b)))))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = (18.0 * x) * y;
	double t_2 = (((c * b) + ((t * (z * t_1)) - ((4.0 * a) * t))) - (i * (4.0 * x))) - (k * (27.0 * j));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = ((fma(fma(-4.0, i, (((z * y) * t) * 18.0)), x, fma((-27.0 * k), j, ((a * t) * -4.0))) / b) + c) * b;
	} else if (t_2 <= 8e+305) {
		tmp = fma((k * j), -27.0, fma((i * x), -4.0, fma(fma(z, t_1, (-4.0 * a)), t, (c * b))));
	} else {
		tmp = fma((-27.0 * k), j, fma(fma(((t * 18.0) * z), y, (-4.0 * i)), x, (c * b)));
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = Float64(Float64(18.0 * x) * y)
	t_2 = Float64(Float64(Float64(Float64(c * b) + Float64(Float64(t * Float64(z * t_1)) - Float64(Float64(4.0 * a) * t))) - Float64(i * Float64(4.0 * x))) - Float64(k * Float64(27.0 * j)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(fma(fma(-4.0, i, Float64(Float64(Float64(z * y) * t) * 18.0)), x, fma(Float64(-27.0 * k), j, Float64(Float64(a * t) * -4.0))) / b) + c) * b);
	elseif (t_2 <= 8e+305)
		tmp = fma(Float64(k * j), -27.0, fma(Float64(i * x), -4.0, fma(fma(z, t_1, Float64(-4.0 * a)), t, Float64(c * b))));
	else
		tmp = fma(Float64(-27.0 * k), j, fma(fma(Float64(Float64(t * 18.0) * z), y, Float64(-4.0 * i)), x, Float64(c * b)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(N[(18.0 * x), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(c * b), $MachinePrecision] + N[(N[(t * N[(z * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(i * N[(4.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(k * N[(27.0 * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[(N[(N[(-4.0 * i + N[(N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] * 18.0), $MachinePrecision]), $MachinePrecision] * x + N[(N[(-27.0 * k), $MachinePrecision] * j + N[(N[(a * t), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] + c), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[t$95$2, 8e+305], N[(N[(k * j), $MachinePrecision] * -27.0 + N[(N[(i * x), $MachinePrecision] * -4.0 + N[(N[(z * t$95$1 + N[(-4.0 * a), $MachinePrecision]), $MachinePrecision] * t + N[(c * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-27.0 * k), $MachinePrecision] * j + N[(N[(N[(N[(t * 18.0), $MachinePrecision] * z), $MachinePrecision] * y + N[(-4.0 * i), $MachinePrecision]), $MachinePrecision] * x + N[(c * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x #s(literal 18 binary64)) y) z) t) (*.f64 (*.f64 a #s(literal 4 binary64)) t)) (*.f64 b c)) (*.f64 (*.f64 x #s(literal 4 binary64)) i)) (*.f64 (*.f64 j #s(literal 27 binary64)) k)) < -inf.0

   1. Initial program 91.5%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(\left(c + 18 \cdot \frac{t \cdot \left(x \cdot \left(y \cdot z\right)\right)}{b}\right) - \left(4 \cdot \frac{a \cdot t}{b} + \left(4 \cdot \frac{i \cdot x}{b} + 27 \cdot \frac{j \cdot k}{b}\right)\right)\right)} \]

   5. Applied rewrites 95.0%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-4, i, \left(\left(z \cdot y\right) \cdot t\right) \cdot 18\right), x, \mathsf{fma}\left(-27 \cdot k, j, -4 \cdot \left(a \cdot t\right)\right)\right)}{b} + c\right) \cdot b} \]

   if -inf.0 < (-.f64 (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x #s(literal 18 binary64)) y) z) t) (*.f64 (*.f64 a #s(literal 4 binary64)) t)) (*.f64 b c)) (*.f64 (*.f64 x #s(literal 4 binary64)) i)) (*.f64 (*.f64 j #s(literal 27 binary64)) k)) < 7.9999999999999995e305

   1. Initial program 99.8%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) + \left(\mathsf{neg}\left(\left(j \cdot 27\right) \cdot k\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(j \cdot 27\right) \cdot k\right)\right) + \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(j \cdot 27\right) \cdot k}\right)\right) + \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{k \cdot \left(j \cdot 27\right)}\right)\right) + \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(k \cdot \color{blue}{\left(j \cdot 27\right)}\right)\right) + \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(k \cdot j\right) \cdot 27}\right)\right) + \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) \]

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

         \[\leadsto \color{blue}{\left(k \cdot j\right) \cdot \left(\mathsf{neg}\left(27\right)\right)} + \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(k \cdot j, \mathsf{neg}\left(27\right), \left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{k \cdot j}, \mathsf{neg}\left(27\right), \left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) \]

      11. metadata-eval 99.8

          \[\leadsto \mathsf{fma}\left(k \cdot j, \color{blue}{-27}, \left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) \]

      12. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(k \cdot j, -27, \color{blue}{\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i}\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(k \cdot j, -27, \color{blue}{\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) + \left(\mathsf{neg}\left(\left(x \cdot 4\right) \cdot i\right)\right)}\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(k \cdot j, -27, \color{blue}{\left(\mathsf{neg}\left(\left(x \cdot 4\right) \cdot i\right)\right) + \left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right)}\right) \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k \cdot j, -27, \mathsf{fma}\left(i \cdot x, -4, \mathsf{fma}\left(\mathsf{fma}\left(z, y \cdot \left(18 \cdot x\right), -4 \cdot a\right), t, c \cdot b\right)\right)\right)} \]

   if 7.9999999999999995e305 < (-.f64 (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x #s(literal 18 binary64)) y) z) t) (*.f64 (*.f64 a #s(literal 4 binary64)) t)) (*.f64 b c)) (*.f64 (*.f64 x #s(literal 4 binary64)) i)) (*.f64 (*.f64 j #s(literal 27 binary64)) k))

   1. Initial program 63.0%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

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

         \[\leadsto \color{blue}{\left(b \cdot c + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot t\right)\right)} - \left(j \cdot 27\right) \cdot k \]

      2. metadata-eval N/A

         \[\leadsto \left(b \cdot c + \color{blue}{-4} \cdot \left(a \cdot t\right)\right) - \left(j \cdot 27\right) \cdot k \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(-4 \cdot \left(a \cdot t\right) + b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-4 \cdot a\right) \cdot t} + b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-4 \cdot a}, t, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, \color{blue}{c \cdot b}\right) - \left(j \cdot 27\right) \cdot k \]

      8. lower-*.f64 50.0

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, \color{blue}{c \cdot b}\right) - \left(j \cdot 27\right) \cdot k \]

   5. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)} - \left(j \cdot 27\right) \cdot k \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) - \left(j \cdot 27\right) \cdot k} \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) - \color{blue}{\left(j \cdot 27\right) \cdot k} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \left(\mathsf{neg}\left(j \cdot 27\right)\right) \cdot k} \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \left(\mathsf{neg}\left(\color{blue}{j \cdot 27}\right)\right) \cdot k \]

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

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \color{blue}{\left(j \cdot \left(\mathsf{neg}\left(27\right)\right)\right)} \cdot k \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \left(j \cdot \color{blue}{-27}\right) \cdot k \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \color{blue}{j \cdot \left(-27 \cdot k\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + j \cdot \color{blue}{\left(-27 \cdot k\right)} \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \color{blue}{\left(-27 \cdot k\right) \cdot j} \]

      10. +-commutative N/A

          \[\leadsto \color{blue}{\left(-27 \cdot k\right) \cdot j + \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)} \]

      11. lower-fma.f64 51.2

          \[\leadsto \color{blue}{\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)\right)} \]

   7. Applied rewrites 51.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)\right)} \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 4 \cdot \left(i \cdot x\right)}\right) \]
   9. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + \left(b \cdot c - 4 \cdot \left(i \cdot x\right)\right)}\right) \]

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

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + \color{blue}{\left(b \cdot c + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(i \cdot x\right)\right)}\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + \left(b \cdot c + \color{blue}{-4} \cdot \left(i \cdot x\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + \color{blue}{\left(-4 \cdot \left(i \cdot x\right) + b \cdot c\right)}\right) \]

      5. associate-+r+ N/A

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + -4 \cdot \left(i \cdot x\right)\right) + b \cdot c}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \left(\color{blue}{\left(18 \cdot t\right) \cdot \left(x \cdot \left(y \cdot z\right)\right)} + -4 \cdot \left(i \cdot x\right)\right) + b \cdot c\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \left(\left(18 \cdot t\right) \cdot \color{blue}{\left(\left(y \cdot z\right) \cdot x\right)} + -4 \cdot \left(i \cdot x\right)\right) + b \cdot c\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \left(\color{blue}{\left(\left(18 \cdot t\right) \cdot \left(y \cdot z\right)\right) \cdot x} + -4 \cdot \left(i \cdot x\right)\right) + b \cdot c\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \left(\color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right)} \cdot x + -4 \cdot \left(i \cdot x\right)\right) + b \cdot c\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \left(\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right) \cdot x + \color{blue}{\left(-4 \cdot i\right) \cdot x}\right) + b \cdot c\right) \]

      11. distribute-rgt-in N/A

          \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{x \cdot \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + -4 \cdot i\right)} + b \cdot c\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, x \cdot \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + \color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot i\right) + b \cdot c\right) \]

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

          \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, x \cdot \color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right)} + b \cdot c\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right) \cdot x} + b \cdot c\right) \]

      15. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{\mathsf{fma}\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i, x, b \cdot c\right)}\right) \]

   10. Applied rewrites 94.0%

       \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(18 \cdot t\right) \cdot z, y, -4 \cdot i\right), x, c \cdot b\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(c \cdot b + \left(t \cdot \left(z \cdot \left(\left(18 \cdot x\right) \cdot y\right)\right) - \left(4 \cdot a\right) \cdot t\right)\right) - i \cdot \left(4 \cdot x\right)\right) - k \cdot \left(27 \cdot j\right) \leq -\infty:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-4, i, \left(\left(z \cdot y\right) \cdot t\right) \cdot 18\right), x, \mathsf{fma}\left(-27 \cdot k, j, \left(a \cdot t\right) \cdot -4\right)\right)}{b} + c\right) \cdot b\\ \mathbf{elif}\;\left(\left(c \cdot b + \left(t \cdot \left(z \cdot \left(\left(18 \cdot x\right) \cdot y\right)\right) - \left(4 \cdot a\right) \cdot t\right)\right) - i \cdot \left(4 \cdot x\right)\right) - k \cdot \left(27 \cdot j\right) \leq 8 \cdot 10^{+305}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot j, -27, \mathsf{fma}\left(i \cdot x, -4, \mathsf{fma}\left(\mathsf{fma}\left(z, \left(18 \cdot x\right) \cdot y, -4 \cdot a\right), t, c \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(\mathsf{fma}\left(\left(t \cdot 18\right) \cdot z, y, -4 \cdot i\right), x, c \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\\\ [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(i, x, a \cdot t\right)\\ t_2 := \mathsf{fma}\left(t\_1, -4, \mathsf{fma}\left(c, b, \left(\left(\left(z \cdot y\right) \cdot x\right) \cdot t\right) \cdot 18\right)\right)\\ t_3 := \left(c \cdot b + \left(t \cdot \left(z \cdot \left(\left(18 \cdot x\right) \cdot y\right)\right) - \left(4 \cdot a\right) \cdot t\right)\right) - i \cdot \left(4 \cdot x\right)\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+287}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 10^{+285}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \mathsf{fma}\left(t\_1, -4, \left(k \cdot j\right) \cdot -27\right)\right)\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, i, \left(\left(z \cdot y\right) \cdot t\right) \cdot 18\right) \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (fma i x (* a t)))
        (t_2 (fma t_1 -4.0 (fma c b (* (* (* (* z y) x) t) 18.0))))
        (t_3
         (-
          (+ (* c b) (- (* t (* z (* (* 18.0 x) y))) (* (* 4.0 a) t)))
          (* i (* 4.0 x)))))
   (if (<= t_3 -2e+287)
     t_2
     (if (<= t_3 1e+285)
       (fma c b (fma t_1 -4.0 (* (* k j) -27.0)))
       (if (<= t_3 INFINITY) t_2 (* (fma -4.0 i (* (* (* z y) t) 18.0)) x))))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = fma(i, x, (a * t));
	double t_2 = fma(t_1, -4.0, fma(c, b, ((((z * y) * x) * t) * 18.0)));
	double t_3 = ((c * b) + ((t * (z * ((18.0 * x) * y))) - ((4.0 * a) * t))) - (i * (4.0 * x));
	double tmp;
	if (t_3 <= -2e+287) {
		tmp = t_2;
	} else if (t_3 <= 1e+285) {
		tmp = fma(c, b, fma(t_1, -4.0, ((k * j) * -27.0)));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = fma(-4.0, i, (((z * y) * t) * 18.0)) * x;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = fma(i, x, Float64(a * t))
	t_2 = fma(t_1, -4.0, fma(c, b, Float64(Float64(Float64(Float64(z * y) * x) * t) * 18.0)))
	t_3 = Float64(Float64(Float64(c * b) + Float64(Float64(t * Float64(z * Float64(Float64(18.0 * x) * y))) - Float64(Float64(4.0 * a) * t))) - Float64(i * Float64(4.0 * x)))
	tmp = 0.0
	if (t_3 <= -2e+287)
		tmp = t_2;
	elseif (t_3 <= 1e+285)
		tmp = fma(c, b, fma(t_1, -4.0, Float64(Float64(k * j) * -27.0)));
	elseif (t_3 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(fma(-4.0, i, Float64(Float64(Float64(z * y) * t) * 18.0)) * x);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(i * x + N[(a * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * -4.0 + N[(c * b + N[(N[(N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision] * t), $MachinePrecision] * 18.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(c * b), $MachinePrecision] + N[(N[(t * N[(z * N[(N[(18.0 * x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(i * N[(4.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -2e+287], t$95$2, If[LessEqual[t$95$3, 1e+285], N[(c * b + N[(t$95$1 * -4.0 + N[(N[(k * j), $MachinePrecision] * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$2, N[(N[(-4.0 * i + N[(N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] * 18.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x #s(literal 18 binary64)) y) z) t) (*.f64 (*.f64 a #s(literal 4 binary64)) t)) (*.f64 b c)) (*.f64 (*.f64 x #s(literal 4 binary64)) i)) < -2.0000000000000002e287 or 9.9999999999999998e284 < (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x #s(literal 18 binary64)) y) z) t) (*.f64 (*.f64 a #s(literal 4 binary64)) t)) (*.f64 b c)) (*.f64 (*.f64 x #s(literal 4 binary64)) i)) < +inf.0

   1. Initial program 84.4%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) + \left(\mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right)\right) + \left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right)} \]

      3. distribute-lft-out N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{4 \cdot \left(a \cdot t + i \cdot x\right)}\right)\right) + \left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(a \cdot t + i \cdot x\right) \cdot 4}\right)\right) + \left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) \]

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

         \[\leadsto \color{blue}{\left(a \cdot t + i \cdot x\right) \cdot \left(\mathsf{neg}\left(4\right)\right)} + \left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(a \cdot t + i \cdot x\right) \cdot \color{blue}{-4} + \left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot t + i \cdot x, -4, 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right)} \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{i \cdot x + a \cdot t}, -4, 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(i, x, a \cdot t\right)}, -4, 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(i, x, \color{blue}{a \cdot t}\right), -4, 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(i, x, a \cdot t\right), -4, \color{blue}{b \cdot c + 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right)}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(i, x, a \cdot t\right), -4, \color{blue}{c \cdot b} + 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right)\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(i, x, a \cdot t\right), -4, \color{blue}{\mathsf{fma}\left(c, b, 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right)\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(i, x, a \cdot t\right), -4, \mathsf{fma}\left(c, b, \color{blue}{\left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot 18}\right)\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(i, x, a \cdot t\right), -4, \mathsf{fma}\left(c, b, \color{blue}{\left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot 18}\right)\right) \]

   5. Applied rewrites 89.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(i, x, a \cdot t\right), -4, \mathsf{fma}\left(c, b, \left(\left(\left(z \cdot y\right) \cdot x\right) \cdot t\right) \cdot 18\right)\right)} \]

   if -2.0000000000000002e287 < (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x #s(literal 18 binary64)) y) z) t) (*.f64 (*.f64 a #s(literal 4 binary64)) t)) (*.f64 b c)) (*.f64 (*.f64 x #s(literal 4 binary64)) i)) < 9.9999999999999998e284

   1. Initial program 99.9%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

         \[\leadsto \color{blue}{b \cdot c + \left(\mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + \left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{c \cdot b} + \left(\mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + \left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + \left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{neg}\left(\color{blue}{\left(\left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right) + 27 \cdot \left(j \cdot k\right)\right)}\right)\right) \]

      5. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(\mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right)\right) + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{fma}\left(c, b, \left(\mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot \left(j \cdot k\right)}\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(\mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right)\right) + \color{blue}{-27} \cdot \left(j \cdot k\right)\right) \]

      8. distribute-lft-out N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(\mathsf{neg}\left(\color{blue}{4 \cdot \left(a \cdot t + i \cdot x\right)}\right)\right) + -27 \cdot \left(j \cdot k\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(\mathsf{neg}\left(\color{blue}{\left(a \cdot t + i \cdot x\right) \cdot 4}\right)\right) + -27 \cdot \left(j \cdot k\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(a \cdot t + i \cdot x\right) \cdot \left(\mathsf{neg}\left(4\right)\right)} + -27 \cdot \left(j \cdot k\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(c, b, \left(a \cdot t + i \cdot x\right) \cdot \color{blue}{-4} + -27 \cdot \left(j \cdot k\right)\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\mathsf{fma}\left(a \cdot t + i \cdot x, -4, -27 \cdot \left(j \cdot k\right)\right)}\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{i \cdot x + a \cdot t}, -4, -27 \cdot \left(j \cdot k\right)\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(i, x, a \cdot t\right)}, -4, -27 \cdot \left(j \cdot k\right)\right)\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\mathsf{fma}\left(i, x, \color{blue}{a \cdot t}\right), -4, -27 \cdot \left(j \cdot k\right)\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\mathsf{fma}\left(i, x, a \cdot t\right), -4, \color{blue}{-27 \cdot \left(j \cdot k\right)}\right)\right) \]

   5. Applied rewrites 93.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{fma}\left(\mathsf{fma}\left(i, x, a \cdot t\right), -4, -27 \cdot \left(k \cdot j\right)\right)\right)} \]

   if +inf.0 < (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x #s(literal 18 binary64)) y) z) t) (*.f64 (*.f64 a #s(literal 4 binary64)) t)) (*.f64 b c)) (*.f64 (*.f64 x #s(literal 4 binary64)) i))

   1. Initial program 0.0%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right) \cdot x} \]

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

         \[\leadsto \color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot i\right)} \cdot x \]

      4. metadata-eval N/A

         \[\leadsto \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + \color{blue}{-4} \cdot i\right) \cdot x \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-4 \cdot i + 18 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right)} \cdot x \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4, i, 18 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right)} \cdot x \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-4, i, \color{blue}{\left(t \cdot \left(y \cdot z\right)\right) \cdot 18}\right) \cdot x \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4, i, \color{blue}{\left(t \cdot \left(y \cdot z\right)\right) \cdot 18}\right) \cdot x \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-4, i, \color{blue}{\left(\left(y \cdot z\right) \cdot t\right)} \cdot 18\right) \cdot x \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-4, i, \color{blue}{\left(\left(y \cdot z\right) \cdot t\right)} \cdot 18\right) \cdot x \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-4, i, \left(\color{blue}{\left(z \cdot y\right)} \cdot t\right) \cdot 18\right) \cdot x \]

      12. lower-*.f64 78.9

          \[\leadsto \mathsf{fma}\left(-4, i, \left(\color{blue}{\left(z \cdot y\right)} \cdot t\right) \cdot 18\right) \cdot x \]

   5. Applied rewrites 78.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4, i, \left(\left(z \cdot y\right) \cdot t\right) \cdot 18\right) \cdot x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot b + \left(t \cdot \left(z \cdot \left(\left(18 \cdot x\right) \cdot y\right)\right) - \left(4 \cdot a\right) \cdot t\right)\right) - i \cdot \left(4 \cdot x\right) \leq -2 \cdot 10^{+287}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(i, x, a \cdot t\right), -4, \mathsf{fma}\left(c, b, \left(\left(\left(z \cdot y\right) \cdot x\right) \cdot t\right) \cdot 18\right)\right)\\ \mathbf{elif}\;\left(c \cdot b + \left(t \cdot \left(z \cdot \left(\left(18 \cdot x\right) \cdot y\right)\right) - \left(4 \cdot a\right) \cdot t\right)\right) - i \cdot \left(4 \cdot x\right) \leq 10^{+285}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \mathsf{fma}\left(\mathsf{fma}\left(i, x, a \cdot t\right), -4, \left(k \cdot j\right) \cdot -27\right)\right)\\ \mathbf{elif}\;\left(c \cdot b + \left(t \cdot \left(z \cdot \left(\left(18 \cdot x\right) \cdot y\right)\right) - \left(4 \cdot a\right) \cdot t\right)\right) - i \cdot \left(4 \cdot x\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(i, x, a \cdot t\right), -4, \mathsf{fma}\left(c, b, \left(\left(\left(z \cdot y\right) \cdot x\right) \cdot t\right) \cdot 18\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, i, \left(\left(z \cdot y\right) \cdot t\right) \cdot 18\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\\\ [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(\mathsf{fma}\left(\left(t \cdot 18\right) \cdot z, y, -4 \cdot i\right), x, c \cdot b\right)\right)\\ t_2 := \left(c \cdot b + \left(t \cdot \left(z \cdot \left(\left(18 \cdot x\right) \cdot y\right)\right) - \left(4 \cdot a\right) \cdot t\right)\right) - i \cdot \left(4 \cdot x\right)\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+303}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \mathsf{fma}\left(\mathsf{fma}\left(i, x, a \cdot t\right), -4, \left(k \cdot j\right) \cdot -27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1
         (fma
          (* -27.0 k)
          j
          (fma (fma (* (* t 18.0) z) y (* -4.0 i)) x (* c b))))
        (t_2
         (-
          (+ (* c b) (- (* t (* z (* (* 18.0 x) y))) (* (* 4.0 a) t)))
          (* i (* 4.0 x)))))
   (if (<= t_2 -5e+303)
     t_1
     (if (<= t_2 1e+306)
       (fma c b (fma (fma i x (* a t)) -4.0 (* (* k j) -27.0)))
       t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = fma((-27.0 * k), j, fma(fma(((t * 18.0) * z), y, (-4.0 * i)), x, (c * b)));
	double t_2 = ((c * b) + ((t * (z * ((18.0 * x) * y))) - ((4.0 * a) * t))) - (i * (4.0 * x));
	double tmp;
	if (t_2 <= -5e+303) {
		tmp = t_1;
	} else if (t_2 <= 1e+306) {
		tmp = fma(c, b, fma(fma(i, x, (a * t)), -4.0, ((k * j) * -27.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = fma(Float64(-27.0 * k), j, fma(fma(Float64(Float64(t * 18.0) * z), y, Float64(-4.0 * i)), x, Float64(c * b)))
	t_2 = Float64(Float64(Float64(c * b) + Float64(Float64(t * Float64(z * Float64(Float64(18.0 * x) * y))) - Float64(Float64(4.0 * a) * t))) - Float64(i * Float64(4.0 * x)))
	tmp = 0.0
	if (t_2 <= -5e+303)
		tmp = t_1;
	elseif (t_2 <= 1e+306)
		tmp = fma(c, b, fma(fma(i, x, Float64(a * t)), -4.0, Float64(Float64(k * j) * -27.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(N[(-27.0 * k), $MachinePrecision] * j + N[(N[(N[(N[(t * 18.0), $MachinePrecision] * z), $MachinePrecision] * y + N[(-4.0 * i), $MachinePrecision]), $MachinePrecision] * x + N[(c * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(c * b), $MachinePrecision] + N[(N[(t * N[(z * N[(N[(18.0 * x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(i * N[(4.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+303], t$95$1, If[LessEqual[t$95$2, 1e+306], N[(c * b + N[(N[(i * x + N[(a * t), $MachinePrecision]), $MachinePrecision] * -4.0 + N[(N[(k * j), $MachinePrecision] * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x #s(literal 18 binary64)) y) z) t) (*.f64 (*.f64 a #s(literal 4 binary64)) t)) (*.f64 b c)) (*.f64 (*.f64 x #s(literal 4 binary64)) i)) < -4.9999999999999997e303 or 1.00000000000000002e306 < (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x #s(literal 18 binary64)) y) z) t) (*.f64 (*.f64 a #s(literal 4 binary64)) t)) (*.f64 b c)) (*.f64 (*.f64 x #s(literal 4 binary64)) i))

   1. Initial program 68.6%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

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

         \[\leadsto \color{blue}{\left(b \cdot c + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot t\right)\right)} - \left(j \cdot 27\right) \cdot k \]

      2. metadata-eval N/A

         \[\leadsto \left(b \cdot c + \color{blue}{-4} \cdot \left(a \cdot t\right)\right) - \left(j \cdot 27\right) \cdot k \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(-4 \cdot \left(a \cdot t\right) + b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-4 \cdot a\right) \cdot t} + b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-4 \cdot a}, t, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, \color{blue}{c \cdot b}\right) - \left(j \cdot 27\right) \cdot k \]

      8. lower-*.f64 41.3

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, \color{blue}{c \cdot b}\right) - \left(j \cdot 27\right) \cdot k \]

   5. Applied rewrites 41.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)} - \left(j \cdot 27\right) \cdot k \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) - \left(j \cdot 27\right) \cdot k} \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) - \color{blue}{\left(j \cdot 27\right) \cdot k} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \left(\mathsf{neg}\left(j \cdot 27\right)\right) \cdot k} \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \left(\mathsf{neg}\left(\color{blue}{j \cdot 27}\right)\right) \cdot k \]

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

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \color{blue}{\left(j \cdot \left(\mathsf{neg}\left(27\right)\right)\right)} \cdot k \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \left(j \cdot \color{blue}{-27}\right) \cdot k \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \color{blue}{j \cdot \left(-27 \cdot k\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + j \cdot \color{blue}{\left(-27 \cdot k\right)} \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \color{blue}{\left(-27 \cdot k\right) \cdot j} \]

      10. +-commutative N/A

          \[\leadsto \color{blue}{\left(-27 \cdot k\right) \cdot j + \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)} \]

      11. lower-fma.f64 42.2

          \[\leadsto \color{blue}{\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)\right)} \]

   7. Applied rewrites 42.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)\right)} \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 4 \cdot \left(i \cdot x\right)}\right) \]
   9. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + \left(b \cdot c - 4 \cdot \left(i \cdot x\right)\right)}\right) \]

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

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + \color{blue}{\left(b \cdot c + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(i \cdot x\right)\right)}\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + \left(b \cdot c + \color{blue}{-4} \cdot \left(i \cdot x\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + \color{blue}{\left(-4 \cdot \left(i \cdot x\right) + b \cdot c\right)}\right) \]

      5. associate-+r+ N/A

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + -4 \cdot \left(i \cdot x\right)\right) + b \cdot c}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \left(\color{blue}{\left(18 \cdot t\right) \cdot \left(x \cdot \left(y \cdot z\right)\right)} + -4 \cdot \left(i \cdot x\right)\right) + b \cdot c\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \left(\left(18 \cdot t\right) \cdot \color{blue}{\left(\left(y \cdot z\right) \cdot x\right)} + -4 \cdot \left(i \cdot x\right)\right) + b \cdot c\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \left(\color{blue}{\left(\left(18 \cdot t\right) \cdot \left(y \cdot z\right)\right) \cdot x} + -4 \cdot \left(i \cdot x\right)\right) + b \cdot c\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \left(\color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right)} \cdot x + -4 \cdot \left(i \cdot x\right)\right) + b \cdot c\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \left(\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right) \cdot x + \color{blue}{\left(-4 \cdot i\right) \cdot x}\right) + b \cdot c\right) \]

      11. distribute-rgt-in N/A

          \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{x \cdot \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + -4 \cdot i\right)} + b \cdot c\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, x \cdot \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + \color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot i\right) + b \cdot c\right) \]

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

          \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, x \cdot \color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right)} + b \cdot c\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right) \cdot x} + b \cdot c\right) \]

      15. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{\mathsf{fma}\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i, x, b \cdot c\right)}\right) \]

   10. Applied rewrites 91.7%

       \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(18 \cdot t\right) \cdot z, y, -4 \cdot i\right), x, c \cdot b\right)}\right) \]

   if -4.9999999999999997e303 < (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x #s(literal 18 binary64)) y) z) t) (*.f64 (*.f64 a #s(literal 4 binary64)) t)) (*.f64 b c)) (*.f64 (*.f64 x #s(literal 4 binary64)) i)) < 1.00000000000000002e306

   1. Initial program 99.8%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

         \[\leadsto \color{blue}{b \cdot c + \left(\mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + \left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{c \cdot b} + \left(\mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + \left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + \left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{neg}\left(\color{blue}{\left(\left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right) + 27 \cdot \left(j \cdot k\right)\right)}\right)\right) \]

      5. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(\mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right)\right) + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{fma}\left(c, b, \left(\mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot \left(j \cdot k\right)}\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(\mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right)\right) + \color{blue}{-27} \cdot \left(j \cdot k\right)\right) \]

      8. distribute-lft-out N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(\mathsf{neg}\left(\color{blue}{4 \cdot \left(a \cdot t + i \cdot x\right)}\right)\right) + -27 \cdot \left(j \cdot k\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(\mathsf{neg}\left(\color{blue}{\left(a \cdot t + i \cdot x\right) \cdot 4}\right)\right) + -27 \cdot \left(j \cdot k\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(a \cdot t + i \cdot x\right) \cdot \left(\mathsf{neg}\left(4\right)\right)} + -27 \cdot \left(j \cdot k\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(c, b, \left(a \cdot t + i \cdot x\right) \cdot \color{blue}{-4} + -27 \cdot \left(j \cdot k\right)\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\mathsf{fma}\left(a \cdot t + i \cdot x, -4, -27 \cdot \left(j \cdot k\right)\right)}\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{i \cdot x + a \cdot t}, -4, -27 \cdot \left(j \cdot k\right)\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(i, x, a \cdot t\right)}, -4, -27 \cdot \left(j \cdot k\right)\right)\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\mathsf{fma}\left(i, x, \color{blue}{a \cdot t}\right), -4, -27 \cdot \left(j \cdot k\right)\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\mathsf{fma}\left(i, x, a \cdot t\right), -4, \color{blue}{-27 \cdot \left(j \cdot k\right)}\right)\right) \]

   5. Applied rewrites 93.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{fma}\left(\mathsf{fma}\left(i, x, a \cdot t\right), -4, -27 \cdot \left(k \cdot j\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot b + \left(t \cdot \left(z \cdot \left(\left(18 \cdot x\right) \cdot y\right)\right) - \left(4 \cdot a\right) \cdot t\right)\right) - i \cdot \left(4 \cdot x\right) \leq -5 \cdot 10^{+303}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(\mathsf{fma}\left(\left(t \cdot 18\right) \cdot z, y, -4 \cdot i\right), x, c \cdot b\right)\right)\\ \mathbf{elif}\;\left(c \cdot b + \left(t \cdot \left(z \cdot \left(\left(18 \cdot x\right) \cdot y\right)\right) - \left(4 \cdot a\right) \cdot t\right)\right) - i \cdot \left(4 \cdot x\right) \leq 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \mathsf{fma}\left(\mathsf{fma}\left(i, x, a \cdot t\right), -4, \left(k \cdot j\right) \cdot -27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(\mathsf{fma}\left(\left(t \cdot 18\right) \cdot z, y, -4 \cdot i\right), x, c \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\\\ [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \left(c \cdot b + \left(t \cdot \left(z \cdot \left(\left(18 \cdot x\right) \cdot y\right)\right) - \left(4 \cdot a\right) \cdot t\right)\right) - i \cdot \left(4 \cdot x\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+287}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-4, a, \left(\left(z \cdot y\right) \cdot x\right) \cdot 18\right), t, c \cdot b\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(c, b, \mathsf{fma}\left(\mathsf{fma}\left(i, x, a \cdot t\right), -4, \left(k \cdot j\right) \cdot -27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, i, \left(\left(z \cdot y\right) \cdot t\right) \cdot 18\right) \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1
         (-
          (+ (* c b) (- (* t (* z (* (* 18.0 x) y))) (* (* 4.0 a) t)))
          (* i (* 4.0 x)))))
   (if (<= t_1 -2e+287)
     (fma (fma -4.0 a (* (* (* z y) x) 18.0)) t (* c b))
     (if (<= t_1 INFINITY)
       (fma c b (fma (fma i x (* a t)) -4.0 (* (* k j) -27.0)))
       (* (fma -4.0 i (* (* (* z y) t) 18.0)) x)))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = ((c * b) + ((t * (z * ((18.0 * x) * y))) - ((4.0 * a) * t))) - (i * (4.0 * x));
	double tmp;
	if (t_1 <= -2e+287) {
		tmp = fma(fma(-4.0, a, (((z * y) * x) * 18.0)), t, (c * b));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma(c, b, fma(fma(i, x, (a * t)), -4.0, ((k * j) * -27.0)));
	} else {
		tmp = fma(-4.0, i, (((z * y) * t) * 18.0)) * x;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = Float64(Float64(Float64(c * b) + Float64(Float64(t * Float64(z * Float64(Float64(18.0 * x) * y))) - Float64(Float64(4.0 * a) * t))) - Float64(i * Float64(4.0 * x)))
	tmp = 0.0
	if (t_1 <= -2e+287)
		tmp = fma(fma(-4.0, a, Float64(Float64(Float64(z * y) * x) * 18.0)), t, Float64(c * b));
	elseif (t_1 <= Inf)
		tmp = fma(c, b, fma(fma(i, x, Float64(a * t)), -4.0, Float64(Float64(k * j) * -27.0)));
	else
		tmp = Float64(fma(-4.0, i, Float64(Float64(Float64(z * y) * t) * 18.0)) * x);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(N[(N[(c * b), $MachinePrecision] + N[(N[(t * N[(z * N[(N[(18.0 * x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(i * N[(4.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+287], N[(N[(-4.0 * a + N[(N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision] * 18.0), $MachinePrecision]), $MachinePrecision] * t + N[(c * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(c * b + N[(N[(i * x + N[(a * t), $MachinePrecision]), $MachinePrecision] * -4.0 + N[(N[(k * j), $MachinePrecision] * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 * i + N[(N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] * 18.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x #s(literal 18 binary64)) y) z) t) (*.f64 (*.f64 a #s(literal 4 binary64)) t)) (*.f64 b c)) (*.f64 (*.f64 x #s(literal 4 binary64)) i)) < -2.0000000000000002e287

   1. Initial program 84.8%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - \left(4 \cdot \left(a \cdot t\right) + 27 \cdot \left(j \cdot k\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - \color{blue}{\left(27 \cdot \left(j \cdot k\right) + 4 \cdot \left(a \cdot t\right)\right)} \]

      2. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 27 \cdot \left(j \cdot k\right)\right) - 4 \cdot \left(a \cdot t\right)} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right)} - 4 \cdot \left(a \cdot t\right) \]

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

         \[\leadsto \left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) + \color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot \left(j \cdot k\right)}\right) - 4 \cdot \left(a \cdot t\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) + \color{blue}{-27} \cdot \left(j \cdot k\right)\right) - 4 \cdot \left(a \cdot t\right) \]

      6. +-commutative N/A

         \[\leadsto \color{blue}{\left(-27 \cdot \left(j \cdot k\right) + \left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right)\right)} - 4 \cdot \left(a \cdot t\right) \]

      7. associate--l+ N/A

         \[\leadsto \color{blue}{-27 \cdot \left(j \cdot k\right) + \left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 4 \cdot \left(a \cdot t\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto -27 \cdot \color{blue}{\left(k \cdot j\right)} + \left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 4 \cdot \left(a \cdot t\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-27 \cdot k\right) \cdot j} + \left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 4 \cdot \left(a \cdot t\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-27 \cdot k, j, \left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 4 \cdot \left(a \cdot t\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-27 \cdot k}, j, \left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 4 \cdot \left(a \cdot t\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{\left(b \cdot c + 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right)\right)} - 4 \cdot \left(a \cdot t\right)\right) \]

   5. Applied rewrites 81.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(\mathsf{fma}\left(-4, a, \left(\left(z \cdot y\right) \cdot x\right) \cdot 18\right), t, c \cdot b\right)\right)} \]

   6. Taylor expanded in k around 0

      \[\leadsto b \cdot c + \color{blue}{t \cdot \left(-4 \cdot a + 18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 76.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-4, a, \left(\left(z \cdot y\right) \cdot x\right) \cdot 18\right), \color{blue}{t}, c \cdot b\right) \]

   if -2.0000000000000002e287 < (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x #s(literal 18 binary64)) y) z) t) (*.f64 (*.f64 a #s(literal 4 binary64)) t)) (*.f64 b c)) (*.f64 (*.f64 x #s(literal 4 binary64)) i)) < +inf.0

   1. Initial program 95.6%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

         \[\leadsto \color{blue}{b \cdot c + \left(\mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + \left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{c \cdot b} + \left(\mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + \left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + \left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{neg}\left(\color{blue}{\left(\left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right) + 27 \cdot \left(j \cdot k\right)\right)}\right)\right) \]

      5. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(\mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right)\right) + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{fma}\left(c, b, \left(\mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot \left(j \cdot k\right)}\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(\mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right)\right) + \color{blue}{-27} \cdot \left(j \cdot k\right)\right) \]

      8. distribute-lft-out N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(\mathsf{neg}\left(\color{blue}{4 \cdot \left(a \cdot t + i \cdot x\right)}\right)\right) + -27 \cdot \left(j \cdot k\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(\mathsf{neg}\left(\color{blue}{\left(a \cdot t + i \cdot x\right) \cdot 4}\right)\right) + -27 \cdot \left(j \cdot k\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(a \cdot t + i \cdot x\right) \cdot \left(\mathsf{neg}\left(4\right)\right)} + -27 \cdot \left(j \cdot k\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(c, b, \left(a \cdot t + i \cdot x\right) \cdot \color{blue}{-4} + -27 \cdot \left(j \cdot k\right)\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\mathsf{fma}\left(a \cdot t + i \cdot x, -4, -27 \cdot \left(j \cdot k\right)\right)}\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{i \cdot x + a \cdot t}, -4, -27 \cdot \left(j \cdot k\right)\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(i, x, a \cdot t\right)}, -4, -27 \cdot \left(j \cdot k\right)\right)\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\mathsf{fma}\left(i, x, \color{blue}{a \cdot t}\right), -4, -27 \cdot \left(j \cdot k\right)\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\mathsf{fma}\left(i, x, a \cdot t\right), -4, \color{blue}{-27 \cdot \left(j \cdot k\right)}\right)\right) \]

   5. Applied rewrites 87.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{fma}\left(\mathsf{fma}\left(i, x, a \cdot t\right), -4, -27 \cdot \left(k \cdot j\right)\right)\right)} \]

   if +inf.0 < (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x #s(literal 18 binary64)) y) z) t) (*.f64 (*.f64 a #s(literal 4 binary64)) t)) (*.f64 b c)) (*.f64 (*.f64 x #s(literal 4 binary64)) i))

   1. Initial program 0.0%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right) \cdot x} \]

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

         \[\leadsto \color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot i\right)} \cdot x \]

      4. metadata-eval N/A

         \[\leadsto \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + \color{blue}{-4} \cdot i\right) \cdot x \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-4 \cdot i + 18 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right)} \cdot x \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4, i, 18 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right)} \cdot x \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-4, i, \color{blue}{\left(t \cdot \left(y \cdot z\right)\right) \cdot 18}\right) \cdot x \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4, i, \color{blue}{\left(t \cdot \left(y \cdot z\right)\right) \cdot 18}\right) \cdot x \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-4, i, \color{blue}{\left(\left(y \cdot z\right) \cdot t\right)} \cdot 18\right) \cdot x \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-4, i, \color{blue}{\left(\left(y \cdot z\right) \cdot t\right)} \cdot 18\right) \cdot x \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-4, i, \left(\color{blue}{\left(z \cdot y\right)} \cdot t\right) \cdot 18\right) \cdot x \]

      12. lower-*.f64 78.9

          \[\leadsto \mathsf{fma}\left(-4, i, \left(\color{blue}{\left(z \cdot y\right)} \cdot t\right) \cdot 18\right) \cdot x \]

   5. Applied rewrites 78.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4, i, \left(\left(z \cdot y\right) \cdot t\right) \cdot 18\right) \cdot x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot b + \left(t \cdot \left(z \cdot \left(\left(18 \cdot x\right) \cdot y\right)\right) - \left(4 \cdot a\right) \cdot t\right)\right) - i \cdot \left(4 \cdot x\right) \leq -2 \cdot 10^{+287}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-4, a, \left(\left(z \cdot y\right) \cdot x\right) \cdot 18\right), t, c \cdot b\right)\\ \mathbf{elif}\;\left(c \cdot b + \left(t \cdot \left(z \cdot \left(\left(18 \cdot x\right) \cdot y\right)\right) - \left(4 \cdot a\right) \cdot t\right)\right) - i \cdot \left(4 \cdot x\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(c, b, \mathsf{fma}\left(\mathsf{fma}\left(i, x, a \cdot t\right), -4, \left(k \cdot j\right) \cdot -27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, i, \left(\left(z \cdot y\right) \cdot t\right) \cdot 18\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\\\ [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := i \cdot \left(4 \cdot x\right)\\ t_2 := k \cdot \left(27 \cdot j\right)\\ t_3 := \left(4 \cdot a\right) \cdot t\\ \mathbf{if}\;\left(\left(c \cdot b + \left(t \cdot \left(z \cdot \left(\left(18 \cdot x\right) \cdot y\right)\right) - t\_3\right)\right) - t\_1\right) - t\_2 \leq 8 \cdot 10^{+305}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(18 \cdot x\right) \cdot z\right) \cdot y\right) \cdot t - t\_3\right) + c \cdot b\right) - t\_1\right) - t\_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(\mathsf{fma}\left(\left(t \cdot 18\right) \cdot z, y, -4 \cdot i\right), x, c \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (* i (* 4.0 x))) (t_2 (* k (* 27.0 j))) (t_3 (* (* 4.0 a) t)))
   (if (<=
        (- (- (+ (* c b) (- (* t (* z (* (* 18.0 x) y))) t_3)) t_1) t_2)
        8e+305)
     (- (- (+ (- (* (* (* (* 18.0 x) z) y) t) t_3) (* c b)) t_1) t_2)
     (fma (* -27.0 k) j (fma (fma (* (* t 18.0) z) y (* -4.0 i)) x (* c b))))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = i * (4.0 * x);
	double t_2 = k * (27.0 * j);
	double t_3 = (4.0 * a) * t;
	double tmp;
	if (((((c * b) + ((t * (z * ((18.0 * x) * y))) - t_3)) - t_1) - t_2) <= 8e+305) {
		tmp = (((((((18.0 * x) * z) * y) * t) - t_3) + (c * b)) - t_1) - t_2;
	} else {
		tmp = fma((-27.0 * k), j, fma(fma(((t * 18.0) * z), y, (-4.0 * i)), x, (c * b)));
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = Float64(i * Float64(4.0 * x))
	t_2 = Float64(k * Float64(27.0 * j))
	t_3 = Float64(Float64(4.0 * a) * t)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(c * b) + Float64(Float64(t * Float64(z * Float64(Float64(18.0 * x) * y))) - t_3)) - t_1) - t_2) <= 8e+305)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(18.0 * x) * z) * y) * t) - t_3) + Float64(c * b)) - t_1) - t_2);
	else
		tmp = fma(Float64(-27.0 * k), j, fma(fma(Float64(Float64(t * 18.0) * z), y, Float64(-4.0 * i)), x, Float64(c * b)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(i * N[(4.0 * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(k * N[(27.0 * j), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(4.0 * a), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(c * b), $MachinePrecision] + N[(N[(t * N[(z * N[(N[(18.0 * x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision], 8e+305], N[(N[(N[(N[(N[(N[(N[(N[(18.0 * x), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision] * t), $MachinePrecision] - t$95$3), $MachinePrecision] + N[(c * b), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision], N[(N[(-27.0 * k), $MachinePrecision] * j + N[(N[(N[(N[(t * 18.0), $MachinePrecision] * z), $MachinePrecision] * y + N[(-4.0 * i), $MachinePrecision]), $MachinePrecision] * x + N[(c * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x #s(literal 18 binary64)) y) z) t) (*.f64 (*.f64 a #s(literal 4 binary64)) t)) (*.f64 b c)) (*.f64 (*.f64 x #s(literal 4 binary64)) i)) (*.f64 (*.f64 j #s(literal 27 binary64)) k)) < 7.9999999999999995e305

   1. Initial program 97.1%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right)} \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{\left(z \cdot \left(\left(x \cdot 18\right) \cdot y\right)\right)} \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(z \cdot \color{blue}{\left(\left(x \cdot 18\right) \cdot y\right)}\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\left(\color{blue}{\left(\left(z \cdot \left(x \cdot 18\right)\right) \cdot y\right)} \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\left(\left(z \cdot \left(x \cdot 18\right)\right) \cdot y\right)} \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]

      6. lower-*.f64 96.0

         \[\leadsto \left(\left(\left(\left(\color{blue}{\left(z \cdot \left(x \cdot 18\right)\right)} \cdot y\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(\left(z \cdot \color{blue}{\left(x \cdot 18\right)}\right) \cdot y\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]

      8. *-commutative N/A

         \[\leadsto \left(\left(\left(\left(\left(z \cdot \color{blue}{\left(18 \cdot x\right)}\right) \cdot y\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]

      9. lower-*.f64 96.0

         \[\leadsto \left(\left(\left(\left(\left(z \cdot \color{blue}{\left(18 \cdot x\right)}\right) \cdot y\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]

   4. Applied rewrites 96.0%

      \[\leadsto \left(\left(\left(\color{blue}{\left(\left(z \cdot \left(18 \cdot x\right)\right) \cdot y\right)} \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]

   if 7.9999999999999995e305 < (-.f64 (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x #s(literal 18 binary64)) y) z) t) (*.f64 (*.f64 a #s(literal 4 binary64)) t)) (*.f64 b c)) (*.f64 (*.f64 x #s(literal 4 binary64)) i)) (*.f64 (*.f64 j #s(literal 27 binary64)) k))

   1. Initial program 63.0%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

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

         \[\leadsto \color{blue}{\left(b \cdot c + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot t\right)\right)} - \left(j \cdot 27\right) \cdot k \]

      2. metadata-eval N/A

         \[\leadsto \left(b \cdot c + \color{blue}{-4} \cdot \left(a \cdot t\right)\right) - \left(j \cdot 27\right) \cdot k \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(-4 \cdot \left(a \cdot t\right) + b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-4 \cdot a\right) \cdot t} + b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-4 \cdot a}, t, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, \color{blue}{c \cdot b}\right) - \left(j \cdot 27\right) \cdot k \]

      8. lower-*.f64 50.0

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, \color{blue}{c \cdot b}\right) - \left(j \cdot 27\right) \cdot k \]

   5. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)} - \left(j \cdot 27\right) \cdot k \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) - \left(j \cdot 27\right) \cdot k} \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) - \color{blue}{\left(j \cdot 27\right) \cdot k} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \left(\mathsf{neg}\left(j \cdot 27\right)\right) \cdot k} \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \left(\mathsf{neg}\left(\color{blue}{j \cdot 27}\right)\right) \cdot k \]

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

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \color{blue}{\left(j \cdot \left(\mathsf{neg}\left(27\right)\right)\right)} \cdot k \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \left(j \cdot \color{blue}{-27}\right) \cdot k \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \color{blue}{j \cdot \left(-27 \cdot k\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + j \cdot \color{blue}{\left(-27 \cdot k\right)} \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \color{blue}{\left(-27 \cdot k\right) \cdot j} \]

      10. +-commutative N/A

          \[\leadsto \color{blue}{\left(-27 \cdot k\right) \cdot j + \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)} \]

      11. lower-fma.f64 51.2

          \[\leadsto \color{blue}{\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)\right)} \]

   7. Applied rewrites 51.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)\right)} \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 4 \cdot \left(i \cdot x\right)}\right) \]
   9. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + \left(b \cdot c - 4 \cdot \left(i \cdot x\right)\right)}\right) \]

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

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + \color{blue}{\left(b \cdot c + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(i \cdot x\right)\right)}\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + \left(b \cdot c + \color{blue}{-4} \cdot \left(i \cdot x\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + \color{blue}{\left(-4 \cdot \left(i \cdot x\right) + b \cdot c\right)}\right) \]

      5. associate-+r+ N/A

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + -4 \cdot \left(i \cdot x\right)\right) + b \cdot c}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \left(\color{blue}{\left(18 \cdot t\right) \cdot \left(x \cdot \left(y \cdot z\right)\right)} + -4 \cdot \left(i \cdot x\right)\right) + b \cdot c\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \left(\left(18 \cdot t\right) \cdot \color{blue}{\left(\left(y \cdot z\right) \cdot x\right)} + -4 \cdot \left(i \cdot x\right)\right) + b \cdot c\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \left(\color{blue}{\left(\left(18 \cdot t\right) \cdot \left(y \cdot z\right)\right) \cdot x} + -4 \cdot \left(i \cdot x\right)\right) + b \cdot c\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \left(\color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right)} \cdot x + -4 \cdot \left(i \cdot x\right)\right) + b \cdot c\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \left(\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right) \cdot x + \color{blue}{\left(-4 \cdot i\right) \cdot x}\right) + b \cdot c\right) \]

      11. distribute-rgt-in N/A

          \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{x \cdot \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + -4 \cdot i\right)} + b \cdot c\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, x \cdot \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + \color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot i\right) + b \cdot c\right) \]

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

          \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, x \cdot \color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right)} + b \cdot c\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right) \cdot x} + b \cdot c\right) \]

      15. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{\mathsf{fma}\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i, x, b \cdot c\right)}\right) \]

   10. Applied rewrites 94.0%

       \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(18 \cdot t\right) \cdot z, y, -4 \cdot i\right), x, c \cdot b\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(c \cdot b + \left(t \cdot \left(z \cdot \left(\left(18 \cdot x\right) \cdot y\right)\right) - \left(4 \cdot a\right) \cdot t\right)\right) - i \cdot \left(4 \cdot x\right)\right) - k \cdot \left(27 \cdot j\right) \leq 8 \cdot 10^{+305}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(18 \cdot x\right) \cdot z\right) \cdot y\right) \cdot t - \left(4 \cdot a\right) \cdot t\right) + c \cdot b\right) - i \cdot \left(4 \cdot x\right)\right) - k \cdot \left(27 \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(\mathsf{fma}\left(\left(t \cdot 18\right) \cdot z, y, -4 \cdot i\right), x, c \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\\\ [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \left(18 \cdot x\right) \cdot y\\ \mathbf{if}\;\left(\left(c \cdot b + \left(t \cdot \left(z \cdot t\_1\right) - \left(4 \cdot a\right) \cdot t\right)\right) - i \cdot \left(4 \cdot x\right)\right) - k \cdot \left(27 \cdot j\right) \leq 8 \cdot 10^{+305}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot j, -27, \mathsf{fma}\left(i \cdot x, -4, \mathsf{fma}\left(\mathsf{fma}\left(z, t\_1, -4 \cdot a\right), t, c \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(\mathsf{fma}\left(\left(t \cdot 18\right) \cdot z, y, -4 \cdot i\right), x, c \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (* (* 18.0 x) y)))
   (if (<=
        (-
         (- (+ (* c b) (- (* t (* z t_1)) (* (* 4.0 a) t))) (* i (* 4.0 x)))
         (* k (* 27.0 j)))
        8e+305)
     (fma
      (* k j)
      -27.0
      (fma (* i x) -4.0 (fma (fma z t_1 (* -4.0 a)) t (* c b))))
     (fma (* -27.0 k) j (fma (fma (* (* t 18.0) z) y (* -4.0 i)) x (* c b))))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = (18.0 * x) * y;
	double tmp;
	if (((((c * b) + ((t * (z * t_1)) - ((4.0 * a) * t))) - (i * (4.0 * x))) - (k * (27.0 * j))) <= 8e+305) {
		tmp = fma((k * j), -27.0, fma((i * x), -4.0, fma(fma(z, t_1, (-4.0 * a)), t, (c * b))));
	} else {
		tmp = fma((-27.0 * k), j, fma(fma(((t * 18.0) * z), y, (-4.0 * i)), x, (c * b)));
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = Float64(Float64(18.0 * x) * y)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(c * b) + Float64(Float64(t * Float64(z * t_1)) - Float64(Float64(4.0 * a) * t))) - Float64(i * Float64(4.0 * x))) - Float64(k * Float64(27.0 * j))) <= 8e+305)
		tmp = fma(Float64(k * j), -27.0, fma(Float64(i * x), -4.0, fma(fma(z, t_1, Float64(-4.0 * a)), t, Float64(c * b))));
	else
		tmp = fma(Float64(-27.0 * k), j, fma(fma(Float64(Float64(t * 18.0) * z), y, Float64(-4.0 * i)), x, Float64(c * b)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(N[(18.0 * x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(c * b), $MachinePrecision] + N[(N[(t * N[(z * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(i * N[(4.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(k * N[(27.0 * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 8e+305], N[(N[(k * j), $MachinePrecision] * -27.0 + N[(N[(i * x), $MachinePrecision] * -4.0 + N[(N[(z * t$95$1 + N[(-4.0 * a), $MachinePrecision]), $MachinePrecision] * t + N[(c * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-27.0 * k), $MachinePrecision] * j + N[(N[(N[(N[(t * 18.0), $MachinePrecision] * z), $MachinePrecision] * y + N[(-4.0 * i), $MachinePrecision]), $MachinePrecision] * x + N[(c * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x #s(literal 18 binary64)) y) z) t) (*.f64 (*.f64 a #s(literal 4 binary64)) t)) (*.f64 b c)) (*.f64 (*.f64 x #s(literal 4 binary64)) i)) (*.f64 (*.f64 j #s(literal 27 binary64)) k)) < 7.9999999999999995e305

   1. Initial program 97.1%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) + \left(\mathsf{neg}\left(\left(j \cdot 27\right) \cdot k\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(j \cdot 27\right) \cdot k\right)\right) + \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(j \cdot 27\right) \cdot k}\right)\right) + \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{k \cdot \left(j \cdot 27\right)}\right)\right) + \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(k \cdot \color{blue}{\left(j \cdot 27\right)}\right)\right) + \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(k \cdot j\right) \cdot 27}\right)\right) + \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) \]

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

         \[\leadsto \color{blue}{\left(k \cdot j\right) \cdot \left(\mathsf{neg}\left(27\right)\right)} + \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(k \cdot j, \mathsf{neg}\left(27\right), \left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{k \cdot j}, \mathsf{neg}\left(27\right), \left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) \]

      11. metadata-eval 97.1

          \[\leadsto \mathsf{fma}\left(k \cdot j, \color{blue}{-27}, \left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) \]

      12. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(k \cdot j, -27, \color{blue}{\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i}\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(k \cdot j, -27, \color{blue}{\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) + \left(\mathsf{neg}\left(\left(x \cdot 4\right) \cdot i\right)\right)}\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(k \cdot j, -27, \color{blue}{\left(\mathsf{neg}\left(\left(x \cdot 4\right) \cdot i\right)\right) + \left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right)}\right) \]

   4. Applied rewrites 97.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k \cdot j, -27, \mathsf{fma}\left(i \cdot x, -4, \mathsf{fma}\left(\mathsf{fma}\left(z, y \cdot \left(18 \cdot x\right), -4 \cdot a\right), t, c \cdot b\right)\right)\right)} \]

   if 7.9999999999999995e305 < (-.f64 (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x #s(literal 18 binary64)) y) z) t) (*.f64 (*.f64 a #s(literal 4 binary64)) t)) (*.f64 b c)) (*.f64 (*.f64 x #s(literal 4 binary64)) i)) (*.f64 (*.f64 j #s(literal 27 binary64)) k))

   1. Initial program 63.0%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

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

         \[\leadsto \color{blue}{\left(b \cdot c + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot t\right)\right)} - \left(j \cdot 27\right) \cdot k \]

      2. metadata-eval N/A

         \[\leadsto \left(b \cdot c + \color{blue}{-4} \cdot \left(a \cdot t\right)\right) - \left(j \cdot 27\right) \cdot k \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(-4 \cdot \left(a \cdot t\right) + b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-4 \cdot a\right) \cdot t} + b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-4 \cdot a}, t, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, \color{blue}{c \cdot b}\right) - \left(j \cdot 27\right) \cdot k \]

      8. lower-*.f64 50.0

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, \color{blue}{c \cdot b}\right) - \left(j \cdot 27\right) \cdot k \]

   5. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)} - \left(j \cdot 27\right) \cdot k \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) - \left(j \cdot 27\right) \cdot k} \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) - \color{blue}{\left(j \cdot 27\right) \cdot k} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \left(\mathsf{neg}\left(j \cdot 27\right)\right) \cdot k} \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \left(\mathsf{neg}\left(\color{blue}{j \cdot 27}\right)\right) \cdot k \]

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

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \color{blue}{\left(j \cdot \left(\mathsf{neg}\left(27\right)\right)\right)} \cdot k \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \left(j \cdot \color{blue}{-27}\right) \cdot k \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \color{blue}{j \cdot \left(-27 \cdot k\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + j \cdot \color{blue}{\left(-27 \cdot k\right)} \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \color{blue}{\left(-27 \cdot k\right) \cdot j} \]

      10. +-commutative N/A

          \[\leadsto \color{blue}{\left(-27 \cdot k\right) \cdot j + \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)} \]

      11. lower-fma.f64 51.2

          \[\leadsto \color{blue}{\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)\right)} \]

   7. Applied rewrites 51.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)\right)} \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 4 \cdot \left(i \cdot x\right)}\right) \]
   9. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + \left(b \cdot c - 4 \cdot \left(i \cdot x\right)\right)}\right) \]

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

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + \color{blue}{\left(b \cdot c + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(i \cdot x\right)\right)}\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + \left(b \cdot c + \color{blue}{-4} \cdot \left(i \cdot x\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + \color{blue}{\left(-4 \cdot \left(i \cdot x\right) + b \cdot c\right)}\right) \]

      5. associate-+r+ N/A

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + -4 \cdot \left(i \cdot x\right)\right) + b \cdot c}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \left(\color{blue}{\left(18 \cdot t\right) \cdot \left(x \cdot \left(y \cdot z\right)\right)} + -4 \cdot \left(i \cdot x\right)\right) + b \cdot c\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \left(\left(18 \cdot t\right) \cdot \color{blue}{\left(\left(y \cdot z\right) \cdot x\right)} + -4 \cdot \left(i \cdot x\right)\right) + b \cdot c\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \left(\color{blue}{\left(\left(18 \cdot t\right) \cdot \left(y \cdot z\right)\right) \cdot x} + -4 \cdot \left(i \cdot x\right)\right) + b \cdot c\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \left(\color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right)} \cdot x + -4 \cdot \left(i \cdot x\right)\right) + b \cdot c\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \left(\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right) \cdot x + \color{blue}{\left(-4 \cdot i\right) \cdot x}\right) + b \cdot c\right) \]

      11. distribute-rgt-in N/A

          \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{x \cdot \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + -4 \cdot i\right)} + b \cdot c\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, x \cdot \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + \color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot i\right) + b \cdot c\right) \]

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

          \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, x \cdot \color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right)} + b \cdot c\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right) \cdot x} + b \cdot c\right) \]

      15. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{\mathsf{fma}\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i, x, b \cdot c\right)}\right) \]

   10. Applied rewrites 94.0%

       \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(18 \cdot t\right) \cdot z, y, -4 \cdot i\right), x, c \cdot b\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(c \cdot b + \left(t \cdot \left(z \cdot \left(\left(18 \cdot x\right) \cdot y\right)\right) - \left(4 \cdot a\right) \cdot t\right)\right) - i \cdot \left(4 \cdot x\right)\right) - k \cdot \left(27 \cdot j\right) \leq 8 \cdot 10^{+305}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot j, -27, \mathsf{fma}\left(i \cdot x, -4, \mathsf{fma}\left(\mathsf{fma}\left(z, \left(18 \cdot x\right) \cdot y, -4 \cdot a\right), t, c \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(\mathsf{fma}\left(\left(t \cdot 18\right) \cdot z, y, -4 \cdot i\right), x, c \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\\\ [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := k \cdot \left(27 \cdot j\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot k, j, \left(-4 \cdot x\right) \cdot i\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \left(i \cdot x\right) \cdot -4\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+20}:\\ \;\;\;\;\left(\left(t \cdot x\right) \cdot 18\right) \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot k, j, c \cdot b\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (* k (* 27.0 j))))
   (if (<= t_1 -5e+132)
     (fma (* -27.0 k) j (* (* -4.0 x) i))
     (if (<= t_1 1e-9)
       (fma c b (* (* i x) -4.0))
       (if (<= t_1 5e+20)
         (* (* (* t x) 18.0) (* z y))
         (fma (* -27.0 k) j (* c b)))))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = k * (27.0 * j);
	double tmp;
	if (t_1 <= -5e+132) {
		tmp = fma((-27.0 * k), j, ((-4.0 * x) * i));
	} else if (t_1 <= 1e-9) {
		tmp = fma(c, b, ((i * x) * -4.0));
	} else if (t_1 <= 5e+20) {
		tmp = ((t * x) * 18.0) * (z * y);
	} else {
		tmp = fma((-27.0 * k), j, (c * b));
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = Float64(k * Float64(27.0 * j))
	tmp = 0.0
	if (t_1 <= -5e+132)
		tmp = fma(Float64(-27.0 * k), j, Float64(Float64(-4.0 * x) * i));
	elseif (t_1 <= 1e-9)
		tmp = fma(c, b, Float64(Float64(i * x) * -4.0));
	elseif (t_1 <= 5e+20)
		tmp = Float64(Float64(Float64(t * x) * 18.0) * Float64(z * y));
	else
		tmp = fma(Float64(-27.0 * k), j, Float64(c * b));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(k * N[(27.0 * j), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+132], N[(N[(-27.0 * k), $MachinePrecision] * j + N[(N[(-4.0 * x), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-9], N[(c * b + N[(N[(i * x), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+20], N[(N[(N[(t * x), $MachinePrecision] * 18.0), $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision], N[(N[(-27.0 * k), $MachinePrecision] * j + N[(c * b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 j #s(literal 27 binary64)) k) < -5.0000000000000001e132

   1. Initial program 87.1%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) - \left(4 \cdot \left(a \cdot t\right) + \left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)} \]

   5. Applied rewrites 81.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-4, i, \left(\left(z \cdot y\right) \cdot t\right) \cdot 18\right), x, \mathsf{fma}\left(-27 \cdot k, j, -4 \cdot \left(a \cdot t\right)\right)\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto -27 \cdot \left(j \cdot k\right) + \color{blue}{-4 \cdot \left(i \cdot x\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 71.5%

      \[\leadsto \mathsf{fma}\left(-4 \cdot i, \color{blue}{x}, -27 \cdot \left(k \cdot j\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 78.8%

       \[\leadsto \mathsf{fma}\left(k \cdot -27, j, \left(x \cdot -4\right) \cdot i\right) \]

   if -5.0000000000000001e132 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 1.00000000000000006e-9

   1. Initial program 88.2%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

         \[\leadsto \color{blue}{b \cdot c + \left(\mathsf{neg}\left(\left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{c \cdot b} + \left(\mathsf{neg}\left(\left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{neg}\left(\left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)} \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(\mathsf{neg}\left(4 \cdot \left(i \cdot x\right)\right)\right) + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(i \cdot x\right)} + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{-4} \cdot \left(i \cdot x\right) + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, -4 \cdot \color{blue}{\left(x \cdot i\right)} + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(-4 \cdot x\right) \cdot i} + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(c, b, \left(-4 \cdot x\right) \cdot i + \color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot \left(j \cdot k\right)}\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(c, b, \left(-4 \cdot x\right) \cdot i + \color{blue}{-27} \cdot \left(j \cdot k\right)\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \left(j \cdot k\right)\right)}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{-4 \cdot x}, i, -27 \cdot \left(j \cdot k\right)\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, \color{blue}{-27 \cdot \left(j \cdot k\right)}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \color{blue}{\left(k \cdot j\right)}\right)\right) \]

      15. lower-*.f64 61.8

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \color{blue}{\left(k \cdot j\right)}\right)\right) \]

   5. Applied rewrites 61.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \left(k \cdot j\right)\right)\right)} \]

   6. Taylor expanded in k around 0

      \[\leadsto \mathsf{fma}\left(c, b, -4 \cdot \left(i \cdot x\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 58.6%

      \[\leadsto \mathsf{fma}\left(c, b, \left(i \cdot x\right) \cdot -4\right) \]

   if 1.00000000000000006e-9 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 5e20

   1. Initial program 72.0%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot 18} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot 18} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot \left(y \cdot z\right)\right) \cdot t\right)} \cdot 18 \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot \left(y \cdot z\right)\right) \cdot t\right)} \cdot 18 \]

      5. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(y \cdot z\right) \cdot x\right)} \cdot t\right) \cdot 18 \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(y \cdot z\right) \cdot x\right)} \cdot t\right) \cdot 18 \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(z \cdot y\right)} \cdot x\right) \cdot t\right) \cdot 18 \]

      8. lower-*.f64 65.3

         \[\leadsto \left(\left(\color{blue}{\left(z \cdot y\right)} \cdot x\right) \cdot t\right) \cdot 18 \]

   5. Applied rewrites 65.3%

      \[\leadsto \color{blue}{\left(\left(\left(z \cdot y\right) \cdot x\right) \cdot t\right) \cdot 18} \]
   6. Step-by-step derivation

   7. Applied rewrites 78.6%

      \[\leadsto \left(y \cdot z\right) \cdot \color{blue}{\left(\left(t \cdot x\right) \cdot 18\right)} \]

   if 5e20 < (*.f64 (*.f64 j #s(literal 27 binary64)) k)

   1. Initial program 80.9%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

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

         \[\leadsto \color{blue}{\left(b \cdot c + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot t\right)\right)} - \left(j \cdot 27\right) \cdot k \]

      2. metadata-eval N/A

         \[\leadsto \left(b \cdot c + \color{blue}{-4} \cdot \left(a \cdot t\right)\right) - \left(j \cdot 27\right) \cdot k \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(-4 \cdot \left(a \cdot t\right) + b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-4 \cdot a\right) \cdot t} + b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-4 \cdot a}, t, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, \color{blue}{c \cdot b}\right) - \left(j \cdot 27\right) \cdot k \]

      8. lower-*.f64 78.8

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, \color{blue}{c \cdot b}\right) - \left(j \cdot 27\right) \cdot k \]

   5. Applied rewrites 78.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)} - \left(j \cdot 27\right) \cdot k \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) - \left(j \cdot 27\right) \cdot k} \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) - \color{blue}{\left(j \cdot 27\right) \cdot k} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \left(\mathsf{neg}\left(j \cdot 27\right)\right) \cdot k} \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \left(\mathsf{neg}\left(\color{blue}{j \cdot 27}\right)\right) \cdot k \]

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

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \color{blue}{\left(j \cdot \left(\mathsf{neg}\left(27\right)\right)\right)} \cdot k \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \left(j \cdot \color{blue}{-27}\right) \cdot k \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \color{blue}{j \cdot \left(-27 \cdot k\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + j \cdot \color{blue}{\left(-27 \cdot k\right)} \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \color{blue}{\left(-27 \cdot k\right) \cdot j} \]

      10. +-commutative N/A

          \[\leadsto \color{blue}{\left(-27 \cdot k\right) \cdot j + \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)} \]

      11. lower-fma.f64 80.7

          \[\leadsto \color{blue}{\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)\right)} \]

   7. Applied rewrites 80.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)\right)} \]

   8. Taylor expanded in c around inf

      \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{b \cdot c}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{c \cdot b}\right) \]

      2. lower-*.f64 67.6

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{c \cdot b}\right) \]

   10. Applied rewrites 67.6%

       \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{c \cdot b}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \cdot \left(27 \cdot j\right) \leq -5 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot k, j, \left(-4 \cdot x\right) \cdot i\right)\\ \mathbf{elif}\;k \cdot \left(27 \cdot j\right) \leq 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \left(i \cdot x\right) \cdot -4\right)\\ \mathbf{elif}\;k \cdot \left(27 \cdot j\right) \leq 5 \cdot 10^{+20}:\\ \;\;\;\;\left(\left(t \cdot x\right) \cdot 18\right) \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot k, j, c \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\\\ [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := k \cdot \left(27 \cdot j\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+132}:\\ \;\;\;\;c \cdot b - t\_1\\ \mathbf{elif}\;t\_1 \leq 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \left(i \cdot x\right) \cdot -4\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+20}:\\ \;\;\;\;\left(\left(t \cdot x\right) \cdot 18\right) \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot k, j, c \cdot b\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (* k (* 27.0 j))))
   (if (<= t_1 -5e+132)
     (- (* c b) t_1)
     (if (<= t_1 1e-9)
       (fma c b (* (* i x) -4.0))
       (if (<= t_1 5e+20)
         (* (* (* t x) 18.0) (* z y))
         (fma (* -27.0 k) j (* c b)))))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = k * (27.0 * j);
	double tmp;
	if (t_1 <= -5e+132) {
		tmp = (c * b) - t_1;
	} else if (t_1 <= 1e-9) {
		tmp = fma(c, b, ((i * x) * -4.0));
	} else if (t_1 <= 5e+20) {
		tmp = ((t * x) * 18.0) * (z * y);
	} else {
		tmp = fma((-27.0 * k), j, (c * b));
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = Float64(k * Float64(27.0 * j))
	tmp = 0.0
	if (t_1 <= -5e+132)
		tmp = Float64(Float64(c * b) - t_1);
	elseif (t_1 <= 1e-9)
		tmp = fma(c, b, Float64(Float64(i * x) * -4.0));
	elseif (t_1 <= 5e+20)
		tmp = Float64(Float64(Float64(t * x) * 18.0) * Float64(z * y));
	else
		tmp = fma(Float64(-27.0 * k), j, Float64(c * b));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(k * N[(27.0 * j), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+132], N[(N[(c * b), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[t$95$1, 1e-9], N[(c * b + N[(N[(i * x), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+20], N[(N[(N[(t * x), $MachinePrecision] * 18.0), $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision], N[(N[(-27.0 * k), $MachinePrecision] * j + N[(c * b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 j #s(literal 27 binary64)) k) < -5.0000000000000001e132

   1. Initial program 87.1%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{c \cdot b} - \left(j \cdot 27\right) \cdot k \]

      2. lower-*.f64 73.2

         \[\leadsto \color{blue}{c \cdot b} - \left(j \cdot 27\right) \cdot k \]

   5. Applied rewrites 73.2%

      \[\leadsto \color{blue}{c \cdot b} - \left(j \cdot 27\right) \cdot k \]

   if -5.0000000000000001e132 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 1.00000000000000006e-9

   1. Initial program 88.2%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

         \[\leadsto \color{blue}{b \cdot c + \left(\mathsf{neg}\left(\left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{c \cdot b} + \left(\mathsf{neg}\left(\left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{neg}\left(\left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)} \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(\mathsf{neg}\left(4 \cdot \left(i \cdot x\right)\right)\right) + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(i \cdot x\right)} + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{-4} \cdot \left(i \cdot x\right) + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, -4 \cdot \color{blue}{\left(x \cdot i\right)} + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(-4 \cdot x\right) \cdot i} + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(c, b, \left(-4 \cdot x\right) \cdot i + \color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot \left(j \cdot k\right)}\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(c, b, \left(-4 \cdot x\right) \cdot i + \color{blue}{-27} \cdot \left(j \cdot k\right)\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \left(j \cdot k\right)\right)}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{-4 \cdot x}, i, -27 \cdot \left(j \cdot k\right)\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, \color{blue}{-27 \cdot \left(j \cdot k\right)}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \color{blue}{\left(k \cdot j\right)}\right)\right) \]

      15. lower-*.f64 61.8

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \color{blue}{\left(k \cdot j\right)}\right)\right) \]

   5. Applied rewrites 61.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \left(k \cdot j\right)\right)\right)} \]

   6. Taylor expanded in k around 0

      \[\leadsto \mathsf{fma}\left(c, b, -4 \cdot \left(i \cdot x\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 58.6%

      \[\leadsto \mathsf{fma}\left(c, b, \left(i \cdot x\right) \cdot -4\right) \]

   if 1.00000000000000006e-9 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 5e20

   1. Initial program 72.0%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot 18} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot 18} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot \left(y \cdot z\right)\right) \cdot t\right)} \cdot 18 \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot \left(y \cdot z\right)\right) \cdot t\right)} \cdot 18 \]

      5. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(y \cdot z\right) \cdot x\right)} \cdot t\right) \cdot 18 \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(y \cdot z\right) \cdot x\right)} \cdot t\right) \cdot 18 \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(z \cdot y\right)} \cdot x\right) \cdot t\right) \cdot 18 \]

      8. lower-*.f64 65.3

         \[\leadsto \left(\left(\color{blue}{\left(z \cdot y\right)} \cdot x\right) \cdot t\right) \cdot 18 \]

   5. Applied rewrites 65.3%

      \[\leadsto \color{blue}{\left(\left(\left(z \cdot y\right) \cdot x\right) \cdot t\right) \cdot 18} \]
   6. Step-by-step derivation

   7. Applied rewrites 78.6%

      \[\leadsto \left(y \cdot z\right) \cdot \color{blue}{\left(\left(t \cdot x\right) \cdot 18\right)} \]

   if 5e20 < (*.f64 (*.f64 j #s(literal 27 binary64)) k)

   1. Initial program 80.9%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

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

         \[\leadsto \color{blue}{\left(b \cdot c + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot t\right)\right)} - \left(j \cdot 27\right) \cdot k \]

      2. metadata-eval N/A

         \[\leadsto \left(b \cdot c + \color{blue}{-4} \cdot \left(a \cdot t\right)\right) - \left(j \cdot 27\right) \cdot k \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(-4 \cdot \left(a \cdot t\right) + b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-4 \cdot a\right) \cdot t} + b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-4 \cdot a}, t, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, \color{blue}{c \cdot b}\right) - \left(j \cdot 27\right) \cdot k \]

      8. lower-*.f64 78.8

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, \color{blue}{c \cdot b}\right) - \left(j \cdot 27\right) \cdot k \]

   5. Applied rewrites 78.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)} - \left(j \cdot 27\right) \cdot k \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) - \left(j \cdot 27\right) \cdot k} \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) - \color{blue}{\left(j \cdot 27\right) \cdot k} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \left(\mathsf{neg}\left(j \cdot 27\right)\right) \cdot k} \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \left(\mathsf{neg}\left(\color{blue}{j \cdot 27}\right)\right) \cdot k \]

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

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \color{blue}{\left(j \cdot \left(\mathsf{neg}\left(27\right)\right)\right)} \cdot k \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \left(j \cdot \color{blue}{-27}\right) \cdot k \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \color{blue}{j \cdot \left(-27 \cdot k\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + j \cdot \color{blue}{\left(-27 \cdot k\right)} \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \color{blue}{\left(-27 \cdot k\right) \cdot j} \]

      10. +-commutative N/A

          \[\leadsto \color{blue}{\left(-27 \cdot k\right) \cdot j + \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)} \]

      11. lower-fma.f64 80.7

          \[\leadsto \color{blue}{\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)\right)} \]

   7. Applied rewrites 80.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)\right)} \]

   8. Taylor expanded in c around inf

      \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{b \cdot c}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{c \cdot b}\right) \]

      2. lower-*.f64 67.6

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{c \cdot b}\right) \]

   10. Applied rewrites 67.6%

       \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{c \cdot b}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \cdot \left(27 \cdot j\right) \leq -5 \cdot 10^{+132}:\\ \;\;\;\;c \cdot b - k \cdot \left(27 \cdot j\right)\\ \mathbf{elif}\;k \cdot \left(27 \cdot j\right) \leq 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \left(i \cdot x\right) \cdot -4\right)\\ \mathbf{elif}\;k \cdot \left(27 \cdot j\right) \leq 5 \cdot 10^{+20}:\\ \;\;\;\;\left(\left(t \cdot x\right) \cdot 18\right) \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot k, j, c \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\\\ [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-27 \cdot k, j, c \cdot b\right)\\ \mathbf{if}\;c \cdot b \leq -50:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \cdot b \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(i, x, a \cdot t\right), -4, \left(k \cdot j\right) \cdot -27\right)\\ \mathbf{elif}\;c \cdot b \leq 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(-4, i, \left(\left(z \cdot y\right) \cdot t\right) \cdot 18\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (fma (* -27.0 k) j (* c b))))
   (if (<= (* c b) -50.0)
     t_1
     (if (<= (* c b) 5e-21)
       (fma (fma i x (* a t)) -4.0 (* (* k j) -27.0))
       (if (<= (* c b) 1e+36)
         (* (fma -4.0 i (* (* (* z y) t) 18.0)) x)
         t_1)))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = fma((-27.0 * k), j, (c * b));
	double tmp;
	if ((c * b) <= -50.0) {
		tmp = t_1;
	} else if ((c * b) <= 5e-21) {
		tmp = fma(fma(i, x, (a * t)), -4.0, ((k * j) * -27.0));
	} else if ((c * b) <= 1e+36) {
		tmp = fma(-4.0, i, (((z * y) * t) * 18.0)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = fma(Float64(-27.0 * k), j, Float64(c * b))
	tmp = 0.0
	if (Float64(c * b) <= -50.0)
		tmp = t_1;
	elseif (Float64(c * b) <= 5e-21)
		tmp = fma(fma(i, x, Float64(a * t)), -4.0, Float64(Float64(k * j) * -27.0));
	elseif (Float64(c * b) <= 1e+36)
		tmp = Float64(fma(-4.0, i, Float64(Float64(Float64(z * y) * t) * 18.0)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(N[(-27.0 * k), $MachinePrecision] * j + N[(c * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(c * b), $MachinePrecision], -50.0], t$95$1, If[LessEqual[N[(c * b), $MachinePrecision], 5e-21], N[(N[(i * x + N[(a * t), $MachinePrecision]), $MachinePrecision] * -4.0 + N[(N[(k * j), $MachinePrecision] * -27.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(c * b), $MachinePrecision], 1e+36], N[(N[(-4.0 * i + N[(N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] * 18.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 b c) < -50 or 1.00000000000000004e36 < (*.f64 b c)

   1. Initial program 82.9%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

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

         \[\leadsto \color{blue}{\left(b \cdot c + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot t\right)\right)} - \left(j \cdot 27\right) \cdot k \]

      2. metadata-eval N/A

         \[\leadsto \left(b \cdot c + \color{blue}{-4} \cdot \left(a \cdot t\right)\right) - \left(j \cdot 27\right) \cdot k \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(-4 \cdot \left(a \cdot t\right) + b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-4 \cdot a\right) \cdot t} + b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-4 \cdot a}, t, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, \color{blue}{c \cdot b}\right) - \left(j \cdot 27\right) \cdot k \]

      8. lower-*.f64 69.4

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, \color{blue}{c \cdot b}\right) - \left(j \cdot 27\right) \cdot k \]

   5. Applied rewrites 69.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)} - \left(j \cdot 27\right) \cdot k \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) - \left(j \cdot 27\right) \cdot k} \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) - \color{blue}{\left(j \cdot 27\right) \cdot k} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \left(\mathsf{neg}\left(j \cdot 27\right)\right) \cdot k} \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \left(\mathsf{neg}\left(\color{blue}{j \cdot 27}\right)\right) \cdot k \]

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

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \color{blue}{\left(j \cdot \left(\mathsf{neg}\left(27\right)\right)\right)} \cdot k \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \left(j \cdot \color{blue}{-27}\right) \cdot k \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \color{blue}{j \cdot \left(-27 \cdot k\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + j \cdot \color{blue}{\left(-27 \cdot k\right)} \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \color{blue}{\left(-27 \cdot k\right) \cdot j} \]

      10. +-commutative N/A

          \[\leadsto \color{blue}{\left(-27 \cdot k\right) \cdot j + \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)} \]

      11. lower-fma.f64 70.1

          \[\leadsto \color{blue}{\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)\right)} \]

   7. Applied rewrites 70.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)\right)} \]

   8. Taylor expanded in c around inf

      \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{b \cdot c}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{c \cdot b}\right) \]

      2. lower-*.f64 64.4

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{c \cdot b}\right) \]

   10. Applied rewrites 64.4%

       \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{c \cdot b}\right) \]

   if -50 < (*.f64 b c) < 4.99999999999999973e-21

   1. Initial program 89.0%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) - \left(4 \cdot \left(a \cdot t\right) + \left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)} \]

   5. Applied rewrites 92.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-4, i, \left(\left(z \cdot y\right) \cdot t\right) \cdot 18\right), x, \mathsf{fma}\left(-27 \cdot k, j, -4 \cdot \left(a \cdot t\right)\right)\right)} \]

   6. Taylor expanded in z around 0

      \[\leadsto -27 \cdot \left(j \cdot k\right) + \color{blue}{\left(-4 \cdot \left(a \cdot t\right) + -4 \cdot \left(i \cdot x\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 80.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(i, x, a \cdot t\right), \color{blue}{-4}, -27 \cdot \left(k \cdot j\right)\right) \]

   if 4.99999999999999973e-21 < (*.f64 b c) < 1.00000000000000004e36

   1. Initial program 92.9%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right) \cdot x} \]

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

         \[\leadsto \color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot i\right)} \cdot x \]

      4. metadata-eval N/A

         \[\leadsto \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + \color{blue}{-4} \cdot i\right) \cdot x \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-4 \cdot i + 18 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right)} \cdot x \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4, i, 18 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right)} \cdot x \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-4, i, \color{blue}{\left(t \cdot \left(y \cdot z\right)\right) \cdot 18}\right) \cdot x \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4, i, \color{blue}{\left(t \cdot \left(y \cdot z\right)\right) \cdot 18}\right) \cdot x \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-4, i, \color{blue}{\left(\left(y \cdot z\right) \cdot t\right)} \cdot 18\right) \cdot x \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-4, i, \color{blue}{\left(\left(y \cdot z\right) \cdot t\right)} \cdot 18\right) \cdot x \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-4, i, \left(\color{blue}{\left(z \cdot y\right)} \cdot t\right) \cdot 18\right) \cdot x \]

      12. lower-*.f64 79.1

          \[\leadsto \mathsf{fma}\left(-4, i, \left(\color{blue}{\left(z \cdot y\right)} \cdot t\right) \cdot 18\right) \cdot x \]

   5. Applied rewrites 79.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4, i, \left(\left(z \cdot y\right) \cdot t\right) \cdot 18\right) \cdot x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot b \leq -50:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot k, j, c \cdot b\right)\\ \mathbf{elif}\;c \cdot b \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(i, x, a \cdot t\right), -4, \left(k \cdot j\right) \cdot -27\right)\\ \mathbf{elif}\;c \cdot b \leq 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(-4, i, \left(\left(z \cdot y\right) \cdot t\right) \cdot 18\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot k, j, c \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 83.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\\\ [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(\mathsf{fma}\left(-4, a, \left(\left(z \cdot y\right) \cdot x\right) \cdot 18\right), t, c \cdot b\right)\right)\\ \mathbf{if}\;z \leq -7.8 \cdot 10^{-107}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \mathsf{fma}\left(\mathsf{fma}\left(i, x, a \cdot t\right), -4, \left(k \cdot j\right) \cdot -27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1
         (fma
          (* -27.0 k)
          j
          (fma (fma -4.0 a (* (* (* z y) x) 18.0)) t (* c b)))))
   (if (<= z -7.8e-107)
     t_1
     (if (<= z 9.2e+94)
       (fma c b (fma (fma i x (* a t)) -4.0 (* (* k j) -27.0)))
       t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = fma((-27.0 * k), j, fma(fma(-4.0, a, (((z * y) * x) * 18.0)), t, (c * b)));
	double tmp;
	if (z <= -7.8e-107) {
		tmp = t_1;
	} else if (z <= 9.2e+94) {
		tmp = fma(c, b, fma(fma(i, x, (a * t)), -4.0, ((k * j) * -27.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = fma(Float64(-27.0 * k), j, fma(fma(-4.0, a, Float64(Float64(Float64(z * y) * x) * 18.0)), t, Float64(c * b)))
	tmp = 0.0
	if (z <= -7.8e-107)
		tmp = t_1;
	elseif (z <= 9.2e+94)
		tmp = fma(c, b, fma(fma(i, x, Float64(a * t)), -4.0, Float64(Float64(k * j) * -27.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(N[(-27.0 * k), $MachinePrecision] * j + N[(N[(-4.0 * a + N[(N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision] * 18.0), $MachinePrecision]), $MachinePrecision] * t + N[(c * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -7.8e-107], t$95$1, If[LessEqual[z, 9.2e+94], N[(c * b + N[(N[(i * x + N[(a * t), $MachinePrecision]), $MachinePrecision] * -4.0 + N[(N[(k * j), $MachinePrecision] * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.8000000000000002e-107 or 9.1999999999999999e94 < z

   1. Initial program 84.1%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - \left(4 \cdot \left(a \cdot t\right) + 27 \cdot \left(j \cdot k\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - \color{blue}{\left(27 \cdot \left(j \cdot k\right) + 4 \cdot \left(a \cdot t\right)\right)} \]

      2. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 27 \cdot \left(j \cdot k\right)\right) - 4 \cdot \left(a \cdot t\right)} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right)} - 4 \cdot \left(a \cdot t\right) \]

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

         \[\leadsto \left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) + \color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot \left(j \cdot k\right)}\right) - 4 \cdot \left(a \cdot t\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) + \color{blue}{-27} \cdot \left(j \cdot k\right)\right) - 4 \cdot \left(a \cdot t\right) \]

      6. +-commutative N/A

         \[\leadsto \color{blue}{\left(-27 \cdot \left(j \cdot k\right) + \left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right)\right)} - 4 \cdot \left(a \cdot t\right) \]

      7. associate--l+ N/A

         \[\leadsto \color{blue}{-27 \cdot \left(j \cdot k\right) + \left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 4 \cdot \left(a \cdot t\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto -27 \cdot \color{blue}{\left(k \cdot j\right)} + \left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 4 \cdot \left(a \cdot t\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-27 \cdot k\right) \cdot j} + \left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 4 \cdot \left(a \cdot t\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-27 \cdot k, j, \left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 4 \cdot \left(a \cdot t\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-27 \cdot k}, j, \left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 4 \cdot \left(a \cdot t\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{\left(b \cdot c + 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right)\right)} - 4 \cdot \left(a \cdot t\right)\right) \]

   5. Applied rewrites 85.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(\mathsf{fma}\left(-4, a, \left(\left(z \cdot y\right) \cdot x\right) \cdot 18\right), t, c \cdot b\right)\right)} \]

   if -7.8000000000000002e-107 < z < 9.1999999999999999e94

   1. Initial program 87.9%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

         \[\leadsto \color{blue}{b \cdot c + \left(\mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + \left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{c \cdot b} + \left(\mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + \left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + \left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{neg}\left(\color{blue}{\left(\left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right) + 27 \cdot \left(j \cdot k\right)\right)}\right)\right) \]

      5. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(\mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right)\right) + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{fma}\left(c, b, \left(\mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot \left(j \cdot k\right)}\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(\mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right)\right) + \color{blue}{-27} \cdot \left(j \cdot k\right)\right) \]

      8. distribute-lft-out N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(\mathsf{neg}\left(\color{blue}{4 \cdot \left(a \cdot t + i \cdot x\right)}\right)\right) + -27 \cdot \left(j \cdot k\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(\mathsf{neg}\left(\color{blue}{\left(a \cdot t + i \cdot x\right) \cdot 4}\right)\right) + -27 \cdot \left(j \cdot k\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(a \cdot t + i \cdot x\right) \cdot \left(\mathsf{neg}\left(4\right)\right)} + -27 \cdot \left(j \cdot k\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(c, b, \left(a \cdot t + i \cdot x\right) \cdot \color{blue}{-4} + -27 \cdot \left(j \cdot k\right)\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\mathsf{fma}\left(a \cdot t + i \cdot x, -4, -27 \cdot \left(j \cdot k\right)\right)}\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{i \cdot x + a \cdot t}, -4, -27 \cdot \left(j \cdot k\right)\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(i, x, a \cdot t\right)}, -4, -27 \cdot \left(j \cdot k\right)\right)\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\mathsf{fma}\left(i, x, \color{blue}{a \cdot t}\right), -4, -27 \cdot \left(j \cdot k\right)\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\mathsf{fma}\left(i, x, a \cdot t\right), -4, \color{blue}{-27 \cdot \left(j \cdot k\right)}\right)\right) \]

   5. Applied rewrites 88.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{fma}\left(\mathsf{fma}\left(i, x, a \cdot t\right), -4, -27 \cdot \left(k \cdot j\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{-107}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(\mathsf{fma}\left(-4, a, \left(\left(z \cdot y\right) \cdot x\right) \cdot 18\right), t, c \cdot b\right)\right)\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \mathsf{fma}\left(\mathsf{fma}\left(i, x, a \cdot t\right), -4, \left(k \cdot j\right) \cdot -27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(\mathsf{fma}\left(-4, a, \left(\left(z \cdot y\right) \cdot x\right) \cdot 18\right), t, c \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\\\ [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(c, b, \left(\left(\left(z \cdot y\right) \cdot x\right) \cdot t\right) \cdot 18\right)\right)\\ \mathbf{if}\;z \leq -3.6 \cdot 10^{-66}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \mathsf{fma}\left(\mathsf{fma}\left(i, x, a \cdot t\right), -4, \left(k \cdot j\right) \cdot -27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (fma (* -27.0 k) j (fma c b (* (* (* (* z y) x) t) 18.0)))))
   (if (<= z -3.6e-66)
     t_1
     (if (<= z 1.35e+140)
       (fma c b (fma (fma i x (* a t)) -4.0 (* (* k j) -27.0)))
       t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = fma((-27.0 * k), j, fma(c, b, ((((z * y) * x) * t) * 18.0)));
	double tmp;
	if (z <= -3.6e-66) {
		tmp = t_1;
	} else if (z <= 1.35e+140) {
		tmp = fma(c, b, fma(fma(i, x, (a * t)), -4.0, ((k * j) * -27.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = fma(Float64(-27.0 * k), j, fma(c, b, Float64(Float64(Float64(Float64(z * y) * x) * t) * 18.0)))
	tmp = 0.0
	if (z <= -3.6e-66)
		tmp = t_1;
	elseif (z <= 1.35e+140)
		tmp = fma(c, b, fma(fma(i, x, Float64(a * t)), -4.0, Float64(Float64(k * j) * -27.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(N[(-27.0 * k), $MachinePrecision] * j + N[(c * b + N[(N[(N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision] * t), $MachinePrecision] * 18.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.6e-66], t$95$1, If[LessEqual[z, 1.35e+140], N[(c * b + N[(N[(i * x + N[(a * t), $MachinePrecision]), $MachinePrecision] * -4.0 + N[(N[(k * j), $MachinePrecision] * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.60000000000000012e-66 or 1.35000000000000009e140 < z

   1. Initial program 82.5%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - \left(4 \cdot \left(a \cdot t\right) + 27 \cdot \left(j \cdot k\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - \color{blue}{\left(27 \cdot \left(j \cdot k\right) + 4 \cdot \left(a \cdot t\right)\right)} \]

      2. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 27 \cdot \left(j \cdot k\right)\right) - 4 \cdot \left(a \cdot t\right)} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right)} - 4 \cdot \left(a \cdot t\right) \]

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

         \[\leadsto \left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) + \color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot \left(j \cdot k\right)}\right) - 4 \cdot \left(a \cdot t\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) + \color{blue}{-27} \cdot \left(j \cdot k\right)\right) - 4 \cdot \left(a \cdot t\right) \]

      6. +-commutative N/A

         \[\leadsto \color{blue}{\left(-27 \cdot \left(j \cdot k\right) + \left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right)\right)} - 4 \cdot \left(a \cdot t\right) \]

      7. associate--l+ N/A

         \[\leadsto \color{blue}{-27 \cdot \left(j \cdot k\right) + \left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 4 \cdot \left(a \cdot t\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto -27 \cdot \color{blue}{\left(k \cdot j\right)} + \left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 4 \cdot \left(a \cdot t\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-27 \cdot k\right) \cdot j} + \left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 4 \cdot \left(a \cdot t\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-27 \cdot k, j, \left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 4 \cdot \left(a \cdot t\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-27 \cdot k}, j, \left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 4 \cdot \left(a \cdot t\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{\left(b \cdot c + 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right)\right)} - 4 \cdot \left(a \cdot t\right)\right) \]

   5. Applied rewrites 85.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(\mathsf{fma}\left(-4, a, \left(\left(z \cdot y\right) \cdot x\right) \cdot 18\right), t, c \cdot b\right)\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 74.9%

      \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(c, b, \left(\left(\left(z \cdot y\right) \cdot x\right) \cdot t\right) \cdot 18\right)\right) \]

   if -3.60000000000000012e-66 < z < 1.35000000000000009e140

   1. Initial program 88.4%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

         \[\leadsto \color{blue}{b \cdot c + \left(\mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + \left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{c \cdot b} + \left(\mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + \left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + \left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{neg}\left(\color{blue}{\left(\left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right) + 27 \cdot \left(j \cdot k\right)\right)}\right)\right) \]

      5. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(\mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right)\right) + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{fma}\left(c, b, \left(\mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot \left(j \cdot k\right)}\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(\mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right)\right) + \color{blue}{-27} \cdot \left(j \cdot k\right)\right) \]

      8. distribute-lft-out N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(\mathsf{neg}\left(\color{blue}{4 \cdot \left(a \cdot t + i \cdot x\right)}\right)\right) + -27 \cdot \left(j \cdot k\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(\mathsf{neg}\left(\color{blue}{\left(a \cdot t + i \cdot x\right) \cdot 4}\right)\right) + -27 \cdot \left(j \cdot k\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(a \cdot t + i \cdot x\right) \cdot \left(\mathsf{neg}\left(4\right)\right)} + -27 \cdot \left(j \cdot k\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(c, b, \left(a \cdot t + i \cdot x\right) \cdot \color{blue}{-4} + -27 \cdot \left(j \cdot k\right)\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\mathsf{fma}\left(a \cdot t + i \cdot x, -4, -27 \cdot \left(j \cdot k\right)\right)}\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{i \cdot x + a \cdot t}, -4, -27 \cdot \left(j \cdot k\right)\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(i, x, a \cdot t\right)}, -4, -27 \cdot \left(j \cdot k\right)\right)\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\mathsf{fma}\left(i, x, \color{blue}{a \cdot t}\right), -4, -27 \cdot \left(j \cdot k\right)\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\mathsf{fma}\left(i, x, a \cdot t\right), -4, \color{blue}{-27 \cdot \left(j \cdot k\right)}\right)\right) \]

   5. Applied rewrites 86.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{fma}\left(\mathsf{fma}\left(i, x, a \cdot t\right), -4, -27 \cdot \left(k \cdot j\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{-66}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(c, b, \left(\left(\left(z \cdot y\right) \cdot x\right) \cdot t\right) \cdot 18\right)\right)\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \mathsf{fma}\left(\mathsf{fma}\left(i, x, a \cdot t\right), -4, \left(k \cdot j\right) \cdot -27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(c, b, \left(\left(\left(z \cdot y\right) \cdot x\right) \cdot t\right) \cdot 18\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 54.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\\\ [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := k \cdot \left(27 \cdot j\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+132}:\\ \;\;\;\;c \cdot b - t\_1\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+189}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \left(i \cdot x\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot k, j, c \cdot b\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (* k (* 27.0 j))))
   (if (<= t_1 -5e+132)
     (- (* c b) t_1)
     (if (<= t_1 5e+189)
       (fma c b (* (* i x) -4.0))
       (fma (* -27.0 k) j (* c b))))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = k * (27.0 * j);
	double tmp;
	if (t_1 <= -5e+132) {
		tmp = (c * b) - t_1;
	} else if (t_1 <= 5e+189) {
		tmp = fma(c, b, ((i * x) * -4.0));
	} else {
		tmp = fma((-27.0 * k), j, (c * b));
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = Float64(k * Float64(27.0 * j))
	tmp = 0.0
	if (t_1 <= -5e+132)
		tmp = Float64(Float64(c * b) - t_1);
	elseif (t_1 <= 5e+189)
		tmp = fma(c, b, Float64(Float64(i * x) * -4.0));
	else
		tmp = fma(Float64(-27.0 * k), j, Float64(c * b));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(k * N[(27.0 * j), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+132], N[(N[(c * b), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[t$95$1, 5e+189], N[(c * b + N[(N[(i * x), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], N[(N[(-27.0 * k), $MachinePrecision] * j + N[(c * b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 j #s(literal 27 binary64)) k) < -5.0000000000000001e132

   1. Initial program 87.1%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{c \cdot b} - \left(j \cdot 27\right) \cdot k \]

      2. lower-*.f64 73.2

         \[\leadsto \color{blue}{c \cdot b} - \left(j \cdot 27\right) \cdot k \]

   5. Applied rewrites 73.2%

      \[\leadsto \color{blue}{c \cdot b} - \left(j \cdot 27\right) \cdot k \]

   if -5.0000000000000001e132 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 5.0000000000000004e189

   1. Initial program 85.8%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

         \[\leadsto \color{blue}{b \cdot c + \left(\mathsf{neg}\left(\left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{c \cdot b} + \left(\mathsf{neg}\left(\left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{neg}\left(\left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)} \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(\mathsf{neg}\left(4 \cdot \left(i \cdot x\right)\right)\right) + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(i \cdot x\right)} + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{-4} \cdot \left(i \cdot x\right) + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, -4 \cdot \color{blue}{\left(x \cdot i\right)} + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(-4 \cdot x\right) \cdot i} + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(c, b, \left(-4 \cdot x\right) \cdot i + \color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot \left(j \cdot k\right)}\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(c, b, \left(-4 \cdot x\right) \cdot i + \color{blue}{-27} \cdot \left(j \cdot k\right)\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \left(j \cdot k\right)\right)}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{-4 \cdot x}, i, -27 \cdot \left(j \cdot k\right)\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, \color{blue}{-27 \cdot \left(j \cdot k\right)}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \color{blue}{\left(k \cdot j\right)}\right)\right) \]

      15. lower-*.f64 58.9

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \color{blue}{\left(k \cdot j\right)}\right)\right) \]

   5. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \left(k \cdot j\right)\right)\right)} \]

   6. Taylor expanded in k around 0

      \[\leadsto \mathsf{fma}\left(c, b, -4 \cdot \left(i \cdot x\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 55.0%

      \[\leadsto \mathsf{fma}\left(c, b, \left(i \cdot x\right) \cdot -4\right) \]

   if 5.0000000000000004e189 < (*.f64 (*.f64 j #s(literal 27 binary64)) k)

   1. Initial program 85.0%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

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

         \[\leadsto \color{blue}{\left(b \cdot c + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot t\right)\right)} - \left(j \cdot 27\right) \cdot k \]

      2. metadata-eval N/A

         \[\leadsto \left(b \cdot c + \color{blue}{-4} \cdot \left(a \cdot t\right)\right) - \left(j \cdot 27\right) \cdot k \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(-4 \cdot \left(a \cdot t\right) + b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-4 \cdot a\right) \cdot t} + b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-4 \cdot a}, t, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, \color{blue}{c \cdot b}\right) - \left(j \cdot 27\right) \cdot k \]

      8. lower-*.f64 89.0

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, \color{blue}{c \cdot b}\right) - \left(j \cdot 27\right) \cdot k \]

   5. Applied rewrites 89.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)} - \left(j \cdot 27\right) \cdot k \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) - \left(j \cdot 27\right) \cdot k} \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) - \color{blue}{\left(j \cdot 27\right) \cdot k} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \left(\mathsf{neg}\left(j \cdot 27\right)\right) \cdot k} \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \left(\mathsf{neg}\left(\color{blue}{j \cdot 27}\right)\right) \cdot k \]

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

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \color{blue}{\left(j \cdot \left(\mathsf{neg}\left(27\right)\right)\right)} \cdot k \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \left(j \cdot \color{blue}{-27}\right) \cdot k \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \color{blue}{j \cdot \left(-27 \cdot k\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + j \cdot \color{blue}{\left(-27 \cdot k\right)} \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \color{blue}{\left(-27 \cdot k\right) \cdot j} \]

      10. +-commutative N/A

          \[\leadsto \color{blue}{\left(-27 \cdot k\right) \cdot j + \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)} \]

      11. lower-fma.f64 92.7

          \[\leadsto \color{blue}{\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)\right)} \]

   7. Applied rewrites 92.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)\right)} \]

   8. Taylor expanded in c around inf

      \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{b \cdot c}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{c \cdot b}\right) \]

      2. lower-*.f64 89.0

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{c \cdot b}\right) \]

   10. Applied rewrites 89.0%

       \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{c \cdot b}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \cdot \left(27 \cdot j\right) \leq -5 \cdot 10^{+132}:\\ \;\;\;\;c \cdot b - k \cdot \left(27 \cdot j\right)\\ \mathbf{elif}\;k \cdot \left(27 \cdot j\right) \leq 5 \cdot 10^{+189}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \left(i \cdot x\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot k, j, c \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 55.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\\\ [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := k \cdot \left(27 \cdot j\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \left(k \cdot j\right) \cdot -27\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+189}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \left(i \cdot x\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot k, j, c \cdot b\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (* k (* 27.0 j))))
   (if (<= t_1 -5e+132)
     (fma c b (* (* k j) -27.0))
     (if (<= t_1 5e+189)
       (fma c b (* (* i x) -4.0))
       (fma (* -27.0 k) j (* c b))))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = k * (27.0 * j);
	double tmp;
	if (t_1 <= -5e+132) {
		tmp = fma(c, b, ((k * j) * -27.0));
	} else if (t_1 <= 5e+189) {
		tmp = fma(c, b, ((i * x) * -4.0));
	} else {
		tmp = fma((-27.0 * k), j, (c * b));
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = Float64(k * Float64(27.0 * j))
	tmp = 0.0
	if (t_1 <= -5e+132)
		tmp = fma(c, b, Float64(Float64(k * j) * -27.0));
	elseif (t_1 <= 5e+189)
		tmp = fma(c, b, Float64(Float64(i * x) * -4.0));
	else
		tmp = fma(Float64(-27.0 * k), j, Float64(c * b));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(k * N[(27.0 * j), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+132], N[(c * b + N[(N[(k * j), $MachinePrecision] * -27.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+189], N[(c * b + N[(N[(i * x), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], N[(N[(-27.0 * k), $MachinePrecision] * j + N[(c * b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 j #s(literal 27 binary64)) k) < -5.0000000000000001e132

   1. Initial program 87.1%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

         \[\leadsto \color{blue}{b \cdot c + \left(\mathsf{neg}\left(\left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{c \cdot b} + \left(\mathsf{neg}\left(\left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{neg}\left(\left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)} \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(\mathsf{neg}\left(4 \cdot \left(i \cdot x\right)\right)\right) + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(i \cdot x\right)} + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{-4} \cdot \left(i \cdot x\right) + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, -4 \cdot \color{blue}{\left(x \cdot i\right)} + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(-4 \cdot x\right) \cdot i} + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(c, b, \left(-4 \cdot x\right) \cdot i + \color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot \left(j \cdot k\right)}\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(c, b, \left(-4 \cdot x\right) \cdot i + \color{blue}{-27} \cdot \left(j \cdot k\right)\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \left(j \cdot k\right)\right)}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{-4 \cdot x}, i, -27 \cdot \left(j \cdot k\right)\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, \color{blue}{-27 \cdot \left(j \cdot k\right)}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \color{blue}{\left(k \cdot j\right)}\right)\right) \]

      15. lower-*.f64 78.5

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \color{blue}{\left(k \cdot j\right)}\right)\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \left(k \cdot j\right)\right)\right)} \]

   6. Taylor expanded in k around inf

      \[\leadsto \mathsf{fma}\left(c, b, -27 \cdot \left(j \cdot k\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 73.2%

      \[\leadsto \mathsf{fma}\left(c, b, \left(k \cdot j\right) \cdot -27\right) \]

   if -5.0000000000000001e132 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 5.0000000000000004e189

   1. Initial program 85.8%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

         \[\leadsto \color{blue}{b \cdot c + \left(\mathsf{neg}\left(\left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{c \cdot b} + \left(\mathsf{neg}\left(\left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{neg}\left(\left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)} \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(\mathsf{neg}\left(4 \cdot \left(i \cdot x\right)\right)\right) + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(i \cdot x\right)} + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{-4} \cdot \left(i \cdot x\right) + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, -4 \cdot \color{blue}{\left(x \cdot i\right)} + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(-4 \cdot x\right) \cdot i} + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(c, b, \left(-4 \cdot x\right) \cdot i + \color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot \left(j \cdot k\right)}\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(c, b, \left(-4 \cdot x\right) \cdot i + \color{blue}{-27} \cdot \left(j \cdot k\right)\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \left(j \cdot k\right)\right)}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{-4 \cdot x}, i, -27 \cdot \left(j \cdot k\right)\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, \color{blue}{-27 \cdot \left(j \cdot k\right)}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \color{blue}{\left(k \cdot j\right)}\right)\right) \]

      15. lower-*.f64 58.9

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \color{blue}{\left(k \cdot j\right)}\right)\right) \]

   5. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \left(k \cdot j\right)\right)\right)} \]

   6. Taylor expanded in k around 0

      \[\leadsto \mathsf{fma}\left(c, b, -4 \cdot \left(i \cdot x\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 55.0%

      \[\leadsto \mathsf{fma}\left(c, b, \left(i \cdot x\right) \cdot -4\right) \]

   if 5.0000000000000004e189 < (*.f64 (*.f64 j #s(literal 27 binary64)) k)

   1. Initial program 85.0%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

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

         \[\leadsto \color{blue}{\left(b \cdot c + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot t\right)\right)} - \left(j \cdot 27\right) \cdot k \]

      2. metadata-eval N/A

         \[\leadsto \left(b \cdot c + \color{blue}{-4} \cdot \left(a \cdot t\right)\right) - \left(j \cdot 27\right) \cdot k \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(-4 \cdot \left(a \cdot t\right) + b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-4 \cdot a\right) \cdot t} + b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-4 \cdot a}, t, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, \color{blue}{c \cdot b}\right) - \left(j \cdot 27\right) \cdot k \]

      8. lower-*.f64 89.0

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, \color{blue}{c \cdot b}\right) - \left(j \cdot 27\right) \cdot k \]

   5. Applied rewrites 89.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)} - \left(j \cdot 27\right) \cdot k \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) - \left(j \cdot 27\right) \cdot k} \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) - \color{blue}{\left(j \cdot 27\right) \cdot k} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \left(\mathsf{neg}\left(j \cdot 27\right)\right) \cdot k} \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \left(\mathsf{neg}\left(\color{blue}{j \cdot 27}\right)\right) \cdot k \]

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

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \color{blue}{\left(j \cdot \left(\mathsf{neg}\left(27\right)\right)\right)} \cdot k \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \left(j \cdot \color{blue}{-27}\right) \cdot k \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \color{blue}{j \cdot \left(-27 \cdot k\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + j \cdot \color{blue}{\left(-27 \cdot k\right)} \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \color{blue}{\left(-27 \cdot k\right) \cdot j} \]

      10. +-commutative N/A

          \[\leadsto \color{blue}{\left(-27 \cdot k\right) \cdot j + \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)} \]

      11. lower-fma.f64 92.7

          \[\leadsto \color{blue}{\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)\right)} \]

   7. Applied rewrites 92.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)\right)} \]

   8. Taylor expanded in c around inf

      \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{b \cdot c}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{c \cdot b}\right) \]

      2. lower-*.f64 89.0

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{c \cdot b}\right) \]

   10. Applied rewrites 89.0%

       \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{c \cdot b}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \cdot \left(27 \cdot j\right) \leq -5 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \left(k \cdot j\right) \cdot -27\right)\\ \mathbf{elif}\;k \cdot \left(27 \cdot j\right) \leq 5 \cdot 10^{+189}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \left(i \cdot x\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot k, j, c \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 55.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\\\ [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(c, b, \left(k \cdot j\right) \cdot -27\right)\\ t_2 := k \cdot \left(27 \cdot j\right)\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+132}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+189}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \left(i \cdot x\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (fma c b (* (* k j) -27.0))) (t_2 (* k (* 27.0 j))))
   (if (<= t_2 -5e+132)
     t_1
     (if (<= t_2 5e+189) (fma c b (* (* i x) -4.0)) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = fma(c, b, ((k * j) * -27.0));
	double t_2 = k * (27.0 * j);
	double tmp;
	if (t_2 <= -5e+132) {
		tmp = t_1;
	} else if (t_2 <= 5e+189) {
		tmp = fma(c, b, ((i * x) * -4.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = fma(c, b, Float64(Float64(k * j) * -27.0))
	t_2 = Float64(k * Float64(27.0 * j))
	tmp = 0.0
	if (t_2 <= -5e+132)
		tmp = t_1;
	elseif (t_2 <= 5e+189)
		tmp = fma(c, b, Float64(Float64(i * x) * -4.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(c * b + N[(N[(k * j), $MachinePrecision] * -27.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(k * N[(27.0 * j), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+132], t$95$1, If[LessEqual[t$95$2, 5e+189], N[(c * b + N[(N[(i * x), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 j #s(literal 27 binary64)) k) < -5.0000000000000001e132 or 5.0000000000000004e189 < (*.f64 (*.f64 j #s(literal 27 binary64)) k)

   1. Initial program 86.4%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

         \[\leadsto \color{blue}{b \cdot c + \left(\mathsf{neg}\left(\left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{c \cdot b} + \left(\mathsf{neg}\left(\left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{neg}\left(\left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)} \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(\mathsf{neg}\left(4 \cdot \left(i \cdot x\right)\right)\right) + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(i \cdot x\right)} + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{-4} \cdot \left(i \cdot x\right) + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, -4 \cdot \color{blue}{\left(x \cdot i\right)} + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(-4 \cdot x\right) \cdot i} + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(c, b, \left(-4 \cdot x\right) \cdot i + \color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot \left(j \cdot k\right)}\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(c, b, \left(-4 \cdot x\right) \cdot i + \color{blue}{-27} \cdot \left(j \cdot k\right)\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \left(j \cdot k\right)\right)}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{-4 \cdot x}, i, -27 \cdot \left(j \cdot k\right)\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, \color{blue}{-27 \cdot \left(j \cdot k\right)}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \color{blue}{\left(k \cdot j\right)}\right)\right) \]

      15. lower-*.f64 80.7

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \color{blue}{\left(k \cdot j\right)}\right)\right) \]

   5. Applied rewrites 80.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \left(k \cdot j\right)\right)\right)} \]

   6. Taylor expanded in k around inf

      \[\leadsto \mathsf{fma}\left(c, b, -27 \cdot \left(j \cdot k\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 77.1%

      \[\leadsto \mathsf{fma}\left(c, b, \left(k \cdot j\right) \cdot -27\right) \]

   if -5.0000000000000001e132 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 5.0000000000000004e189

   1. Initial program 85.8%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

         \[\leadsto \color{blue}{b \cdot c + \left(\mathsf{neg}\left(\left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{c \cdot b} + \left(\mathsf{neg}\left(\left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{neg}\left(\left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)} \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(\mathsf{neg}\left(4 \cdot \left(i \cdot x\right)\right)\right) + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(i \cdot x\right)} + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{-4} \cdot \left(i \cdot x\right) + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, -4 \cdot \color{blue}{\left(x \cdot i\right)} + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(-4 \cdot x\right) \cdot i} + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(c, b, \left(-4 \cdot x\right) \cdot i + \color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot \left(j \cdot k\right)}\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(c, b, \left(-4 \cdot x\right) \cdot i + \color{blue}{-27} \cdot \left(j \cdot k\right)\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \left(j \cdot k\right)\right)}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{-4 \cdot x}, i, -27 \cdot \left(j \cdot k\right)\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, \color{blue}{-27 \cdot \left(j \cdot k\right)}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \color{blue}{\left(k \cdot j\right)}\right)\right) \]

      15. lower-*.f64 58.9

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \color{blue}{\left(k \cdot j\right)}\right)\right) \]

   5. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \left(k \cdot j\right)\right)\right)} \]

   6. Taylor expanded in k around 0

      \[\leadsto \mathsf{fma}\left(c, b, -4 \cdot \left(i \cdot x\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 55.0%

      \[\leadsto \mathsf{fma}\left(c, b, \left(i \cdot x\right) \cdot -4\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \cdot \left(27 \cdot j\right) \leq -5 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \left(k \cdot j\right) \cdot -27\right)\\ \mathbf{elif}\;k \cdot \left(27 \cdot j\right) \leq 5 \cdot 10^{+189}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \left(i \cdot x\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \left(k \cdot j\right) \cdot -27\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 53.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\\\ [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := k \cdot \left(27 \cdot j\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+179}:\\ \;\;\;\;\left(-27 \cdot j\right) \cdot k\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+189}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \left(i \cdot x\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot j\right) \cdot -27\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (* k (* 27.0 j))))
   (if (<= t_1 -1e+179)
     (* (* -27.0 j) k)
     (if (<= t_1 5e+189) (fma c b (* (* i x) -4.0)) (* (* k j) -27.0)))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = k * (27.0 * j);
	double tmp;
	if (t_1 <= -1e+179) {
		tmp = (-27.0 * j) * k;
	} else if (t_1 <= 5e+189) {
		tmp = fma(c, b, ((i * x) * -4.0));
	} else {
		tmp = (k * j) * -27.0;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = Float64(k * Float64(27.0 * j))
	tmp = 0.0
	if (t_1 <= -1e+179)
		tmp = Float64(Float64(-27.0 * j) * k);
	elseif (t_1 <= 5e+189)
		tmp = fma(c, b, Float64(Float64(i * x) * -4.0));
	else
		tmp = Float64(Float64(k * j) * -27.0);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(k * N[(27.0 * j), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+179], N[(N[(-27.0 * j), $MachinePrecision] * k), $MachinePrecision], If[LessEqual[t$95$1, 5e+189], N[(c * b + N[(N[(i * x), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], N[(N[(k * j), $MachinePrecision] * -27.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 j #s(literal 27 binary64)) k) < -9.9999999999999998e178

   1. Initial program 84.9%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in k around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{-27 \cdot \left(j \cdot k\right)} \]

      2. *-commutative N/A

         \[\leadsto -27 \cdot \color{blue}{\left(k \cdot j\right)} \]

      3. lower-*.f64 70.9

         \[\leadsto -27 \cdot \color{blue}{\left(k \cdot j\right)} \]

   5. Applied rewrites 70.9%

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

   7. Applied rewrites 70.9%

      \[\leadsto \left(j \cdot -27\right) \cdot \color{blue}{k} \]

   if -9.9999999999999998e178 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 5.0000000000000004e189

   1. Initial program 86.5%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

         \[\leadsto \color{blue}{b \cdot c + \left(\mathsf{neg}\left(\left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{c \cdot b} + \left(\mathsf{neg}\left(\left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{neg}\left(\left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)} \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(\mathsf{neg}\left(4 \cdot \left(i \cdot x\right)\right)\right) + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(i \cdot x\right)} + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{-4} \cdot \left(i \cdot x\right) + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, -4 \cdot \color{blue}{\left(x \cdot i\right)} + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(-4 \cdot x\right) \cdot i} + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(c, b, \left(-4 \cdot x\right) \cdot i + \color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot \left(j \cdot k\right)}\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(c, b, \left(-4 \cdot x\right) \cdot i + \color{blue}{-27} \cdot \left(j \cdot k\right)\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \left(j \cdot k\right)\right)}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{-4 \cdot x}, i, -27 \cdot \left(j \cdot k\right)\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, \color{blue}{-27 \cdot \left(j \cdot k\right)}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \color{blue}{\left(k \cdot j\right)}\right)\right) \]

      15. lower-*.f64 59.6

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \color{blue}{\left(k \cdot j\right)}\right)\right) \]

   5. Applied rewrites 59.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \left(k \cdot j\right)\right)\right)} \]

   6. Taylor expanded in k around 0

      \[\leadsto \mathsf{fma}\left(c, b, -4 \cdot \left(i \cdot x\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 54.2%

      \[\leadsto \mathsf{fma}\left(c, b, \left(i \cdot x\right) \cdot -4\right) \]

   if 5.0000000000000004e189 < (*.f64 (*.f64 j #s(literal 27 binary64)) k)

   1. Initial program 85.0%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in k around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{-27 \cdot \left(j \cdot k\right)} \]

      2. *-commutative N/A

         \[\leadsto -27 \cdot \color{blue}{\left(k \cdot j\right)} \]

      3. lower-*.f64 81.6

         \[\leadsto -27 \cdot \color{blue}{\left(k \cdot j\right)} \]

   5. Applied rewrites 81.6%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \cdot \left(27 \cdot j\right) \leq -1 \cdot 10^{+179}:\\ \;\;\;\;\left(-27 \cdot j\right) \cdot k\\ \mathbf{elif}\;k \cdot \left(27 \cdot j\right) \leq 5 \cdot 10^{+189}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \left(i \cdot x\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot j\right) \cdot -27\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 71.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\\\ [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-4, i, \left(\left(z \cdot y\right) \cdot t\right) \cdot 18\right) \cdot x\\ \mathbf{if}\;x \leq -1.08 \cdot 10^{+186}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)\right)\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+209}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, \left(k \cdot j\right) \cdot -27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (* (fma -4.0 i (* (* (* z y) t) 18.0)) x)))
   (if (<= x -1.08e+186)
     t_1
     (if (<= x 2.75e+20)
       (fma (* -27.0 k) j (fma (* -4.0 a) t (* c b)))
       (if (<= x 1.9e+209)
         (fma c b (fma (* -4.0 x) i (* (* k j) -27.0)))
         t_1)))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = fma(-4.0, i, (((z * y) * t) * 18.0)) * x;
	double tmp;
	if (x <= -1.08e+186) {
		tmp = t_1;
	} else if (x <= 2.75e+20) {
		tmp = fma((-27.0 * k), j, fma((-4.0 * a), t, (c * b)));
	} else if (x <= 1.9e+209) {
		tmp = fma(c, b, fma((-4.0 * x), i, ((k * j) * -27.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = Float64(fma(-4.0, i, Float64(Float64(Float64(z * y) * t) * 18.0)) * x)
	tmp = 0.0
	if (x <= -1.08e+186)
		tmp = t_1;
	elseif (x <= 2.75e+20)
		tmp = fma(Float64(-27.0 * k), j, fma(Float64(-4.0 * a), t, Float64(c * b)));
	elseif (x <= 1.9e+209)
		tmp = fma(c, b, fma(Float64(-4.0 * x), i, Float64(Float64(k * j) * -27.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(N[(-4.0 * i + N[(N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] * 18.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -1.08e+186], t$95$1, If[LessEqual[x, 2.75e+20], N[(N[(-27.0 * k), $MachinePrecision] * j + N[(N[(-4.0 * a), $MachinePrecision] * t + N[(c * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.9e+209], N[(c * b + N[(N[(-4.0 * x), $MachinePrecision] * i + N[(N[(k * j), $MachinePrecision] * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.08000000000000003e186 or 1.89999999999999992e209 < x

   1. Initial program 63.5%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right) \cdot x} \]

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

         \[\leadsto \color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot i\right)} \cdot x \]

      4. metadata-eval N/A

         \[\leadsto \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + \color{blue}{-4} \cdot i\right) \cdot x \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-4 \cdot i + 18 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right)} \cdot x \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4, i, 18 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right)} \cdot x \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-4, i, \color{blue}{\left(t \cdot \left(y \cdot z\right)\right) \cdot 18}\right) \cdot x \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4, i, \color{blue}{\left(t \cdot \left(y \cdot z\right)\right) \cdot 18}\right) \cdot x \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-4, i, \color{blue}{\left(\left(y \cdot z\right) \cdot t\right)} \cdot 18\right) \cdot x \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-4, i, \color{blue}{\left(\left(y \cdot z\right) \cdot t\right)} \cdot 18\right) \cdot x \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-4, i, \left(\color{blue}{\left(z \cdot y\right)} \cdot t\right) \cdot 18\right) \cdot x \]

      12. lower-*.f64 84.8

          \[\leadsto \mathsf{fma}\left(-4, i, \left(\color{blue}{\left(z \cdot y\right)} \cdot t\right) \cdot 18\right) \cdot x \]

   5. Applied rewrites 84.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4, i, \left(\left(z \cdot y\right) \cdot t\right) \cdot 18\right) \cdot x} \]

   if -1.08000000000000003e186 < x < 2.75e20

   1. Initial program 95.0%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

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

         \[\leadsto \color{blue}{\left(b \cdot c + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot t\right)\right)} - \left(j \cdot 27\right) \cdot k \]

      2. metadata-eval N/A

         \[\leadsto \left(b \cdot c + \color{blue}{-4} \cdot \left(a \cdot t\right)\right) - \left(j \cdot 27\right) \cdot k \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(-4 \cdot \left(a \cdot t\right) + b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-4 \cdot a\right) \cdot t} + b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-4 \cdot a}, t, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, \color{blue}{c \cdot b}\right) - \left(j \cdot 27\right) \cdot k \]

      8. lower-*.f64 77.0

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, \color{blue}{c \cdot b}\right) - \left(j \cdot 27\right) \cdot k \]

   5. Applied rewrites 77.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)} - \left(j \cdot 27\right) \cdot k \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) - \left(j \cdot 27\right) \cdot k} \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) - \color{blue}{\left(j \cdot 27\right) \cdot k} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \left(\mathsf{neg}\left(j \cdot 27\right)\right) \cdot k} \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \left(\mathsf{neg}\left(\color{blue}{j \cdot 27}\right)\right) \cdot k \]

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

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \color{blue}{\left(j \cdot \left(\mathsf{neg}\left(27\right)\right)\right)} \cdot k \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \left(j \cdot \color{blue}{-27}\right) \cdot k \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \color{blue}{j \cdot \left(-27 \cdot k\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + j \cdot \color{blue}{\left(-27 \cdot k\right)} \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \color{blue}{\left(-27 \cdot k\right) \cdot j} \]

      10. +-commutative N/A

          \[\leadsto \color{blue}{\left(-27 \cdot k\right) \cdot j + \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)} \]

      11. lower-fma.f64 77.6

          \[\leadsto \color{blue}{\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)\right)} \]

   7. Applied rewrites 77.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)\right)} \]

   if 2.75e20 < x < 1.89999999999999992e209

   1. Initial program 80.1%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

         \[\leadsto \color{blue}{b \cdot c + \left(\mathsf{neg}\left(\left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{c \cdot b} + \left(\mathsf{neg}\left(\left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{neg}\left(\left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)} \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(\mathsf{neg}\left(4 \cdot \left(i \cdot x\right)\right)\right) + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(i \cdot x\right)} + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{-4} \cdot \left(i \cdot x\right) + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, -4 \cdot \color{blue}{\left(x \cdot i\right)} + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(-4 \cdot x\right) \cdot i} + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(c, b, \left(-4 \cdot x\right) \cdot i + \color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot \left(j \cdot k\right)}\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(c, b, \left(-4 \cdot x\right) \cdot i + \color{blue}{-27} \cdot \left(j \cdot k\right)\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \left(j \cdot k\right)\right)}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{-4 \cdot x}, i, -27 \cdot \left(j \cdot k\right)\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, \color{blue}{-27 \cdot \left(j \cdot k\right)}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \color{blue}{\left(k \cdot j\right)}\right)\right) \]

      15. lower-*.f64 75.9

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \color{blue}{\left(k \cdot j\right)}\right)\right) \]

   5. Applied rewrites 75.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \left(k \cdot j\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.08 \cdot 10^{+186}:\\ \;\;\;\;\mathsf{fma}\left(-4, i, \left(\left(z \cdot y\right) \cdot t\right) \cdot 18\right) \cdot x\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)\right)\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+209}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, \left(k \cdot j\right) \cdot -27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, i, \left(\left(z \cdot y\right) \cdot t\right) \cdot 18\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 72.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\\\ [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-4, i, \left(\left(z \cdot y\right) \cdot t\right) \cdot 18\right) \cdot x\\ \mathbf{if}\;x \leq -1.08 \cdot 10^{+186}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \mathsf{fma}\left(-27 \cdot k, j, \left(a \cdot t\right) \cdot -4\right)\right)\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+209}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, \left(k \cdot j\right) \cdot -27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (* (fma -4.0 i (* (* (* z y) t) 18.0)) x)))
   (if (<= x -1.08e+186)
     t_1
     (if (<= x 2.75e+20)
       (fma c b (fma (* -27.0 k) j (* (* a t) -4.0)))
       (if (<= x 1.9e+209)
         (fma c b (fma (* -4.0 x) i (* (* k j) -27.0)))
         t_1)))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = fma(-4.0, i, (((z * y) * t) * 18.0)) * x;
	double tmp;
	if (x <= -1.08e+186) {
		tmp = t_1;
	} else if (x <= 2.75e+20) {
		tmp = fma(c, b, fma((-27.0 * k), j, ((a * t) * -4.0)));
	} else if (x <= 1.9e+209) {
		tmp = fma(c, b, fma((-4.0 * x), i, ((k * j) * -27.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = Float64(fma(-4.0, i, Float64(Float64(Float64(z * y) * t) * 18.0)) * x)
	tmp = 0.0
	if (x <= -1.08e+186)
		tmp = t_1;
	elseif (x <= 2.75e+20)
		tmp = fma(c, b, fma(Float64(-27.0 * k), j, Float64(Float64(a * t) * -4.0)));
	elseif (x <= 1.9e+209)
		tmp = fma(c, b, fma(Float64(-4.0 * x), i, Float64(Float64(k * j) * -27.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(N[(-4.0 * i + N[(N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] * 18.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -1.08e+186], t$95$1, If[LessEqual[x, 2.75e+20], N[(c * b + N[(N[(-27.0 * k), $MachinePrecision] * j + N[(N[(a * t), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.9e+209], N[(c * b + N[(N[(-4.0 * x), $MachinePrecision] * i + N[(N[(k * j), $MachinePrecision] * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.08000000000000003e186 or 1.89999999999999992e209 < x

   1. Initial program 63.5%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right) \cdot x} \]

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

         \[\leadsto \color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot i\right)} \cdot x \]

      4. metadata-eval N/A

         \[\leadsto \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + \color{blue}{-4} \cdot i\right) \cdot x \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-4 \cdot i + 18 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right)} \cdot x \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4, i, 18 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right)} \cdot x \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-4, i, \color{blue}{\left(t \cdot \left(y \cdot z\right)\right) \cdot 18}\right) \cdot x \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4, i, \color{blue}{\left(t \cdot \left(y \cdot z\right)\right) \cdot 18}\right) \cdot x \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-4, i, \color{blue}{\left(\left(y \cdot z\right) \cdot t\right)} \cdot 18\right) \cdot x \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-4, i, \color{blue}{\left(\left(y \cdot z\right) \cdot t\right)} \cdot 18\right) \cdot x \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-4, i, \left(\color{blue}{\left(z \cdot y\right)} \cdot t\right) \cdot 18\right) \cdot x \]

      12. lower-*.f64 84.8

          \[\leadsto \mathsf{fma}\left(-4, i, \left(\color{blue}{\left(z \cdot y\right)} \cdot t\right) \cdot 18\right) \cdot x \]

   5. Applied rewrites 84.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4, i, \left(\left(z \cdot y\right) \cdot t\right) \cdot 18\right) \cdot x} \]

   if -1.08000000000000003e186 < x < 2.75e20

   1. Initial program 95.0%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

         \[\leadsto \color{blue}{b \cdot c + \left(\mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{c \cdot b} + \left(\mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{neg}\left(\color{blue}{\left(27 \cdot \left(j \cdot k\right) + 4 \cdot \left(a \cdot t\right)\right)}\right)\right) \]

      5. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right) + \left(\mathsf{neg}\left(4 \cdot \left(a \cdot t\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot \left(j \cdot k\right)} + \left(\mathsf{neg}\left(4 \cdot \left(a \cdot t\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{-27} \cdot \left(j \cdot k\right) + \left(\mathsf{neg}\left(4 \cdot \left(a \cdot t\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, -27 \cdot \color{blue}{\left(k \cdot j\right)} + \left(\mathsf{neg}\left(4 \cdot \left(a \cdot t\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(-27 \cdot k\right) \cdot j} + \left(\mathsf{neg}\left(4 \cdot \left(a \cdot t\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(c, b, \left(-27 \cdot k\right) \cdot j + \color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot t\right)}\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(c, b, \left(-27 \cdot k\right) \cdot j + \color{blue}{-4} \cdot \left(a \cdot t\right)\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\mathsf{fma}\left(-27 \cdot k, j, -4 \cdot \left(a \cdot t\right)\right)}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{-27 \cdot k}, j, -4 \cdot \left(a \cdot t\right)\right)\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{-4 \cdot \left(a \cdot t\right)}\right)\right) \]

      15. lower-*.f64 77.0

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(-27 \cdot k, j, -4 \cdot \color{blue}{\left(a \cdot t\right)}\right)\right) \]

   5. Applied rewrites 77.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{fma}\left(-27 \cdot k, j, -4 \cdot \left(a \cdot t\right)\right)\right)} \]

   if 2.75e20 < x < 1.89999999999999992e209

   1. Initial program 80.1%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

         \[\leadsto \color{blue}{b \cdot c + \left(\mathsf{neg}\left(\left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{c \cdot b} + \left(\mathsf{neg}\left(\left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{neg}\left(\left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)} \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(\mathsf{neg}\left(4 \cdot \left(i \cdot x\right)\right)\right) + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(i \cdot x\right)} + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{-4} \cdot \left(i \cdot x\right) + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, -4 \cdot \color{blue}{\left(x \cdot i\right)} + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(-4 \cdot x\right) \cdot i} + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(c, b, \left(-4 \cdot x\right) \cdot i + \color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot \left(j \cdot k\right)}\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(c, b, \left(-4 \cdot x\right) \cdot i + \color{blue}{-27} \cdot \left(j \cdot k\right)\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \left(j \cdot k\right)\right)}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{-4 \cdot x}, i, -27 \cdot \left(j \cdot k\right)\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, \color{blue}{-27 \cdot \left(j \cdot k\right)}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \color{blue}{\left(k \cdot j\right)}\right)\right) \]

      15. lower-*.f64 75.9

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \color{blue}{\left(k \cdot j\right)}\right)\right) \]

   5. Applied rewrites 75.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \left(k \cdot j\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.08 \cdot 10^{+186}:\\ \;\;\;\;\mathsf{fma}\left(-4, i, \left(\left(z \cdot y\right) \cdot t\right) \cdot 18\right) \cdot x\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \mathsf{fma}\left(-27 \cdot k, j, \left(a \cdot t\right) \cdot -4\right)\right)\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+209}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, \left(k \cdot j\right) \cdot -27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, i, \left(\left(z \cdot y\right) \cdot t\right) \cdot 18\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 76.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\\\ [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(-4, a, \left(\left(z \cdot y\right) \cdot x\right) \cdot 18\right), t, c \cdot b\right)\\ \mathbf{if}\;t \leq -4.8 \cdot 10^{+112}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, \left(k \cdot j\right) \cdot -27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (fma (fma -4.0 a (* (* (* z y) x) 18.0)) t (* c b))))
   (if (<= t -4.8e+112)
     t_1
     (if (<= t 0.005) (fma c b (fma (* -4.0 x) i (* (* k j) -27.0))) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = fma(fma(-4.0, a, (((z * y) * x) * 18.0)), t, (c * b));
	double tmp;
	if (t <= -4.8e+112) {
		tmp = t_1;
	} else if (t <= 0.005) {
		tmp = fma(c, b, fma((-4.0 * x), i, ((k * j) * -27.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = fma(fma(-4.0, a, Float64(Float64(Float64(z * y) * x) * 18.0)), t, Float64(c * b))
	tmp = 0.0
	if (t <= -4.8e+112)
		tmp = t_1;
	elseif (t <= 0.005)
		tmp = fma(c, b, fma(Float64(-4.0 * x), i, Float64(Float64(k * j) * -27.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(N[(-4.0 * a + N[(N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision] * 18.0), $MachinePrecision]), $MachinePrecision] * t + N[(c * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4.8e+112], t$95$1, If[LessEqual[t, 0.005], N[(c * b + N[(N[(-4.0 * x), $MachinePrecision] * i + N[(N[(k * j), $MachinePrecision] * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.8e112 or 0.0050000000000000001 < t

   1. Initial program 84.1%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - \left(4 \cdot \left(a \cdot t\right) + 27 \cdot \left(j \cdot k\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - \color{blue}{\left(27 \cdot \left(j \cdot k\right) + 4 \cdot \left(a \cdot t\right)\right)} \]

      2. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 27 \cdot \left(j \cdot k\right)\right) - 4 \cdot \left(a \cdot t\right)} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right)} - 4 \cdot \left(a \cdot t\right) \]

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

         \[\leadsto \left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) + \color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot \left(j \cdot k\right)}\right) - 4 \cdot \left(a \cdot t\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) + \color{blue}{-27} \cdot \left(j \cdot k\right)\right) - 4 \cdot \left(a \cdot t\right) \]

      6. +-commutative N/A

         \[\leadsto \color{blue}{\left(-27 \cdot \left(j \cdot k\right) + \left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right)\right)} - 4 \cdot \left(a \cdot t\right) \]

      7. associate--l+ N/A

         \[\leadsto \color{blue}{-27 \cdot \left(j \cdot k\right) + \left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 4 \cdot \left(a \cdot t\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto -27 \cdot \color{blue}{\left(k \cdot j\right)} + \left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 4 \cdot \left(a \cdot t\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-27 \cdot k\right) \cdot j} + \left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 4 \cdot \left(a \cdot t\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-27 \cdot k, j, \left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 4 \cdot \left(a \cdot t\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-27 \cdot k}, j, \left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 4 \cdot \left(a \cdot t\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{\left(b \cdot c + 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right)\right)} - 4 \cdot \left(a \cdot t\right)\right) \]

   5. Applied rewrites 88.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(\mathsf{fma}\left(-4, a, \left(\left(z \cdot y\right) \cdot x\right) \cdot 18\right), t, c \cdot b\right)\right)} \]

   6. Taylor expanded in k around 0

      \[\leadsto b \cdot c + \color{blue}{t \cdot \left(-4 \cdot a + 18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 78.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-4, a, \left(\left(z \cdot y\right) \cdot x\right) \cdot 18\right), \color{blue}{t}, c \cdot b\right) \]

   if -4.8e112 < t < 0.0050000000000000001

   1. Initial program 87.3%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

         \[\leadsto \color{blue}{b \cdot c + \left(\mathsf{neg}\left(\left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{c \cdot b} + \left(\mathsf{neg}\left(\left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{neg}\left(\left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)} \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(\mathsf{neg}\left(4 \cdot \left(i \cdot x\right)\right)\right) + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(i \cdot x\right)} + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{-4} \cdot \left(i \cdot x\right) + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, -4 \cdot \color{blue}{\left(x \cdot i\right)} + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(-4 \cdot x\right) \cdot i} + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(c, b, \left(-4 \cdot x\right) \cdot i + \color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot \left(j \cdot k\right)}\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(c, b, \left(-4 \cdot x\right) \cdot i + \color{blue}{-27} \cdot \left(j \cdot k\right)\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \left(j \cdot k\right)\right)}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{-4 \cdot x}, i, -27 \cdot \left(j \cdot k\right)\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, \color{blue}{-27 \cdot \left(j \cdot k\right)}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \color{blue}{\left(k \cdot j\right)}\right)\right) \]

      15. lower-*.f64 77.8

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \color{blue}{\left(k \cdot j\right)}\right)\right) \]

   5. Applied rewrites 77.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, -27 \cdot \left(k \cdot j\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{+112}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-4, a, \left(\left(z \cdot y\right) \cdot x\right) \cdot 18\right), t, c \cdot b\right)\\ \mathbf{elif}\;t \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, \left(k \cdot j\right) \cdot -27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-4, a, \left(\left(z \cdot y\right) \cdot x\right) \cdot 18\right), t, c \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 37.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\\\ [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \left(-27 \cdot j\right) \cdot k\\ t_2 := k \cdot \left(27 \cdot j\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+101}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+90}:\\ \;\;\;\;c \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (* (* -27.0 j) k)) (t_2 (* k (* 27.0 j))))
   (if (<= t_2 -1e+101) t_1 (if (<= t_2 1e+90) (* c b) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = (-27.0 * j) * k;
	double t_2 = k * (27.0 * j);
	double tmp;
	if (t_2 <= -1e+101) {
		tmp = t_1;
	} else if (t_2 <= 1e+90) {
		tmp = c * b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b, c, i, j, k)
    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), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((-27.0d0) * j) * k
    t_2 = k * (27.0d0 * j)
    if (t_2 <= (-1d+101)) then
        tmp = t_1
    else if (t_2 <= 1d+90) then
        tmp = c * b
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k;
assert x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k;
public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = (-27.0 * j) * k;
	double t_2 = k * (27.0 * j);
	double tmp;
	if (t_2 <= -1e+101) {
		tmp = t_1;
	} else if (t_2 <= 1e+90) {
		tmp = c * b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a, b, c, i, j, k] = sort([x, y, z, t, a, b, c, i, j, k])
[x, y, z, t, a, b, c, i, j, k] = sort([x, y, z, t, a, b, c, i, j, k])
def code(x, y, z, t, a, b, c, i, j, k):
	t_1 = (-27.0 * j) * k
	t_2 = k * (27.0 * j)
	tmp = 0
	if t_2 <= -1e+101:
		tmp = t_1
	elif t_2 <= 1e+90:
		tmp = c * b
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = Float64(Float64(-27.0 * j) * k)
	t_2 = Float64(k * Float64(27.0 * j))
	tmp = 0.0
	if (t_2 <= -1e+101)
		tmp = t_1;
	elseif (t_2 <= 1e+90)
		tmp = Float64(c * b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t, a, b, c, i, j, k = num2cell(sort([x, y, z, t, a, b, c, i, j, k])){:}
x, y, z, t, a, b, c, i, j, k = num2cell(sort([x, y, z, t, a, b, c, i, j, k])){:}
function tmp_2 = code(x, y, z, t, a, b, c, i, j, k)
	t_1 = (-27.0 * j) * k;
	t_2 = k * (27.0 * j);
	tmp = 0.0;
	if (t_2 <= -1e+101)
		tmp = t_1;
	elseif (t_2 <= 1e+90)
		tmp = c * b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(N[(-27.0 * j), $MachinePrecision] * k), $MachinePrecision]}, Block[{t$95$2 = N[(k * N[(27.0 * j), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+101], t$95$1, If[LessEqual[t$95$2, 1e+90], N[(c * b), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 j #s(literal 27 binary64)) k) < -9.9999999999999998e100 or 9.99999999999999966e89 < (*.f64 (*.f64 j #s(literal 27 binary64)) k)

   1. Initial program 86.4%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in k around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{-27 \cdot \left(j \cdot k\right)} \]

      2. *-commutative N/A

         \[\leadsto -27 \cdot \color{blue}{\left(k \cdot j\right)} \]

      3. lower-*.f64 63.0

         \[\leadsto -27 \cdot \color{blue}{\left(k \cdot j\right)} \]

   5. Applied rewrites 63.0%

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

   7. Applied rewrites 63.0%

      \[\leadsto \left(j \cdot -27\right) \cdot \color{blue}{k} \]

   if -9.9999999999999998e100 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 9.99999999999999966e89

   1. Initial program 85.8%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{c \cdot b} \]

      2. lower-*.f64 35.7

         \[\leadsto \color{blue}{c \cdot b} \]

   5. Applied rewrites 35.7%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \cdot \left(27 \cdot j\right) \leq -1 \cdot 10^{+101}:\\ \;\;\;\;\left(-27 \cdot j\right) \cdot k\\ \mathbf{elif}\;k \cdot \left(27 \cdot j\right) \leq 10^{+90}:\\ \;\;\;\;c \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(-27 \cdot j\right) \cdot k\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 37.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\\\ [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \left(k \cdot j\right) \cdot -27\\ t_2 := k \cdot \left(27 \cdot j\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+101}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+90}:\\ \;\;\;\;c \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (* (* k j) -27.0)) (t_2 (* k (* 27.0 j))))
   (if (<= t_2 -1e+101) t_1 (if (<= t_2 1e+90) (* c b) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = (k * j) * -27.0;
	double t_2 = k * (27.0 * j);
	double tmp;
	if (t_2 <= -1e+101) {
		tmp = t_1;
	} else if (t_2 <= 1e+90) {
		tmp = c * b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b, c, i, j, k)
    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), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (k * j) * (-27.0d0)
    t_2 = k * (27.0d0 * j)
    if (t_2 <= (-1d+101)) then
        tmp = t_1
    else if (t_2 <= 1d+90) then
        tmp = c * b
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k;
assert x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k;
public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = (k * j) * -27.0;
	double t_2 = k * (27.0 * j);
	double tmp;
	if (t_2 <= -1e+101) {
		tmp = t_1;
	} else if (t_2 <= 1e+90) {
		tmp = c * b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a, b, c, i, j, k] = sort([x, y, z, t, a, b, c, i, j, k])
[x, y, z, t, a, b, c, i, j, k] = sort([x, y, z, t, a, b, c, i, j, k])
def code(x, y, z, t, a, b, c, i, j, k):
	t_1 = (k * j) * -27.0
	t_2 = k * (27.0 * j)
	tmp = 0
	if t_2 <= -1e+101:
		tmp = t_1
	elif t_2 <= 1e+90:
		tmp = c * b
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = Float64(Float64(k * j) * -27.0)
	t_2 = Float64(k * Float64(27.0 * j))
	tmp = 0.0
	if (t_2 <= -1e+101)
		tmp = t_1;
	elseif (t_2 <= 1e+90)
		tmp = Float64(c * b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t, a, b, c, i, j, k = num2cell(sort([x, y, z, t, a, b, c, i, j, k])){:}
x, y, z, t, a, b, c, i, j, k = num2cell(sort([x, y, z, t, a, b, c, i, j, k])){:}
function tmp_2 = code(x, y, z, t, a, b, c, i, j, k)
	t_1 = (k * j) * -27.0;
	t_2 = k * (27.0 * j);
	tmp = 0.0;
	if (t_2 <= -1e+101)
		tmp = t_1;
	elseif (t_2 <= 1e+90)
		tmp = c * b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(N[(k * j), $MachinePrecision] * -27.0), $MachinePrecision]}, Block[{t$95$2 = N[(k * N[(27.0 * j), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+101], t$95$1, If[LessEqual[t$95$2, 1e+90], N[(c * b), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 j #s(literal 27 binary64)) k) < -9.9999999999999998e100 or 9.99999999999999966e89 < (*.f64 (*.f64 j #s(literal 27 binary64)) k)

   1. Initial program 86.4%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in k around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{-27 \cdot \left(j \cdot k\right)} \]

      2. *-commutative N/A

         \[\leadsto -27 \cdot \color{blue}{\left(k \cdot j\right)} \]

      3. lower-*.f64 63.0

         \[\leadsto -27 \cdot \color{blue}{\left(k \cdot j\right)} \]

   5. Applied rewrites 63.0%

      \[\leadsto \color{blue}{-27 \cdot \left(k \cdot j\right)} \]

   if -9.9999999999999998e100 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 9.99999999999999966e89

   1. Initial program 85.8%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{c \cdot b} \]

      2. lower-*.f64 35.7

         \[\leadsto \color{blue}{c \cdot b} \]

   5. Applied rewrites 35.7%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \cdot \left(27 \cdot j\right) \leq -1 \cdot 10^{+101}:\\ \;\;\;\;\left(k \cdot j\right) \cdot -27\\ \mathbf{elif}\;k \cdot \left(27 \cdot j\right) \leq 10^{+90}:\\ \;\;\;\;c \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot j\right) \cdot -27\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 71.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\\\ [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-4, i, \left(\left(z \cdot y\right) \cdot t\right) \cdot 18\right) \cdot x\\ \mathbf{if}\;x \leq -1.08 \cdot 10^{+186}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+161}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \mathsf{fma}\left(-27 \cdot k, j, \left(a \cdot t\right) \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (* (fma -4.0 i (* (* (* z y) t) 18.0)) x)))
   (if (<= x -1.08e+186)
     t_1
     (if (<= x 2.2e+161) (fma c b (fma (* -27.0 k) j (* (* a t) -4.0))) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = fma(-4.0, i, (((z * y) * t) * 18.0)) * x;
	double tmp;
	if (x <= -1.08e+186) {
		tmp = t_1;
	} else if (x <= 2.2e+161) {
		tmp = fma(c, b, fma((-27.0 * k), j, ((a * t) * -4.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = Float64(fma(-4.0, i, Float64(Float64(Float64(z * y) * t) * 18.0)) * x)
	tmp = 0.0
	if (x <= -1.08e+186)
		tmp = t_1;
	elseif (x <= 2.2e+161)
		tmp = fma(c, b, fma(Float64(-27.0 * k), j, Float64(Float64(a * t) * -4.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(N[(-4.0 * i + N[(N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] * 18.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -1.08e+186], t$95$1, If[LessEqual[x, 2.2e+161], N[(c * b + N[(N[(-27.0 * k), $MachinePrecision] * j + N[(N[(a * t), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.08000000000000003e186 or 2.2e161 < x

   1. Initial program 61.7%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right) \cdot x} \]

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

         \[\leadsto \color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot i\right)} \cdot x \]

      4. metadata-eval N/A

         \[\leadsto \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + \color{blue}{-4} \cdot i\right) \cdot x \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-4 \cdot i + 18 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right)} \cdot x \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4, i, 18 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right)} \cdot x \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-4, i, \color{blue}{\left(t \cdot \left(y \cdot z\right)\right) \cdot 18}\right) \cdot x \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4, i, \color{blue}{\left(t \cdot \left(y \cdot z\right)\right) \cdot 18}\right) \cdot x \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-4, i, \color{blue}{\left(\left(y \cdot z\right) \cdot t\right)} \cdot 18\right) \cdot x \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-4, i, \color{blue}{\left(\left(y \cdot z\right) \cdot t\right)} \cdot 18\right) \cdot x \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-4, i, \left(\color{blue}{\left(z \cdot y\right)} \cdot t\right) \cdot 18\right) \cdot x \]

      12. lower-*.f64 80.5

          \[\leadsto \mathsf{fma}\left(-4, i, \left(\color{blue}{\left(z \cdot y\right)} \cdot t\right) \cdot 18\right) \cdot x \]

   5. Applied rewrites 80.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4, i, \left(\left(z \cdot y\right) \cdot t\right) \cdot 18\right) \cdot x} \]

   if -1.08000000000000003e186 < x < 2.2e161

   1. Initial program 94.7%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

         \[\leadsto \color{blue}{b \cdot c + \left(\mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{c \cdot b} + \left(\mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{neg}\left(\color{blue}{\left(27 \cdot \left(j \cdot k\right) + 4 \cdot \left(a \cdot t\right)\right)}\right)\right) \]

      5. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right) + \left(\mathsf{neg}\left(4 \cdot \left(a \cdot t\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot \left(j \cdot k\right)} + \left(\mathsf{neg}\left(4 \cdot \left(a \cdot t\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{-27} \cdot \left(j \cdot k\right) + \left(\mathsf{neg}\left(4 \cdot \left(a \cdot t\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, -27 \cdot \color{blue}{\left(k \cdot j\right)} + \left(\mathsf{neg}\left(4 \cdot \left(a \cdot t\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(-27 \cdot k\right) \cdot j} + \left(\mathsf{neg}\left(4 \cdot \left(a \cdot t\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(c, b, \left(-27 \cdot k\right) \cdot j + \color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot t\right)}\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(c, b, \left(-27 \cdot k\right) \cdot j + \color{blue}{-4} \cdot \left(a \cdot t\right)\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\mathsf{fma}\left(-27 \cdot k, j, -4 \cdot \left(a \cdot t\right)\right)}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{-27 \cdot k}, j, -4 \cdot \left(a \cdot t\right)\right)\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{-4 \cdot \left(a \cdot t\right)}\right)\right) \]

      15. lower-*.f64 74.1

          \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(-27 \cdot k, j, -4 \cdot \color{blue}{\left(a \cdot t\right)}\right)\right) \]

   5. Applied rewrites 74.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{fma}\left(-27 \cdot k, j, -4 \cdot \left(a \cdot t\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.08 \cdot 10^{+186}:\\ \;\;\;\;\mathsf{fma}\left(-4, i, \left(\left(z \cdot y\right) \cdot t\right) \cdot 18\right) \cdot x\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+161}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \mathsf{fma}\left(-27 \cdot k, j, \left(a \cdot t\right) \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, i, \left(\left(z \cdot y\right) \cdot t\right) \cdot 18\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 58.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\\\ [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-4, i, \left(\left(z \cdot y\right) \cdot t\right) \cdot 18\right) \cdot x\\ \mathbf{if}\;x \leq -1.16 \cdot 10^{+27}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot k, j, c \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (* (fma -4.0 i (* (* (* z y) t) 18.0)) x)))
   (if (<= x -1.16e+27)
     t_1
     (if (<= x 1e+155) (fma (* -27.0 k) j (* c b)) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = fma(-4.0, i, (((z * y) * t) * 18.0)) * x;
	double tmp;
	if (x <= -1.16e+27) {
		tmp = t_1;
	} else if (x <= 1e+155) {
		tmp = fma((-27.0 * k), j, (c * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = Float64(fma(-4.0, i, Float64(Float64(Float64(z * y) * t) * 18.0)) * x)
	tmp = 0.0
	if (x <= -1.16e+27)
		tmp = t_1;
	elseif (x <= 1e+155)
		tmp = fma(Float64(-27.0 * k), j, Float64(c * b));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(N[(-4.0 * i + N[(N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] * 18.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -1.16e+27], t$95$1, If[LessEqual[x, 1e+155], N[(N[(-27.0 * k), $MachinePrecision] * j + N[(c * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.16e27 or 1.00000000000000001e155 < x

   1. Initial program 67.5%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right) \cdot x} \]

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

         \[\leadsto \color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot i\right)} \cdot x \]

      4. metadata-eval N/A

         \[\leadsto \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + \color{blue}{-4} \cdot i\right) \cdot x \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-4 \cdot i + 18 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right)} \cdot x \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4, i, 18 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right)} \cdot x \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-4, i, \color{blue}{\left(t \cdot \left(y \cdot z\right)\right) \cdot 18}\right) \cdot x \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4, i, \color{blue}{\left(t \cdot \left(y \cdot z\right)\right) \cdot 18}\right) \cdot x \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-4, i, \color{blue}{\left(\left(y \cdot z\right) \cdot t\right)} \cdot 18\right) \cdot x \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-4, i, \color{blue}{\left(\left(y \cdot z\right) \cdot t\right)} \cdot 18\right) \cdot x \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-4, i, \left(\color{blue}{\left(z \cdot y\right)} \cdot t\right) \cdot 18\right) \cdot x \]

      12. lower-*.f64 71.6

          \[\leadsto \mathsf{fma}\left(-4, i, \left(\color{blue}{\left(z \cdot y\right)} \cdot t\right) \cdot 18\right) \cdot x \]

   5. Applied rewrites 71.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4, i, \left(\left(z \cdot y\right) \cdot t\right) \cdot 18\right) \cdot x} \]

   if -1.16e27 < x < 1.00000000000000001e155

   1. Initial program 96.3%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

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

         \[\leadsto \color{blue}{\left(b \cdot c + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot t\right)\right)} - \left(j \cdot 27\right) \cdot k \]

      2. metadata-eval N/A

         \[\leadsto \left(b \cdot c + \color{blue}{-4} \cdot \left(a \cdot t\right)\right) - \left(j \cdot 27\right) \cdot k \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(-4 \cdot \left(a \cdot t\right) + b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-4 \cdot a\right) \cdot t} + b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-4 \cdot a}, t, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, \color{blue}{c \cdot b}\right) - \left(j \cdot 27\right) \cdot k \]

      8. lower-*.f64 78.0

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, \color{blue}{c \cdot b}\right) - \left(j \cdot 27\right) \cdot k \]

   5. Applied rewrites 78.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)} - \left(j \cdot 27\right) \cdot k \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) - \left(j \cdot 27\right) \cdot k} \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) - \color{blue}{\left(j \cdot 27\right) \cdot k} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \left(\mathsf{neg}\left(j \cdot 27\right)\right) \cdot k} \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \left(\mathsf{neg}\left(\color{blue}{j \cdot 27}\right)\right) \cdot k \]

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

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \color{blue}{\left(j \cdot \left(\mathsf{neg}\left(27\right)\right)\right)} \cdot k \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \left(j \cdot \color{blue}{-27}\right) \cdot k \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \color{blue}{j \cdot \left(-27 \cdot k\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + j \cdot \color{blue}{\left(-27 \cdot k\right)} \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right) + \color{blue}{\left(-27 \cdot k\right) \cdot j} \]

      10. +-commutative N/A

          \[\leadsto \color{blue}{\left(-27 \cdot k\right) \cdot j + \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)} \]

      11. lower-fma.f64 78.7

          \[\leadsto \color{blue}{\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)\right)} \]

   7. Applied rewrites 78.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(-4 \cdot a, t, c \cdot b\right)\right)} \]

   8. Taylor expanded in c around inf

      \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{b \cdot c}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{c \cdot b}\right) \]

      2. lower-*.f64 61.5

         \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{c \cdot b}\right) \]

   10. Applied rewrites 61.5%

       \[\leadsto \mathsf{fma}\left(-27 \cdot k, j, \color{blue}{c \cdot b}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 23.9% accurate, 11.3× speedup

Math

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

FPCore

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

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	return c * b;
}

Fortran

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

Java

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

Python

[x, y, z, t, a, b, c, i, j, k] = sort([x, y, z, t, a, b, c, i, j, k])
[x, y, z, t, a, b, c, i, j, k] = sort([x, y, z, t, a, b, c, i, j, k])
def code(x, y, z, t, a, b, c, i, j, k):
	return c * b

Julia

x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
function code(x, y, z, t, a, b, c, i, j, k)
	return Float64(c * b)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.0%

   \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
2. Add Preprocessing

3. Taylor expanded in c around inf

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{c \cdot b} \]

   2. lower-*.f64 26.9

      \[\leadsto \color{blue}{c \cdot b} \]

5. Applied rewrites 26.9%

   \[\leadsto \color{blue}{c \cdot b} \]
6. Add Preprocessing

Developer Target 1: 89.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a \cdot t + i \cdot x\right) \cdot 4\\ t_2 := \left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - t\_1\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \mathbf{if}\;t < -1.6210815397541398 \cdot 10^{-69}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t < 165.68027943805222:\\ \;\;\;\;\left(\left(18 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - t\_1\right) + \left(c \cdot b - 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (* (+ (* a t) (* i x)) 4.0))
        (t_2
         (-
          (- (* (* 18.0 t) (* (* x y) z)) t_1)
          (- (* (* k j) 27.0) (* c b)))))
   (if (< t -1.6210815397541398e-69)
     t_2
     (if (< t 165.68027943805222)
       (+ (- (* (* 18.0 y) (* x (* z t))) t_1) (- (* c b) (* 27.0 (* k j))))
       t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = ((a * t) + (i * x)) * 4.0;
	double t_2 = (((18.0 * t) * ((x * y) * z)) - t_1) - (((k * j) * 27.0) - (c * b));
	double tmp;
	if (t < -1.6210815397541398e-69) {
		tmp = t_2;
	} else if (t < 165.68027943805222) {
		tmp = (((18.0 * y) * (x * (z * t))) - t_1) + ((c * b) - (27.0 * (k * j)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j, k)
    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), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((a * t) + (i * x)) * 4.0d0
    t_2 = (((18.0d0 * t) * ((x * y) * z)) - t_1) - (((k * j) * 27.0d0) - (c * b))
    if (t < (-1.6210815397541398d-69)) then
        tmp = t_2
    else if (t < 165.68027943805222d0) then
        tmp = (((18.0d0 * y) * (x * (z * t))) - t_1) + ((c * b) - (27.0d0 * (k * j)))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = ((a * t) + (i * x)) * 4.0;
	double t_2 = (((18.0 * t) * ((x * y) * z)) - t_1) - (((k * j) * 27.0) - (c * b));
	double tmp;
	if (t < -1.6210815397541398e-69) {
		tmp = t_2;
	} else if (t < 165.68027943805222) {
		tmp = (((18.0 * y) * (x * (z * t))) - t_1) + ((c * b) - (27.0 * (k * j)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j, k):
	t_1 = ((a * t) + (i * x)) * 4.0
	t_2 = (((18.0 * t) * ((x * y) * z)) - t_1) - (((k * j) * 27.0) - (c * b))
	tmp = 0
	if t < -1.6210815397541398e-69:
		tmp = t_2
	elif t < 165.68027943805222:
		tmp = (((18.0 * y) * (x * (z * t))) - t_1) + ((c * b) - (27.0 * (k * j)))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = Float64(Float64(Float64(a * t) + Float64(i * x)) * 4.0)
	t_2 = Float64(Float64(Float64(Float64(18.0 * t) * Float64(Float64(x * y) * z)) - t_1) - Float64(Float64(Float64(k * j) * 27.0) - Float64(c * b)))
	tmp = 0.0
	if (t < -1.6210815397541398e-69)
		tmp = t_2;
	elseif (t < 165.68027943805222)
		tmp = Float64(Float64(Float64(Float64(18.0 * y) * Float64(x * Float64(z * t))) - t_1) + Float64(Float64(c * b) - Float64(27.0 * Float64(k * j))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j, k)
	t_1 = ((a * t) + (i * x)) * 4.0;
	t_2 = (((18.0 * t) * ((x * y) * z)) - t_1) - (((k * j) * 27.0) - (c * b));
	tmp = 0.0;
	if (t < -1.6210815397541398e-69)
		tmp = t_2;
	elseif (t < 165.68027943805222)
		tmp = (((18.0 * y) * (x * (z * t))) - t_1) + ((c * b) - (27.0 * (k * j)));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(N[(N[(a * t), $MachinePrecision] + N[(i * x), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(18.0 * t), $MachinePrecision] * N[(N[(x * y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] - N[(N[(N[(k * j), $MachinePrecision] * 27.0), $MachinePrecision] - N[(c * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t, -1.6210815397541398e-69], t$95$2, If[Less[t, 165.68027943805222], N[(N[(N[(N[(18.0 * y), $MachinePrecision] * N[(x * N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] + N[(N[(c * b), $MachinePrecision] - N[(27.0 * N[(k * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Reproduce

herbie shell --seed 2024268 
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, E"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< t -8105407698770699/5000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (- (* (* 18 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4)) (- (* (* k j) 27) (* c b))) (if (< t 8284013971902611/50000000000000) (+ (- (* (* 18 y) (* x (* z t))) (* (+ (* a t) (* i x)) 4)) (- (* c b) (* 27 (* k j)))) (- (- (* (* 18 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4)) (- (* (* k j) 27) (* c b))))))

  (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)))