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

Percentage Accurate: 84.7% → 93.1%

Time: 31.1s

Alternatives: 21

Speedup: 1.2×

• 50.8% 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: 84.7% 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: 93.1% accurate, 0.3× speedup

Math

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

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
         (-
          (-
           (+ (* c b) (- (* t (* z (* (* 18.0 x) y))) (* (* 4.0 a) t)))
           (* i (* 4.0 x)))
          (* k (* 27.0 j)))))
   (if (<= t_1 (- INFINITY))
     (*
      (fma
       i
       4.0
       (fma
        (* (* -18.0 t) y)
        z
        (/ (fma c b (fma (* -27.0 k) j (* (* a t) -4.0))) (- x))))
      (- x))
     (if (<= t_1 INFINITY)
       t_1
       (fma
        (* -27.0 k)
        j
        (fma (fma (* (* z y) t) 18.0 (* -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 = (((c * b) + ((t * (z * ((18.0 * x) * y))) - ((4.0 * a) * t))) - (i * (4.0 * x))) - (k * (27.0 * j));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(i, 4.0, fma(((-18.0 * t) * y), z, (fma(c, b, fma((-27.0 * k), j, ((a * t) * -4.0))) / -x))) * -x;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = fma((-27.0 * k), j, fma(fma(((z * y) * t), 18.0, (-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(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))) - Float64(k * Float64(27.0 * j)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(i, 4.0, fma(Float64(Float64(-18.0 * t) * y), z, Float64(fma(c, b, fma(Float64(-27.0 * k), j, Float64(Float64(a * t) * -4.0))) / Float64(-x)))) * Float64(-x));
	elseif (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = fma(Float64(-27.0 * k), j, fma(fma(Float64(Float64(z * y) * t), 18.0, 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[(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] - N[(k * N[(27.0 * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(i * 4.0 + N[(N[(N[(-18.0 * t), $MachinePrecision] * y), $MachinePrecision] * z + N[(N[(c * b + N[(N[(-27.0 * k), $MachinePrecision] * j + N[(N[(a * t), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-x)), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$1, N[(N[(-27.0 * k), $MachinePrecision] * j + N[(N[(N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] * 18.0 + 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 84.3%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\left(-18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + -1 \cdot \frac{b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 27 \cdot \left(j \cdot k\right)\right)}{x}\right) - -4 \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot x\right) \cdot \left(\left(-18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + -1 \cdot \frac{b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 27 \cdot \left(j \cdot k\right)\right)}{x}\right) - -4 \cdot i\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot x\right) \cdot \left(\left(-18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + -1 \cdot \frac{b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 27 \cdot \left(j \cdot k\right)\right)}{x}\right) - -4 \cdot i\right)} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \left(\left(-18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + -1 \cdot \frac{b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 27 \cdot \left(j \cdot k\right)\right)}{x}\right) - -4 \cdot i\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-x\right)} \cdot \left(\left(-18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + -1 \cdot \frac{b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 27 \cdot \left(j \cdot k\right)\right)}{x}\right) - -4 \cdot i\right) \]

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

         \[\leadsto \left(-x\right) \cdot \color{blue}{\left(\left(-18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + -1 \cdot \frac{b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 27 \cdot \left(j \cdot k\right)\right)}{x}\right) + \left(\mathsf{neg}\left(-4\right)\right) \cdot i\right)} \]

      6. metadata-eval N/A

         \[\leadsto \left(-x\right) \cdot \left(\left(-18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + -1 \cdot \frac{b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 27 \cdot \left(j \cdot k\right)\right)}{x}\right) + \color{blue}{4} \cdot i\right) \]

      7. +-commutative N/A

         \[\leadsto \left(-x\right) \cdot \color{blue}{\left(4 \cdot i + \left(-18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + -1 \cdot \frac{b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 27 \cdot \left(j \cdot k\right)\right)}{x}\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \left(-x\right) \cdot \left(\color{blue}{i \cdot 4} + \left(-18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + -1 \cdot \frac{b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 27 \cdot \left(j \cdot k\right)\right)}{x}\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(-x\right) \cdot \color{blue}{\mathsf{fma}\left(i, 4, -18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + -1 \cdot \frac{b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 27 \cdot \left(j \cdot k\right)\right)}{x}\right)} \]

   5. Applied rewrites 98.4%

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

   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)) < +inf.0

   1. Initial program 98.1%

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

   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))

   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 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 26.3

         \[\leadsto \color{blue}{c \cdot b} \]

   5. Applied rewrites 26.3%

      \[\leadsto \color{blue}{c \cdot b} \]

   6. Taylor expanded in a around 0

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

      1. associate--r+ N/A

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

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

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   8. Applied rewrites 82.1%

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

4. Final simplification 96.5%

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

Alternative 2: 49.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\\\ [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \left(c \cdot b + \left(t \cdot \left(z \cdot \left(\left(18 \cdot x\right) \cdot y\right)\right) - \left(4 \cdot a\right) \cdot t\right)\right) - i \cdot \left(4 \cdot x\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+240}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \left(i \cdot x\right) \cdot -4\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+176}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot j, -27, \left(a \cdot t\right) \cdot -4\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+255}:\\ \;\;\;\;\mathsf{fma}\left(i, x, a \cdot t\right) \cdot -4\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(-4 \cdot t, a, c \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t \cdot x\right) \cdot z\right) \cdot y\right) \cdot 18\\ \end{array} \end{array} \]

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+240)
     (fma c b (* (* i x) -4.0))
     (if (<= t_1 2e+176)
       (fma (* k j) -27.0 (* (* a t) -4.0))
       (if (<= t_1 5e+255)
         (* (fma i x (* a t)) -4.0)
         (if (<= t_1 INFINITY)
           (fma (* -4.0 t) a (* c b))
           (* (* (* (* t x) z) y) 18.0)))))))

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 = ((c * b) + ((t * (z * ((18.0 * x) * y))) - ((4.0 * a) * t))) - (i * (4.0 * x));
	double tmp;
	if (t_1 <= -2e+240) {
		tmp = fma(c, b, ((i * x) * -4.0));
	} else if (t_1 <= 2e+176) {
		tmp = fma((k * j), -27.0, ((a * t) * -4.0));
	} else if (t_1 <= 5e+255) {
		tmp = fma(i, x, (a * t)) * -4.0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma((-4.0 * t), a, (c * b));
	} else {
		tmp = (((t * x) * z) * y) * 18.0;
	}
	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(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+240)
		tmp = fma(c, b, Float64(Float64(i * x) * -4.0));
	elseif (t_1 <= 2e+176)
		tmp = fma(Float64(k * j), -27.0, Float64(Float64(a * t) * -4.0));
	elseif (t_1 <= 5e+255)
		tmp = Float64(fma(i, x, Float64(a * t)) * -4.0);
	elseif (t_1 <= Inf)
		tmp = fma(Float64(-4.0 * t), a, Float64(c * b));
	else
		tmp = Float64(Float64(Float64(Float64(t * x) * z) * y) * 18.0);
	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[(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+240], N[(c * b + N[(N[(i * x), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+176], N[(N[(k * j), $MachinePrecision] * -27.0 + N[(N[(a * t), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+255], N[(N[(i * x + N[(a * t), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(-4.0 * t), $MachinePrecision] * a + N[(c * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t * x), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision] * 18.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 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)) < -2.00000000000000003e240

   1. Initial program 80.2%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   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 63.4

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

   5. Applied rewrites 63.4%

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

   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 60.9%

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

   if -2.00000000000000003e240 < (-.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)) < 2e176

   1. Initial program 98.7%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing
   3. 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 99.7

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

      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%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 65.2

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

   7. Applied rewrites 65.2%

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

   if 2e176 < (-.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)) < 5.0000000000000002e255

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 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 90.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 90.7%

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

   9. Taylor expanded in c around 0

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

   11. Applied rewrites 86.3%

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

   if 5.0000000000000002e255 < (-.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 92.8%

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

   3. Taylor expanded in 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 87.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 58.6%

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

   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 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 71.5

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

   5. Applied rewrites 71.5%

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

   7. Applied rewrites 71.5%

      \[\leadsto \left(y \cdot \left(z \cdot \left(t \cdot x\right)\right)\right) \cdot 18 \]
3. Recombined 5 regimes into one program.

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot b + \left(t \cdot \left(z \cdot \left(\left(18 \cdot x\right) \cdot y\right)\right) - \left(4 \cdot a\right) \cdot t\right)\right) - i \cdot \left(4 \cdot x\right) \leq -2 \cdot 10^{+240}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \left(i \cdot x\right) \cdot -4\right)\\ \mathbf{elif}\;\left(c \cdot b + \left(t \cdot \left(z \cdot \left(\left(18 \cdot x\right) \cdot y\right)\right) - \left(4 \cdot a\right) \cdot t\right)\right) - i \cdot \left(4 \cdot x\right) \leq 2 \cdot 10^{+176}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot j, -27, \left(a \cdot t\right) \cdot -4\right)\\ \mathbf{elif}\;\left(c \cdot b + \left(t \cdot \left(z \cdot \left(\left(18 \cdot x\right) \cdot y\right)\right) - \left(4 \cdot a\right) \cdot t\right)\right) - i \cdot \left(4 \cdot x\right) \leq 5 \cdot 10^{+255}:\\ \;\;\;\;\mathsf{fma}\left(i, x, a \cdot t\right) \cdot -4\\ \mathbf{elif}\;\left(c \cdot b + \left(t \cdot \left(z \cdot \left(\left(18 \cdot x\right) \cdot y\right)\right) - \left(4 \cdot a\right) \cdot t\right)\right) - i \cdot \left(4 \cdot x\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(-4 \cdot t, a, c \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t \cdot x\right) \cdot z\right) \cdot y\right) \cdot 18\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\\\ [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \left(c \cdot b + \left(t \cdot \left(z \cdot \left(\left(18 \cdot x\right) \cdot y\right)\right) - \left(4 \cdot a\right) \cdot t\right)\right) - i \cdot \left(4 \cdot x\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+244}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \left(i \cdot x\right) \cdot -4\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot j, k, c \cdot b\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+255}:\\ \;\;\;\;\mathsf{fma}\left(i, x, a \cdot t\right) \cdot -4\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(-4 \cdot t, a, c \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t \cdot x\right) \cdot z\right) \cdot y\right) \cdot 18\\ \end{array} \end{array} \]

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+244)
     (fma c b (* (* i x) -4.0))
     (if (<= t_1 5e+114)
       (fma (* -27.0 j) k (* c b))
       (if (<= t_1 5e+255)
         (* (fma i x (* a t)) -4.0)
         (if (<= t_1 INFINITY)
           (fma (* -4.0 t) a (* c b))
           (* (* (* (* t x) z) y) 18.0)))))))

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 = ((c * b) + ((t * (z * ((18.0 * x) * y))) - ((4.0 * a) * t))) - (i * (4.0 * x));
	double tmp;
	if (t_1 <= -2e+244) {
		tmp = fma(c, b, ((i * x) * -4.0));
	} else if (t_1 <= 5e+114) {
		tmp = fma((-27.0 * j), k, (c * b));
	} else if (t_1 <= 5e+255) {
		tmp = fma(i, x, (a * t)) * -4.0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma((-4.0 * t), a, (c * b));
	} else {
		tmp = (((t * x) * z) * y) * 18.0;
	}
	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(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+244)
		tmp = fma(c, b, Float64(Float64(i * x) * -4.0));
	elseif (t_1 <= 5e+114)
		tmp = fma(Float64(-27.0 * j), k, Float64(c * b));
	elseif (t_1 <= 5e+255)
		tmp = Float64(fma(i, x, Float64(a * t)) * -4.0);
	elseif (t_1 <= Inf)
		tmp = fma(Float64(-4.0 * t), a, Float64(c * b));
	else
		tmp = Float64(Float64(Float64(Float64(t * x) * z) * y) * 18.0);
	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[(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+244], N[(c * b + N[(N[(i * x), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+114], N[(N[(-27.0 * j), $MachinePrecision] * k + N[(c * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+255], N[(N[(i * x + N[(a * t), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(-4.0 * t), $MachinePrecision] * a + N[(c * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t * x), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision] * 18.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 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)) < -2.00000000000000015e244

   1. Initial program 79.9%

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

   3. Taylor expanded in 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 62.9

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

   5. Applied rewrites 62.9%

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

   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 60.3%

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

   if -2.00000000000000015e244 < (-.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)) < 5.0000000000000001e114

   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 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 75.2

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

   5. Applied rewrites 75.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 63.1%

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

   if 5.0000000000000001e114 < (-.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)) < 5.0000000000000002e255

   1. Initial program 95.1%

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

   3. Taylor expanded in 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 71.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 71.3%

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

   9. Taylor expanded in c around 0

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

   11. Applied rewrites 68.5%

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

   if 5.0000000000000002e255 < (-.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 92.8%

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

   3. Taylor expanded in 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 87.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 58.6%

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

   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 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 71.5

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

   5. Applied rewrites 71.5%

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

   7. Applied rewrites 71.5%

      \[\leadsto \left(y \cdot \left(z \cdot \left(t \cdot x\right)\right)\right) \cdot 18 \]
3. Recombined 5 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot b + \left(t \cdot \left(z \cdot \left(\left(18 \cdot x\right) \cdot y\right)\right) - \left(4 \cdot a\right) \cdot t\right)\right) - i \cdot \left(4 \cdot x\right) \leq -2 \cdot 10^{+244}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \left(i \cdot x\right) \cdot -4\right)\\ \mathbf{elif}\;\left(c \cdot b + \left(t \cdot \left(z \cdot \left(\left(18 \cdot x\right) \cdot y\right)\right) - \left(4 \cdot a\right) \cdot t\right)\right) - i \cdot \left(4 \cdot x\right) \leq 5 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot j, k, c \cdot b\right)\\ \mathbf{elif}\;\left(c \cdot b + \left(t \cdot \left(z \cdot \left(\left(18 \cdot x\right) \cdot y\right)\right) - \left(4 \cdot a\right) \cdot t\right)\right) - i \cdot \left(4 \cdot x\right) \leq 5 \cdot 10^{+255}:\\ \;\;\;\;\mathsf{fma}\left(i, x, a \cdot t\right) \cdot -4\\ \mathbf{elif}\;\left(c \cdot b + \left(t \cdot \left(z \cdot \left(\left(18 \cdot x\right) \cdot y\right)\right) - \left(4 \cdot a\right) \cdot t\right)\right) - i \cdot \left(4 \cdot x\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(-4 \cdot t, a, c \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t \cdot x\right) \cdot z\right) \cdot y\right) \cdot 18\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 49.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\\\ [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \left(c \cdot b + \left(t \cdot \left(z \cdot \left(\left(18 \cdot x\right) \cdot y\right)\right) - \left(4 \cdot a\right) \cdot t\right)\right) - i \cdot \left(4 \cdot x\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+244}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \left(i \cdot x\right) \cdot -4\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot j, k, c \cdot b\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+255}:\\ \;\;\;\;\mathsf{fma}\left(i, x, a \cdot t\right) \cdot -4\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(-4 \cdot t, a, c \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t \cdot z\right) \cdot \left(y \cdot 18\right)\right) \cdot x\\ \end{array} \end{array} \]

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+244)
     (fma c b (* (* i x) -4.0))
     (if (<= t_1 5e+114)
       (fma (* -27.0 j) k (* c b))
       (if (<= t_1 5e+255)
         (* (fma i x (* a t)) -4.0)
         (if (<= t_1 INFINITY)
           (fma (* -4.0 t) a (* c b))
           (* (* (* t z) (* y 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+244) {
		tmp = fma(c, b, ((i * x) * -4.0));
	} else if (t_1 <= 5e+114) {
		tmp = fma((-27.0 * j), k, (c * b));
	} else if (t_1 <= 5e+255) {
		tmp = fma(i, x, (a * t)) * -4.0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma((-4.0 * t), a, (c * b));
	} else {
		tmp = ((t * z) * (y * 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+244)
		tmp = fma(c, b, Float64(Float64(i * x) * -4.0));
	elseif (t_1 <= 5e+114)
		tmp = fma(Float64(-27.0 * j), k, Float64(c * b));
	elseif (t_1 <= 5e+255)
		tmp = Float64(fma(i, x, Float64(a * t)) * -4.0);
	elseif (t_1 <= Inf)
		tmp = fma(Float64(-4.0 * t), a, Float64(c * b));
	else
		tmp = Float64(Float64(Float64(t * z) * Float64(y * 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+244], N[(c * b + N[(N[(i * x), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+114], N[(N[(-27.0 * j), $MachinePrecision] * k + N[(c * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+255], N[(N[(i * x + N[(a * t), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(-4.0 * t), $MachinePrecision] * a + N[(c * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t * z), $MachinePrecision] * N[(y * 18.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 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)) < -2.00000000000000015e244

   1. Initial program 79.9%

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

   3. Taylor expanded in 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 62.9

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

   5. Applied rewrites 62.9%

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

   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 60.3%

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

   if -2.00000000000000015e244 < (-.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)) < 5.0000000000000001e114

   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 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 75.2

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

   5. Applied rewrites 75.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 63.1%

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

   if 5.0000000000000001e114 < (-.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)) < 5.0000000000000002e255

   1. Initial program 95.1%

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

   3. Taylor expanded in 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 71.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 71.3%

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

   9. Taylor expanded in c around 0

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

   11. Applied rewrites 68.5%

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

   if 5.0000000000000002e255 < (-.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 92.8%

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

   3. Taylor expanded in 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 87.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 58.6%

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

   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 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 71.5

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

   5. Applied rewrites 71.5%

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

   7. Applied rewrites 71.5%

      \[\leadsto x \cdot \color{blue}{\left(\left(18 \cdot y\right) \cdot \left(t \cdot z\right)\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot b + \left(t \cdot \left(z \cdot \left(\left(18 \cdot x\right) \cdot y\right)\right) - \left(4 \cdot a\right) \cdot t\right)\right) - i \cdot \left(4 \cdot x\right) \leq -2 \cdot 10^{+244}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \left(i \cdot x\right) \cdot -4\right)\\ \mathbf{elif}\;\left(c \cdot b + \left(t \cdot \left(z \cdot \left(\left(18 \cdot x\right) \cdot y\right)\right) - \left(4 \cdot a\right) \cdot t\right)\right) - i \cdot \left(4 \cdot x\right) \leq 5 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot j, k, c \cdot b\right)\\ \mathbf{elif}\;\left(c \cdot b + \left(t \cdot \left(z \cdot \left(\left(18 \cdot x\right) \cdot y\right)\right) - \left(4 \cdot a\right) \cdot t\right)\right) - i \cdot \left(4 \cdot x\right) \leq 5 \cdot 10^{+255}:\\ \;\;\;\;\mathsf{fma}\left(i, x, a \cdot t\right) \cdot -4\\ \mathbf{elif}\;\left(c \cdot b + \left(t \cdot \left(z \cdot \left(\left(18 \cdot x\right) \cdot y\right)\right) - \left(4 \cdot a\right) \cdot t\right)\right) - i \cdot \left(4 \cdot x\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(-4 \cdot t, a, c \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t \cdot z\right) \cdot \left(y \cdot 18\right)\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.0% accurate, 0.3× speedup

Math

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

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 (* k (* 27.0 j)))
        (t_2
         (-
          (-
           (+ (* c b) (- (* t (* z (* (* 18.0 x) y))) (* (* 4.0 a) t)))
           (* i (* 4.0 x)))
          t_1)))
   (if (<= t_2 (- INFINITY))
     (-
      (fma
       x
       (* (* t z) (* y 18.0))
       (fma (* a t) -4.0 (fma c b (* (* i x) -4.0))))
      t_1)
     (if (<= t_2 INFINITY)
       t_2
       (fma
        (* -27.0 k)
        j
        (fma (fma (* (* z y) t) 18.0 (* -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 = k * (27.0 * j);
	double t_2 = (((c * b) + ((t * (z * ((18.0 * x) * y))) - ((4.0 * a) * t))) - (i * (4.0 * x))) - t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = fma(x, ((t * z) * (y * 18.0)), fma((a * t), -4.0, fma(c, b, ((i * x) * -4.0)))) - t_1;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = fma((-27.0 * k), j, fma(fma(((z * y) * t), 18.0, (-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(k * Float64(27.0 * j))
	t_2 = Float64(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))) - t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(fma(x, Float64(Float64(t * z) * Float64(y * 18.0)), fma(Float64(a * t), -4.0, fma(c, b, Float64(Float64(i * x) * -4.0)))) - t_1);
	elseif (t_2 <= Inf)
		tmp = t_2;
	else
		tmp = fma(Float64(-27.0 * k), j, fma(fma(Float64(Float64(z * y) * t), 18.0, 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[(k * N[(27.0 * j), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(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] - t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(x * N[(N[(t * z), $MachinePrecision] * N[(y * 18.0), $MachinePrecision]), $MachinePrecision] + N[(N[(a * t), $MachinePrecision] * -4.0 + N[(c * b + N[(N[(i * x), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$2, N[(N[(-27.0 * k), $MachinePrecision] * j + N[(N[(N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] * 18.0 + 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 84.3%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing
   3. 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. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 95.7%

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

   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)) < +inf.0

   1. Initial program 98.1%

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

   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))

   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 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 26.3

         \[\leadsto \color{blue}{c \cdot b} \]

   5. Applied rewrites 26.3%

      \[\leadsto \color{blue}{c \cdot b} \]

   6. Taylor expanded in a around 0

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

      1. associate--r+ N/A

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

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

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   8. Applied rewrites 82.1%

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

4. Final simplification 95.8%

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

Alternative 6: 93.0% accurate, 0.3× speedup

Math

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

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 (* k (* 27.0 j)))
        (t_2 (* (* 18.0 x) y))
        (t_3
         (-
          (- (+ (* c b) (- (* t (* z t_2)) (* (* 4.0 a) t))) (* i (* 4.0 x)))
          t_1)))
   (if (<= t_3 (- INFINITY))
     (-
      (fma
       x
       (* (* t z) (* y 18.0))
       (fma (* a t) -4.0 (fma c b (* (* i x) -4.0))))
      t_1)
     (if (<= t_3 INFINITY)
       (fma
        (* k j)
        -27.0
        (fma (* i x) -4.0 (fma (fma z t_2 (* -4.0 a)) t (* c b))))
       (fma
        (* -27.0 k)
        j
        (fma (fma (* (* z y) t) 18.0 (* -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 = k * (27.0 * j);
	double t_2 = (18.0 * x) * y;
	double t_3 = (((c * b) + ((t * (z * t_2)) - ((4.0 * a) * t))) - (i * (4.0 * x))) - t_1;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = fma(x, ((t * z) * (y * 18.0)), fma((a * t), -4.0, fma(c, b, ((i * x) * -4.0)))) - t_1;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = fma((k * j), -27.0, fma((i * x), -4.0, fma(fma(z, t_2, (-4.0 * a)), t, (c * b))));
	} else {
		tmp = fma((-27.0 * k), j, fma(fma(((z * y) * t), 18.0, (-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(k * Float64(27.0 * j))
	t_2 = Float64(Float64(18.0 * x) * y)
	t_3 = Float64(Float64(Float64(Float64(c * b) + Float64(Float64(t * Float64(z * t_2)) - Float64(Float64(4.0 * a) * t))) - Float64(i * Float64(4.0 * x))) - t_1)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(fma(x, Float64(Float64(t * z) * Float64(y * 18.0)), fma(Float64(a * t), -4.0, fma(c, b, Float64(Float64(i * x) * -4.0)))) - t_1);
	elseif (t_3 <= Inf)
		tmp = fma(Float64(k * j), -27.0, fma(Float64(i * x), -4.0, fma(fma(z, t_2, Float64(-4.0 * a)), t, Float64(c * b))));
	else
		tmp = fma(Float64(-27.0 * k), j, fma(fma(Float64(Float64(z * y) * t), 18.0, 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[(k * N[(27.0 * j), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(18.0 * x), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(c * b), $MachinePrecision] + N[(N[(t * N[(z * t$95$2), $MachinePrecision]), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(i * N[(4.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(x * N[(N[(t * z), $MachinePrecision] * N[(y * 18.0), $MachinePrecision]), $MachinePrecision] + N[(N[(a * t), $MachinePrecision] * -4.0 + N[(c * b + N[(N[(i * x), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(k * j), $MachinePrecision] * -27.0 + N[(N[(i * x), $MachinePrecision] * -4.0 + N[(N[(z * t$95$2 + 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[(z * y), $MachinePrecision] * t), $MachinePrecision] * 18.0 + 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 84.3%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing
   3. 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. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 95.7%

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

   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)) < +inf.0

   1. Initial program 98.1%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   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 98.2

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

      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 98.2%

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

   if +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))

   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 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 26.3

         \[\leadsto \color{blue}{c \cdot b} \]

   5. Applied rewrites 26.3%

      \[\leadsto \color{blue}{c \cdot b} \]

   6. Taylor expanded in a around 0

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

      1. associate--r+ N/A

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

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

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   8. Applied rewrites 82.1%

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

4. Final simplification 95.8%

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

Alternative 7: 91.3% accurate, 0.5× speedup

Math

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

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
 (if (<=
      (-
       (-
        (+ (* c b) (- (* t (* z (* (* 18.0 x) y))) (* (* 4.0 a) t)))
        (* i (* 4.0 x)))
       (* k (* 27.0 j)))
      INFINITY)
   (fma
    (fma i x (* a t))
    -4.0
    (fma -27.0 (* k j) (fma c b (* (* (* (* z y) x) t) 18.0))))
   (fma (* -27.0 k) j (fma (fma (* (* z y) t) 18.0 (* -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 tmp;
	if (((((c * b) + ((t * (z * ((18.0 * x) * y))) - ((4.0 * a) * t))) - (i * (4.0 * x))) - (k * (27.0 * j))) <= ((double) INFINITY)) {
		tmp = fma(fma(i, x, (a * t)), -4.0, fma(-27.0, (k * j), fma(c, b, ((((z * y) * x) * t) * 18.0))));
	} else {
		tmp = fma((-27.0 * k), j, fma(fma(((z * y) * t), 18.0, (-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)
	tmp = 0.0
	if (Float64(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))) - Float64(k * Float64(27.0 * j))) <= Inf)
		tmp = fma(fma(i, x, Float64(a * t)), -4.0, fma(-27.0, Float64(k * j), fma(c, b, Float64(Float64(Float64(Float64(z * y) * x) * t) * 18.0))));
	else
		tmp = fma(Float64(-27.0 * k), j, fma(fma(Float64(Float64(z * y) * t), 18.0, 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_] := If[LessEqual[N[(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] - N[(k * N[(27.0 * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(i * x + N[(a * t), $MachinePrecision]), $MachinePrecision] * -4.0 + N[(-27.0 * N[(k * j), $MachinePrecision] + N[(c * b + N[(N[(N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision] * t), $MachinePrecision] * 18.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-27.0 * k), $MachinePrecision] * j + N[(N[(N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] * 18.0 + 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)) < +inf.0

   1. Initial program 94.5%

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

   3. Taylor expanded in c around 0

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

      1. associate-+r+ N/A

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

      2. associate--r+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

   5. Applied rewrites 93.2%

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

   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))

   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 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 26.3

         \[\leadsto \color{blue}{c \cdot b} \]

   5. Applied rewrites 26.3%

      \[\leadsto \color{blue}{c \cdot b} \]

   6. Taylor expanded in a around 0

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

      1. associate--r+ N/A

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

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

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   8. Applied rewrites 82.1%

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

4. Final simplification 92.0%

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

Alternative 8: 79.1% accurate, 0.6× speedup

Math

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

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
 (if (<=
      (-
       (-
        (+ (* c b) (- (* t (* z (* (* 18.0 x) y))) (* (* 4.0 a) t)))
        (* i (* 4.0 x)))
       (* k (* 27.0 j)))
      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 tmp;
	if (((((c * b) + ((t * (z * ((18.0 * x) * y))) - ((4.0 * a) * t))) - (i * (4.0 * x))) - (k * (27.0 * j))) <= ((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)
	tmp = 0.0
	if (Float64(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))) - Float64(k * Float64(27.0 * j))) <= 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_] := If[LessEqual[N[(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] - N[(k * N[(27.0 * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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 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)) < +inf.0

   1. Initial program 94.5%

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

   3. Taylor expanded in 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 85.2%

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

   if +inf.0 < (-.f64 (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.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 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 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%

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

4. Final simplification 83.7%

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

Alternative 9: 92.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\\\ [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(\mathsf{fma}\left(\left(z \cdot y\right) \cdot t, 18, -4 \cdot i\right), x, c \cdot b\right)\right)\\ \mathbf{if}\;x \leq -4.3 \cdot 10^{+130}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+149}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot j, -27, \mathsf{fma}\left(i \cdot x, -4, \mathsf{fma}\left(\mathsf{fma}\left(z, \left(18 \cdot x\right) \cdot y, -4 \cdot a\right), t, c \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* (* z y) t) 18.0 (* -4.0 i)) x (* c b)))))
   (if (<= x -4.3e+130)
     t_1
     (if (<= x 7e+149)
       (fma
        (* k j)
        -27.0
        (fma (* i x) -4.0 (fma (fma z (* (* 18.0 x) y) (* -4.0 a)) t (* 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((-27.0 * k), j, fma(fma(((z * y) * t), 18.0, (-4.0 * i)), x, (c * b)));
	double tmp;
	if (x <= -4.3e+130) {
		tmp = t_1;
	} else if (x <= 7e+149) {
		tmp = fma((k * j), -27.0, fma((i * x), -4.0, fma(fma(z, ((18.0 * x) * y), (-4.0 * a)), t, (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 = fma(Float64(-27.0 * k), j, fma(fma(Float64(Float64(z * y) * t), 18.0, Float64(-4.0 * i)), x, Float64(c * b)))
	tmp = 0.0
	if (x <= -4.3e+130)
		tmp = t_1;
	elseif (x <= 7e+149)
		tmp = fma(Float64(k * j), -27.0, fma(Float64(i * x), -4.0, fma(fma(z, Float64(Float64(18.0 * x) * y), Float64(-4.0 * a)), t, 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[(-27.0 * k), $MachinePrecision] * j + N[(N[(N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] * 18.0 + N[(-4.0 * i), $MachinePrecision]), $MachinePrecision] * x + N[(c * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.3e+130], t$95$1, If[LessEqual[x, 7e+149], N[(N[(k * j), $MachinePrecision] * -27.0 + N[(N[(i * x), $MachinePrecision] * -4.0 + N[(N[(z * N[(N[(18.0 * x), $MachinePrecision] * y), $MachinePrecision] + N[(-4.0 * a), $MachinePrecision]), $MachinePrecision] * t + N[(c * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.29999999999999984e130 or 7.00000000000000023e149 < x

   1. Initial program 65.3%

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

   3. Taylor expanded in 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 16.5

         \[\leadsto \color{blue}{c \cdot b} \]

   5. Applied rewrites 16.5%

      \[\leadsto \color{blue}{c \cdot b} \]

   6. Taylor expanded in a around 0

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

      1. associate--r+ N/A

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

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

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   8. Applied rewrites 93.8%

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

   if -4.29999999999999984e130 < x < 7.00000000000000023e149

   1. Initial program 92.6%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing
   3. 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 93.1

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

      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%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{+130}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(\mathsf{fma}\left(\left(z \cdot y\right) \cdot t, 18, -4 \cdot i\right), x, c \cdot b\right)\right)\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+149}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot j, -27, \mathsf{fma}\left(i \cdot x, -4, \mathsf{fma}\left(\mathsf{fma}\left(z, \left(18 \cdot x\right) \cdot y, -4 \cdot a\right), t, c \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(\mathsf{fma}\left(\left(z \cdot y\right) \cdot t, 18, -4 \cdot i\right), x, c \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\\\ [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{+89}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(\mathsf{fma}\left(\left(z \cdot y\right) \cdot t, 18, -4 \cdot i\right), x, c \cdot b\right)\right)\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-129}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \mathsf{fma}\left(\mathsf{fma}\left(i, x, a \cdot t\right), -4, \left(k \cdot j\right) \cdot -27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(\mathsf{fma}\left(-4, a, \left(\left(z \cdot y\right) \cdot x\right) \cdot 18\right), t, c \cdot b\right)\right)\\ \end{array} \end{array} \]

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
 (if (<= x -1.95e+89)
   (fma (* -27.0 k) j (fma (fma (* (* z y) t) 18.0 (* -4.0 i)) x (* c b)))
   (if (<= x 8.2e-129)
     (fma c b (fma (fma i x (* a t)) -4.0 (* (* k j) -27.0)))
     (fma (* -27.0 k) j (fma (fma -4.0 a (* (* (* z y) x) 18.0)) t (* 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 tmp;
	if (x <= -1.95e+89) {
		tmp = fma((-27.0 * k), j, fma(fma(((z * y) * t), 18.0, (-4.0 * i)), x, (c * b)));
	} else if (x <= 8.2e-129) {
		tmp = fma(c, b, fma(fma(i, x, (a * t)), -4.0, ((k * j) * -27.0)));
	} else {
		tmp = fma((-27.0 * k), j, fma(fma(-4.0, a, (((z * y) * x) * 18.0)), t, (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)
	tmp = 0.0
	if (x <= -1.95e+89)
		tmp = fma(Float64(-27.0 * k), j, fma(fma(Float64(Float64(z * y) * t), 18.0, Float64(-4.0 * i)), x, Float64(c * b)));
	elseif (x <= 8.2e-129)
		tmp = fma(c, b, fma(fma(i, x, Float64(a * t)), -4.0, Float64(Float64(k * j) * -27.0)));
	else
		tmp = fma(Float64(-27.0 * k), j, fma(fma(-4.0, a, Float64(Float64(Float64(z * y) * x) * 18.0)), t, 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_] := If[LessEqual[x, -1.95e+89], N[(N[(-27.0 * k), $MachinePrecision] * j + N[(N[(N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision] * 18.0 + N[(-4.0 * i), $MachinePrecision]), $MachinePrecision] * x + N[(c * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.2e-129], 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[(-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]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.95000000000000005e89

   1. Initial program 72.8%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   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 21.0

         \[\leadsto \color{blue}{c \cdot b} \]

   5. Applied rewrites 21.0%

      \[\leadsto \color{blue}{c \cdot b} \]

   6. Taylor expanded in a around 0

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

      1. associate--r+ N/A

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

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

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   8. Applied rewrites 97.7%

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

   if -1.95000000000000005e89 < x < 8.1999999999999999e-129

   1. Initial program 93.0%

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

   3. Taylor expanded in z around 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 88.4%

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

   if 8.1999999999999999e-129 < x

   1. Initial program 79.2%

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

   3. Taylor expanded in 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 88.0%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{+89}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(\mathsf{fma}\left(\left(z \cdot y\right) \cdot t, 18, -4 \cdot i\right), x, c \cdot b\right)\right)\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-129}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \mathsf{fma}\left(\mathsf{fma}\left(i, x, a \cdot t\right), -4, \left(k \cdot j\right) \cdot -27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(\mathsf{fma}\left(-4, a, \left(\left(z \cdot y\right) \cdot x\right) \cdot 18\right), t, c \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\\\ [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(\mathsf{fma}\left(-4, a, \left(\left(z \cdot y\right) \cdot x\right) \cdot 18\right), t, c \cdot b\right)\right)\\ \mathbf{if}\;t \leq -3.6 \cdot 10^{+89}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 5.1 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \mathsf{fma}\left(\mathsf{fma}\left(i, x, a \cdot t\right), -4, \left(k \cdot j\right) \cdot -27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (<= t -3.6e+89)
     t_1
     (if (<= t 5.1e+32)
       (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 (t <= -3.6e+89) {
		tmp = t_1;
	} else if (t <= 5.1e+32) {
		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 (t <= -3.6e+89)
		tmp = t_1;
	elseif (t <= 5.1e+32)
		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[t, -3.6e+89], t$95$1, If[LessEqual[t, 5.1e+32], 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 t < -3.6e89 or 5.10000000000000004e32 < t

   1. Initial program 77.7%

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

   3. Taylor expanded in 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 92.6%

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

   if -3.6e89 < t < 5.10000000000000004e32

   1. Initial program 88.9%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   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 87.6%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{+89}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(\mathsf{fma}\left(-4, a, \left(\left(z \cdot y\right) \cdot x\right) \cdot 18\right), t, c \cdot b\right)\right)\\ \mathbf{elif}\;t \leq 5.1 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \mathsf{fma}\left(\mathsf{fma}\left(i, x, a \cdot t\right), -4, \left(k \cdot j\right) \cdot -27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot k, j, \mathsf{fma}\left(\mathsf{fma}\left(-4, a, \left(\left(z \cdot y\right) \cdot x\right) \cdot 18\right), t, c \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\\\ [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := k \cdot \left(27 \cdot j\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+294}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot j, k, c \cdot b\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, t, i \cdot x\right), -4, c \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot j, -27, \left(i \cdot x\right) \cdot -4\right)\\ \end{array} \end{array} \]

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+294)
     (fma (* -27.0 j) k (* c b))
     (if (<= t_1 2e+87)
       (fma (fma a t (* i x)) -4.0 (* c b))
       (fma (* k j) -27.0 (* (* i x) -4.0))))))

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 <= -5e+294) {
		tmp = fma((-27.0 * j), k, (c * b));
	} else if (t_1 <= 2e+87) {
		tmp = fma(fma(a, t, (i * x)), -4.0, (c * b));
	} else {
		tmp = fma((k * j), -27.0, ((i * x) * -4.0));
	}
	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 <= -5e+294)
		tmp = fma(Float64(-27.0 * j), k, Float64(c * b));
	elseif (t_1 <= 2e+87)
		tmp = fma(fma(a, t, Float64(i * x)), -4.0, Float64(c * b));
	else
		tmp = fma(Float64(k * j), -27.0, Float64(Float64(i * x) * -4.0));
	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, -5e+294], N[(N[(-27.0 * j), $MachinePrecision] * k + N[(c * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+87], N[(N[(a * t + N[(i * x), $MachinePrecision]), $MachinePrecision] * -4.0 + N[(c * b), $MachinePrecision]), $MachinePrecision], N[(N[(k * j), $MachinePrecision] * -27.0 + N[(N[(i * x), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 j #s(literal 27 binary64)) k) < -4.9999999999999999e294

   1. Initial program 77.4%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   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 80.5

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

   5. Applied rewrites 80.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 90.3%

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

   if -4.9999999999999999e294 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 1.9999999999999999e87

   1. Initial program 86.9%

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

   3. Taylor expanded in k around 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 80.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 72.0%

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

   if 1.9999999999999999e87 < (*.f64 (*.f64 j #s(literal 27 binary64)) k)

   1. Initial program 78.0%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing
   3. 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 80.0

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

      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 86.7%

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

   5. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 75.7

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

   7. Applied rewrites 75.7%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \cdot \left(27 \cdot j\right) \leq -5 \cdot 10^{+294}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot j, k, c \cdot b\right)\\ \mathbf{elif}\;k \cdot \left(27 \cdot j\right) \leq 2 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, t, i \cdot x\right), -4, c \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot j, -27, \left(i \cdot x\right) \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 54.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\\\ [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := k \cdot \left(27 \cdot j\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot j, k, c \cdot b\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(-4 \cdot t, a, c \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot j, -27, \left(i \cdot x\right) \cdot -4\right)\\ \end{array} \end{array} \]

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 -4e+60)
     (fma (* -27.0 j) k (* c b))
     (if (<= t_1 5e+52)
       (fma (* -4.0 t) a (* c b))
       (fma (* k j) -27.0 (* (* i x) -4.0))))))

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 <= -4e+60) {
		tmp = fma((-27.0 * j), k, (c * b));
	} else if (t_1 <= 5e+52) {
		tmp = fma((-4.0 * t), a, (c * b));
	} else {
		tmp = fma((k * j), -27.0, ((i * x) * -4.0));
	}
	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 <= -4e+60)
		tmp = fma(Float64(-27.0 * j), k, Float64(c * b));
	elseif (t_1 <= 5e+52)
		tmp = fma(Float64(-4.0 * t), a, Float64(c * b));
	else
		tmp = fma(Float64(k * j), -27.0, Float64(Float64(i * x) * -4.0));
	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, -4e+60], N[(N[(-27.0 * j), $MachinePrecision] * k + N[(c * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+52], N[(N[(-4.0 * t), $MachinePrecision] * a + N[(c * b), $MachinePrecision]), $MachinePrecision], N[(N[(k * j), $MachinePrecision] * -27.0 + N[(N[(i * x), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 j #s(literal 27 binary64)) k) < -3.9999999999999998e60

   1. Initial program 85.7%

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

   3. Taylor expanded in 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 72.0

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

   5. Applied rewrites 72.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 68.7%

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

   if -3.9999999999999998e60 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 5e52

   1. Initial program 85.5%

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

   3. Taylor expanded in 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 84.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 56.0%

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

   if 5e52 < (*.f64 (*.f64 j #s(literal 27 binary64)) k)

   1. Initial program 78.6%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing
   3. 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 80.4

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

      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 88.2%

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

   5. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 74.3

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

   7. Applied rewrites 74.3%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \cdot \left(27 \cdot j\right) \leq -4 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot j, k, c \cdot b\right)\\ \mathbf{elif}\;k \cdot \left(27 \cdot j\right) \leq 5 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(-4 \cdot t, a, c \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot j, -27, \left(i \cdot x\right) \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 54.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\\\ [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-27 \cdot j, k, c \cdot b\right)\\ t_2 := k \cdot \left(27 \cdot j\right)\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(-4 \cdot t, a, c \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* -27.0 j) k (* c b))) (t_2 (* k (* 27.0 j))))
   (if (<= t_2 -4e+60)
     t_1
     (if (<= t_2 5e+64) (fma (* -4.0 t) a (* 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((-27.0 * j), k, (c * b));
	double t_2 = k * (27.0 * j);
	double tmp;
	if (t_2 <= -4e+60) {
		tmp = t_1;
	} else if (t_2 <= 5e+64) {
		tmp = fma((-4.0 * t), a, (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 = fma(Float64(-27.0 * j), k, Float64(c * b))
	t_2 = Float64(k * Float64(27.0 * j))
	tmp = 0.0
	if (t_2 <= -4e+60)
		tmp = t_1;
	elseif (t_2 <= 5e+64)
		tmp = fma(Float64(-4.0 * t), a, 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[(-27.0 * j), $MachinePrecision] * k + N[(c * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(k * N[(27.0 * j), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+60], t$95$1, If[LessEqual[t$95$2, 5e+64], N[(N[(-4.0 * t), $MachinePrecision] * a + N[(c * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 j #s(literal 27 binary64)) k) < -3.9999999999999998e60 or 5e64 < (*.f64 (*.f64 j #s(literal 27 binary64)) k)

   1. Initial program 82.8%

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

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 67.0%

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

   if -3.9999999999999998e60 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 5e64

   1. Initial program 85.2%

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

   3. Taylor expanded in 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 84.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 55.2%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \cdot \left(27 \cdot j\right) \leq -4 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot j, k, c \cdot b\right)\\ \mathbf{elif}\;k \cdot \left(27 \cdot j\right) \leq 5 \cdot 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(-4 \cdot t, a, c \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot j, k, c \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 71.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\\\ [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-4, i, \left(\left(z \cdot y\right) \cdot t\right) \cdot 18\right) \cdot x\\ \mathbf{if}\;x \leq -2.2 \cdot 10^{+131}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, t, i \cdot x\right), -4, c \cdot b\right)\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \mathsf{fma}\left(-27 \cdot k, j, \left(a \cdot t\right) \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -2.2e+131)
     t_1
     (if (<= x -1.75e+22)
       (fma (fma a t (* i x)) -4.0 (* c b))
       (if (<= x 9e-33) (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 <= -2.2e+131) {
		tmp = t_1;
	} else if (x <= -1.75e+22) {
		tmp = fma(fma(a, t, (i * x)), -4.0, (c * b));
	} else if (x <= 9e-33) {
		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 <= -2.2e+131)
		tmp = t_1;
	elseif (x <= -1.75e+22)
		tmp = fma(fma(a, t, Float64(i * x)), -4.0, Float64(c * b));
	elseif (x <= 9e-33)
		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, -2.2e+131], t$95$1, If[LessEqual[x, -1.75e+22], N[(N[(a * t + N[(i * x), $MachinePrecision]), $MachinePrecision] * -4.0 + N[(c * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9e-33], 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 3 regimes

2. if x < -2.1999999999999999e131 or 8.99999999999999982e-33 < x

   1. Initial program 71.7%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   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 73.6

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

   5. Applied rewrites 73.6%

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

   if -2.1999999999999999e131 < x < -1.75e22

   1. Initial program 87.5%

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

   3. Taylor expanded in 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 83.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 83.9%

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

   if -1.75e22 < x < 8.99999999999999982e-33

   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 81.1

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

   5. Applied rewrites 81.1%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+131}:\\ \;\;\;\;\mathsf{fma}\left(-4, i, \left(\left(z \cdot y\right) \cdot t\right) \cdot 18\right) \cdot x\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, t, i \cdot x\right), -4, c \cdot b\right)\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \mathsf{fma}\left(-27 \cdot k, j, \left(a \cdot t\right) \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, i, \left(\left(z \cdot y\right) \cdot t\right) \cdot 18\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 36.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\\\ [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \left(-27 \cdot j\right) \cdot k\\ t_2 := k \cdot \left(27 \cdot j\right)\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+87}:\\ \;\;\;\;c \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -4e+60) t_1 (if (<= t_2 2e+87) (* 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 <= -4e+60) {
		tmp = t_1;
	} else if (t_2 <= 2e+87) {
		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 <= (-4d+60)) then
        tmp = t_1
    else if (t_2 <= 2d+87) 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 <= -4e+60) {
		tmp = t_1;
	} else if (t_2 <= 2e+87) {
		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 <= -4e+60:
		tmp = t_1
	elif t_2 <= 2e+87:
		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 <= -4e+60)
		tmp = t_1;
	elseif (t_2 <= 2e+87)
		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 <= -4e+60)
		tmp = t_1;
	elseif (t_2 <= 2e+87)
		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, -4e+60], t$95$1, If[LessEqual[t$95$2, 2e+87], N[(c * b), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 j #s(literal 27 binary64)) k) < -3.9999999999999998e60 or 1.9999999999999999e87 < (*.f64 (*.f64 j #s(literal 27 binary64)) k)

   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 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 16.0

         \[\leadsto \color{blue}{c \cdot b} \]

   5. Applied rewrites 16.0%

      \[\leadsto \color{blue}{c \cdot b} \]

   6. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 57.2

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

   8. Applied rewrites 57.2%

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

   10. Applied rewrites 57.3%

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

   if -3.9999999999999998e60 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 1.9999999999999999e87

   1. Initial program 85.4%

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

   3. Taylor expanded in 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 30.8

         \[\leadsto \color{blue}{c \cdot b} \]

   5. Applied rewrites 30.8%

      \[\leadsto \color{blue}{c \cdot b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \cdot \left(27 \cdot j\right) \leq -4 \cdot 10^{+60}:\\ \;\;\;\;\left(-27 \cdot j\right) \cdot k\\ \mathbf{elif}\;k \cdot \left(27 \cdot j\right) \leq 2 \cdot 10^{+87}:\\ \;\;\;\;c \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(-27 \cdot j\right) \cdot k\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 36.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\\\ [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \left(k \cdot j\right) \cdot -27\\ t_2 := k \cdot \left(27 \cdot j\right)\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+87}:\\ \;\;\;\;c \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -4e+60) t_1 (if (<= t_2 2e+87) (* 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 <= -4e+60) {
		tmp = t_1;
	} else if (t_2 <= 2e+87) {
		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 <= (-4d+60)) then
        tmp = t_1
    else if (t_2 <= 2d+87) 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 <= -4e+60) {
		tmp = t_1;
	} else if (t_2 <= 2e+87) {
		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 <= -4e+60:
		tmp = t_1
	elif t_2 <= 2e+87:
		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 <= -4e+60)
		tmp = t_1;
	elseif (t_2 <= 2e+87)
		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 <= -4e+60)
		tmp = t_1;
	elseif (t_2 <= 2e+87)
		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, -4e+60], t$95$1, If[LessEqual[t$95$2, 2e+87], N[(c * b), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 j #s(literal 27 binary64)) k) < -3.9999999999999998e60 or 1.9999999999999999e87 < (*.f64 (*.f64 j #s(literal 27 binary64)) k)

   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 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 57.2

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

   5. Applied rewrites 57.2%

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

   if -3.9999999999999998e60 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 1.9999999999999999e87

   1. Initial program 85.4%

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

   3. Taylor expanded in 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 30.8

         \[\leadsto \color{blue}{c \cdot b} \]

   5. Applied rewrites 30.8%

      \[\leadsto \color{blue}{c \cdot b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \cdot \left(27 \cdot j\right) \leq -4 \cdot 10^{+60}:\\ \;\;\;\;\left(k \cdot j\right) \cdot -27\\ \mathbf{elif}\;k \cdot \left(27 \cdot j\right) \leq 2 \cdot 10^{+87}:\\ \;\;\;\;c \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot j\right) \cdot -27\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 73.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\\\ [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \left(z \cdot y\right) \cdot x\\ \mathbf{if}\;t \leq -1.16 \cdot 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, 18, -4 \cdot a\right) \cdot t\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, \left(k \cdot j\right) \cdot -27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, a, t\_1 \cdot 18\right) \cdot t\\ \end{array} \end{array} \]

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 (* (* z y) x)))
   (if (<= t -1.16e+85)
     (* (fma t_1 18.0 (* -4.0 a)) t)
     (if (<= t 6.2e+34)
       (fma c b (fma (* -4.0 x) i (* (* k j) -27.0)))
       (* (fma -4.0 a (* t_1 18.0)) t)))))

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 = (z * y) * x;
	double tmp;
	if (t <= -1.16e+85) {
		tmp = fma(t_1, 18.0, (-4.0 * a)) * t;
	} else if (t <= 6.2e+34) {
		tmp = fma(c, b, fma((-4.0 * x), i, ((k * j) * -27.0)));
	} else {
		tmp = fma(-4.0, a, (t_1 * 18.0)) * t;
	}
	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(z * y) * x)
	tmp = 0.0
	if (t <= -1.16e+85)
		tmp = Float64(fma(t_1, 18.0, Float64(-4.0 * a)) * t);
	elseif (t <= 6.2e+34)
		tmp = fma(c, b, fma(Float64(-4.0 * x), i, Float64(Float64(k * j) * -27.0)));
	else
		tmp = Float64(fma(-4.0, a, Float64(t_1 * 18.0)) * t);
	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[(z * y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t, -1.16e+85], N[(N[(t$95$1 * 18.0 + N[(-4.0 * a), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t, 6.2e+34], N[(c * b + N[(N[(-4.0 * x), $MachinePrecision] * i + N[(N[(k * j), $MachinePrecision] * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 * a + N[(t$95$1 * 18.0), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.15999999999999995e85

   1. Initial program 72.2%

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

   3. Taylor expanded in 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 18.7

         \[\leadsto \color{blue}{c \cdot b} \]

   5. Applied rewrites 18.7%

      \[\leadsto \color{blue}{c \cdot b} \]

   6. Taylor expanded in t around inf

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 81.2

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

   8. Applied rewrites 81.2%

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

   if -1.15999999999999995e85 < t < 6.19999999999999955e34

   1. Initial program 89.4%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   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 81.6

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

   5. Applied rewrites 81.6%

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

   if 6.19999999999999955e34 < t

   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 t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 70.5

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

   5. Applied rewrites 70.5%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.16 \cdot 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(\left(z \cdot y\right) \cdot x, 18, -4 \cdot a\right) \cdot t\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \mathsf{fma}\left(-4 \cdot x, i, \left(k \cdot j\right) \cdot -27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, a, \left(\left(z \cdot y\right) \cdot x\right) \cdot 18\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 52.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\\\ [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-27 \cdot j, k, c \cdot b\right)\\ \mathbf{if}\;c \cdot b \leq -5 \cdot 10^{+126}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \cdot b \leq 5 \cdot 10^{+232}:\\ \;\;\;\;\mathsf{fma}\left(i, x, a \cdot t\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 j) k (* c b))))
   (if (<= (* c b) -5e+126)
     t_1
     (if (<= (* c b) 5e+232) (* (fma i x (* 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((-27.0 * j), k, (c * b));
	double tmp;
	if ((c * b) <= -5e+126) {
		tmp = t_1;
	} else if ((c * b) <= 5e+232) {
		tmp = fma(i, x, (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 = fma(Float64(-27.0 * j), k, Float64(c * b))
	tmp = 0.0
	if (Float64(c * b) <= -5e+126)
		tmp = t_1;
	elseif (Float64(c * b) <= 5e+232)
		tmp = Float64(fma(i, x, 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[(-27.0 * j), $MachinePrecision] * k + N[(c * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(c * b), $MachinePrecision], -5e+126], t$95$1, If[LessEqual[N[(c * b), $MachinePrecision], 5e+232], N[(N[(i * x + N[(a * t), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b c) < -4.99999999999999977e126 or 4.99999999999999987e232 < (*.f64 b c)

   1. Initial program 81.4%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   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 82.4

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

   5. Applied rewrites 82.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 81.8%

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

   if -4.99999999999999977e126 < (*.f64 b c) < 4.99999999999999987e232

   1. Initial program 85.3%

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

   3. Taylor expanded in 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 66.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 56.3%

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

   9. Taylor expanded in c around 0

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

   11. Applied rewrites 49.5%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot b \leq -5 \cdot 10^{+126}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot j, k, c \cdot b\right)\\ \mathbf{elif}\;c \cdot b \leq 5 \cdot 10^{+232}:\\ \;\;\;\;\mathsf{fma}\left(i, x, a \cdot t\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot j, k, c \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 46.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\\\ [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \left(-4 \cdot i\right) \cdot x\\ \mathbf{if}\;i \leq -2.6 \cdot 10^{+131}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 2.6 \cdot 10^{+175}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot j, k, c \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* (* -4.0 i) x)))
   (if (<= i -2.6e+131)
     t_1
     (if (<= i 2.6e+175) (fma (* -27.0 j) k (* 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 = (-4.0 * i) * x;
	double tmp;
	if (i <= -2.6e+131) {
		tmp = t_1;
	} else if (i <= 2.6e+175) {
		tmp = fma((-27.0 * j), k, (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(-4.0 * i) * x)
	tmp = 0.0
	if (i <= -2.6e+131)
		tmp = t_1;
	elseif (i <= 2.6e+175)
		tmp = fma(Float64(-27.0 * j), k, 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), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[i, -2.6e+131], t$95$1, If[LessEqual[i, 2.6e+175], N[(N[(-27.0 * j), $MachinePrecision] * k + N[(c * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if i < -2.6e131 or 2.6e175 < i

   1. Initial program 82.7%

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

   3. Taylor expanded in i around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 55.1

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

   5. Applied rewrites 55.1%

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

   if -2.6e131 < i < 2.6e175

   1. Initial program 84.6%

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

   3. Taylor expanded in 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 58.1

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

   5. Applied rewrites 58.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 51.4%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2.6 \cdot 10^{+131}:\\ \;\;\;\;\left(-4 \cdot i\right) \cdot x\\ \mathbf{elif}\;i \leq 2.6 \cdot 10^{+175}:\\ \;\;\;\;\mathsf{fma}\left(-27 \cdot j, k, c \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-4 \cdot i\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 23.1% 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 84.2%

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

3. Taylor expanded in 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 24.6

      \[\leadsto \color{blue}{c \cdot b} \]

5. Applied rewrites 24.6%

   \[\leadsto \color{blue}{c \cdot b} \]
6. Add Preprocessing

Developer Target 1: 88.9% 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 2024244 
(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)))