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

Percentage Accurate: 85.4% → 93.5%

Time: 18.7s

Alternatives: 18

Speedup: 1.2×

• 50.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(N[(N[(N[(N[(N[(N[(N[(x * 18.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] - N[(t * N[(a * 4.0), $MachinePrecision]), $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]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(j * N[(x * N[(N[(y * N[(z * N[(18.0 * t), $MachinePrecision]), $MachinePrecision] + N[(i * -4.0), $MachinePrecision]), $MachinePrecision] / j), $MachinePrecision] + N[(k * -27.0), $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 96.7%

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

   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

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

      1. lower-*.f64 N/A

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

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 54.8

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

   5. Simplified 54.8%

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

   6. Taylor expanded in j around inf

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

      1. lower-*.f64 N/A

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

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

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

   8. Simplified 73.0%

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

4. Final simplification 94.7%

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

Alternative 2: 55.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := j \cdot \left(k \cdot -27\right)\\ \mathbf{if}\;b \cdot c \leq -2 \cdot 10^{+119}:\\ \;\;\;\;\mathsf{fma}\left(b, c, t\_1\right)\\ \mathbf{elif}\;b \cdot c \leq -1 \cdot 10^{-133}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(x, z \cdot \left(18 \cdot y\right), a \cdot -4\right)\\ \mathbf{elif}\;b \cdot c \leq -2 \cdot 10^{-316}:\\ \;\;\;\;\mathsf{fma}\left(x, i \cdot -4, t\_1\right)\\ \mathbf{elif}\;b \cdot c \leq 2 \cdot 10^{+84}:\\ \;\;\;\;t \cdot \left(a \cdot -4\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a \cdot -4, b \cdot c\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (* j (* k -27.0))))
   (if (<= (* b c) -2e+119)
     (fma b c t_1)
     (if (<= (* b c) -1e-133)
       (* t (fma x (* z (* 18.0 y)) (* a -4.0)))
       (if (<= (* b c) -2e-316)
         (fma x (* i -4.0) t_1)
         (if (<= (* b c) 2e+84)
           (- (* t (* a -4.0)) (* (* j 27.0) k))
           (fma t (* a -4.0) (* b c))))))))

C

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 = j * (k * -27.0);
	double tmp;
	if ((b * c) <= -2e+119) {
		tmp = fma(b, c, t_1);
	} else if ((b * c) <= -1e-133) {
		tmp = t * fma(x, (z * (18.0 * y)), (a * -4.0));
	} else if ((b * c) <= -2e-316) {
		tmp = fma(x, (i * -4.0), t_1);
	} else if ((b * c) <= 2e+84) {
		tmp = (t * (a * -4.0)) - ((j * 27.0) * k);
	} else {
		tmp = fma(t, (a * -4.0), (b * c));
	}
	return tmp;
}

Julia

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(j * Float64(k * -27.0))
	tmp = 0.0
	if (Float64(b * c) <= -2e+119)
		tmp = fma(b, c, t_1);
	elseif (Float64(b * c) <= -1e-133)
		tmp = Float64(t * fma(x, Float64(z * Float64(18.0 * y)), Float64(a * -4.0)));
	elseif (Float64(b * c) <= -2e-316)
		tmp = fma(x, Float64(i * -4.0), t_1);
	elseif (Float64(b * c) <= 2e+84)
		tmp = Float64(Float64(t * Float64(a * -4.0)) - Float64(Float64(j * 27.0) * k));
	else
		tmp = fma(t, Float64(a * -4.0), Float64(b * c));
	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.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(j * N[(k * -27.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(b * c), $MachinePrecision], -2e+119], N[(b * c + t$95$1), $MachinePrecision], If[LessEqual[N[(b * c), $MachinePrecision], -1e-133], N[(t * N[(x * N[(z * N[(18.0 * y), $MachinePrecision]), $MachinePrecision] + N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(b * c), $MachinePrecision], -2e-316], N[(x * N[(i * -4.0), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[N[(b * c), $MachinePrecision], 2e+84], N[(N[(t * N[(a * -4.0), $MachinePrecision]), $MachinePrecision] - N[(N[(j * 27.0), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(t * N[(a * -4.0), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 b c) < -1.99999999999999989e119

   1. Initial program 91.4%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out-- N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 91.5

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

   5. Simplified 91.5%

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

   6. Taylor expanded in t around 0

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

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

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 83.1

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

   8. Simplified 83.1%

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

   if -1.99999999999999989e119 < (*.f64 b c) < -1.0000000000000001e-133

   1. Initial program 82.3%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out-- N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 82.7

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

   5. Simplified 82.7%

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

   6. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 59.7

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

   8. Simplified 59.7%

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

   if -1.0000000000000001e-133 < (*.f64 b c) < -2.000000017e-316

   1. Initial program 93.3%

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

   3. Taylor expanded in i around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 68.4

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

   5. Simplified 68.4%

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

   6. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 68.4

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

   8. Simplified 68.4%

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

   if -2.000000017e-316 < (*.f64 b c) < 2.00000000000000012e84

   1. Initial program 87.7%

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

   3. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 62.3

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

   5. Simplified 62.3%

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

   if 2.00000000000000012e84 < (*.f64 b c)

   1. Initial program 90.8%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out-- N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 81.4

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

   5. Simplified 81.4%

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

   6. Taylor expanded in z around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 76.0

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

   8. Simplified 76.0%

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

   9. Taylor expanded in b around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 70.6

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

   11. Simplified 70.6%

       \[\leadsto \mathsf{fma}\left(t, -4 \cdot a, \color{blue}{c \cdot b}\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot c \leq -2 \cdot 10^{+119}:\\ \;\;\;\;\mathsf{fma}\left(b, c, j \cdot \left(k \cdot -27\right)\right)\\ \mathbf{elif}\;b \cdot c \leq -1 \cdot 10^{-133}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(x, z \cdot \left(18 \cdot y\right), a \cdot -4\right)\\ \mathbf{elif}\;b \cdot c \leq -2 \cdot 10^{-316}:\\ \;\;\;\;\mathsf{fma}\left(x, i \cdot -4, j \cdot \left(k \cdot -27\right)\right)\\ \mathbf{elif}\;b \cdot c \leq 2 \cdot 10^{+84}:\\ \;\;\;\;t \cdot \left(a \cdot -4\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a \cdot -4, b \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, c, j \cdot \left(k \cdot -27\right)\right)\\ t_2 := x \cdot \mathsf{fma}\left(y, z \cdot \left(18 \cdot t\right), i \cdot -4\right)\\ \mathbf{if}\;x \leq -8 \cdot 10^{+198}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq -85000000000000:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot \left(18 \cdot \left(z \cdot t\right)\right), t\_1\right)\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(t, a \cdot -4, t\_1\right)\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+108}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), a \cdot -4\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (fma b c (* j (* k -27.0))))
        (t_2 (* x (fma y (* z (* 18.0 t)) (* i -4.0)))))
   (if (<= x -8e+198)
     t_2
     (if (<= x -85000000000000.0)
       (fma x (* y (* 18.0 (* z t))) t_1)
       (if (<= x 8e-58)
         (fma t (* a -4.0) t_1)
         (if (<= x 2.9e+108)
           (- (* t (fma 18.0 (* z (* x y)) (* a -4.0))) (* (* j 27.0) k))
           t_2))))))

C

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(b, c, (j * (k * -27.0)));
	double t_2 = x * fma(y, (z * (18.0 * t)), (i * -4.0));
	double tmp;
	if (x <= -8e+198) {
		tmp = t_2;
	} else if (x <= -85000000000000.0) {
		tmp = fma(x, (y * (18.0 * (z * t))), t_1);
	} else if (x <= 8e-58) {
		tmp = fma(t, (a * -4.0), t_1);
	} else if (x <= 2.9e+108) {
		tmp = (t * fma(18.0, (z * (x * y)), (a * -4.0))) - ((j * 27.0) * k);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

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(b, c, Float64(j * Float64(k * -27.0)))
	t_2 = Float64(x * fma(y, Float64(z * Float64(18.0 * t)), Float64(i * -4.0)))
	tmp = 0.0
	if (x <= -8e+198)
		tmp = t_2;
	elseif (x <= -85000000000000.0)
		tmp = fma(x, Float64(y * Float64(18.0 * Float64(z * t))), t_1);
	elseif (x <= 8e-58)
		tmp = fma(t, Float64(a * -4.0), t_1);
	elseif (x <= 2.9e+108)
		tmp = Float64(Float64(t * fma(18.0, Float64(z * Float64(x * y)), Float64(a * -4.0))) - Float64(Float64(j * 27.0) * k));
	else
		tmp = t_2;
	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.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(b * c + N[(j * N[(k * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(y * N[(z * N[(18.0 * t), $MachinePrecision]), $MachinePrecision] + N[(i * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8e+198], t$95$2, If[LessEqual[x, -85000000000000.0], N[(x * N[(y * N[(18.0 * N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[x, 8e-58], N[(t * N[(a * -4.0), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[x, 2.9e+108], N[(N[(t * N[(18.0 * N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision] + N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(j * 27.0), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -8.00000000000000014e198 or 2.90000000000000007e108 < x

   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 x around inf

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

      1. lower-*.f64 N/A

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

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 90.7

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

   5. Simplified 90.7%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 82.8

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

   8. Simplified 82.8%

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

   if -8.00000000000000014e198 < x < -8.5e13

   1. Initial program 74.5%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out-- N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 71.1

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

   5. Simplified 71.1%

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

   6. Taylor expanded in a around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

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

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 N/A

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

      20. *-commutative N/A

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

      21. associate-*l* N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 75.3

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

   8. Simplified 75.3%

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

   9. Taylor expanded in z around 0

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 75.3

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

   11. Simplified 75.3%

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

   if -8.5e13 < x < 8.0000000000000002e-58

   1. Initial program 96.8%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out-- N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 95.4

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

   5. Simplified 95.4%

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

   6. Taylor expanded in z around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 89.0

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

   8. Simplified 89.0%

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

   if 8.0000000000000002e-58 < x < 2.90000000000000007e108

   1. Initial program 88.5%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 77.5

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

   5. Simplified 77.5%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+198}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, z \cdot \left(18 \cdot t\right), i \cdot -4\right)\\ \mathbf{elif}\;x \leq -85000000000000:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot \left(18 \cdot \left(z \cdot t\right)\right), \mathsf{fma}\left(b, c, j \cdot \left(k \cdot -27\right)\right)\right)\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(t, a \cdot -4, \mathsf{fma}\left(b, c, j \cdot \left(k \cdot -27\right)\right)\right)\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+108}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), a \cdot -4\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, z \cdot \left(18 \cdot t\right), i \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \left(j \cdot 27\right) \cdot k\\ t_2 := \mathsf{fma}\left(x, \mathsf{fma}\left(y \cdot z, 18 \cdot t, i \cdot -4\right), b \cdot c\right) - t\_1\\ \mathbf{if}\;x \leq -1.95 \cdot 10^{+62}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), a \cdot -4\right), b \cdot c\right) - t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (* (* j 27.0) k))
        (t_2 (- (fma x (fma (* y z) (* 18.0 t) (* i -4.0)) (* b c)) t_1)))
   (if (<= x -1.95e+62)
     t_2
     (if (<= x 1.85e+74)
       (- (fma t (fma 18.0 (* z (* x y)) (* a -4.0)) (* b c)) t_1)
       t_2))))

C

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 = (j * 27.0) * k;
	double t_2 = fma(x, fma((y * z), (18.0 * t), (i * -4.0)), (b * c)) - t_1;
	double tmp;
	if (x <= -1.95e+62) {
		tmp = t_2;
	} else if (x <= 1.85e+74) {
		tmp = fma(t, fma(18.0, (z * (x * y)), (a * -4.0)), (b * c)) - t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

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(j * 27.0) * k)
	t_2 = Float64(fma(x, fma(Float64(y * z), Float64(18.0 * t), Float64(i * -4.0)), Float64(b * c)) - t_1)
	tmp = 0.0
	if (x <= -1.95e+62)
		tmp = t_2;
	elseif (x <= 1.85e+74)
		tmp = Float64(fma(t, fma(18.0, Float64(z * Float64(x * y)), Float64(a * -4.0)), Float64(b * c)) - t_1);
	else
		tmp = t_2;
	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.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(N[(j * 27.0), $MachinePrecision] * k), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[(N[(y * z), $MachinePrecision] * N[(18.0 * t), $MachinePrecision] + N[(i * -4.0), $MachinePrecision]), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]}, If[LessEqual[x, -1.95e+62], t$95$2, If[LessEqual[x, 1.85e+74], N[(N[(t * N[(18.0 * N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision] + N[(a * -4.0), $MachinePrecision]), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.95e62 or 1.8500000000000001e74 < 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 a around 0

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

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

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. associate-+l+ N/A

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

      5. metadata-eval N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out-- N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

   5. Simplified 89.9%

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

   if -1.95e62 < x < 1.8500000000000001e74

   1. Initial program 94.2%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out-- N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 93.8

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

   5. Simplified 93.8%

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

Alternative 5: 81.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \left(j \cdot 27\right) \cdot k\\ t_2 := x \cdot \mathsf{fma}\left(y \cdot z, 18 \cdot t, i \cdot -4\right) - t\_1\\ \mathbf{if}\;x \leq -4.7 \cdot 10^{+176}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+79}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), a \cdot -4\right), b \cdot c\right) - t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (* (* j 27.0) k))
        (t_2 (- (* x (fma (* y z) (* 18.0 t) (* i -4.0))) t_1)))
   (if (<= x -4.7e+176)
     t_2
     (if (<= x 1.6e+79)
       (- (fma t (fma 18.0 (* z (* x y)) (* a -4.0)) (* b c)) t_1)
       t_2))))

C

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 = (j * 27.0) * k;
	double t_2 = (x * fma((y * z), (18.0 * t), (i * -4.0))) - t_1;
	double tmp;
	if (x <= -4.7e+176) {
		tmp = t_2;
	} else if (x <= 1.6e+79) {
		tmp = fma(t, fma(18.0, (z * (x * y)), (a * -4.0)), (b * c)) - t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

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(j * 27.0) * k)
	t_2 = Float64(Float64(x * fma(Float64(y * z), Float64(18.0 * t), Float64(i * -4.0))) - t_1)
	tmp = 0.0
	if (x <= -4.7e+176)
		tmp = t_2;
	elseif (x <= 1.6e+79)
		tmp = Float64(fma(t, fma(18.0, Float64(z * Float64(x * y)), Float64(a * -4.0)), Float64(b * c)) - t_1);
	else
		tmp = t_2;
	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.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(N[(j * 27.0), $MachinePrecision] * k), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[(N[(y * z), $MachinePrecision] * N[(18.0 * t), $MachinePrecision] + N[(i * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]}, If[LessEqual[x, -4.7e+176], t$95$2, If[LessEqual[x, 1.6e+79], N[(N[(t * N[(18.0 * N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision] + N[(a * -4.0), $MachinePrecision]), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.69999999999999981e176 or 1.60000000000000001e79 < x

   1. Initial program 78.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 90.5

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

   5. Simplified 90.5%

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

   if -4.69999999999999981e176 < x < 1.60000000000000001e79

   1. Initial program 92.1%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out-- N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 90.2

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

   5. Simplified 90.2%

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

Alternative 6: 50.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, a \cdot -4, b \cdot c\right)\\ \mathbf{if}\;a \cdot 4 \leq -6 \cdot 10^{+56}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \cdot 4 \leq -5 \cdot 10^{-91}:\\ \;\;\;\;b \cdot c - \left(j \cdot 27\right) \cdot k\\ \mathbf{elif}\;a \cdot 4 \leq 2 \cdot 10^{+61}:\\ \;\;\;\;\mathsf{fma}\left(x, i \cdot -4, j \cdot \left(k \cdot -27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (fma t (* a -4.0) (* b c))))
   (if (<= (* a 4.0) -6e+56)
     t_1
     (if (<= (* a 4.0) -5e-91)
       (- (* b c) (* (* j 27.0) k))
       (if (<= (* a 4.0) 2e+61) (fma x (* i -4.0) (* j (* k -27.0))) t_1)))))

C

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(t, (a * -4.0), (b * c));
	double tmp;
	if ((a * 4.0) <= -6e+56) {
		tmp = t_1;
	} else if ((a * 4.0) <= -5e-91) {
		tmp = (b * c) - ((j * 27.0) * k);
	} else if ((a * 4.0) <= 2e+61) {
		tmp = fma(x, (i * -4.0), (j * (k * -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])
function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = fma(t, Float64(a * -4.0), Float64(b * c))
	tmp = 0.0
	if (Float64(a * 4.0) <= -6e+56)
		tmp = t_1;
	elseif (Float64(a * 4.0) <= -5e-91)
		tmp = Float64(Float64(b * c) - Float64(Float64(j * 27.0) * k));
	elseif (Float64(a * 4.0) <= 2e+61)
		tmp = fma(x, Float64(i * -4.0), Float64(j * Float64(k * -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.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(t * N[(a * -4.0), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * 4.0), $MachinePrecision], -6e+56], t$95$1, If[LessEqual[N[(a * 4.0), $MachinePrecision], -5e-91], N[(N[(b * c), $MachinePrecision] - N[(N[(j * 27.0), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * 4.0), $MachinePrecision], 2e+61], N[(x * N[(i * -4.0), $MachinePrecision] + N[(j * N[(k * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a #s(literal 4 binary64)) < -6.00000000000000012e56 or 1.9999999999999999e61 < (*.f64 a #s(literal 4 binary64))

   1. Initial program 85.6%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out-- N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 90.0

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

   5. Simplified 90.0%

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

   6. Taylor expanded in z around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 71.9

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

   8. Simplified 71.9%

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

   9. Taylor expanded in b around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 63.9

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

   11. Simplified 63.9%

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

   if -6.00000000000000012e56 < (*.f64 a #s(literal 4 binary64)) < -4.99999999999999997e-91

   1. Initial program 90.3%

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

   3. Taylor expanded in b around inf

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

      1. lower-*.f64 60.2

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

   5. Simplified 60.2%

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

   if -4.99999999999999997e-91 < (*.f64 a #s(literal 4 binary64)) < 1.9999999999999999e61

   1. Initial program 90.1%

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

   3. Taylor expanded in i around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 58.9

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

   5. Simplified 58.9%

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

   6. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 59.0

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

   8. Simplified 59.0%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 4 \leq -6 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(t, a \cdot -4, b \cdot c\right)\\ \mathbf{elif}\;a \cdot 4 \leq -5 \cdot 10^{-91}:\\ \;\;\;\;b \cdot c - \left(j \cdot 27\right) \cdot k\\ \mathbf{elif}\;a \cdot 4 \leq 2 \cdot 10^{+61}:\\ \;\;\;\;\mathsf{fma}\left(x, i \cdot -4, j \cdot \left(k \cdot -27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a \cdot -4, b \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(y \cdot z, 18 \cdot t, i \cdot -4\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{+88}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-55}:\\ \;\;\;\;\mathsf{fma}\left(t, a \cdot -4, \mathsf{fma}\left(b, c, j \cdot \left(k \cdot -27\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (- (* x (fma (* y z) (* 18.0 t) (* i -4.0))) (* (* j 27.0) k))))
   (if (<= x -6.2e+88)
     t_1
     (if (<= x 5.7e-55) (fma t (* a -4.0) (fma b c (* j (* k -27.0)))) t_1))))

C

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 = (x * fma((y * z), (18.0 * t), (i * -4.0))) - ((j * 27.0) * k);
	double tmp;
	if (x <= -6.2e+88) {
		tmp = t_1;
	} else if (x <= 5.7e-55) {
		tmp = fma(t, (a * -4.0), fma(b, c, (j * (k * -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])
function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = Float64(Float64(x * fma(Float64(y * z), Float64(18.0 * t), Float64(i * -4.0))) - Float64(Float64(j * 27.0) * k))
	tmp = 0.0
	if (x <= -6.2e+88)
		tmp = t_1;
	elseif (x <= 5.7e-55)
		tmp = fma(t, Float64(a * -4.0), fma(b, c, Float64(j * Float64(k * -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.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(N[(x * N[(N[(y * z), $MachinePrecision] * N[(18.0 * t), $MachinePrecision] + N[(i * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(j * 27.0), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.2e+88], t$95$1, If[LessEqual[x, 5.7e-55], N[(t * N[(a * -4.0), $MachinePrecision] + N[(b * c + N[(j * N[(k * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.2000000000000003e88 or 5.7000000000000002e-55 < x

   1. Initial program 82.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 83.2

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

   5. Simplified 83.2%

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

   if -6.2000000000000003e88 < x < 5.7000000000000002e-55

   1. Initial program 93.7%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out-- N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 93.1

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

   5. Simplified 93.1%

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

   6. Taylor expanded in z around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 87.8

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

   8. Simplified 87.8%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+88}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot z, 18 \cdot t, i \cdot -4\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-55}:\\ \;\;\;\;\mathsf{fma}\left(t, a \cdot -4, \mathsf{fma}\left(b, c, j \cdot \left(k \cdot -27\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot z, 18 \cdot t, i \cdot -4\right) - \left(j \cdot 27\right) \cdot k\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, c, j \cdot \left(k \cdot -27\right)\right)\\ t_2 := x \cdot \mathsf{fma}\left(y, z \cdot \left(18 \cdot t\right), i \cdot -4\right)\\ \mathbf{if}\;x \leq -8 \cdot 10^{+198}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq -85000000000000:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot \left(18 \cdot \left(z \cdot t\right)\right), t\_1\right)\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(t, a \cdot -4, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (fma b c (* j (* k -27.0))))
        (t_2 (* x (fma y (* z (* 18.0 t)) (* i -4.0)))))
   (if (<= x -8e+198)
     t_2
     (if (<= x -85000000000000.0)
       (fma x (* y (* 18.0 (* z t))) t_1)
       (if (<= x 2.4e-40) (fma t (* a -4.0) t_1) t_2)))))

C

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(b, c, (j * (k * -27.0)));
	double t_2 = x * fma(y, (z * (18.0 * t)), (i * -4.0));
	double tmp;
	if (x <= -8e+198) {
		tmp = t_2;
	} else if (x <= -85000000000000.0) {
		tmp = fma(x, (y * (18.0 * (z * t))), t_1);
	} else if (x <= 2.4e-40) {
		tmp = fma(t, (a * -4.0), t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

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(b, c, Float64(j * Float64(k * -27.0)))
	t_2 = Float64(x * fma(y, Float64(z * Float64(18.0 * t)), Float64(i * -4.0)))
	tmp = 0.0
	if (x <= -8e+198)
		tmp = t_2;
	elseif (x <= -85000000000000.0)
		tmp = fma(x, Float64(y * Float64(18.0 * Float64(z * t))), t_1);
	elseif (x <= 2.4e-40)
		tmp = fma(t, Float64(a * -4.0), t_1);
	else
		tmp = t_2;
	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.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(b * c + N[(j * N[(k * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(y * N[(z * N[(18.0 * t), $MachinePrecision]), $MachinePrecision] + N[(i * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8e+198], t$95$2, If[LessEqual[x, -85000000000000.0], N[(x * N[(y * N[(18.0 * N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[x, 2.4e-40], N[(t * N[(a * -4.0), $MachinePrecision] + t$95$1), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.00000000000000014e198 or 2.39999999999999991e-40 < x

   1. Initial program 82.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 86.2

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

   5. Simplified 86.2%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 76.0

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

   8. Simplified 76.0%

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

   if -8.00000000000000014e198 < x < -8.5e13

   1. Initial program 74.5%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out-- N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 71.1

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

   5. Simplified 71.1%

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

   6. Taylor expanded in a around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

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

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 N/A

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

      20. *-commutative N/A

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

      21. associate-*l* N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 75.3

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

   8. Simplified 75.3%

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

   9. Taylor expanded in z around 0

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 75.3

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

   11. Simplified 75.3%

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

   if -8.5e13 < x < 2.39999999999999991e-40

   1. Initial program 96.9%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out-- N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 95.5

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

   5. Simplified 95.5%

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

   6. Taylor expanded in z around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 88.4

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

   8. Simplified 88.4%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+198}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, z \cdot \left(18 \cdot t\right), i \cdot -4\right)\\ \mathbf{elif}\;x \leq -85000000000000:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot \left(18 \cdot \left(z \cdot t\right)\right), \mathsf{fma}\left(b, c, j \cdot \left(k \cdot -27\right)\right)\right)\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(t, a \cdot -4, \mathsf{fma}\left(b, c, j \cdot \left(k \cdot -27\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, z \cdot \left(18 \cdot t\right), i \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 53.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \left(j \cdot 27\right) \cdot k\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+14}:\\ \;\;\;\;b \cdot c - t\_1\\ \mathbf{elif}\;t\_1 \leq 20000000:\\ \;\;\;\;\mathsf{fma}\left(t, a \cdot -4, b \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, c, j \cdot \left(k \cdot -27\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (* (* j 27.0) k)))
   (if (<= t_1 -2e+14)
     (- (* b c) t_1)
     (if (<= t_1 20000000.0)
       (fma t (* a -4.0) (* b c))
       (fma b c (* j (* k -27.0)))))))

C

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 = (j * 27.0) * k;
	double tmp;
	if (t_1 <= -2e+14) {
		tmp = (b * c) - t_1;
	} else if (t_1 <= 20000000.0) {
		tmp = fma(t, (a * -4.0), (b * c));
	} else {
		tmp = fma(b, c, (j * (k * -27.0)));
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = Float64(Float64(j * 27.0) * k)
	tmp = 0.0
	if (t_1 <= -2e+14)
		tmp = Float64(Float64(b * c) - t_1);
	elseif (t_1 <= 20000000.0)
		tmp = fma(t, Float64(a * -4.0), Float64(b * c));
	else
		tmp = fma(b, c, Float64(j * Float64(k * -27.0)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(N[(j * 27.0), $MachinePrecision] * k), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+14], N[(N[(b * c), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[t$95$1, 20000000.0], N[(t * N[(a * -4.0), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision], N[(b * c + N[(j * N[(k * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 j #s(literal 27 binary64)) k) < -2e14

   1. Initial program 88.7%

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

   3. Taylor expanded in b around inf

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

      1. lower-*.f64 63.6

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

   5. Simplified 63.6%

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

   if -2e14 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 2e7

   1. Initial program 89.7%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out-- N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 77.3

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

   5. Simplified 77.3%

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

   6. Taylor expanded in z around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 58.0

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

   8. Simplified 58.0%

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

   9. Taylor expanded in b around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 55.7

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

   11. Simplified 55.7%

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

   if 2e7 < (*.f64 (*.f64 j #s(literal 27 binary64)) k)

   1. Initial program 85.8%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out-- N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 86.9

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

   5. Simplified 86.9%

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

   6. Taylor expanded in t around 0

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

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

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 56.6

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

   8. Simplified 56.6%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(j \cdot 27\right) \cdot k \leq -2 \cdot 10^{+14}:\\ \;\;\;\;b \cdot c - \left(j \cdot 27\right) \cdot k\\ \mathbf{elif}\;\left(j \cdot 27\right) \cdot k \leq 20000000:\\ \;\;\;\;\mathsf{fma}\left(t, a \cdot -4, b \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, c, j \cdot \left(k \cdot -27\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 54.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} \mathbf{if}\;b \cdot c \leq -2 \cdot 10^{+119}:\\ \;\;\;\;\mathsf{fma}\left(b, c, j \cdot \left(k \cdot -27\right)\right)\\ \mathbf{elif}\;b \cdot c \leq 2 \cdot 10^{+84}:\\ \;\;\;\;t \cdot \left(a \cdot -4\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a \cdot -4, b \cdot c\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i j k)
 :precision binary64
 (if (<= (* b c) -2e+119)
   (fma b c (* j (* k -27.0)))
   (if (<= (* b c) 2e+84)
     (- (* t (* a -4.0)) (* (* j 27.0) k))
     (fma t (* a -4.0) (* b c)))))

C

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 ((b * c) <= -2e+119) {
		tmp = fma(b, c, (j * (k * -27.0)));
	} else if ((b * c) <= 2e+84) {
		tmp = (t * (a * -4.0)) - ((j * 27.0) * k);
	} else {
		tmp = fma(t, (a * -4.0), (b * c));
	}
	return tmp;
}

Julia

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(b * c) <= -2e+119)
		tmp = fma(b, c, Float64(j * Float64(k * -27.0)));
	elseif (Float64(b * c) <= 2e+84)
		tmp = Float64(Float64(t * Float64(a * -4.0)) - Float64(Float64(j * 27.0) * k));
	else
		tmp = fma(t, Float64(a * -4.0), Float64(b * c));
	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.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := If[LessEqual[N[(b * c), $MachinePrecision], -2e+119], N[(b * c + N[(j * N[(k * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(b * c), $MachinePrecision], 2e+84], N[(N[(t * N[(a * -4.0), $MachinePrecision]), $MachinePrecision] - N[(N[(j * 27.0), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(t * N[(a * -4.0), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 b c) < -1.99999999999999989e119

   1. Initial program 91.4%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out-- N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 91.5

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

   5. Simplified 91.5%

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

   6. Taylor expanded in t around 0

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

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

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 83.1

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

   8. Simplified 83.1%

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

   if -1.99999999999999989e119 < (*.f64 b c) < 2.00000000000000012e84

   1. Initial program 87.1%

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

   3. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 55.2

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

   5. Simplified 55.2%

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

   if 2.00000000000000012e84 < (*.f64 b c)

   1. Initial program 90.8%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out-- N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 81.4

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

   5. Simplified 81.4%

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

   6. Taylor expanded in z around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(t, -4 \cdot a, \mathsf{fma}\left(b, c, \color{blue}{j \cdot \left(k \cdot -27\right)}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t, -4 \cdot a, \mathsf{fma}\left(b, c, j \cdot \color{blue}{\left(-27 \cdot k\right)}\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, -4 \cdot a, \mathsf{fma}\left(b, c, \color{blue}{j \cdot \left(-27 \cdot k\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t, -4 \cdot a, \mathsf{fma}\left(b, c, j \cdot \color{blue}{\left(k \cdot -27\right)}\right)\right) \]

      14. lower-*.f64 76.0

          \[\leadsto \mathsf{fma}\left(t, -4 \cdot a, \mathsf{fma}\left(b, c, j \cdot \color{blue}{\left(k \cdot -27\right)}\right)\right) \]

   8. Simplified 76.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, -4 \cdot a, \mathsf{fma}\left(b, c, j \cdot \left(k \cdot -27\right)\right)\right)} \]

   9. Taylor expanded in b around inf

      \[\leadsto \mathsf{fma}\left(t, -4 \cdot a, \color{blue}{b \cdot c}\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t, -4 \cdot a, \color{blue}{c \cdot b}\right) \]

       2. lower-*.f64 70.6

          \[\leadsto \mathsf{fma}\left(t, -4 \cdot a, \color{blue}{c \cdot b}\right) \]

   11. Simplified 70.6%

       \[\leadsto \mathsf{fma}\left(t, -4 \cdot a, \color{blue}{c \cdot b}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot c \leq -2 \cdot 10^{+119}:\\ \;\;\;\;\mathsf{fma}\left(b, c, j \cdot \left(k \cdot -27\right)\right)\\ \mathbf{elif}\;b \cdot c \leq 2 \cdot 10^{+84}:\\ \;\;\;\;t \cdot \left(a \cdot -4\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a \cdot -4, b \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 37.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} \mathbf{if}\;b \cdot c \leq -3.2 \cdot 10^{+115}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;b \cdot c \leq -9.5 \cdot 10^{-127}:\\ \;\;\;\;t \cdot \left(a \cdot -4\right)\\ \mathbf{elif}\;b \cdot c \leq 5.5 \cdot 10^{+87}:\\ \;\;\;\;j \cdot \left(k \cdot -27\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot c\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i j k)
 :precision binary64
 (if (<= (* b c) -3.2e+115)
   (* b c)
   (if (<= (* b c) -9.5e-127)
     (* t (* a -4.0))
     (if (<= (* b c) 5.5e+87) (* j (* k -27.0)) (* b c)))))

C

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 ((b * c) <= -3.2e+115) {
		tmp = b * c;
	} else if ((b * c) <= -9.5e-127) {
		tmp = t * (a * -4.0);
	} else if ((b * c) <= 5.5e+87) {
		tmp = j * (k * -27.0);
	} else {
		tmp = b * c;
	}
	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.
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) :: tmp
    if ((b * c) <= (-3.2d+115)) then
        tmp = b * c
    else if ((b * c) <= (-9.5d-127)) then
        tmp = t * (a * (-4.0d0))
    else if ((b * c) <= 5.5d+87) then
        tmp = j * (k * (-27.0d0))
    else
        tmp = b * c
    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;
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 tmp;
	if ((b * c) <= -3.2e+115) {
		tmp = b * c;
	} else if ((b * c) <= -9.5e-127) {
		tmp = t * (a * -4.0);
	} else if ((b * c) <= 5.5e+87) {
		tmp = j * (k * -27.0);
	} else {
		tmp = b * c;
	}
	return tmp;
}

Python

[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):
	tmp = 0
	if (b * c) <= -3.2e+115:
		tmp = b * c
	elif (b * c) <= -9.5e-127:
		tmp = t * (a * -4.0)
	elif (b * c) <= 5.5e+87:
		tmp = j * (k * -27.0)
	else:
		tmp = b * c
	return tmp

Julia

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(b * c) <= -3.2e+115)
		tmp = Float64(b * c);
	elseif (Float64(b * c) <= -9.5e-127)
		tmp = Float64(t * Float64(a * -4.0));
	elseif (Float64(b * c) <= 5.5e+87)
		tmp = Float64(j * Float64(k * -27.0));
	else
		tmp = Float64(b * c);
	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])){:}
function tmp_2 = code(x, y, z, t, a, b, c, i, j, k)
	tmp = 0.0;
	if ((b * c) <= -3.2e+115)
		tmp = b * c;
	elseif ((b * c) <= -9.5e-127)
		tmp = t * (a * -4.0);
	elseif ((b * c) <= 5.5e+87)
		tmp = j * (k * -27.0);
	else
		tmp = b * c;
	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.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := If[LessEqual[N[(b * c), $MachinePrecision], -3.2e+115], N[(b * c), $MachinePrecision], If[LessEqual[N[(b * c), $MachinePrecision], -9.5e-127], N[(t * N[(a * -4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(b * c), $MachinePrecision], 5.5e+87], N[(j * N[(k * -27.0), $MachinePrecision]), $MachinePrecision], N[(b * c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 b c) < -3.2e115 or 5.50000000000000022e87 < (*.f64 b c)

   1. Initial program 91.0%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 4 \cdot \left(a \cdot t\right)\right)} - \left(j \cdot 27\right) \cdot k \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(b \cdot c + 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right)\right)} - 4 \cdot \left(a \cdot t\right)\right) - \left(j \cdot 27\right) \cdot k \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(b \cdot c + \left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) - 4 \cdot \left(a \cdot t\right)\right)\right)} - \left(j \cdot 27\right) \cdot k \]

      3. *-commutative N/A

         \[\leadsto \left(b \cdot c + \left(18 \cdot \color{blue}{\left(\left(x \cdot \left(y \cdot z\right)\right) \cdot t\right)} - 4 \cdot \left(a \cdot t\right)\right)\right) - \left(j \cdot 27\right) \cdot k \]

      4. associate-*r* N/A

         \[\leadsto \left(b \cdot c + \left(\color{blue}{\left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot t} - 4 \cdot \left(a \cdot t\right)\right)\right) - \left(j \cdot 27\right) \cdot k \]

      5. associate-*r* N/A

         \[\leadsto \left(b \cdot c + \left(\left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot t - \color{blue}{\left(4 \cdot a\right) \cdot t}\right)\right) - \left(j \cdot 27\right) \cdot k \]

      6. distribute-rgt-out-- N/A

         \[\leadsto \left(b \cdot c + \color{blue}{t \cdot \left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a\right)}\right) - \left(j \cdot 27\right) \cdot k \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot \left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a\right) + b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, 18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a, b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{18 \cdot \left(x \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot a}, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t, 18 \cdot \left(x \cdot \left(y \cdot z\right)\right) + \color{blue}{-4} \cdot a, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{fma}\left(18, x \cdot \left(y \cdot z\right), -4 \cdot a\right)}, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{\left(x \cdot y\right) \cdot z}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{z \cdot \left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{z \cdot \left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \color{blue}{\left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), \color{blue}{a \cdot -4}\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      17. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), \color{blue}{a \cdot -4}\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      18. lower-*.f64 85.5

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), a \cdot -4\right), \color{blue}{b \cdot c}\right) - \left(j \cdot 27\right) \cdot k \]

   5. Simplified 85.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), a \cdot -4\right), b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot c} \]
   7. Step-by-step derivation

      1. lower-*.f64 64.4

         \[\leadsto \color{blue}{b \cdot c} \]

   8. Simplified 64.4%

      \[\leadsto \color{blue}{b \cdot c} \]

   if -3.2e115 < (*.f64 b c) < -9.4999999999999997e-127

   1. Initial program 83.6%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 4 \cdot \left(a \cdot t\right)\right)} - \left(j \cdot 27\right) \cdot k \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(b \cdot c + 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right)\right)} - 4 \cdot \left(a \cdot t\right)\right) - \left(j \cdot 27\right) \cdot k \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(b \cdot c + \left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) - 4 \cdot \left(a \cdot t\right)\right)\right)} - \left(j \cdot 27\right) \cdot k \]

      3. *-commutative N/A

         \[\leadsto \left(b \cdot c + \left(18 \cdot \color{blue}{\left(\left(x \cdot \left(y \cdot z\right)\right) \cdot t\right)} - 4 \cdot \left(a \cdot t\right)\right)\right) - \left(j \cdot 27\right) \cdot k \]

      4. associate-*r* N/A

         \[\leadsto \left(b \cdot c + \left(\color{blue}{\left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot t} - 4 \cdot \left(a \cdot t\right)\right)\right) - \left(j \cdot 27\right) \cdot k \]

      5. associate-*r* N/A

         \[\leadsto \left(b \cdot c + \left(\left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot t - \color{blue}{\left(4 \cdot a\right) \cdot t}\right)\right) - \left(j \cdot 27\right) \cdot k \]

      6. distribute-rgt-out-- N/A

         \[\leadsto \left(b \cdot c + \color{blue}{t \cdot \left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a\right)}\right) - \left(j \cdot 27\right) \cdot k \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot \left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a\right) + b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, 18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a, b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{18 \cdot \left(x \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot a}, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t, 18 \cdot \left(x \cdot \left(y \cdot z\right)\right) + \color{blue}{-4} \cdot a, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{fma}\left(18, x \cdot \left(y \cdot z\right), -4 \cdot a\right)}, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{\left(x \cdot y\right) \cdot z}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{z \cdot \left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{z \cdot \left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \color{blue}{\left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), \color{blue}{a \cdot -4}\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      17. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), \color{blue}{a \cdot -4}\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      18. lower-*.f64 81.9

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), a \cdot -4\right), \color{blue}{b \cdot c}\right) - \left(j \cdot 27\right) \cdot k \]

   5. Simplified 81.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), a \cdot -4\right), b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

   6. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-4 \cdot \left(a \cdot t\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-4 \cdot a\right) \cdot t} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{t \cdot \left(-4 \cdot a\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{t \cdot \left(-4 \cdot a\right)} \]

      4. lower-*.f64 34.1

         \[\leadsto t \cdot \color{blue}{\left(-4 \cdot a\right)} \]

   8. Simplified 34.1%

      \[\leadsto \color{blue}{t \cdot \left(-4 \cdot a\right)} \]

   if -9.4999999999999997e-127 < (*.f64 b c) < 5.50000000000000022e87

   1. Initial program 88.5%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 4 \cdot \left(a \cdot t\right)\right)} - \left(j \cdot 27\right) \cdot k \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(b \cdot c + 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right)\right)} - 4 \cdot \left(a \cdot t\right)\right) - \left(j \cdot 27\right) \cdot k \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(b \cdot c + \left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) - 4 \cdot \left(a \cdot t\right)\right)\right)} - \left(j \cdot 27\right) \cdot k \]

      3. *-commutative N/A

         \[\leadsto \left(b \cdot c + \left(18 \cdot \color{blue}{\left(\left(x \cdot \left(y \cdot z\right)\right) \cdot t\right)} - 4 \cdot \left(a \cdot t\right)\right)\right) - \left(j \cdot 27\right) \cdot k \]

      4. associate-*r* N/A

         \[\leadsto \left(b \cdot c + \left(\color{blue}{\left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot t} - 4 \cdot \left(a \cdot t\right)\right)\right) - \left(j \cdot 27\right) \cdot k \]

      5. associate-*r* N/A

         \[\leadsto \left(b \cdot c + \left(\left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot t - \color{blue}{\left(4 \cdot a\right) \cdot t}\right)\right) - \left(j \cdot 27\right) \cdot k \]

      6. distribute-rgt-out-- N/A

         \[\leadsto \left(b \cdot c + \color{blue}{t \cdot \left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a\right)}\right) - \left(j \cdot 27\right) \cdot k \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot \left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a\right) + b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, 18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a, b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{18 \cdot \left(x \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot a}, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t, 18 \cdot \left(x \cdot \left(y \cdot z\right)\right) + \color{blue}{-4} \cdot a, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{fma}\left(18, x \cdot \left(y \cdot z\right), -4 \cdot a\right)}, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{\left(x \cdot y\right) \cdot z}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{z \cdot \left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{z \cdot \left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \color{blue}{\left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), \color{blue}{a \cdot -4}\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      17. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), \color{blue}{a \cdot -4}\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      18. lower-*.f64 79.9

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), a \cdot -4\right), \color{blue}{b \cdot c}\right) - \left(j \cdot 27\right) \cdot k \]

   5. Simplified 79.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), a \cdot -4\right), b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

   6. Taylor expanded in j 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. associate-*l* N/A

         \[\leadsto \color{blue}{j \cdot \left(k \cdot -27\right)} \]

      3. *-commutative N/A

         \[\leadsto j \cdot \color{blue}{\left(-27 \cdot k\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{j \cdot \left(-27 \cdot k\right)} \]

      5. *-commutative N/A

         \[\leadsto j \cdot \color{blue}{\left(k \cdot -27\right)} \]

      6. lower-*.f64 36.3

         \[\leadsto j \cdot \color{blue}{\left(k \cdot -27\right)} \]

   8. Simplified 36.3%

      \[\leadsto \color{blue}{j \cdot \left(k \cdot -27\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot c \leq -3.2 \cdot 10^{+115}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;b \cdot c \leq -9.5 \cdot 10^{-127}:\\ \;\;\;\;t \cdot \left(a \cdot -4\right)\\ \mathbf{elif}\;b \cdot c \leq 5.5 \cdot 10^{+87}:\\ \;\;\;\;j \cdot \left(k \cdot -27\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 71.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(y, z \cdot \left(18 \cdot t\right), i \cdot -4\right)\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{+89}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(t, a \cdot -4, \mathsf{fma}\left(b, c, j \cdot \left(k \cdot -27\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (* x (fma y (* z (* 18.0 t)) (* i -4.0)))))
   (if (<= x -1.65e+89)
     t_1
     (if (<= x 2.4e-40) (fma t (* a -4.0) (fma b c (* j (* k -27.0)))) t_1))))

C

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 = x * fma(y, (z * (18.0 * t)), (i * -4.0));
	double tmp;
	if (x <= -1.65e+89) {
		tmp = t_1;
	} else if (x <= 2.4e-40) {
		tmp = fma(t, (a * -4.0), fma(b, c, (j * (k * -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])
function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = Float64(x * fma(y, Float64(z * Float64(18.0 * t)), Float64(i * -4.0)))
	tmp = 0.0
	if (x <= -1.65e+89)
		tmp = t_1;
	elseif (x <= 2.4e-40)
		tmp = fma(t, Float64(a * -4.0), fma(b, c, Float64(j * Float64(k * -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.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(x * N[(y * N[(z * N[(18.0 * t), $MachinePrecision]), $MachinePrecision] + N[(i * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.65e+89], t$95$1, If[LessEqual[x, 2.4e-40], N[(t * N[(a * -4.0), $MachinePrecision] + N[(b * c + N[(j * N[(k * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.64999999999999987e89 or 2.39999999999999991e-40 < x

   1. Initial program 81.7%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right)} - \left(j \cdot 27\right) \cdot k \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right)} - \left(j \cdot 27\right) \cdot k \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto x \cdot \color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot i\right)} - \left(j \cdot 27\right) \cdot k \]

      3. associate-*r* N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(18 \cdot t\right) \cdot \left(y \cdot z\right)} + \left(\mathsf{neg}\left(4\right)\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(y \cdot z\right) \cdot \left(18 \cdot t\right)} + \left(\mathsf{neg}\left(4\right)\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]

      5. metadata-eval N/A

         \[\leadsto x \cdot \left(\left(y \cdot z\right) \cdot \left(18 \cdot t\right) + \color{blue}{-4} \cdot i\right) - \left(j \cdot 27\right) \cdot k \]

      6. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y \cdot z, 18 \cdot t, -4 \cdot i\right)} - \left(j \cdot 27\right) \cdot k \]

      7. lower-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{y \cdot z}, 18 \cdot t, -4 \cdot i\right) - \left(j \cdot 27\right) \cdot k \]

      8. lower-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y \cdot z, \color{blue}{18 \cdot t}, -4 \cdot i\right) - \left(j \cdot 27\right) \cdot k \]

      9. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y \cdot z, 18 \cdot t, \color{blue}{i \cdot -4}\right) - \left(j \cdot 27\right) \cdot k \]

      10. lower-*.f64 82.9

          \[\leadsto x \cdot \mathsf{fma}\left(y \cdot z, 18 \cdot t, \color{blue}{i \cdot -4}\right) - \left(j \cdot 27\right) \cdot k \]

   5. Simplified 82.9%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(y \cdot z, 18 \cdot t, i \cdot -4\right)} - \left(j \cdot 27\right) \cdot k \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(-4 \cdot i + 18 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(-4 \cdot i + 18 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + -4 \cdot i\right)} \]

      3. *-commutative N/A

         \[\leadsto x \cdot \left(18 \cdot \left(t \cdot \color{blue}{\left(z \cdot y\right)}\right) + -4 \cdot i\right) \]

      4. associate-*r* N/A

         \[\leadsto x \cdot \left(18 \cdot \color{blue}{\left(\left(t \cdot z\right) \cdot y\right)} + -4 \cdot i\right) \]

      5. associate-*l* N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(18 \cdot \left(t \cdot z\right)\right) \cdot y} + -4 \cdot i\right) \]

      6. *-commutative N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot \left(18 \cdot \left(t \cdot z\right)\right)} + -4 \cdot i\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, 18 \cdot \left(t \cdot z\right), -4 \cdot i\right)} \]

      8. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\left(18 \cdot t\right) \cdot z}, -4 \cdot i\right) \]

      9. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{z \cdot \left(18 \cdot t\right)}, -4 \cdot i\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{z \cdot \left(18 \cdot t\right)}, -4 \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, z \cdot \color{blue}{\left(t \cdot 18\right)}, -4 \cdot i\right) \]

      12. lower-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, z \cdot \color{blue}{\left(t \cdot 18\right)}, -4 \cdot i\right) \]

      13. lower-*.f64 72.7

          \[\leadsto x \cdot \mathsf{fma}\left(y, z \cdot \left(t \cdot 18\right), \color{blue}{-4 \cdot i}\right) \]

   8. Simplified 72.7%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(y, z \cdot \left(t \cdot 18\right), -4 \cdot i\right)} \]

   if -1.64999999999999987e89 < x < 2.39999999999999991e-40

   1. Initial program 93.8%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 4 \cdot \left(a \cdot t\right)\right)} - \left(j \cdot 27\right) \cdot k \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(b \cdot c + 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right)\right)} - 4 \cdot \left(a \cdot t\right)\right) - \left(j \cdot 27\right) \cdot k \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(b \cdot c + \left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) - 4 \cdot \left(a \cdot t\right)\right)\right)} - \left(j \cdot 27\right) \cdot k \]

      3. *-commutative N/A

         \[\leadsto \left(b \cdot c + \left(18 \cdot \color{blue}{\left(\left(x \cdot \left(y \cdot z\right)\right) \cdot t\right)} - 4 \cdot \left(a \cdot t\right)\right)\right) - \left(j \cdot 27\right) \cdot k \]

      4. associate-*r* N/A

         \[\leadsto \left(b \cdot c + \left(\color{blue}{\left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot t} - 4 \cdot \left(a \cdot t\right)\right)\right) - \left(j \cdot 27\right) \cdot k \]

      5. associate-*r* N/A

         \[\leadsto \left(b \cdot c + \left(\left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot t - \color{blue}{\left(4 \cdot a\right) \cdot t}\right)\right) - \left(j \cdot 27\right) \cdot k \]

      6. distribute-rgt-out-- N/A

         \[\leadsto \left(b \cdot c + \color{blue}{t \cdot \left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a\right)}\right) - \left(j \cdot 27\right) \cdot k \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot \left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a\right) + b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, 18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a, b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{18 \cdot \left(x \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot a}, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t, 18 \cdot \left(x \cdot \left(y \cdot z\right)\right) + \color{blue}{-4} \cdot a, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{fma}\left(18, x \cdot \left(y \cdot z\right), -4 \cdot a\right)}, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{\left(x \cdot y\right) \cdot z}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{z \cdot \left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{z \cdot \left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \color{blue}{\left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), \color{blue}{a \cdot -4}\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      17. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), \color{blue}{a \cdot -4}\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      18. lower-*.f64 93.2

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), a \cdot -4\right), \color{blue}{b \cdot c}\right) - \left(j \cdot 27\right) \cdot k \]

   5. Simplified 93.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), a \cdot -4\right), b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(-4 \cdot \left(a \cdot t\right) + b \cdot c\right) - 27 \cdot \left(j \cdot k\right)} \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{-4 \cdot \left(a \cdot t\right) + \left(b \cdot c - 27 \cdot \left(j \cdot k\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-4 \cdot a\right) \cdot t} + \left(b \cdot c - 27 \cdot \left(j \cdot k\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{t \cdot \left(-4 \cdot a\right)} + \left(b \cdot c - 27 \cdot \left(j \cdot k\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -4 \cdot a, b \cdot c - 27 \cdot \left(j \cdot k\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{-4 \cdot a}, b \cdot c - 27 \cdot \left(j \cdot k\right)\right) \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(t, -4 \cdot a, \color{blue}{b \cdot c + \left(\mathsf{neg}\left(27\right)\right) \cdot \left(j \cdot k\right)}\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(t, -4 \cdot a, b \cdot c + \color{blue}{-27} \cdot \left(j \cdot k\right)\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, -4 \cdot a, \color{blue}{\mathsf{fma}\left(b, c, -27 \cdot \left(j \cdot k\right)\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(t, -4 \cdot a, \mathsf{fma}\left(b, c, \color{blue}{\left(j \cdot k\right) \cdot -27}\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(t, -4 \cdot a, \mathsf{fma}\left(b, c, \color{blue}{j \cdot \left(k \cdot -27\right)}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t, -4 \cdot a, \mathsf{fma}\left(b, c, j \cdot \color{blue}{\left(-27 \cdot k\right)}\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, -4 \cdot a, \mathsf{fma}\left(b, c, \color{blue}{j \cdot \left(-27 \cdot k\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t, -4 \cdot a, \mathsf{fma}\left(b, c, j \cdot \color{blue}{\left(k \cdot -27\right)}\right)\right) \]

      14. lower-*.f64 87.3

          \[\leadsto \mathsf{fma}\left(t, -4 \cdot a, \mathsf{fma}\left(b, c, j \cdot \color{blue}{\left(k \cdot -27\right)}\right)\right) \]

   8. Simplified 87.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, -4 \cdot a, \mathsf{fma}\left(b, c, j \cdot \left(k \cdot -27\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+89}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, z \cdot \left(18 \cdot t\right), i \cdot -4\right)\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(t, a \cdot -4, \mathsf{fma}\left(b, c, j \cdot \left(k \cdot -27\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, z \cdot \left(18 \cdot t\right), i \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 59.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(y, z \cdot \left(18 \cdot t\right), i \cdot -4\right)\\ \mathbf{if}\;x \leq -1.05 \cdot 10^{+89}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-55}:\\ \;\;\;\;\mathsf{fma}\left(b, c, j \cdot \left(k \cdot -27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (* x (fma y (* z (* 18.0 t)) (* i -4.0)))))
   (if (<= x -1.05e+89)
     t_1
     (if (<= x 5.8e-55) (fma b c (* j (* k -27.0))) t_1))))

C

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 = x * fma(y, (z * (18.0 * t)), (i * -4.0));
	double tmp;
	if (x <= -1.05e+89) {
		tmp = t_1;
	} else if (x <= 5.8e-55) {
		tmp = fma(b, c, (j * (k * -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])
function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = Float64(x * fma(y, Float64(z * Float64(18.0 * t)), Float64(i * -4.0)))
	tmp = 0.0
	if (x <= -1.05e+89)
		tmp = t_1;
	elseif (x <= 5.8e-55)
		tmp = fma(b, c, Float64(j * Float64(k * -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.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(x * N[(y * N[(z * N[(18.0 * t), $MachinePrecision]), $MachinePrecision] + N[(i * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.05e+89], t$95$1, If[LessEqual[x, 5.8e-55], N[(b * c + N[(j * N[(k * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.04999999999999993e89 or 5.8e-55 < x

   1. Initial program 82.0%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right)} - \left(j \cdot 27\right) \cdot k \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right)} - \left(j \cdot 27\right) \cdot k \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto x \cdot \color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot i\right)} - \left(j \cdot 27\right) \cdot k \]

      3. associate-*r* N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(18 \cdot t\right) \cdot \left(y \cdot z\right)} + \left(\mathsf{neg}\left(4\right)\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(y \cdot z\right) \cdot \left(18 \cdot t\right)} + \left(\mathsf{neg}\left(4\right)\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]

      5. metadata-eval N/A

         \[\leadsto x \cdot \left(\left(y \cdot z\right) \cdot \left(18 \cdot t\right) + \color{blue}{-4} \cdot i\right) - \left(j \cdot 27\right) \cdot k \]

      6. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y \cdot z, 18 \cdot t, -4 \cdot i\right)} - \left(j \cdot 27\right) \cdot k \]

      7. lower-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{y \cdot z}, 18 \cdot t, -4 \cdot i\right) - \left(j \cdot 27\right) \cdot k \]

      8. lower-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y \cdot z, \color{blue}{18 \cdot t}, -4 \cdot i\right) - \left(j \cdot 27\right) \cdot k \]

      9. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y \cdot z, 18 \cdot t, \color{blue}{i \cdot -4}\right) - \left(j \cdot 27\right) \cdot k \]

      10. lower-*.f64 83.2

          \[\leadsto x \cdot \mathsf{fma}\left(y \cdot z, 18 \cdot t, \color{blue}{i \cdot -4}\right) - \left(j \cdot 27\right) \cdot k \]

   5. Simplified 83.2%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(y \cdot z, 18 \cdot t, i \cdot -4\right)} - \left(j \cdot 27\right) \cdot k \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(-4 \cdot i + 18 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(-4 \cdot i + 18 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + -4 \cdot i\right)} \]

      3. *-commutative N/A

         \[\leadsto x \cdot \left(18 \cdot \left(t \cdot \color{blue}{\left(z \cdot y\right)}\right) + -4 \cdot i\right) \]

      4. associate-*r* N/A

         \[\leadsto x \cdot \left(18 \cdot \color{blue}{\left(\left(t \cdot z\right) \cdot y\right)} + -4 \cdot i\right) \]

      5. associate-*l* N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(18 \cdot \left(t \cdot z\right)\right) \cdot y} + -4 \cdot i\right) \]

      6. *-commutative N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot \left(18 \cdot \left(t \cdot z\right)\right)} + -4 \cdot i\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, 18 \cdot \left(t \cdot z\right), -4 \cdot i\right)} \]

      8. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\left(18 \cdot t\right) \cdot z}, -4 \cdot i\right) \]

      9. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{z \cdot \left(18 \cdot t\right)}, -4 \cdot i\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{z \cdot \left(18 \cdot t\right)}, -4 \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, z \cdot \color{blue}{\left(t \cdot 18\right)}, -4 \cdot i\right) \]

      12. lower-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, z \cdot \color{blue}{\left(t \cdot 18\right)}, -4 \cdot i\right) \]

      13. lower-*.f64 72.3

          \[\leadsto x \cdot \mathsf{fma}\left(y, z \cdot \left(t \cdot 18\right), \color{blue}{-4 \cdot i}\right) \]

   8. Simplified 72.3%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(y, z \cdot \left(t \cdot 18\right), -4 \cdot i\right)} \]

   if -1.04999999999999993e89 < x < 5.8e-55

   1. Initial program 93.7%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 4 \cdot \left(a \cdot t\right)\right)} - \left(j \cdot 27\right) \cdot k \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(b \cdot c + 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right)\right)} - 4 \cdot \left(a \cdot t\right)\right) - \left(j \cdot 27\right) \cdot k \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(b \cdot c + \left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) - 4 \cdot \left(a \cdot t\right)\right)\right)} - \left(j \cdot 27\right) \cdot k \]

      3. *-commutative N/A

         \[\leadsto \left(b \cdot c + \left(18 \cdot \color{blue}{\left(\left(x \cdot \left(y \cdot z\right)\right) \cdot t\right)} - 4 \cdot \left(a \cdot t\right)\right)\right) - \left(j \cdot 27\right) \cdot k \]

      4. associate-*r* N/A

         \[\leadsto \left(b \cdot c + \left(\color{blue}{\left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot t} - 4 \cdot \left(a \cdot t\right)\right)\right) - \left(j \cdot 27\right) \cdot k \]

      5. associate-*r* N/A

         \[\leadsto \left(b \cdot c + \left(\left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot t - \color{blue}{\left(4 \cdot a\right) \cdot t}\right)\right) - \left(j \cdot 27\right) \cdot k \]

      6. distribute-rgt-out-- N/A

         \[\leadsto \left(b \cdot c + \color{blue}{t \cdot \left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a\right)}\right) - \left(j \cdot 27\right) \cdot k \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot \left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a\right) + b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, 18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a, b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{18 \cdot \left(x \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot a}, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t, 18 \cdot \left(x \cdot \left(y \cdot z\right)\right) + \color{blue}{-4} \cdot a, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{fma}\left(18, x \cdot \left(y \cdot z\right), -4 \cdot a\right)}, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{\left(x \cdot y\right) \cdot z}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{z \cdot \left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{z \cdot \left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \color{blue}{\left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), \color{blue}{a \cdot -4}\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      17. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), \color{blue}{a \cdot -4}\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      18. lower-*.f64 93.1

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), a \cdot -4\right), \color{blue}{b \cdot c}\right) - \left(j \cdot 27\right) \cdot k \]

   5. Simplified 93.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), a \cdot -4\right), b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{b \cdot c - 27 \cdot \left(j \cdot k\right)} \]
   7. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{b \cdot c + \left(\mathsf{neg}\left(27\right)\right) \cdot \left(j \cdot k\right)} \]

      2. metadata-eval N/A

         \[\leadsto b \cdot c + \color{blue}{-27} \cdot \left(j \cdot k\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(b, c, -27 \cdot \left(j \cdot k\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, c, \color{blue}{\left(j \cdot k\right) \cdot -27}\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(b, c, \color{blue}{j \cdot \left(k \cdot -27\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, c, j \cdot \color{blue}{\left(-27 \cdot k\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(b, c, \color{blue}{j \cdot \left(-27 \cdot k\right)}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, c, j \cdot \color{blue}{\left(k \cdot -27\right)}\right) \]

      9. lower-*.f64 62.4

         \[\leadsto \mathsf{fma}\left(b, c, j \cdot \color{blue}{\left(k \cdot -27\right)}\right) \]

   8. Simplified 62.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, c, j \cdot \left(k \cdot -27\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+89}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, z \cdot \left(18 \cdot t\right), i \cdot -4\right)\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-55}:\\ \;\;\;\;\mathsf{fma}\left(b, c, j \cdot \left(k \cdot -27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, z \cdot \left(18 \cdot t\right), i \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 52.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, a \cdot -4, b \cdot c\right)\\ \mathbf{if}\;a \cdot 4 \leq -6 \cdot 10^{+56}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \cdot 4 \leq 2 \cdot 10^{+61}:\\ \;\;\;\;\mathsf{fma}\left(b, c, j \cdot \left(k \cdot -27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (fma t (* a -4.0) (* b c))))
   (if (<= (* a 4.0) -6e+56)
     t_1
     (if (<= (* a 4.0) 2e+61) (fma b c (* j (* k -27.0))) t_1))))

C

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(t, (a * -4.0), (b * c));
	double tmp;
	if ((a * 4.0) <= -6e+56) {
		tmp = t_1;
	} else if ((a * 4.0) <= 2e+61) {
		tmp = fma(b, c, (j * (k * -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])
function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = fma(t, Float64(a * -4.0), Float64(b * c))
	tmp = 0.0
	if (Float64(a * 4.0) <= -6e+56)
		tmp = t_1;
	elseif (Float64(a * 4.0) <= 2e+61)
		tmp = fma(b, c, Float64(j * Float64(k * -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.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(t * N[(a * -4.0), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * 4.0), $MachinePrecision], -6e+56], t$95$1, If[LessEqual[N[(a * 4.0), $MachinePrecision], 2e+61], N[(b * c + N[(j * N[(k * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a #s(literal 4 binary64)) < -6.00000000000000012e56 or 1.9999999999999999e61 < (*.f64 a #s(literal 4 binary64))

   1. Initial program 85.6%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 4 \cdot \left(a \cdot t\right)\right)} - \left(j \cdot 27\right) \cdot k \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(b \cdot c + 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right)\right)} - 4 \cdot \left(a \cdot t\right)\right) - \left(j \cdot 27\right) \cdot k \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(b \cdot c + \left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) - 4 \cdot \left(a \cdot t\right)\right)\right)} - \left(j \cdot 27\right) \cdot k \]

      3. *-commutative N/A

         \[\leadsto \left(b \cdot c + \left(18 \cdot \color{blue}{\left(\left(x \cdot \left(y \cdot z\right)\right) \cdot t\right)} - 4 \cdot \left(a \cdot t\right)\right)\right) - \left(j \cdot 27\right) \cdot k \]

      4. associate-*r* N/A

         \[\leadsto \left(b \cdot c + \left(\color{blue}{\left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot t} - 4 \cdot \left(a \cdot t\right)\right)\right) - \left(j \cdot 27\right) \cdot k \]

      5. associate-*r* N/A

         \[\leadsto \left(b \cdot c + \left(\left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot t - \color{blue}{\left(4 \cdot a\right) \cdot t}\right)\right) - \left(j \cdot 27\right) \cdot k \]

      6. distribute-rgt-out-- N/A

         \[\leadsto \left(b \cdot c + \color{blue}{t \cdot \left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a\right)}\right) - \left(j \cdot 27\right) \cdot k \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot \left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a\right) + b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, 18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a, b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{18 \cdot \left(x \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot a}, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t, 18 \cdot \left(x \cdot \left(y \cdot z\right)\right) + \color{blue}{-4} \cdot a, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{fma}\left(18, x \cdot \left(y \cdot z\right), -4 \cdot a\right)}, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{\left(x \cdot y\right) \cdot z}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{z \cdot \left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{z \cdot \left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \color{blue}{\left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), \color{blue}{a \cdot -4}\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      17. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), \color{blue}{a \cdot -4}\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      18. lower-*.f64 90.0

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), a \cdot -4\right), \color{blue}{b \cdot c}\right) - \left(j \cdot 27\right) \cdot k \]

   5. Simplified 90.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), a \cdot -4\right), b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(-4 \cdot \left(a \cdot t\right) + b \cdot c\right) - 27 \cdot \left(j \cdot k\right)} \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{-4 \cdot \left(a \cdot t\right) + \left(b \cdot c - 27 \cdot \left(j \cdot k\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-4 \cdot a\right) \cdot t} + \left(b \cdot c - 27 \cdot \left(j \cdot k\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{t \cdot \left(-4 \cdot a\right)} + \left(b \cdot c - 27 \cdot \left(j \cdot k\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -4 \cdot a, b \cdot c - 27 \cdot \left(j \cdot k\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{-4 \cdot a}, b \cdot c - 27 \cdot \left(j \cdot k\right)\right) \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(t, -4 \cdot a, \color{blue}{b \cdot c + \left(\mathsf{neg}\left(27\right)\right) \cdot \left(j \cdot k\right)}\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(t, -4 \cdot a, b \cdot c + \color{blue}{-27} \cdot \left(j \cdot k\right)\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, -4 \cdot a, \color{blue}{\mathsf{fma}\left(b, c, -27 \cdot \left(j \cdot k\right)\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(t, -4 \cdot a, \mathsf{fma}\left(b, c, \color{blue}{\left(j \cdot k\right) \cdot -27}\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(t, -4 \cdot a, \mathsf{fma}\left(b, c, \color{blue}{j \cdot \left(k \cdot -27\right)}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t, -4 \cdot a, \mathsf{fma}\left(b, c, j \cdot \color{blue}{\left(-27 \cdot k\right)}\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, -4 \cdot a, \mathsf{fma}\left(b, c, \color{blue}{j \cdot \left(-27 \cdot k\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t, -4 \cdot a, \mathsf{fma}\left(b, c, j \cdot \color{blue}{\left(k \cdot -27\right)}\right)\right) \]

      14. lower-*.f64 71.9

          \[\leadsto \mathsf{fma}\left(t, -4 \cdot a, \mathsf{fma}\left(b, c, j \cdot \color{blue}{\left(k \cdot -27\right)}\right)\right) \]

   8. Simplified 71.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, -4 \cdot a, \mathsf{fma}\left(b, c, j \cdot \left(k \cdot -27\right)\right)\right)} \]

   9. Taylor expanded in b around inf

      \[\leadsto \mathsf{fma}\left(t, -4 \cdot a, \color{blue}{b \cdot c}\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t, -4 \cdot a, \color{blue}{c \cdot b}\right) \]

       2. lower-*.f64 63.9

          \[\leadsto \mathsf{fma}\left(t, -4 \cdot a, \color{blue}{c \cdot b}\right) \]

   11. Simplified 63.9%

       \[\leadsto \mathsf{fma}\left(t, -4 \cdot a, \color{blue}{c \cdot b}\right) \]

   if -6.00000000000000012e56 < (*.f64 a #s(literal 4 binary64)) < 1.9999999999999999e61

   1. Initial program 90.1%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 4 \cdot \left(a \cdot t\right)\right)} - \left(j \cdot 27\right) \cdot k \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(b \cdot c + 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right)\right)} - 4 \cdot \left(a \cdot t\right)\right) - \left(j \cdot 27\right) \cdot k \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(b \cdot c + \left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) - 4 \cdot \left(a \cdot t\right)\right)\right)} - \left(j \cdot 27\right) \cdot k \]

      3. *-commutative N/A

         \[\leadsto \left(b \cdot c + \left(18 \cdot \color{blue}{\left(\left(x \cdot \left(y \cdot z\right)\right) \cdot t\right)} - 4 \cdot \left(a \cdot t\right)\right)\right) - \left(j \cdot 27\right) \cdot k \]

      4. associate-*r* N/A

         \[\leadsto \left(b \cdot c + \left(\color{blue}{\left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot t} - 4 \cdot \left(a \cdot t\right)\right)\right) - \left(j \cdot 27\right) \cdot k \]

      5. associate-*r* N/A

         \[\leadsto \left(b \cdot c + \left(\left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot t - \color{blue}{\left(4 \cdot a\right) \cdot t}\right)\right) - \left(j \cdot 27\right) \cdot k \]

      6. distribute-rgt-out-- N/A

         \[\leadsto \left(b \cdot c + \color{blue}{t \cdot \left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a\right)}\right) - \left(j \cdot 27\right) \cdot k \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot \left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a\right) + b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, 18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a, b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{18 \cdot \left(x \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot a}, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t, 18 \cdot \left(x \cdot \left(y \cdot z\right)\right) + \color{blue}{-4} \cdot a, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{fma}\left(18, x \cdot \left(y \cdot z\right), -4 \cdot a\right)}, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{\left(x \cdot y\right) \cdot z}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{z \cdot \left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{z \cdot \left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \color{blue}{\left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), \color{blue}{a \cdot -4}\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      17. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), \color{blue}{a \cdot -4}\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      18. lower-*.f64 77.5

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), a \cdot -4\right), \color{blue}{b \cdot c}\right) - \left(j \cdot 27\right) \cdot k \]

   5. Simplified 77.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), a \cdot -4\right), b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{b \cdot c - 27 \cdot \left(j \cdot k\right)} \]
   7. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{b \cdot c + \left(\mathsf{neg}\left(27\right)\right) \cdot \left(j \cdot k\right)} \]

      2. metadata-eval N/A

         \[\leadsto b \cdot c + \color{blue}{-27} \cdot \left(j \cdot k\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(b, c, -27 \cdot \left(j \cdot k\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, c, \color{blue}{\left(j \cdot k\right) \cdot -27}\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(b, c, \color{blue}{j \cdot \left(k \cdot -27\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, c, j \cdot \color{blue}{\left(-27 \cdot k\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(b, c, \color{blue}{j \cdot \left(-27 \cdot k\right)}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, c, j \cdot \color{blue}{\left(k \cdot -27\right)}\right) \]

      9. lower-*.f64 55.1

         \[\leadsto \mathsf{fma}\left(b, c, j \cdot \color{blue}{\left(k \cdot -27\right)}\right) \]

   8. Simplified 55.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, c, j \cdot \left(k \cdot -27\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 4 \leq -6 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(t, a \cdot -4, b \cdot c\right)\\ \mathbf{elif}\;a \cdot 4 \leq 2 \cdot 10^{+61}:\\ \;\;\;\;\mathsf{fma}\left(b, c, j \cdot \left(k \cdot -27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a \cdot -4, b \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 48.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := t \cdot \left(a \cdot -4\right)\\ \mathbf{if}\;a \cdot 4 \leq -5 \cdot 10^{+135}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \cdot 4 \leq 2 \cdot 10^{+165}:\\ \;\;\;\;\mathsf{fma}\left(b, c, j \cdot \left(k \cdot -27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (* t (* a -4.0))))
   (if (<= (* a 4.0) -5e+135)
     t_1
     (if (<= (* a 4.0) 2e+165) (fma b c (* j (* k -27.0))) t_1))))

C

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 = t * (a * -4.0);
	double tmp;
	if ((a * 4.0) <= -5e+135) {
		tmp = t_1;
	} else if ((a * 4.0) <= 2e+165) {
		tmp = fma(b, c, (j * (k * -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])
function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = Float64(t * Float64(a * -4.0))
	tmp = 0.0
	if (Float64(a * 4.0) <= -5e+135)
		tmp = t_1;
	elseif (Float64(a * 4.0) <= 2e+165)
		tmp = fma(b, c, Float64(j * Float64(k * -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.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(t * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * 4.0), $MachinePrecision], -5e+135], t$95$1, If[LessEqual[N[(a * 4.0), $MachinePrecision], 2e+165], N[(b * c + N[(j * N[(k * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a #s(literal 4 binary64)) < -5.00000000000000029e135 or 1.9999999999999998e165 < (*.f64 a #s(literal 4 binary64))

   1. Initial program 84.1%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 4 \cdot \left(a \cdot t\right)\right)} - \left(j \cdot 27\right) \cdot k \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(b \cdot c + 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right)\right)} - 4 \cdot \left(a \cdot t\right)\right) - \left(j \cdot 27\right) \cdot k \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(b \cdot c + \left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) - 4 \cdot \left(a \cdot t\right)\right)\right)} - \left(j \cdot 27\right) \cdot k \]

      3. *-commutative N/A

         \[\leadsto \left(b \cdot c + \left(18 \cdot \color{blue}{\left(\left(x \cdot \left(y \cdot z\right)\right) \cdot t\right)} - 4 \cdot \left(a \cdot t\right)\right)\right) - \left(j \cdot 27\right) \cdot k \]

      4. associate-*r* N/A

         \[\leadsto \left(b \cdot c + \left(\color{blue}{\left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot t} - 4 \cdot \left(a \cdot t\right)\right)\right) - \left(j \cdot 27\right) \cdot k \]

      5. associate-*r* N/A

         \[\leadsto \left(b \cdot c + \left(\left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot t - \color{blue}{\left(4 \cdot a\right) \cdot t}\right)\right) - \left(j \cdot 27\right) \cdot k \]

      6. distribute-rgt-out-- N/A

         \[\leadsto \left(b \cdot c + \color{blue}{t \cdot \left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a\right)}\right) - \left(j \cdot 27\right) \cdot k \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot \left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a\right) + b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, 18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a, b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{18 \cdot \left(x \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot a}, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t, 18 \cdot \left(x \cdot \left(y \cdot z\right)\right) + \color{blue}{-4} \cdot a, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{fma}\left(18, x \cdot \left(y \cdot z\right), -4 \cdot a\right)}, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{\left(x \cdot y\right) \cdot z}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{z \cdot \left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{z \cdot \left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \color{blue}{\left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), \color{blue}{a \cdot -4}\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      17. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), \color{blue}{a \cdot -4}\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      18. lower-*.f64 90.8

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), a \cdot -4\right), \color{blue}{b \cdot c}\right) - \left(j \cdot 27\right) \cdot k \]

   5. Simplified 90.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), a \cdot -4\right), b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

   6. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-4 \cdot \left(a \cdot t\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-4 \cdot a\right) \cdot t} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{t \cdot \left(-4 \cdot a\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{t \cdot \left(-4 \cdot a\right)} \]

      4. lower-*.f64 55.2

         \[\leadsto t \cdot \color{blue}{\left(-4 \cdot a\right)} \]

   8. Simplified 55.2%

      \[\leadsto \color{blue}{t \cdot \left(-4 \cdot a\right)} \]

   if -5.00000000000000029e135 < (*.f64 a #s(literal 4 binary64)) < 1.9999999999999998e165

   1. Initial program 89.8%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 4 \cdot \left(a \cdot t\right)\right)} - \left(j \cdot 27\right) \cdot k \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(b \cdot c + 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right)\right)} - 4 \cdot \left(a \cdot t\right)\right) - \left(j \cdot 27\right) \cdot k \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(b \cdot c + \left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) - 4 \cdot \left(a \cdot t\right)\right)\right)} - \left(j \cdot 27\right) \cdot k \]

      3. *-commutative N/A

         \[\leadsto \left(b \cdot c + \left(18 \cdot \color{blue}{\left(\left(x \cdot \left(y \cdot z\right)\right) \cdot t\right)} - 4 \cdot \left(a \cdot t\right)\right)\right) - \left(j \cdot 27\right) \cdot k \]

      4. associate-*r* N/A

         \[\leadsto \left(b \cdot c + \left(\color{blue}{\left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot t} - 4 \cdot \left(a \cdot t\right)\right)\right) - \left(j \cdot 27\right) \cdot k \]

      5. associate-*r* N/A

         \[\leadsto \left(b \cdot c + \left(\left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot t - \color{blue}{\left(4 \cdot a\right) \cdot t}\right)\right) - \left(j \cdot 27\right) \cdot k \]

      6. distribute-rgt-out-- N/A

         \[\leadsto \left(b \cdot c + \color{blue}{t \cdot \left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a\right)}\right) - \left(j \cdot 27\right) \cdot k \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot \left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a\right) + b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, 18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a, b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{18 \cdot \left(x \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot a}, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t, 18 \cdot \left(x \cdot \left(y \cdot z\right)\right) + \color{blue}{-4} \cdot a, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{fma}\left(18, x \cdot \left(y \cdot z\right), -4 \cdot a\right)}, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{\left(x \cdot y\right) \cdot z}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{z \cdot \left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{z \cdot \left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \color{blue}{\left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), \color{blue}{a \cdot -4}\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      17. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), \color{blue}{a \cdot -4}\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      18. lower-*.f64 79.5

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), a \cdot -4\right), \color{blue}{b \cdot c}\right) - \left(j \cdot 27\right) \cdot k \]

   5. Simplified 79.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), a \cdot -4\right), b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{b \cdot c - 27 \cdot \left(j \cdot k\right)} \]
   7. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{b \cdot c + \left(\mathsf{neg}\left(27\right)\right) \cdot \left(j \cdot k\right)} \]

      2. metadata-eval N/A

         \[\leadsto b \cdot c + \color{blue}{-27} \cdot \left(j \cdot k\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(b, c, -27 \cdot \left(j \cdot k\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, c, \color{blue}{\left(j \cdot k\right) \cdot -27}\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(b, c, \color{blue}{j \cdot \left(k \cdot -27\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, c, j \cdot \color{blue}{\left(-27 \cdot k\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(b, c, \color{blue}{j \cdot \left(-27 \cdot k\right)}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, c, j \cdot \color{blue}{\left(k \cdot -27\right)}\right) \]

      9. lower-*.f64 53.9

         \[\leadsto \mathsf{fma}\left(b, c, j \cdot \color{blue}{\left(k \cdot -27\right)}\right) \]

   8. Simplified 53.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, c, j \cdot \left(k \cdot -27\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 4 \leq -5 \cdot 10^{+135}:\\ \;\;\;\;t \cdot \left(a \cdot -4\right)\\ \mathbf{elif}\;a \cdot 4 \leq 2 \cdot 10^{+165}:\\ \;\;\;\;\mathsf{fma}\left(b, c, j \cdot \left(k \cdot -27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(a \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 37.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} \mathbf{if}\;b \cdot c \leq -1.45 \cdot 10^{+97}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;b \cdot c \leq 5.5 \cdot 10^{+87}:\\ \;\;\;\;j \cdot \left(k \cdot -27\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot c\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i j k)
 :precision binary64
 (if (<= (* b c) -1.45e+97)
   (* b c)
   (if (<= (* b c) 5.5e+87) (* j (* k -27.0)) (* b c))))

C

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 ((b * c) <= -1.45e+97) {
		tmp = b * c;
	} else if ((b * c) <= 5.5e+87) {
		tmp = j * (k * -27.0);
	} else {
		tmp = b * c;
	}
	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.
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) :: tmp
    if ((b * c) <= (-1.45d+97)) then
        tmp = b * c
    else if ((b * c) <= 5.5d+87) then
        tmp = j * (k * (-27.0d0))
    else
        tmp = b * c
    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;
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 tmp;
	if ((b * c) <= -1.45e+97) {
		tmp = b * c;
	} else if ((b * c) <= 5.5e+87) {
		tmp = j * (k * -27.0);
	} else {
		tmp = b * c;
	}
	return tmp;
}

Python

[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):
	tmp = 0
	if (b * c) <= -1.45e+97:
		tmp = b * c
	elif (b * c) <= 5.5e+87:
		tmp = j * (k * -27.0)
	else:
		tmp = b * c
	return tmp

Julia

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(b * c) <= -1.45e+97)
		tmp = Float64(b * c);
	elseif (Float64(b * c) <= 5.5e+87)
		tmp = Float64(j * Float64(k * -27.0));
	else
		tmp = Float64(b * c);
	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])){:}
function tmp_2 = code(x, y, z, t, a, b, c, i, j, k)
	tmp = 0.0;
	if ((b * c) <= -1.45e+97)
		tmp = b * c;
	elseif ((b * c) <= 5.5e+87)
		tmp = j * (k * -27.0);
	else
		tmp = b * c;
	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.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := If[LessEqual[N[(b * c), $MachinePrecision], -1.45e+97], N[(b * c), $MachinePrecision], If[LessEqual[N[(b * c), $MachinePrecision], 5.5e+87], N[(j * N[(k * -27.0), $MachinePrecision]), $MachinePrecision], N[(b * c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b c) < -1.44999999999999994e97 or 5.50000000000000022e87 < (*.f64 b c)

   1. Initial program 90.1%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 4 \cdot \left(a \cdot t\right)\right)} - \left(j \cdot 27\right) \cdot k \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(b \cdot c + 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right)\right)} - 4 \cdot \left(a \cdot t\right)\right) - \left(j \cdot 27\right) \cdot k \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(b \cdot c + \left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) - 4 \cdot \left(a \cdot t\right)\right)\right)} - \left(j \cdot 27\right) \cdot k \]

      3. *-commutative N/A

         \[\leadsto \left(b \cdot c + \left(18 \cdot \color{blue}{\left(\left(x \cdot \left(y \cdot z\right)\right) \cdot t\right)} - 4 \cdot \left(a \cdot t\right)\right)\right) - \left(j \cdot 27\right) \cdot k \]

      4. associate-*r* N/A

         \[\leadsto \left(b \cdot c + \left(\color{blue}{\left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot t} - 4 \cdot \left(a \cdot t\right)\right)\right) - \left(j \cdot 27\right) \cdot k \]

      5. associate-*r* N/A

         \[\leadsto \left(b \cdot c + \left(\left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot t - \color{blue}{\left(4 \cdot a\right) \cdot t}\right)\right) - \left(j \cdot 27\right) \cdot k \]

      6. distribute-rgt-out-- N/A

         \[\leadsto \left(b \cdot c + \color{blue}{t \cdot \left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a\right)}\right) - \left(j \cdot 27\right) \cdot k \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot \left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a\right) + b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, 18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a, b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{18 \cdot \left(x \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot a}, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t, 18 \cdot \left(x \cdot \left(y \cdot z\right)\right) + \color{blue}{-4} \cdot a, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{fma}\left(18, x \cdot \left(y \cdot z\right), -4 \cdot a\right)}, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{\left(x \cdot y\right) \cdot z}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{z \cdot \left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{z \cdot \left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \color{blue}{\left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), \color{blue}{a \cdot -4}\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      17. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), \color{blue}{a \cdot -4}\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      18. lower-*.f64 84.8

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), a \cdot -4\right), \color{blue}{b \cdot c}\right) - \left(j \cdot 27\right) \cdot k \]

   5. Simplified 84.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), a \cdot -4\right), b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot c} \]
   7. Step-by-step derivation

      1. lower-*.f64 63.0

         \[\leadsto \color{blue}{b \cdot c} \]

   8. Simplified 63.0%

      \[\leadsto \color{blue}{b \cdot c} \]

   if -1.44999999999999994e97 < (*.f64 b c) < 5.50000000000000022e87

   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 i around 0

      \[\leadsto \color{blue}{\left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 4 \cdot \left(a \cdot t\right)\right)} - \left(j \cdot 27\right) \cdot k \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(b \cdot c + 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right)\right)} - 4 \cdot \left(a \cdot t\right)\right) - \left(j \cdot 27\right) \cdot k \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(b \cdot c + \left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) - 4 \cdot \left(a \cdot t\right)\right)\right)} - \left(j \cdot 27\right) \cdot k \]

      3. *-commutative N/A

         \[\leadsto \left(b \cdot c + \left(18 \cdot \color{blue}{\left(\left(x \cdot \left(y \cdot z\right)\right) \cdot t\right)} - 4 \cdot \left(a \cdot t\right)\right)\right) - \left(j \cdot 27\right) \cdot k \]

      4. associate-*r* N/A

         \[\leadsto \left(b \cdot c + \left(\color{blue}{\left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot t} - 4 \cdot \left(a \cdot t\right)\right)\right) - \left(j \cdot 27\right) \cdot k \]

      5. associate-*r* N/A

         \[\leadsto \left(b \cdot c + \left(\left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot t - \color{blue}{\left(4 \cdot a\right) \cdot t}\right)\right) - \left(j \cdot 27\right) \cdot k \]

      6. distribute-rgt-out-- N/A

         \[\leadsto \left(b \cdot c + \color{blue}{t \cdot \left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a\right)}\right) - \left(j \cdot 27\right) \cdot k \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot \left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a\right) + b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, 18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a, b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{18 \cdot \left(x \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot a}, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t, 18 \cdot \left(x \cdot \left(y \cdot z\right)\right) + \color{blue}{-4} \cdot a, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{fma}\left(18, x \cdot \left(y \cdot z\right), -4 \cdot a\right)}, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{\left(x \cdot y\right) \cdot z}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{z \cdot \left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{z \cdot \left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \color{blue}{\left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), \color{blue}{a \cdot -4}\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      17. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), \color{blue}{a \cdot -4}\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      18. lower-*.f64 80.8

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), a \cdot -4\right), \color{blue}{b \cdot c}\right) - \left(j \cdot 27\right) \cdot k \]

   5. Simplified 80.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), a \cdot -4\right), b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

   6. Taylor expanded in j 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. associate-*l* N/A

         \[\leadsto \color{blue}{j \cdot \left(k \cdot -27\right)} \]

      3. *-commutative N/A

         \[\leadsto j \cdot \color{blue}{\left(-27 \cdot k\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{j \cdot \left(-27 \cdot k\right)} \]

      5. *-commutative N/A

         \[\leadsto j \cdot \color{blue}{\left(k \cdot -27\right)} \]

      6. lower-*.f64 31.4

         \[\leadsto j \cdot \color{blue}{\left(k \cdot -27\right)} \]

   8. Simplified 31.4%

      \[\leadsto \color{blue}{j \cdot \left(k \cdot -27\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 37.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} \mathbf{if}\;b \cdot c \leq -1.45 \cdot 10^{+97}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;b \cdot c \leq 5.5 \cdot 10^{+87}:\\ \;\;\;\;-27 \cdot \left(j \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot c\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i j k)
 :precision binary64
 (if (<= (* b c) -1.45e+97)
   (* b c)
   (if (<= (* b c) 5.5e+87) (* -27.0 (* j k)) (* b c))))

C

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 ((b * c) <= -1.45e+97) {
		tmp = b * c;
	} else if ((b * c) <= 5.5e+87) {
		tmp = -27.0 * (j * k);
	} else {
		tmp = b * c;
	}
	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.
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) :: tmp
    if ((b * c) <= (-1.45d+97)) then
        tmp = b * c
    else if ((b * c) <= 5.5d+87) then
        tmp = (-27.0d0) * (j * k)
    else
        tmp = b * c
    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;
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 tmp;
	if ((b * c) <= -1.45e+97) {
		tmp = b * c;
	} else if ((b * c) <= 5.5e+87) {
		tmp = -27.0 * (j * k);
	} else {
		tmp = b * c;
	}
	return tmp;
}

Python

[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):
	tmp = 0
	if (b * c) <= -1.45e+97:
		tmp = b * c
	elif (b * c) <= 5.5e+87:
		tmp = -27.0 * (j * k)
	else:
		tmp = b * c
	return tmp

Julia

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(b * c) <= -1.45e+97)
		tmp = Float64(b * c);
	elseif (Float64(b * c) <= 5.5e+87)
		tmp = Float64(-27.0 * Float64(j * k));
	else
		tmp = Float64(b * c);
	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])){:}
function tmp_2 = code(x, y, z, t, a, b, c, i, j, k)
	tmp = 0.0;
	if ((b * c) <= -1.45e+97)
		tmp = b * c;
	elseif ((b * c) <= 5.5e+87)
		tmp = -27.0 * (j * k);
	else
		tmp = b * c;
	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.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := If[LessEqual[N[(b * c), $MachinePrecision], -1.45e+97], N[(b * c), $MachinePrecision], If[LessEqual[N[(b * c), $MachinePrecision], 5.5e+87], N[(-27.0 * N[(j * k), $MachinePrecision]), $MachinePrecision], N[(b * c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b c) < -1.44999999999999994e97 or 5.50000000000000022e87 < (*.f64 b c)

   1. Initial program 90.1%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 4 \cdot \left(a \cdot t\right)\right)} - \left(j \cdot 27\right) \cdot k \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(b \cdot c + 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right)\right)} - 4 \cdot \left(a \cdot t\right)\right) - \left(j \cdot 27\right) \cdot k \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(b \cdot c + \left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) - 4 \cdot \left(a \cdot t\right)\right)\right)} - \left(j \cdot 27\right) \cdot k \]

      3. *-commutative N/A

         \[\leadsto \left(b \cdot c + \left(18 \cdot \color{blue}{\left(\left(x \cdot \left(y \cdot z\right)\right) \cdot t\right)} - 4 \cdot \left(a \cdot t\right)\right)\right) - \left(j \cdot 27\right) \cdot k \]

      4. associate-*r* N/A

         \[\leadsto \left(b \cdot c + \left(\color{blue}{\left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot t} - 4 \cdot \left(a \cdot t\right)\right)\right) - \left(j \cdot 27\right) \cdot k \]

      5. associate-*r* N/A

         \[\leadsto \left(b \cdot c + \left(\left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot t - \color{blue}{\left(4 \cdot a\right) \cdot t}\right)\right) - \left(j \cdot 27\right) \cdot k \]

      6. distribute-rgt-out-- N/A

         \[\leadsto \left(b \cdot c + \color{blue}{t \cdot \left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a\right)}\right) - \left(j \cdot 27\right) \cdot k \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot \left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a\right) + b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, 18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a, b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{18 \cdot \left(x \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot a}, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t, 18 \cdot \left(x \cdot \left(y \cdot z\right)\right) + \color{blue}{-4} \cdot a, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{fma}\left(18, x \cdot \left(y \cdot z\right), -4 \cdot a\right)}, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{\left(x \cdot y\right) \cdot z}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{z \cdot \left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{z \cdot \left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \color{blue}{\left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), \color{blue}{a \cdot -4}\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      17. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), \color{blue}{a \cdot -4}\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      18. lower-*.f64 84.8

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), a \cdot -4\right), \color{blue}{b \cdot c}\right) - \left(j \cdot 27\right) \cdot k \]

   5. Simplified 84.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), a \cdot -4\right), b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot c} \]
   7. Step-by-step derivation

      1. lower-*.f64 63.0

         \[\leadsto \color{blue}{b \cdot c} \]

   8. Simplified 63.0%

      \[\leadsto \color{blue}{b \cdot c} \]

   if -1.44999999999999994e97 < (*.f64 b c) < 5.50000000000000022e87

   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 i around 0

      \[\leadsto \color{blue}{\left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 4 \cdot \left(a \cdot t\right)\right)} - \left(j \cdot 27\right) \cdot k \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(b \cdot c + 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right)\right)} - 4 \cdot \left(a \cdot t\right)\right) - \left(j \cdot 27\right) \cdot k \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(b \cdot c + \left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) - 4 \cdot \left(a \cdot t\right)\right)\right)} - \left(j \cdot 27\right) \cdot k \]

      3. *-commutative N/A

         \[\leadsto \left(b \cdot c + \left(18 \cdot \color{blue}{\left(\left(x \cdot \left(y \cdot z\right)\right) \cdot t\right)} - 4 \cdot \left(a \cdot t\right)\right)\right) - \left(j \cdot 27\right) \cdot k \]

      4. associate-*r* N/A

         \[\leadsto \left(b \cdot c + \left(\color{blue}{\left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot t} - 4 \cdot \left(a \cdot t\right)\right)\right) - \left(j \cdot 27\right) \cdot k \]

      5. associate-*r* N/A

         \[\leadsto \left(b \cdot c + \left(\left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot t - \color{blue}{\left(4 \cdot a\right) \cdot t}\right)\right) - \left(j \cdot 27\right) \cdot k \]

      6. distribute-rgt-out-- N/A

         \[\leadsto \left(b \cdot c + \color{blue}{t \cdot \left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a\right)}\right) - \left(j \cdot 27\right) \cdot k \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot \left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a\right) + b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, 18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a, b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{18 \cdot \left(x \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot a}, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t, 18 \cdot \left(x \cdot \left(y \cdot z\right)\right) + \color{blue}{-4} \cdot a, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{fma}\left(18, x \cdot \left(y \cdot z\right), -4 \cdot a\right)}, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{\left(x \cdot y\right) \cdot z}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{z \cdot \left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{z \cdot \left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \color{blue}{\left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), \color{blue}{a \cdot -4}\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      17. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), \color{blue}{a \cdot -4}\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

      18. lower-*.f64 80.8

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), a \cdot -4\right), \color{blue}{b \cdot c}\right) - \left(j \cdot 27\right) \cdot k \]

   5. Simplified 80.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), a \cdot -4\right), b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(\left(b + \frac{t \cdot \left(-4 \cdot a + 18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right)}{c}\right) - 27 \cdot \frac{j \cdot k}{c}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(\left(b + \frac{t \cdot \left(-4 \cdot a + 18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right)}{c}\right) - 27 \cdot \frac{j \cdot k}{c}\right)} \]

      2. associate--l+ N/A

         \[\leadsto c \cdot \color{blue}{\left(b + \left(\frac{t \cdot \left(-4 \cdot a + 18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right)}{c} - 27 \cdot \frac{j \cdot k}{c}\right)\right)} \]

      3. associate-*r/ N/A

         \[\leadsto c \cdot \left(b + \left(\frac{t \cdot \left(-4 \cdot a + 18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right)}{c} - \color{blue}{\frac{27 \cdot \left(j \cdot k\right)}{c}}\right)\right) \]

      4. div-sub N/A

         \[\leadsto c \cdot \left(b + \color{blue}{\frac{t \cdot \left(-4 \cdot a + 18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) - 27 \cdot \left(j \cdot k\right)}{c}}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto c \cdot \color{blue}{\left(b + \frac{t \cdot \left(-4 \cdot a + 18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) - 27 \cdot \left(j \cdot k\right)}{c}\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto c \cdot \left(b + \color{blue}{\frac{t \cdot \left(-4 \cdot a + 18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) - 27 \cdot \left(j \cdot k\right)}{c}}\right) \]

   8. Simplified 67.0%

      \[\leadsto \color{blue}{c \cdot \left(b + \frac{\mathsf{fma}\left(t, \mathsf{fma}\left(x, z \cdot \left(y \cdot 18\right), -4 \cdot a\right), j \cdot \left(k \cdot -27\right)\right)}{c}\right)} \]

   9. Taylor expanded in j around inf

      \[\leadsto \color{blue}{-27 \cdot \left(j \cdot k\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{-27 \cdot \left(j \cdot k\right)} \]

       2. lower-*.f64 31.4

          \[\leadsto -27 \cdot \color{blue}{\left(j \cdot k\right)} \]

   11. Simplified 31.4%

       \[\leadsto \color{blue}{-27 \cdot \left(j \cdot k\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 23.5% accurate, 11.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ b \cdot c \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i j k) :precision binary64 (* b c))

C

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 b * c;
}

Fortran

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 = b * c
end function

Java

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 b * c;
}

Python

[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 b * c

Julia

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(b * c)
end

MATLAB

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 = b * c;
end

Wolfram

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[(b * c), $MachinePrecision]

Derivation

1. Initial program 88.4%

   \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
2. Add Preprocessing

3. Taylor expanded in i around 0

   \[\leadsto \color{blue}{\left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - 4 \cdot \left(a \cdot t\right)\right)} - \left(j \cdot 27\right) \cdot k \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\color{blue}{\left(b \cdot c + 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right)\right)} - 4 \cdot \left(a \cdot t\right)\right) - \left(j \cdot 27\right) \cdot k \]

   2. associate--l+ N/A

      \[\leadsto \color{blue}{\left(b \cdot c + \left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) - 4 \cdot \left(a \cdot t\right)\right)\right)} - \left(j \cdot 27\right) \cdot k \]

   3. *-commutative N/A

      \[\leadsto \left(b \cdot c + \left(18 \cdot \color{blue}{\left(\left(x \cdot \left(y \cdot z\right)\right) \cdot t\right)} - 4 \cdot \left(a \cdot t\right)\right)\right) - \left(j \cdot 27\right) \cdot k \]

   4. associate-*r* N/A

      \[\leadsto \left(b \cdot c + \left(\color{blue}{\left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot t} - 4 \cdot \left(a \cdot t\right)\right)\right) - \left(j \cdot 27\right) \cdot k \]

   5. associate-*r* N/A

      \[\leadsto \left(b \cdot c + \left(\left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot t - \color{blue}{\left(4 \cdot a\right) \cdot t}\right)\right) - \left(j \cdot 27\right) \cdot k \]

   6. distribute-rgt-out-- N/A

      \[\leadsto \left(b \cdot c + \color{blue}{t \cdot \left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a\right)}\right) - \left(j \cdot 27\right) \cdot k \]

   7. +-commutative N/A

      \[\leadsto \color{blue}{\left(t \cdot \left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a\right) + b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, 18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a, b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

   9. cancel-sign-sub-inv N/A

      \[\leadsto \mathsf{fma}\left(t, \color{blue}{18 \cdot \left(x \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot a}, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(t, 18 \cdot \left(x \cdot \left(y \cdot z\right)\right) + \color{blue}{-4} \cdot a, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

   11. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{fma}\left(18, x \cdot \left(y \cdot z\right), -4 \cdot a\right)}, b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{\left(x \cdot y\right) \cdot z}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

   13. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{z \cdot \left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

   14. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, \color{blue}{z \cdot \left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

   15. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \color{blue}{\left(x \cdot y\right)}, -4 \cdot a\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

   16. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), \color{blue}{a \cdot -4}\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

   17. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), \color{blue}{a \cdot -4}\right), b \cdot c\right) - \left(j \cdot 27\right) \cdot k \]

   18. lower-*.f64 82.2

       \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), a \cdot -4\right), \color{blue}{b \cdot c}\right) - \left(j \cdot 27\right) \cdot k \]

5. Simplified 82.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(18, z \cdot \left(x \cdot y\right), a \cdot -4\right), b \cdot c\right)} - \left(j \cdot 27\right) \cdot k \]

6. Taylor expanded in b around inf

   \[\leadsto \color{blue}{b \cdot c} \]
7. Step-by-step derivation

   1. lower-*.f64 24.6

      \[\leadsto \color{blue}{b \cdot c} \]

8. Simplified 24.6%

   \[\leadsto \color{blue}{b \cdot c} \]
9. Add Preprocessing

Developer Target 1: 89.4% 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 2024215 
(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)))