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

Percentage Accurate: 85.3% → 93.2%

Time: 35.0s

Alternatives: 23

Speedup: 1.0×

• 50.1% 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: 85.3% 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: 93.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(j \cdot k, -27, \mathsf{fma}\left(x, i \cdot -4, \mathsf{fma}\left(t, \mathsf{fma}\left(x, 18 \cdot \left(z \cdot y\right), a \cdot -4\right), b \cdot c\right)\right)\right)\\ \mathbf{if}\;t \leq -1.5 \cdot 10^{-17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(-4, \mathsf{fma}\left(i, x, t \cdot a\right), \mathsf{fma}\left(b, c, \mathsf{fma}\left(x \cdot \left(z \cdot \left(18 \cdot t\right)\right), y, k \cdot \left(j \cdot -27\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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
         (fma
          (* j k)
          -27.0
          (fma
           x
           (* i -4.0)
           (fma t (fma x (* 18.0 (* z y)) (* a -4.0)) (* b c))))))
   (if (<= t -1.5e-17)
     t_1
     (if (<= t 3.3e+86)
       (fma
        -4.0
        (fma i x (* t a))
        (fma b c (fma (* x (* z (* 18.0 t))) y (* k (* j -27.0)))))
       t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
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((j * k), -27.0, fma(x, (i * -4.0), fma(t, fma(x, (18.0 * (z * y)), (a * -4.0)), (b * c))));
	double tmp;
	if (t <= -1.5e-17) {
		tmp = t_1;
	} else if (t <= 3.3e+86) {
		tmp = fma(-4.0, fma(i, x, (t * a)), fma(b, c, fma((x * (z * (18.0 * t))), y, (k * (j * -27.0)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = fma(Float64(j * k), -27.0, fma(x, Float64(i * -4.0), fma(t, fma(x, Float64(18.0 * Float64(z * y)), Float64(a * -4.0)), Float64(b * c))))
	tmp = 0.0
	if (t <= -1.5e-17)
		tmp = t_1;
	elseif (t <= 3.3e+86)
		tmp = fma(-4.0, fma(i, x, Float64(t * a)), fma(b, c, fma(Float64(x * Float64(z * Float64(18.0 * t))), y, Float64(k * Float64(j * -27.0)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(N[(j * k), $MachinePrecision] * -27.0 + N[(x * N[(i * -4.0), $MachinePrecision] + N[(t * N[(x * N[(18.0 * N[(z * y), $MachinePrecision]), $MachinePrecision] + N[(a * -4.0), $MachinePrecision]), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.5e-17], t$95$1, If[LessEqual[t, 3.3e+86], N[(-4.0 * N[(i * x + N[(t * a), $MachinePrecision]), $MachinePrecision] + N[(b * c + N[(N[(x * N[(z * N[(18.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + N[(k * N[(j * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.50000000000000003e-17 or 3.2999999999999999e86 < t

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 88.4

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. +-commutative N/A

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

   4. Applied rewrites 93.4%

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

   if -1.50000000000000003e-17 < t < 3.2999999999999999e86

   1. Initial program 86.3%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot c} \]
   4. Step-by-step derivation

      1. lower-*.f64 19.7

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

   5. Applied rewrites 19.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.7%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(j \cdot k, -27, \mathsf{fma}\left(x, i \cdot -4, \mathsf{fma}\left(t, \mathsf{fma}\left(x, 18 \cdot \left(z \cdot y\right), a \cdot -4\right), b \cdot c\right)\right)\right)\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(-4, \mathsf{fma}\left(i, x, t \cdot a\right), \mathsf{fma}\left(b, c, \mathsf{fma}\left(x \cdot \left(z \cdot \left(18 \cdot t\right)\right), y, k \cdot \left(j \cdot -27\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(j \cdot k, -27, \mathsf{fma}\left(x, i \cdot -4, \mathsf{fma}\left(t, \mathsf{fma}\left(x, 18 \cdot \left(z \cdot y\right), a \cdot -4\right), b \cdot c\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 68.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \left(b \cdot c + \left(t \cdot \left(z \cdot \left(y \cdot \left(x \cdot 18\right)\right)\right) - t \cdot \left(a \cdot 4\right)\right)\right) - i \cdot \left(x \cdot 4\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+280}:\\ \;\;\;\;\mathsf{fma}\left(b, c, -4 \cdot \mathsf{fma}\left(i, x, t \cdot a\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+299}:\\ \;\;\;\;\mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, t \cdot a, j \cdot \left(k \cdot -27\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(z \cdot \left(18 \cdot t\right), y, i \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1
         (-
          (+ (* b c) (- (* t (* z (* y (* x 18.0)))) (* t (* a 4.0))))
          (* i (* x 4.0)))))
   (if (<= t_1 -1e+280)
     (fma b c (* -4.0 (fma i x (* t a))))
     (if (<= t_1 2e+299)
       (fma b c (fma -4.0 (* t a) (* j (* k -27.0))))
       (* x (fma (* z (* 18.0 t)) y (* i -4.0)))))))

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 = ((b * c) + ((t * (z * (y * (x * 18.0)))) - (t * (a * 4.0)))) - (i * (x * 4.0));
	double tmp;
	if (t_1 <= -1e+280) {
		tmp = fma(b, c, (-4.0 * fma(i, x, (t * a))));
	} else if (t_1 <= 2e+299) {
		tmp = fma(b, c, fma(-4.0, (t * a), (j * (k * -27.0))));
	} else {
		tmp = x * fma((z * (18.0 * t)), y, (i * -4.0));
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = Float64(Float64(Float64(b * c) + Float64(Float64(t * Float64(z * Float64(y * Float64(x * 18.0)))) - Float64(t * Float64(a * 4.0)))) - Float64(i * Float64(x * 4.0)))
	tmp = 0.0
	if (t_1 <= -1e+280)
		tmp = fma(b, c, Float64(-4.0 * fma(i, x, Float64(t * a))));
	elseif (t_1 <= 2e+299)
		tmp = fma(b, c, fma(-4.0, Float64(t * a), Float64(j * Float64(k * -27.0))));
	else
		tmp = Float64(x * fma(Float64(z * Float64(18.0 * t)), y, Float64(i * -4.0)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(N[(N[(b * c), $MachinePrecision] + N[(N[(t * N[(z * N[(y * N[(x * 18.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(i * N[(x * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+280], N[(b * c + N[(-4.0 * N[(i * x + N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+299], N[(b * c + N[(-4.0 * N[(t * a), $MachinePrecision] + N[(j * N[(k * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(z * N[(18.0 * t), $MachinePrecision]), $MachinePrecision] * y + N[(i * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x #s(literal 18 binary64)) y) z) t) (*.f64 (*.f64 a #s(literal 4 binary64)) t)) (*.f64 b c)) (*.f64 (*.f64 x #s(literal 4 binary64)) i)) < -1e280

   1. Initial program 82.2%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. distribute-neg-in N/A

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

      5. distribute-lft-out N/A

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

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

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 71.7

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

   5. Applied rewrites 71.7%

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

   6. Taylor expanded in j around 0

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

   8. Applied rewrites 68.6%

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

   if -1e280 < (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x #s(literal 18 binary64)) y) z) t) (*.f64 (*.f64 a #s(literal 4 binary64)) t)) (*.f64 b c)) (*.f64 (*.f64 x #s(literal 4 binary64)) i)) < 2.0000000000000001e299

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. lower-fma.f64 N/A

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

      3. distribute-neg-in N/A

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

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

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

      5. metadata-eval N/A

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

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

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 79.5

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

   5. Applied rewrites 79.5%

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

   if 2.0000000000000001e299 < (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x #s(literal 18 binary64)) y) z) t) (*.f64 (*.f64 a #s(literal 4 binary64)) t)) (*.f64 b c)) (*.f64 (*.f64 x #s(literal 4 binary64)) i))

   1. Initial program 71.0%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot c} \]
   4. Step-by-step derivation

      1. lower-*.f64 15.4

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

   5. Applied rewrites 15.4%

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

   6. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 76.8

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

   8. Applied rewrites 76.8%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(b \cdot c + \left(t \cdot \left(z \cdot \left(y \cdot \left(x \cdot 18\right)\right)\right) - t \cdot \left(a \cdot 4\right)\right)\right) - i \cdot \left(x \cdot 4\right) \leq -1 \cdot 10^{+280}:\\ \;\;\;\;\mathsf{fma}\left(b, c, -4 \cdot \mathsf{fma}\left(i, x, t \cdot a\right)\right)\\ \mathbf{elif}\;\left(b \cdot c + \left(t \cdot \left(z \cdot \left(y \cdot \left(x \cdot 18\right)\right)\right) - t \cdot \left(a \cdot 4\right)\right)\right) - i \cdot \left(x \cdot 4\right) \leq 2 \cdot 10^{+299}:\\ \;\;\;\;\mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, t \cdot a, j \cdot \left(k \cdot -27\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(z \cdot \left(18 \cdot t\right), y, i \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := k \cdot \left(j \cdot 27\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+185}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot -27, j, t \cdot \left(a \cdot -4\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+58}:\\ \;\;\;\;\mathsf{fma}\left(b, c, -4 \cdot \mathsf{fma}\left(i, x, t \cdot a\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+244}:\\ \;\;\;\;-4 \cdot \left(x \cdot i\right) - t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot -27, j, b \cdot c\right)\\ \end{array} \end{array} \]

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 (* k (* j 27.0))))
   (if (<= t_1 -1e+185)
     (fma (* k -27.0) j (* t (* a -4.0)))
     (if (<= t_1 2e+58)
       (fma b c (* -4.0 (fma i x (* t a))))
       (if (<= t_1 2e+244)
         (- (* -4.0 (* x i)) t_1)
         (fma (* k -27.0) j (* 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 = k * (j * 27.0);
	double tmp;
	if (t_1 <= -1e+185) {
		tmp = fma((k * -27.0), j, (t * (a * -4.0)));
	} else if (t_1 <= 2e+58) {
		tmp = fma(b, c, (-4.0 * fma(i, x, (t * a))));
	} else if (t_1 <= 2e+244) {
		tmp = (-4.0 * (x * i)) - t_1;
	} else {
		tmp = fma((k * -27.0), j, (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(k * Float64(j * 27.0))
	tmp = 0.0
	if (t_1 <= -1e+185)
		tmp = fma(Float64(k * -27.0), j, Float64(t * Float64(a * -4.0)));
	elseif (t_1 <= 2e+58)
		tmp = fma(b, c, Float64(-4.0 * fma(i, x, Float64(t * a))));
	elseif (t_1 <= 2e+244)
		tmp = Float64(Float64(-4.0 * Float64(x * i)) - t_1);
	else
		tmp = fma(Float64(k * -27.0), j, 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[(k * N[(j * 27.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+185], N[(N[(k * -27.0), $MachinePrecision] * j + N[(t * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+58], N[(b * c + N[(-4.0 * N[(i * x + N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+244], N[(N[(-4.0 * N[(x * i), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(k * -27.0), $MachinePrecision] * j + N[(b * c), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 j #s(literal 27 binary64)) k) < -9.9999999999999998e184

   1. Initial program 83.9%

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

   3. Taylor expanded in 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. lower-*.f64 N/A

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

      2. lower-*.f64 73.3

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

   5. Applied rewrites 73.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lift-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. +-commutative N/A

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

   7. Applied rewrites 75.7%

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

   if -9.9999999999999998e184 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 1.99999999999999989e58

   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 y around 0

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

      1. sub-neg N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. distribute-neg-in N/A

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

      5. distribute-lft-out N/A

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

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

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 81.8

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

   5. Applied rewrites 81.8%

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

   6. Taylor expanded in j around 0

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

   8. Applied rewrites 79.1%

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

   if 1.99999999999999989e58 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 2.00000000000000015e244

   1. Initial program 88.3%

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

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

      2. *-commutative N/A

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

      3. lower-*.f64 64.8

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

   5. Applied rewrites 64.8%

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

   if 2.00000000000000015e244 < (*.f64 (*.f64 j #s(literal 27 binary64)) k)

   1. Initial program 84.2%

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

   3. Taylor expanded in b around inf

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

      1. lower-*.f64 84.2

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

   5. Applied rewrites 84.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 88.1

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

   7. Applied rewrites 88.1%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \cdot \left(j \cdot 27\right) \leq -1 \cdot 10^{+185}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot -27, j, t \cdot \left(a \cdot -4\right)\right)\\ \mathbf{elif}\;k \cdot \left(j \cdot 27\right) \leq 2 \cdot 10^{+58}:\\ \;\;\;\;\mathsf{fma}\left(b, c, -4 \cdot \mathsf{fma}\left(i, x, t \cdot a\right)\right)\\ \mathbf{elif}\;k \cdot \left(j \cdot 27\right) \leq 2 \cdot 10^{+244}:\\ \;\;\;\;-4 \cdot \left(x \cdot i\right) - k \cdot \left(j \cdot 27\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot -27, j, b \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := k \cdot \left(j \cdot 27\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+185}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot -27, j, t \cdot \left(a \cdot -4\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+58}:\\ \;\;\;\;\mathsf{fma}\left(b, c, -4 \cdot \mathsf{fma}\left(i, x, t \cdot a\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+244}:\\ \;\;\;\;\mathsf{fma}\left(j \cdot k, -27, -4 \cdot \left(x \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot -27, j, b \cdot c\right)\\ \end{array} \end{array} \]

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 (* k (* j 27.0))))
   (if (<= t_1 -1e+185)
     (fma (* k -27.0) j (* t (* a -4.0)))
     (if (<= t_1 2e+58)
       (fma b c (* -4.0 (fma i x (* t a))))
       (if (<= t_1 2e+244)
         (fma (* j k) -27.0 (* -4.0 (* x i)))
         (fma (* k -27.0) j (* 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 = k * (j * 27.0);
	double tmp;
	if (t_1 <= -1e+185) {
		tmp = fma((k * -27.0), j, (t * (a * -4.0)));
	} else if (t_1 <= 2e+58) {
		tmp = fma(b, c, (-4.0 * fma(i, x, (t * a))));
	} else if (t_1 <= 2e+244) {
		tmp = fma((j * k), -27.0, (-4.0 * (x * i)));
	} else {
		tmp = fma((k * -27.0), j, (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(k * Float64(j * 27.0))
	tmp = 0.0
	if (t_1 <= -1e+185)
		tmp = fma(Float64(k * -27.0), j, Float64(t * Float64(a * -4.0)));
	elseif (t_1 <= 2e+58)
		tmp = fma(b, c, Float64(-4.0 * fma(i, x, Float64(t * a))));
	elseif (t_1 <= 2e+244)
		tmp = fma(Float64(j * k), -27.0, Float64(-4.0 * Float64(x * i)));
	else
		tmp = fma(Float64(k * -27.0), j, 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[(k * N[(j * 27.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+185], N[(N[(k * -27.0), $MachinePrecision] * j + N[(t * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+58], N[(b * c + N[(-4.0 * N[(i * x + N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+244], N[(N[(j * k), $MachinePrecision] * -27.0 + N[(-4.0 * N[(x * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(k * -27.0), $MachinePrecision] * j + N[(b * c), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 j #s(literal 27 binary64)) k) < -9.9999999999999998e184

   1. Initial program 83.9%

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

   3. Taylor expanded in 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. lower-*.f64 N/A

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

      2. lower-*.f64 73.3

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

   5. Applied rewrites 73.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lift-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. +-commutative N/A

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

   7. Applied rewrites 75.7%

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

   if -9.9999999999999998e184 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 1.99999999999999989e58

   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 y around 0

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

      1. sub-neg N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. distribute-neg-in N/A

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

      5. distribute-lft-out N/A

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

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

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 81.8

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

   5. Applied rewrites 81.8%

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

   6. Taylor expanded in j around 0

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

   8. Applied rewrites 79.1%

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

   if 1.99999999999999989e58 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 2.00000000000000015e244

   1. Initial program 88.3%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   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 33.7

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

   5. Applied rewrites 33.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 33.5

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

   7. Applied rewrites 33.5%

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

   8. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 64.7

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

   10. Applied rewrites 64.7%

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

   if 2.00000000000000015e244 < (*.f64 (*.f64 j #s(literal 27 binary64)) k)

   1. Initial program 84.2%

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

   3. Taylor expanded in b around inf

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

      1. lower-*.f64 84.2

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

   5. Applied rewrites 84.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 88.1

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

   7. Applied rewrites 88.1%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \cdot \left(j \cdot 27\right) \leq -1 \cdot 10^{+185}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot -27, j, t \cdot \left(a \cdot -4\right)\right)\\ \mathbf{elif}\;k \cdot \left(j \cdot 27\right) \leq 2 \cdot 10^{+58}:\\ \;\;\;\;\mathsf{fma}\left(b, c, -4 \cdot \mathsf{fma}\left(i, x, t \cdot a\right)\right)\\ \mathbf{elif}\;k \cdot \left(j \cdot 27\right) \leq 2 \cdot 10^{+244}:\\ \;\;\;\;\mathsf{fma}\left(j \cdot k, -27, -4 \cdot \left(x \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot -27, j, b \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := -4 \cdot \left(x \cdot i\right)\\ t_2 := k \cdot \left(j \cdot 27\right)\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+165}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot -27, j, t \cdot \left(a \cdot -4\right)\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+53}:\\ \;\;\;\;\mathsf{fma}\left(b, c, t\_1\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+244}:\\ \;\;\;\;\mathsf{fma}\left(j \cdot k, -27, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot -27, j, b \cdot c\right)\\ \end{array} \end{array} \]

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 (* -4.0 (* x i))) (t_2 (* k (* j 27.0))))
   (if (<= t_2 -4e+165)
     (fma (* k -27.0) j (* t (* a -4.0)))
     (if (<= t_2 1e+53)
       (fma b c t_1)
       (if (<= t_2 2e+244)
         (fma (* j k) -27.0 t_1)
         (fma (* k -27.0) j (* 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 = -4.0 * (x * i);
	double t_2 = k * (j * 27.0);
	double tmp;
	if (t_2 <= -4e+165) {
		tmp = fma((k * -27.0), j, (t * (a * -4.0)));
	} else if (t_2 <= 1e+53) {
		tmp = fma(b, c, t_1);
	} else if (t_2 <= 2e+244) {
		tmp = fma((j * k), -27.0, t_1);
	} else {
		tmp = fma((k * -27.0), j, (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(-4.0 * Float64(x * i))
	t_2 = Float64(k * Float64(j * 27.0))
	tmp = 0.0
	if (t_2 <= -4e+165)
		tmp = fma(Float64(k * -27.0), j, Float64(t * Float64(a * -4.0)));
	elseif (t_2 <= 1e+53)
		tmp = fma(b, c, t_1);
	elseif (t_2 <= 2e+244)
		tmp = fma(Float64(j * k), -27.0, t_1);
	else
		tmp = fma(Float64(k * -27.0), j, 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[(-4.0 * N[(x * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(k * N[(j * 27.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+165], N[(N[(k * -27.0), $MachinePrecision] * j + N[(t * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+53], N[(b * c + t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 2e+244], N[(N[(j * k), $MachinePrecision] * -27.0 + t$95$1), $MachinePrecision], N[(N[(k * -27.0), $MachinePrecision] * j + N[(b * c), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 j #s(literal 27 binary64)) k) < -3.9999999999999996e165

   1. Initial program 84.2%

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

   3. Taylor expanded in 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. lower-*.f64 N/A

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

      2. lower-*.f64 73.9

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

   5. Applied rewrites 73.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lift-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. +-commutative N/A

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

   7. Applied rewrites 76.2%

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

   if -3.9999999999999996e165 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 9.9999999999999999e52

   1. Initial program 88.3%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. distribute-neg-in N/A

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

      5. distribute-lft-out N/A

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

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

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 81.5

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

   5. Applied rewrites 81.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 58.7%

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

   if 9.9999999999999999e52 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 2.00000000000000015e244

   1. Initial program 88.9%

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

   3. Taylor expanded in b around inf

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

      1. lower-*.f64 31.9

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

   5. Applied rewrites 31.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 31.7

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

   7. Applied rewrites 31.7%

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

   8. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 61.2

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

   10. Applied rewrites 61.2%

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

   if 2.00000000000000015e244 < (*.f64 (*.f64 j #s(literal 27 binary64)) k)

   1. Initial program 84.2%

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

   3. Taylor expanded in b around inf

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

      1. lower-*.f64 84.2

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

   5. Applied rewrites 84.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 88.1

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

   7. Applied rewrites 88.1%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \cdot \left(j \cdot 27\right) \leq -4 \cdot 10^{+165}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot -27, j, t \cdot \left(a \cdot -4\right)\right)\\ \mathbf{elif}\;k \cdot \left(j \cdot 27\right) \leq 10^{+53}:\\ \;\;\;\;\mathsf{fma}\left(b, c, -4 \cdot \left(x \cdot i\right)\right)\\ \mathbf{elif}\;k \cdot \left(j \cdot 27\right) \leq 2 \cdot 10^{+244}:\\ \;\;\;\;\mathsf{fma}\left(j \cdot k, -27, -4 \cdot \left(x \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot -27, j, b \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(k \cdot -27, j, b \cdot c\right)\\ t_2 := k \cdot \left(j \cdot 27\right)\\ t_3 := -4 \cdot \left(x \cdot i\right)\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+151}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+53}:\\ \;\;\;\;\mathsf{fma}\left(b, c, t\_3\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+244}:\\ \;\;\;\;\mathsf{fma}\left(j \cdot k, -27, t\_3\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* k -27.0) j (* b c)))
        (t_2 (* k (* j 27.0)))
        (t_3 (* -4.0 (* x i))))
   (if (<= t_2 -5e+151)
     t_1
     (if (<= t_2 1e+53)
       (fma b c t_3)
       (if (<= t_2 2e+244) (fma (* j k) -27.0 t_3) 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((k * -27.0), j, (b * c));
	double t_2 = k * (j * 27.0);
	double t_3 = -4.0 * (x * i);
	double tmp;
	if (t_2 <= -5e+151) {
		tmp = t_1;
	} else if (t_2 <= 1e+53) {
		tmp = fma(b, c, t_3);
	} else if (t_2 <= 2e+244) {
		tmp = fma((j * k), -27.0, t_3);
	} 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(Float64(k * -27.0), j, Float64(b * c))
	t_2 = Float64(k * Float64(j * 27.0))
	t_3 = Float64(-4.0 * Float64(x * i))
	tmp = 0.0
	if (t_2 <= -5e+151)
		tmp = t_1;
	elseif (t_2 <= 1e+53)
		tmp = fma(b, c, t_3);
	elseif (t_2 <= 2e+244)
		tmp = fma(Float64(j * k), -27.0, t_3);
	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[(k * -27.0), $MachinePrecision] * j + N[(b * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(k * N[(j * 27.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-4.0 * N[(x * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+151], t$95$1, If[LessEqual[t$95$2, 1e+53], N[(b * c + t$95$3), $MachinePrecision], If[LessEqual[t$95$2, 2e+244], N[(N[(j * k), $MachinePrecision] * -27.0 + t$95$3), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 j #s(literal 27 binary64)) k) < -5.0000000000000002e151 or 2.00000000000000015e244 < (*.f64 (*.f64 j #s(literal 27 binary64)) k)

   1. Initial program 84.7%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   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 74.4

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

   5. Applied rewrites 74.4%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 75.8

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

   7. Applied rewrites 75.8%

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

   if -5.0000000000000002e151 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 9.9999999999999999e52

   1. Initial program 88.1%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. distribute-neg-in N/A

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

      5. distribute-lft-out N/A

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

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

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 81.8

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

   5. Applied rewrites 81.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 59.0%

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

   if 9.9999999999999999e52 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 2.00000000000000015e244

   1. Initial program 88.9%

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

   3. Taylor expanded in b around inf

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

      1. lower-*.f64 31.9

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

   5. Applied rewrites 31.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 31.7

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

   7. Applied rewrites 31.7%

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

   8. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 61.2

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

   10. Applied rewrites 61.2%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \cdot \left(j \cdot 27\right) \leq -5 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot -27, j, b \cdot c\right)\\ \mathbf{elif}\;k \cdot \left(j \cdot 27\right) \leq 10^{+53}:\\ \;\;\;\;\mathsf{fma}\left(b, c, -4 \cdot \left(x \cdot i\right)\right)\\ \mathbf{elif}\;k \cdot \left(j \cdot 27\right) \leq 2 \cdot 10^{+244}:\\ \;\;\;\;\mathsf{fma}\left(j \cdot k, -27, -4 \cdot \left(x \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot -27, j, b \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 4 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(-4, \mathsf{fma}\left(i, x, t \cdot a\right), \mathsf{fma}\left(b, c, \mathsf{fma}\left(x \cdot \left(z \cdot \left(18 \cdot t\right)\right), y, k \cdot \left(j \cdot -27\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot \left(x \cdot \left(18 \cdot y\right)\right), z, \mathsf{fma}\left(t, a \cdot -4, \mathsf{fma}\left(b, c, x \cdot \left(i \cdot -4\right)\right)\right)\right) - k \cdot \left(j \cdot 27\right)\\ \end{array} \end{array} \]

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
 (if (<= z 4e+30)
   (fma
    -4.0
    (fma i x (* t a))
    (fma b c (fma (* x (* z (* 18.0 t))) y (* k (* j -27.0)))))
   (-
    (fma
     (* t (* x (* 18.0 y)))
     z
     (fma t (* a -4.0) (fma b c (* x (* i -4.0)))))
    (* k (* j 27.0)))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k);
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 (z <= 4e+30) {
		tmp = fma(-4.0, fma(i, x, (t * a)), fma(b, c, fma((x * (z * (18.0 * t))), y, (k * (j * -27.0)))));
	} else {
		tmp = fma((t * (x * (18.0 * y))), z, fma(t, (a * -4.0), fma(b, c, (x * (i * -4.0))))) - (k * (j * 27.0));
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
function code(x, y, z, t, a, b, c, i, j, k)
	tmp = 0.0
	if (z <= 4e+30)
		tmp = fma(-4.0, fma(i, x, Float64(t * a)), fma(b, c, fma(Float64(x * Float64(z * Float64(18.0 * t))), y, Float64(k * Float64(j * -27.0)))));
	else
		tmp = Float64(fma(Float64(t * Float64(x * Float64(18.0 * y))), z, fma(t, Float64(a * -4.0), fma(b, c, Float64(x * Float64(i * -4.0))))) - Float64(k * Float64(j * 27.0)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := If[LessEqual[z, 4e+30], N[(-4.0 * N[(i * x + N[(t * a), $MachinePrecision]), $MachinePrecision] + N[(b * c + N[(N[(x * N[(z * N[(18.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + N[(k * N[(j * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t * N[(x * N[(18.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z + N[(t * N[(a * -4.0), $MachinePrecision] + N[(b * c + N[(x * N[(i * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(k * N[(j * 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.0000000000000001e30

   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 b around inf

      \[\leadsto \color{blue}{b \cdot c} \]
   4. Step-by-step derivation

      1. lower-*.f64 21.2

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

   5. Applied rewrites 21.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 92.4%

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

   if 4.0000000000000001e30 < z

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 93.5%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(-4, \mathsf{fma}\left(i, x, t \cdot a\right), \mathsf{fma}\left(b, c, \mathsf{fma}\left(x \cdot \left(z \cdot \left(18 \cdot t\right)\right), y, k \cdot \left(j \cdot -27\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot \left(x \cdot \left(18 \cdot y\right)\right), z, \mathsf{fma}\left(t, a \cdot -4, \mathsf{fma}\left(b, c, x \cdot \left(i \cdot -4\right)\right)\right)\right) - k \cdot \left(j \cdot 27\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := k \cdot \left(j \cdot -27\right)\\ t_2 := \mathsf{fma}\left(t, \mathsf{fma}\left(x, 18 \cdot \left(z \cdot y\right), a \cdot -4\right), \mathsf{fma}\left(b, c, t\_1\right)\right)\\ \mathbf{if}\;t \leq -5.9 \cdot 10^{+93}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+166}:\\ \;\;\;\;\mathsf{fma}\left(-4, \mathsf{fma}\left(i, x, t \cdot a\right), \mathsf{fma}\left(b, c, \mathsf{fma}\left(x \cdot \left(z \cdot \left(18 \cdot t\right)\right), y, t\_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 (* k (* j -27.0)))
        (t_2 (fma t (fma x (* 18.0 (* z y)) (* a -4.0)) (fma b c t_1))))
   (if (<= t -5.9e+93)
     t_2
     (if (<= t 2.5e+166)
       (fma
        -4.0
        (fma i x (* t a))
        (fma b c (fma (* x (* z (* 18.0 t))) y 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 = k * (j * -27.0);
	double t_2 = fma(t, fma(x, (18.0 * (z * y)), (a * -4.0)), fma(b, c, t_1));
	double tmp;
	if (t <= -5.9e+93) {
		tmp = t_2;
	} else if (t <= 2.5e+166) {
		tmp = fma(-4.0, fma(i, x, (t * a)), fma(b, c, fma((x * (z * (18.0 * t))), y, 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(k * Float64(j * -27.0))
	t_2 = fma(t, fma(x, Float64(18.0 * Float64(z * y)), Float64(a * -4.0)), fma(b, c, t_1))
	tmp = 0.0
	if (t <= -5.9e+93)
		tmp = t_2;
	elseif (t <= 2.5e+166)
		tmp = fma(-4.0, fma(i, x, Float64(t * a)), fma(b, c, fma(Float64(x * Float64(z * Float64(18.0 * t))), y, 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[(k * N[(j * -27.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(x * N[(18.0 * N[(z * y), $MachinePrecision]), $MachinePrecision] + N[(a * -4.0), $MachinePrecision]), $MachinePrecision] + N[(b * c + t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5.9e+93], t$95$2, If[LessEqual[t, 2.5e+166], N[(-4.0 * N[(i * x + N[(t * a), $MachinePrecision]), $MachinePrecision] + N[(b * c + N[(N[(x * N[(z * N[(18.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if t < -5.90000000000000008e93 or 2.5000000000000001e166 < t

   1. Initial program 88.2%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot c} \]
   4. Step-by-step derivation

      1. lower-*.f64 19.2

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

   5. Applied rewrites 19.2%

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

   6. Taylor expanded in i around 0

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

      1. associate--l+ N/A

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

      2. associate--r+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-rgt-in N/A

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

      13. lower-fma.f64 N/A

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

   8. Applied rewrites 89.6%

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

   if -5.90000000000000008e93 < t < 2.5000000000000001e166

   1. Initial program 86.9%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot c} \]
   4. Step-by-step derivation

      1. lower-*.f64 23.0

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

   5. Applied rewrites 23.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 95.1%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.9 \cdot 10^{+93}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(x, 18 \cdot \left(z \cdot y\right), a \cdot -4\right), \mathsf{fma}\left(b, c, k \cdot \left(j \cdot -27\right)\right)\right)\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+166}:\\ \;\;\;\;\mathsf{fma}\left(-4, \mathsf{fma}\left(i, x, t \cdot a\right), \mathsf{fma}\left(b, c, \mathsf{fma}\left(x \cdot \left(z \cdot \left(18 \cdot t\right)\right), y, k \cdot \left(j \cdot -27\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(x, 18 \cdot \left(z \cdot y\right), a \cdot -4\right), \mathsf{fma}\left(b, c, k \cdot \left(j \cdot -27\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := k \cdot \left(j \cdot 27\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(j, k \cdot -27, \mathsf{fma}\left(a, t \cdot -4, b \cdot c\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+58}:\\ \;\;\;\;\mathsf{fma}\left(b, c, -4 \cdot \mathsf{fma}\left(i, x, t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, x \cdot i, j \cdot \left(k \cdot -27\right)\right)\right)\\ \end{array} \end{array} \]

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 (* k (* j 27.0))))
   (if (<= t_1 -5e+151)
     (fma j (* k -27.0) (fma a (* t -4.0) (* b c)))
     (if (<= t_1 2e+58)
       (fma b c (* -4.0 (fma i x (* t a))))
       (fma b c (fma -4.0 (* x i) (* 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 = k * (j * 27.0);
	double tmp;
	if (t_1 <= -5e+151) {
		tmp = fma(j, (k * -27.0), fma(a, (t * -4.0), (b * c)));
	} else if (t_1 <= 2e+58) {
		tmp = fma(b, c, (-4.0 * fma(i, x, (t * a))));
	} else {
		tmp = fma(b, c, fma(-4.0, (x * i), (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(k * Float64(j * 27.0))
	tmp = 0.0
	if (t_1 <= -5e+151)
		tmp = fma(j, Float64(k * -27.0), fma(a, Float64(t * -4.0), Float64(b * c)));
	elseif (t_1 <= 2e+58)
		tmp = fma(b, c, Float64(-4.0 * fma(i, x, Float64(t * a))));
	else
		tmp = fma(b, c, fma(-4.0, Float64(x * i), 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[(k * N[(j * 27.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+151], N[(j * N[(k * -27.0), $MachinePrecision] + N[(a * N[(t * -4.0), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+58], N[(b * c + N[(-4.0 * N[(i * x + N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * c + N[(-4.0 * N[(x * i), $MachinePrecision] + N[(j * N[(k * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 j #s(literal 27 binary64)) k) < -5.0000000000000002e151

   1. Initial program 84.9%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot c} \]
   4. Step-by-step derivation

      1. lower-*.f64 14.3

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

   5. Applied rewrites 14.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 89.2%

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

   9. Taylor expanded in x around 0

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

       1. associate--r+ N/A

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

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

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

       3. metadata-eval N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

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

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

       12. metadata-eval N/A

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

       13. +-commutative N/A

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

       14. *-commutative N/A

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

       15. associate-*r* N/A

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

       16. *-commutative N/A

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

       17. lower-fma.f64 N/A

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

       18. *-commutative N/A

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

       19. lower-*.f64 N/A

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

       20. lower-*.f64 76.0

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

   11. Applied rewrites 76.0%

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

   if -5.0000000000000002e151 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 1.99999999999999989e58

   1. Initial program 88.3%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. distribute-neg-in N/A

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

      5. distribute-lft-out N/A

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

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

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 82.1

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

   5. Applied rewrites 82.1%

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

   6. Taylor expanded in j around 0

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

   8. Applied rewrites 79.6%

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

   if 1.99999999999999989e58 < (*.f64 (*.f64 j #s(literal 27 binary64)) k)

   1. Initial program 86.5%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. lower-fma.f64 N/A

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

      3. distribute-neg-in N/A

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

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

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

      5. metadata-eval N/A

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

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

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 74.1

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

   5. Applied rewrites 74.1%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \cdot \left(j \cdot 27\right) \leq -5 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(j, k \cdot -27, \mathsf{fma}\left(a, t \cdot -4, b \cdot c\right)\right)\\ \mathbf{elif}\;k \cdot \left(j \cdot 27\right) \leq 2 \cdot 10^{+58}:\\ \;\;\;\;\mathsf{fma}\left(b, c, -4 \cdot \mathsf{fma}\left(i, x, t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, x \cdot i, j \cdot \left(k \cdot -27\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := k \cdot \left(j \cdot 27\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+185}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot -27, j, t \cdot \left(a \cdot -4\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+58}:\\ \;\;\;\;\mathsf{fma}\left(b, c, -4 \cdot \mathsf{fma}\left(i, x, t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, x \cdot i, j \cdot \left(k \cdot -27\right)\right)\right)\\ \end{array} \end{array} \]

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 (* k (* j 27.0))))
   (if (<= t_1 -1e+185)
     (fma (* k -27.0) j (* t (* a -4.0)))
     (if (<= t_1 2e+58)
       (fma b c (* -4.0 (fma i x (* t a))))
       (fma b c (fma -4.0 (* x i) (* 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 = k * (j * 27.0);
	double tmp;
	if (t_1 <= -1e+185) {
		tmp = fma((k * -27.0), j, (t * (a * -4.0)));
	} else if (t_1 <= 2e+58) {
		tmp = fma(b, c, (-4.0 * fma(i, x, (t * a))));
	} else {
		tmp = fma(b, c, fma(-4.0, (x * i), (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(k * Float64(j * 27.0))
	tmp = 0.0
	if (t_1 <= -1e+185)
		tmp = fma(Float64(k * -27.0), j, Float64(t * Float64(a * -4.0)));
	elseif (t_1 <= 2e+58)
		tmp = fma(b, c, Float64(-4.0 * fma(i, x, Float64(t * a))));
	else
		tmp = fma(b, c, fma(-4.0, Float64(x * i), 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[(k * N[(j * 27.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+185], N[(N[(k * -27.0), $MachinePrecision] * j + N[(t * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+58], N[(b * c + N[(-4.0 * N[(i * x + N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * c + N[(-4.0 * N[(x * i), $MachinePrecision] + N[(j * N[(k * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 j #s(literal 27 binary64)) k) < -9.9999999999999998e184

   1. Initial program 83.9%

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

   3. Taylor expanded in 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. lower-*.f64 N/A

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

      2. lower-*.f64 73.3

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

   5. Applied rewrites 73.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lift-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. +-commutative N/A

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

   7. Applied rewrites 75.7%

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

   if -9.9999999999999998e184 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 1.99999999999999989e58

   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 y around 0

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

      1. sub-neg N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. distribute-neg-in N/A

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

      5. distribute-lft-out N/A

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

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

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 81.8

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

   5. Applied rewrites 81.8%

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

   6. Taylor expanded in j around 0

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

   8. Applied rewrites 79.1%

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

   if 1.99999999999999989e58 < (*.f64 (*.f64 j #s(literal 27 binary64)) k)

   1. Initial program 86.5%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. lower-fma.f64 N/A

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

      3. distribute-neg-in N/A

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

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

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

      5. metadata-eval N/A

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

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

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 74.1

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

   5. Applied rewrites 74.1%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \cdot \left(j \cdot 27\right) \leq -1 \cdot 10^{+185}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot -27, j, t \cdot \left(a \cdot -4\right)\right)\\ \mathbf{elif}\;k \cdot \left(j \cdot 27\right) \leq 2 \cdot 10^{+58}:\\ \;\;\;\;\mathsf{fma}\left(b, c, -4 \cdot \mathsf{fma}\left(i, x, t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, x \cdot i, j \cdot \left(k \cdot -27\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 48.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := k \cdot \left(j \cdot 27\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+226}:\\ \;\;\;\;j \cdot \left(k \cdot -27\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(b, c, -4 \cdot \left(t \cdot a\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+244}:\\ \;\;\;\;-4 \cdot \left(x \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(j \cdot -27\right)\\ \end{array} \end{array} \]

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 (* k (* j 27.0))))
   (if (<= t_1 -2e+226)
     (* j (* k -27.0))
     (if (<= t_1 1e-37)
       (fma b c (* -4.0 (* t a)))
       (if (<= t_1 2e+244) (* -4.0 (* x i)) (* k (* j -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 = k * (j * 27.0);
	double tmp;
	if (t_1 <= -2e+226) {
		tmp = j * (k * -27.0);
	} else if (t_1 <= 1e-37) {
		tmp = fma(b, c, (-4.0 * (t * a)));
	} else if (t_1 <= 2e+244) {
		tmp = -4.0 * (x * i);
	} else {
		tmp = k * (j * -27.0);
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = Float64(k * Float64(j * 27.0))
	tmp = 0.0
	if (t_1 <= -2e+226)
		tmp = Float64(j * Float64(k * -27.0));
	elseif (t_1 <= 1e-37)
		tmp = fma(b, c, Float64(-4.0 * Float64(t * a)));
	elseif (t_1 <= 2e+244)
		tmp = Float64(-4.0 * Float64(x * i));
	else
		tmp = Float64(k * Float64(j * -27.0));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(k * N[(j * 27.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+226], N[(j * N[(k * -27.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-37], N[(b * c + N[(-4.0 * N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+244], N[(-4.0 * N[(x * i), $MachinePrecision]), $MachinePrecision], N[(k * N[(j * -27.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 j #s(literal 27 binary64)) k) < -1.99999999999999992e226

   1. Initial program 81.3%

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

   3. Taylor expanded in j around inf

      \[\leadsto \color{blue}{-27 \cdot \left(j \cdot k\right)} \]
   4. 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 74.3

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

   5. Applied rewrites 74.3%

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

   if -1.99999999999999992e226 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 1.00000000000000007e-37

   1. Initial program 89.4%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. distribute-neg-in N/A

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

      5. distribute-lft-out N/A

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

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

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 81.2

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

   5. Applied rewrites 81.2%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 51.3%

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

   if 1.00000000000000007e-37 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 2.00000000000000015e244

   1. Initial program 86.8%

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

   3. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 42.5

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

   5. Applied rewrites 42.5%

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

   if 2.00000000000000015e244 < (*.f64 (*.f64 j #s(literal 27 binary64)) k)

   1. Initial program 84.2%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot c} \]
   4. Step-by-step derivation

      1. lower-*.f64 14.7

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

   5. Applied rewrites 14.7%

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

   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. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 75.7

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

   8. Applied rewrites 75.7%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \cdot \left(j \cdot 27\right) \leq -2 \cdot 10^{+226}:\\ \;\;\;\;j \cdot \left(k \cdot -27\right)\\ \mathbf{elif}\;k \cdot \left(j \cdot 27\right) \leq 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(b, c, -4 \cdot \left(t \cdot a\right)\right)\\ \mathbf{elif}\;k \cdot \left(j \cdot 27\right) \leq 2 \cdot 10^{+244}:\\ \;\;\;\;-4 \cdot \left(x \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(j \cdot -27\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 85.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, \mathsf{fma}\left(x, 18 \cdot \left(z \cdot y\right), a \cdot -4\right), \mathsf{fma}\left(b, c, k \cdot \left(j \cdot -27\right)\right)\right)\\ \mathbf{if}\;t \leq -8.6 \cdot 10^{+118}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-36}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot -27, j, \mathsf{fma}\left(-4, \mathsf{fma}\left(t, a, x \cdot i\right), b \cdot c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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
          (fma x (* 18.0 (* z y)) (* a -4.0))
          (fma b c (* k (* j -27.0))))))
   (if (<= t -8.6e+118)
     t_1
     (if (<= t 1.5e-36)
       (fma (* k -27.0) j (fma -4.0 (fma t a (* x i)) (* b c)))
       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, fma(x, (18.0 * (z * y)), (a * -4.0)), fma(b, c, (k * (j * -27.0))));
	double tmp;
	if (t <= -8.6e+118) {
		tmp = t_1;
	} else if (t <= 1.5e-36) {
		tmp = fma((k * -27.0), j, fma(-4.0, fma(t, a, (x * i)), (b * c)));
	} 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, fma(x, Float64(18.0 * Float64(z * y)), Float64(a * -4.0)), fma(b, c, Float64(k * Float64(j * -27.0))))
	tmp = 0.0
	if (t <= -8.6e+118)
		tmp = t_1;
	elseif (t <= 1.5e-36)
		tmp = fma(Float64(k * -27.0), j, fma(-4.0, fma(t, a, Float64(x * i)), Float64(b * c)));
	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[(x * N[(18.0 * N[(z * y), $MachinePrecision]), $MachinePrecision] + N[(a * -4.0), $MachinePrecision]), $MachinePrecision] + N[(b * c + N[(k * N[(j * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -8.6e+118], t$95$1, If[LessEqual[t, 1.5e-36], N[(N[(k * -27.0), $MachinePrecision] * j + N[(-4.0 * N[(t * a + N[(x * i), $MachinePrecision]), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -8.6000000000000006e118 or 1.5000000000000001e-36 < t

   1. Initial program 90.5%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot c} \]
   4. Step-by-step derivation

      1. lower-*.f64 21.7

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

   5. Applied rewrites 21.7%

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

   6. Taylor expanded in i around 0

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

      1. associate--l+ N/A

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

      2. associate--r+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-rgt-in N/A

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

      13. lower-fma.f64 N/A

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

   8. Applied rewrites 89.9%

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

   if -8.6000000000000006e118 < t < 1.5000000000000001e-36

   1. Initial program 84.6%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. distribute-neg-in N/A

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

      5. distribute-lft-out N/A

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

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

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 89.3

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

   5. Applied rewrites 89.3%

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

   7. Applied rewrites 90.0%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.6 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(x, 18 \cdot \left(z \cdot y\right), a \cdot -4\right), \mathsf{fma}\left(b, c, k \cdot \left(j \cdot -27\right)\right)\right)\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-36}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot -27, j, \mathsf{fma}\left(-4, \mathsf{fma}\left(t, a, x \cdot i\right), b \cdot c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(x, 18 \cdot \left(z \cdot y\right), a \cdot -4\right), \mathsf{fma}\left(b, c, k \cdot \left(j \cdot -27\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 78.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+165}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(z \cdot \left(18 \cdot t\right), y, i \cdot -4\right)\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot -27, j, \mathsf{fma}\left(-4, \mathsf{fma}\left(t, a, x \cdot i\right), b \cdot c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot \left(x \cdot \left(18 \cdot y\right)\right), z, b \cdot c\right) - k \cdot \left(j \cdot 27\right)\\ \end{array} \end{array} \]

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
 (if (<= y -7e+165)
   (* x (fma (* z (* 18.0 t)) y (* i -4.0)))
   (if (<= y 7.4e+39)
     (fma (* k -27.0) j (fma -4.0 (fma t a (* x i)) (* b c)))
     (- (fma (* t (* x (* 18.0 y))) z (* b c)) (* k (* j 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 tmp;
	if (y <= -7e+165) {
		tmp = x * fma((z * (18.0 * t)), y, (i * -4.0));
	} else if (y <= 7.4e+39) {
		tmp = fma((k * -27.0), j, fma(-4.0, fma(t, a, (x * i)), (b * c)));
	} else {
		tmp = fma((t * (x * (18.0 * y))), z, (b * c)) - (k * (j * 27.0));
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
function code(x, y, z, t, a, b, c, i, j, k)
	tmp = 0.0
	if (y <= -7e+165)
		tmp = Float64(x * fma(Float64(z * Float64(18.0 * t)), y, Float64(i * -4.0)));
	elseif (y <= 7.4e+39)
		tmp = fma(Float64(k * -27.0), j, fma(-4.0, fma(t, a, Float64(x * i)), Float64(b * c)));
	else
		tmp = Float64(fma(Float64(t * Float64(x * Float64(18.0 * y))), z, Float64(b * c)) - Float64(k * Float64(j * 27.0)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := If[LessEqual[y, -7e+165], N[(x * N[(N[(z * N[(18.0 * t), $MachinePrecision]), $MachinePrecision] * y + N[(i * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.4e+39], N[(N[(k * -27.0), $MachinePrecision] * j + N[(-4.0 * N[(t * a + N[(x * i), $MachinePrecision]), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t * N[(x * N[(18.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z + N[(b * c), $MachinePrecision]), $MachinePrecision] - N[(k * N[(j * 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.99999999999999991e165

   1. Initial program 80.3%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot c} \]
   4. Step-by-step derivation

      1. lower-*.f64 14.5

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

   5. Applied rewrites 14.5%

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

   6. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 69.0

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

   8. Applied rewrites 69.0%

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

   if -6.99999999999999991e165 < y < 7.40000000000000025e39

   1. Initial program 91.7%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. distribute-neg-in N/A

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

      5. distribute-lft-out N/A

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

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

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 87.6

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

   5. Applied rewrites 87.6%

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

   7. Applied rewrites 88.2%

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

   if 7.40000000000000025e39 < y

   1. Initial program 78.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 78.6%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 68.5

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

   7. Applied rewrites 68.5%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+165}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(z \cdot \left(18 \cdot t\right), y, i \cdot -4\right)\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot -27, j, \mathsf{fma}\left(-4, \mathsf{fma}\left(t, a, x \cdot i\right), b \cdot c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot \left(x \cdot \left(18 \cdot y\right)\right), z, b \cdot c\right) - k \cdot \left(j \cdot 27\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 54.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(k \cdot -27, j, b \cdot c\right)\\ t_2 := k \cdot \left(j \cdot 27\right)\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+151}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+249}:\\ \;\;\;\;\mathsf{fma}\left(b, c, -4 \cdot \left(x \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* k -27.0) j (* b c))) (t_2 (* k (* j 27.0))))
   (if (<= t_2 -5e+151)
     t_1
     (if (<= t_2 5e+249) (fma b c (* -4.0 (* x i))) 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((k * -27.0), j, (b * c));
	double t_2 = k * (j * 27.0);
	double tmp;
	if (t_2 <= -5e+151) {
		tmp = t_1;
	} else if (t_2 <= 5e+249) {
		tmp = fma(b, c, (-4.0 * (x * i)));
	} 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(Float64(k * -27.0), j, Float64(b * c))
	t_2 = Float64(k * Float64(j * 27.0))
	tmp = 0.0
	if (t_2 <= -5e+151)
		tmp = t_1;
	elseif (t_2 <= 5e+249)
		tmp = fma(b, c, Float64(-4.0 * Float64(x * i)));
	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[(k * -27.0), $MachinePrecision] * j + N[(b * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(k * N[(j * 27.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+151], t$95$1, If[LessEqual[t$95$2, 5e+249], N[(b * c + N[(-4.0 * N[(x * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 j #s(literal 27 binary64)) k) < -5.0000000000000002e151 or 4.9999999999999996e249 < (*.f64 (*.f64 j #s(literal 27 binary64)) k)

   1. Initial program 84.2%

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

   3. Taylor expanded in b around inf

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

      1. lower-*.f64 75.2

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

   5. Applied rewrites 75.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 76.6

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

   7. Applied rewrites 76.6%

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

   if -5.0000000000000002e151 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 4.9999999999999996e249

   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 y around 0

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

      1. sub-neg N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. distribute-neg-in N/A

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

      5. distribute-lft-out N/A

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

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

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 81.0

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

   5. Applied rewrites 81.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 55.7%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \cdot \left(j \cdot 27\right) \leq -5 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot -27, j, b \cdot c\right)\\ \mathbf{elif}\;k \cdot \left(j \cdot 27\right) \leq 5 \cdot 10^{+249}:\\ \;\;\;\;\mathsf{fma}\left(b, c, -4 \cdot \left(x \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot -27, j, b \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 53.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(j \cdot k, -27, b \cdot c\right)\\ t_2 := k \cdot \left(j \cdot 27\right)\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+151}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+249}:\\ \;\;\;\;\mathsf{fma}\left(b, c, -4 \cdot \left(x \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* j k) -27.0 (* b c))) (t_2 (* k (* j 27.0))))
   (if (<= t_2 -5e+151)
     t_1
     (if (<= t_2 5e+249) (fma b c (* -4.0 (* x i))) 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((j * k), -27.0, (b * c));
	double t_2 = k * (j * 27.0);
	double tmp;
	if (t_2 <= -5e+151) {
		tmp = t_1;
	} else if (t_2 <= 5e+249) {
		tmp = fma(b, c, (-4.0 * (x * i)));
	} 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(Float64(j * k), -27.0, Float64(b * c))
	t_2 = Float64(k * Float64(j * 27.0))
	tmp = 0.0
	if (t_2 <= -5e+151)
		tmp = t_1;
	elseif (t_2 <= 5e+249)
		tmp = fma(b, c, Float64(-4.0 * Float64(x * i)));
	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[(j * k), $MachinePrecision] * -27.0 + N[(b * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(k * N[(j * 27.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+151], t$95$1, If[LessEqual[t$95$2, 5e+249], N[(b * c + N[(-4.0 * N[(x * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 j #s(literal 27 binary64)) k) < -5.0000000000000002e151 or 4.9999999999999996e249 < (*.f64 (*.f64 j #s(literal 27 binary64)) k)

   1. Initial program 84.2%

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

   3. Taylor expanded in b around inf

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

      1. lower-*.f64 75.2

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

   5. Applied rewrites 75.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 75.1

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

   7. Applied rewrites 75.1%

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

   if -5.0000000000000002e151 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 4.9999999999999996e249

   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 y around 0

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

      1. sub-neg N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. distribute-neg-in N/A

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

      5. distribute-lft-out N/A

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

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

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

      7. metadata-eval N/A

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

      8. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{fma}\left(b, c, -4 \cdot \left(a \cdot t + i \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot \left(j \cdot k\right)}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(b, c, -4 \cdot \left(a \cdot t + i \cdot x\right) + \color{blue}{-27} \cdot \left(j \cdot k\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, c, \color{blue}{\mathsf{fma}\left(-4, a \cdot t + i \cdot x, -27 \cdot \left(j \cdot k\right)\right)}\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \color{blue}{\mathsf{fma}\left(a, t, i \cdot x\right)}, -27 \cdot \left(j \cdot k\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \mathsf{fma}\left(a, t, \color{blue}{x \cdot i}\right), -27 \cdot \left(j \cdot k\right)\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \mathsf{fma}\left(a, t, \color{blue}{x \cdot i}\right), -27 \cdot \left(j \cdot k\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \mathsf{fma}\left(a, t, x \cdot i\right), \color{blue}{\left(j \cdot k\right) \cdot -27}\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \mathsf{fma}\left(a, t, x \cdot i\right), \color{blue}{j \cdot \left(k \cdot -27\right)}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \mathsf{fma}\left(a, t, x \cdot i\right), j \cdot \color{blue}{\left(-27 \cdot k\right)}\right)\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \mathsf{fma}\left(a, t, x \cdot i\right), \color{blue}{j \cdot \left(-27 \cdot k\right)}\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \mathsf{fma}\left(a, t, x \cdot i\right), j \cdot \color{blue}{\left(k \cdot -27\right)}\right)\right) \]

      19. lower-*.f64 81.0

          \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \mathsf{fma}\left(a, t, x \cdot i\right), j \cdot \color{blue}{\left(k \cdot -27\right)}\right)\right) \]

   5. Applied rewrites 81.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \mathsf{fma}\left(a, t, x \cdot i\right), j \cdot \left(k \cdot -27\right)\right)\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(b, c, -4 \cdot \left(i \cdot x\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 55.7%

      \[\leadsto \mathsf{fma}\left(b, c, -4 \cdot \left(i \cdot x\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \cdot \left(j \cdot 27\right) \leq -5 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(j \cdot k, -27, b \cdot c\right)\\ \mathbf{elif}\;k \cdot \left(j \cdot 27\right) \leq 5 \cdot 10^{+249}:\\ \;\;\;\;\mathsf{fma}\left(b, c, -4 \cdot \left(x \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(j \cdot k, -27, b \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 52.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := k \cdot \left(j \cdot 27\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+185}:\\ \;\;\;\;j \cdot \left(k \cdot -27\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+249}:\\ \;\;\;\;\mathsf{fma}\left(b, c, -4 \cdot \left(x \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(j \cdot -27\right)\\ \end{array} \end{array} \]

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 (* k (* j 27.0))))
   (if (<= t_1 -1e+185)
     (* j (* k -27.0))
     (if (<= t_1 5e+249) (fma b c (* -4.0 (* x i))) (* k (* j -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 = k * (j * 27.0);
	double tmp;
	if (t_1 <= -1e+185) {
		tmp = j * (k * -27.0);
	} else if (t_1 <= 5e+249) {
		tmp = fma(b, c, (-4.0 * (x * i)));
	} else {
		tmp = k * (j * -27.0);
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = Float64(k * Float64(j * 27.0))
	tmp = 0.0
	if (t_1 <= -1e+185)
		tmp = Float64(j * Float64(k * -27.0));
	elseif (t_1 <= 5e+249)
		tmp = fma(b, c, Float64(-4.0 * Float64(x * i)));
	else
		tmp = Float64(k * Float64(j * -27.0));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(k * N[(j * 27.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+185], N[(j * N[(k * -27.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+249], N[(b * c + N[(-4.0 * N[(x * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(k * N[(j * -27.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 j #s(literal 27 binary64)) k) < -9.9999999999999998e184

   1. Initial program 83.9%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

      \[\leadsto \color{blue}{-27 \cdot \left(j \cdot k\right)} \]
   4. 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 68.2

         \[\leadsto j \cdot \color{blue}{\left(k \cdot -27\right)} \]

   5. Applied rewrites 68.2%

      \[\leadsto \color{blue}{j \cdot \left(k \cdot -27\right)} \]

   if -9.9999999999999998e184 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 4.9999999999999996e249

   1. Initial program 88.6%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{b \cdot c - \left(4 \cdot \left(a \cdot t\right) + \left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{b \cdot c + \left(\mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + \left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(b, c, \mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + \left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)\right)} \]

      3. associate-+r+ N/A

         \[\leadsto \mathsf{fma}\left(b, c, \mathsf{neg}\left(\color{blue}{\left(\left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right) + 27 \cdot \left(j \cdot k\right)\right)}\right)\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(b, c, \color{blue}{\left(\mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right)\right) + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)}\right) \]

      5. distribute-lft-out N/A

         \[\leadsto \mathsf{fma}\left(b, c, \left(\mathsf{neg}\left(\color{blue}{4 \cdot \left(a \cdot t + i \cdot x\right)}\right)\right) + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{fma}\left(b, c, \color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot t + i \cdot x\right)} + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(b, c, \color{blue}{-4} \cdot \left(a \cdot t + i \cdot x\right) + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{fma}\left(b, c, -4 \cdot \left(a \cdot t + i \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot \left(j \cdot k\right)}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(b, c, -4 \cdot \left(a \cdot t + i \cdot x\right) + \color{blue}{-27} \cdot \left(j \cdot k\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, c, \color{blue}{\mathsf{fma}\left(-4, a \cdot t + i \cdot x, -27 \cdot \left(j \cdot k\right)\right)}\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \color{blue}{\mathsf{fma}\left(a, t, i \cdot x\right)}, -27 \cdot \left(j \cdot k\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \mathsf{fma}\left(a, t, \color{blue}{x \cdot i}\right), -27 \cdot \left(j \cdot k\right)\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \mathsf{fma}\left(a, t, \color{blue}{x \cdot i}\right), -27 \cdot \left(j \cdot k\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \mathsf{fma}\left(a, t, x \cdot i\right), \color{blue}{\left(j \cdot k\right) \cdot -27}\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \mathsf{fma}\left(a, t, x \cdot i\right), \color{blue}{j \cdot \left(k \cdot -27\right)}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \mathsf{fma}\left(a, t, x \cdot i\right), j \cdot \color{blue}{\left(-27 \cdot k\right)}\right)\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \mathsf{fma}\left(a, t, x \cdot i\right), \color{blue}{j \cdot \left(-27 \cdot k\right)}\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \mathsf{fma}\left(a, t, x \cdot i\right), j \cdot \color{blue}{\left(k \cdot -27\right)}\right)\right) \]

      19. lower-*.f64 80.8

          \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \mathsf{fma}\left(a, t, x \cdot i\right), j \cdot \color{blue}{\left(k \cdot -27\right)}\right)\right) \]

   5. Applied rewrites 80.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \mathsf{fma}\left(a, t, x \cdot i\right), j \cdot \left(k \cdot -27\right)\right)\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(b, c, -4 \cdot \left(i \cdot x\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 55.2%

      \[\leadsto \mathsf{fma}\left(b, c, -4 \cdot \left(i \cdot x\right)\right) \]

   if 4.9999999999999996e249 < (*.f64 (*.f64 j #s(literal 27 binary64)) k)

   1. Initial program 82.8%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot c} \]
   4. Step-by-step derivation

      1. lower-*.f64 11.4

         \[\leadsto \color{blue}{b \cdot c} \]

   5. Applied rewrites 11.4%

      \[\leadsto \color{blue}{b \cdot c} \]

   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. *-commutative N/A

         \[\leadsto \color{blue}{\left(k \cdot j\right)} \cdot -27 \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{k \cdot \left(j \cdot -27\right)} \]

      4. *-commutative N/A

         \[\leadsto k \cdot \color{blue}{\left(-27 \cdot j\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{k \cdot \left(-27 \cdot j\right)} \]

      6. *-commutative N/A

         \[\leadsto k \cdot \color{blue}{\left(j \cdot -27\right)} \]

      7. lower-*.f64 81.9

         \[\leadsto k \cdot \color{blue}{\left(j \cdot -27\right)} \]

   8. Applied rewrites 81.9%

      \[\leadsto \color{blue}{k \cdot \left(j \cdot -27\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \cdot \left(j \cdot 27\right) \leq -1 \cdot 10^{+185}:\\ \;\;\;\;j \cdot \left(k \cdot -27\right)\\ \mathbf{elif}\;k \cdot \left(j \cdot 27\right) \leq 5 \cdot 10^{+249}:\\ \;\;\;\;\mathsf{fma}\left(b, c, -4 \cdot \left(x \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(j \cdot -27\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 75.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(z \cdot \left(18 \cdot t\right), y, i \cdot -4\right)\\ \mathbf{if}\;y \leq -7 \cdot 10^{+165}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot -27, j, \mathsf{fma}\left(-4, \mathsf{fma}\left(t, a, x \cdot i\right), b \cdot c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* z (* 18.0 t)) y (* i -4.0)))))
   (if (<= y -7e+165)
     t_1
     (if (<= y 2.8e+52)
       (fma (* k -27.0) j (fma -4.0 (fma t a (* x i)) (* b c)))
       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((z * (18.0 * t)), y, (i * -4.0));
	double tmp;
	if (y <= -7e+165) {
		tmp = t_1;
	} else if (y <= 2.8e+52) {
		tmp = fma((k * -27.0), j, fma(-4.0, fma(t, a, (x * i)), (b * c)));
	} 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(Float64(z * Float64(18.0 * t)), y, Float64(i * -4.0)))
	tmp = 0.0
	if (y <= -7e+165)
		tmp = t_1;
	elseif (y <= 2.8e+52)
		tmp = fma(Float64(k * -27.0), j, fma(-4.0, fma(t, a, Float64(x * i)), Float64(b * c)));
	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[(N[(z * N[(18.0 * t), $MachinePrecision]), $MachinePrecision] * y + N[(i * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7e+165], t$95$1, If[LessEqual[y, 2.8e+52], N[(N[(k * -27.0), $MachinePrecision] * j + N[(-4.0 * N[(t * a + N[(x * i), $MachinePrecision]), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.99999999999999991e165 or 2.8e52 < y

   1. Initial program 80.2%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot c} \]
   4. Step-by-step derivation

      1. lower-*.f64 12.5

         \[\leadsto \color{blue}{b \cdot c} \]

   5. Applied rewrites 12.5%

      \[\leadsto \color{blue}{b \cdot c} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right)} \]
   7. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto x \cdot \color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot i\right)} \]

      2. metadata-eval N/A

         \[\leadsto x \cdot \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + \color{blue}{-4} \cdot i\right) \]

      3. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-4 \cdot i + 18 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(-4 \cdot i + 18 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + -4 \cdot i\right)} \]

      6. associate-*r* N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(18 \cdot t\right) \cdot \left(y \cdot z\right)} + -4 \cdot i\right) \]

      7. *-commutative N/A

         \[\leadsto x \cdot \left(\left(18 \cdot t\right) \cdot \color{blue}{\left(z \cdot y\right)} + -4 \cdot i\right) \]

      8. associate-*r* N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(\left(18 \cdot t\right) \cdot z\right) \cdot y} + -4 \cdot i\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(\left(18 \cdot t\right) \cdot z, y, -4 \cdot i\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{\left(18 \cdot t\right) \cdot z}, y, -4 \cdot i\right) \]

      11. lower-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{\left(18 \cdot t\right)} \cdot z, y, -4 \cdot i\right) \]

      12. lower-*.f64 64.3

          \[\leadsto x \cdot \mathsf{fma}\left(\left(18 \cdot t\right) \cdot z, y, \color{blue}{-4 \cdot i}\right) \]

   8. Applied rewrites 64.3%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(\left(18 \cdot t\right) \cdot z, y, -4 \cdot i\right)} \]

   if -6.99999999999999991e165 < y < 2.8e52

   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 y around 0

      \[\leadsto \color{blue}{b \cdot c - \left(4 \cdot \left(a \cdot t\right) + \left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{b \cdot c + \left(\mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + \left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(b, c, \mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + \left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)\right)} \]

      3. associate-+r+ N/A

         \[\leadsto \mathsf{fma}\left(b, c, \mathsf{neg}\left(\color{blue}{\left(\left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right) + 27 \cdot \left(j \cdot k\right)\right)}\right)\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(b, c, \color{blue}{\left(\mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right)\right) + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)}\right) \]

      5. distribute-lft-out N/A

         \[\leadsto \mathsf{fma}\left(b, c, \left(\mathsf{neg}\left(\color{blue}{4 \cdot \left(a \cdot t + i \cdot x\right)}\right)\right) + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{fma}\left(b, c, \color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot t + i \cdot x\right)} + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(b, c, \color{blue}{-4} \cdot \left(a \cdot t + i \cdot x\right) + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{fma}\left(b, c, -4 \cdot \left(a \cdot t + i \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot \left(j \cdot k\right)}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(b, c, -4 \cdot \left(a \cdot t + i \cdot x\right) + \color{blue}{-27} \cdot \left(j \cdot k\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, c, \color{blue}{\mathsf{fma}\left(-4, a \cdot t + i \cdot x, -27 \cdot \left(j \cdot k\right)\right)}\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \color{blue}{\mathsf{fma}\left(a, t, i \cdot x\right)}, -27 \cdot \left(j \cdot k\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \mathsf{fma}\left(a, t, \color{blue}{x \cdot i}\right), -27 \cdot \left(j \cdot k\right)\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \mathsf{fma}\left(a, t, \color{blue}{x \cdot i}\right), -27 \cdot \left(j \cdot k\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \mathsf{fma}\left(a, t, x \cdot i\right), \color{blue}{\left(j \cdot k\right) \cdot -27}\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \mathsf{fma}\left(a, t, x \cdot i\right), \color{blue}{j \cdot \left(k \cdot -27\right)}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \mathsf{fma}\left(a, t, x \cdot i\right), j \cdot \color{blue}{\left(-27 \cdot k\right)}\right)\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \mathsf{fma}\left(a, t, x \cdot i\right), \color{blue}{j \cdot \left(-27 \cdot k\right)}\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \mathsf{fma}\left(a, t, x \cdot i\right), j \cdot \color{blue}{\left(k \cdot -27\right)}\right)\right) \]

      19. lower-*.f64 87.8

          \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \mathsf{fma}\left(a, t, x \cdot i\right), j \cdot \color{blue}{\left(k \cdot -27\right)}\right)\right) \]

   5. Applied rewrites 87.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \mathsf{fma}\left(a, t, x \cdot i\right), j \cdot \left(k \cdot -27\right)\right)\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 88.4%

      \[\leadsto \mathsf{fma}\left(k \cdot -27, \color{blue}{j}, \mathsf{fma}\left(-4, \mathsf{fma}\left(t, a, x \cdot i\right), b \cdot c\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+165}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(z \cdot \left(18 \cdot t\right), y, i \cdot -4\right)\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot -27, j, \mathsf{fma}\left(-4, \mathsf{fma}\left(t, a, x \cdot i\right), b \cdot c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(z \cdot \left(18 \cdot t\right), y, i \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 75.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(z \cdot \left(18 \cdot t\right), y, i \cdot -4\right)\\ \mathbf{if}\;y \leq -7 \cdot 10^{+165}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \mathsf{fma}\left(a, t, x \cdot i\right), j \cdot \left(k \cdot -27\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* z (* 18.0 t)) y (* i -4.0)))))
   (if (<= y -7e+165)
     t_1
     (if (<= y 2.8e+52)
       (fma b c (fma -4.0 (fma a t (* x i)) (* 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((z * (18.0 * t)), y, (i * -4.0));
	double tmp;
	if (y <= -7e+165) {
		tmp = t_1;
	} else if (y <= 2.8e+52) {
		tmp = fma(b, c, fma(-4.0, fma(a, t, (x * i)), (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(Float64(z * Float64(18.0 * t)), y, Float64(i * -4.0)))
	tmp = 0.0
	if (y <= -7e+165)
		tmp = t_1;
	elseif (y <= 2.8e+52)
		tmp = fma(b, c, fma(-4.0, fma(a, t, Float64(x * i)), 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[(N[(z * N[(18.0 * t), $MachinePrecision]), $MachinePrecision] * y + N[(i * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7e+165], t$95$1, If[LessEqual[y, 2.8e+52], N[(b * c + N[(-4.0 * N[(a * t + N[(x * i), $MachinePrecision]), $MachinePrecision] + N[(j * N[(k * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.99999999999999991e165 or 2.8e52 < y

   1. Initial program 80.2%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot c} \]
   4. Step-by-step derivation

      1. lower-*.f64 12.5

         \[\leadsto \color{blue}{b \cdot c} \]

   5. Applied rewrites 12.5%

      \[\leadsto \color{blue}{b \cdot c} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right)} \]
   7. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto x \cdot \color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot i\right)} \]

      2. metadata-eval N/A

         \[\leadsto x \cdot \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + \color{blue}{-4} \cdot i\right) \]

      3. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-4 \cdot i + 18 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(-4 \cdot i + 18 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + -4 \cdot i\right)} \]

      6. associate-*r* N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(18 \cdot t\right) \cdot \left(y \cdot z\right)} + -4 \cdot i\right) \]

      7. *-commutative N/A

         \[\leadsto x \cdot \left(\left(18 \cdot t\right) \cdot \color{blue}{\left(z \cdot y\right)} + -4 \cdot i\right) \]

      8. associate-*r* N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(\left(18 \cdot t\right) \cdot z\right) \cdot y} + -4 \cdot i\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(\left(18 \cdot t\right) \cdot z, y, -4 \cdot i\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{\left(18 \cdot t\right) \cdot z}, y, -4 \cdot i\right) \]

      11. lower-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{\left(18 \cdot t\right)} \cdot z, y, -4 \cdot i\right) \]

      12. lower-*.f64 64.3

          \[\leadsto x \cdot \mathsf{fma}\left(\left(18 \cdot t\right) \cdot z, y, \color{blue}{-4 \cdot i}\right) \]

   8. Applied rewrites 64.3%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(\left(18 \cdot t\right) \cdot z, y, -4 \cdot i\right)} \]

   if -6.99999999999999991e165 < y < 2.8e52

   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 y around 0

      \[\leadsto \color{blue}{b \cdot c - \left(4 \cdot \left(a \cdot t\right) + \left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{b \cdot c + \left(\mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + \left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(b, c, \mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + \left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right)\right)} \]

      3. associate-+r+ N/A

         \[\leadsto \mathsf{fma}\left(b, c, \mathsf{neg}\left(\color{blue}{\left(\left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right) + 27 \cdot \left(j \cdot k\right)\right)}\right)\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(b, c, \color{blue}{\left(\mathsf{neg}\left(\left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right)\right) + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)}\right) \]

      5. distribute-lft-out N/A

         \[\leadsto \mathsf{fma}\left(b, c, \left(\mathsf{neg}\left(\color{blue}{4 \cdot \left(a \cdot t + i \cdot x\right)}\right)\right) + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{fma}\left(b, c, \color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot t + i \cdot x\right)} + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(b, c, \color{blue}{-4} \cdot \left(a \cdot t + i \cdot x\right) + \left(\mathsf{neg}\left(27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{fma}\left(b, c, -4 \cdot \left(a \cdot t + i \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(27\right)\right) \cdot \left(j \cdot k\right)}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(b, c, -4 \cdot \left(a \cdot t + i \cdot x\right) + \color{blue}{-27} \cdot \left(j \cdot k\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, c, \color{blue}{\mathsf{fma}\left(-4, a \cdot t + i \cdot x, -27 \cdot \left(j \cdot k\right)\right)}\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \color{blue}{\mathsf{fma}\left(a, t, i \cdot x\right)}, -27 \cdot \left(j \cdot k\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \mathsf{fma}\left(a, t, \color{blue}{x \cdot i}\right), -27 \cdot \left(j \cdot k\right)\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \mathsf{fma}\left(a, t, \color{blue}{x \cdot i}\right), -27 \cdot \left(j \cdot k\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \mathsf{fma}\left(a, t, x \cdot i\right), \color{blue}{\left(j \cdot k\right) \cdot -27}\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \mathsf{fma}\left(a, t, x \cdot i\right), \color{blue}{j \cdot \left(k \cdot -27\right)}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \mathsf{fma}\left(a, t, x \cdot i\right), j \cdot \color{blue}{\left(-27 \cdot k\right)}\right)\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \mathsf{fma}\left(a, t, x \cdot i\right), \color{blue}{j \cdot \left(-27 \cdot k\right)}\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \mathsf{fma}\left(a, t, x \cdot i\right), j \cdot \color{blue}{\left(k \cdot -27\right)}\right)\right) \]

      19. lower-*.f64 87.8

          \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \mathsf{fma}\left(a, t, x \cdot i\right), j \cdot \color{blue}{\left(k \cdot -27\right)}\right)\right) \]

   5. Applied rewrites 87.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \mathsf{fma}\left(a, t, x \cdot i\right), j \cdot \left(k \cdot -27\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+165}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(z \cdot \left(18 \cdot t\right), y, i \cdot -4\right)\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(b, c, \mathsf{fma}\left(-4, \mathsf{fma}\left(a, t, x \cdot i\right), j \cdot \left(k \cdot -27\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(z \cdot \left(18 \cdot t\right), y, i \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 34.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := k \cdot \left(j \cdot 27\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+165}:\\ \;\;\;\;j \cdot \left(k \cdot -27\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+244}:\\ \;\;\;\;-4 \cdot \left(x \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(j \cdot -27\right)\\ \end{array} \end{array} \]

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 (* k (* j 27.0))))
   (if (<= t_1 -4e+165)
     (* j (* k -27.0))
     (if (<= t_1 2e+244) (* -4.0 (* x i)) (* k (* j -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 = k * (j * 27.0);
	double tmp;
	if (t_1 <= -4e+165) {
		tmp = j * (k * -27.0);
	} else if (t_1 <= 2e+244) {
		tmp = -4.0 * (x * i);
	} else {
		tmp = k * (j * -27.0);
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = k * (j * 27.0d0)
    if (t_1 <= (-4d+165)) then
        tmp = j * (k * (-27.0d0))
    else if (t_1 <= 2d+244) then
        tmp = (-4.0d0) * (x * i)
    else
        tmp = k * (j * (-27.0d0))
    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 t_1 = k * (j * 27.0);
	double tmp;
	if (t_1 <= -4e+165) {
		tmp = j * (k * -27.0);
	} else if (t_1 <= 2e+244) {
		tmp = -4.0 * (x * i);
	} else {
		tmp = k * (j * -27.0);
	}
	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):
	t_1 = k * (j * 27.0)
	tmp = 0
	if t_1 <= -4e+165:
		tmp = j * (k * -27.0)
	elif t_1 <= 2e+244:
		tmp = -4.0 * (x * i)
	else:
		tmp = k * (j * -27.0)
	return tmp

Julia

x, y, z, t, a, b, c, i, j, k = sort([x, y, z, t, a, b, c, i, j, k])
function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = Float64(k * Float64(j * 27.0))
	tmp = 0.0
	if (t_1 <= -4e+165)
		tmp = Float64(j * Float64(k * -27.0));
	elseif (t_1 <= 2e+244)
		tmp = Float64(-4.0 * Float64(x * i));
	else
		tmp = Float64(k * Float64(j * -27.0));
	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)
	t_1 = k * (j * 27.0);
	tmp = 0.0;
	if (t_1 <= -4e+165)
		tmp = j * (k * -27.0);
	elseif (t_1 <= 2e+244)
		tmp = -4.0 * (x * i);
	else
		tmp = k * (j * -27.0);
	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_] := Block[{t$95$1 = N[(k * N[(j * 27.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+165], N[(j * N[(k * -27.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+244], N[(-4.0 * N[(x * i), $MachinePrecision]), $MachinePrecision], N[(k * N[(j * -27.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 j #s(literal 27 binary64)) k) < -3.9999999999999996e165

   1. Initial program 84.2%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

      \[\leadsto \color{blue}{-27 \cdot \left(j \cdot k\right)} \]
   4. 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 66.8

         \[\leadsto j \cdot \color{blue}{\left(k \cdot -27\right)} \]

   5. Applied rewrites 66.8%

      \[\leadsto \color{blue}{j \cdot \left(k \cdot -27\right)} \]

   if -3.9999999999999996e165 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 2.00000000000000015e244

   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 inf

      \[\leadsto \color{blue}{-4 \cdot \left(i \cdot x\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{-4 \cdot \left(i \cdot x\right)} \]

      2. *-commutative N/A

         \[\leadsto -4 \cdot \color{blue}{\left(x \cdot i\right)} \]

      3. lower-*.f64 33.9

         \[\leadsto -4 \cdot \color{blue}{\left(x \cdot i\right)} \]

   5. Applied rewrites 33.9%

      \[\leadsto \color{blue}{-4 \cdot \left(x \cdot i\right)} \]

   if 2.00000000000000015e244 < (*.f64 (*.f64 j #s(literal 27 binary64)) k)

   1. Initial program 84.2%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot c} \]
   4. Step-by-step derivation

      1. lower-*.f64 14.7

         \[\leadsto \color{blue}{b \cdot c} \]

   5. Applied rewrites 14.7%

      \[\leadsto \color{blue}{b \cdot c} \]

   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. *-commutative N/A

         \[\leadsto \color{blue}{\left(k \cdot j\right)} \cdot -27 \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{k \cdot \left(j \cdot -27\right)} \]

      4. *-commutative N/A

         \[\leadsto k \cdot \color{blue}{\left(-27 \cdot j\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{k \cdot \left(-27 \cdot j\right)} \]

      6. *-commutative N/A

         \[\leadsto k \cdot \color{blue}{\left(j \cdot -27\right)} \]

      7. lower-*.f64 75.7

         \[\leadsto k \cdot \color{blue}{\left(j \cdot -27\right)} \]

   8. Applied rewrites 75.7%

      \[\leadsto \color{blue}{k \cdot \left(j \cdot -27\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \cdot \left(j \cdot 27\right) \leq -4 \cdot 10^{+165}:\\ \;\;\;\;j \cdot \left(k \cdot -27\right)\\ \mathbf{elif}\;k \cdot \left(j \cdot 27\right) \leq 2 \cdot 10^{+244}:\\ \;\;\;\;-4 \cdot \left(x \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(j \cdot -27\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 34.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := j \cdot \left(k \cdot -27\right)\\ t_2 := k \cdot \left(j \cdot 27\right)\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+165}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+244}:\\ \;\;\;\;-4 \cdot \left(x \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* j (* k -27.0))) (t_2 (* k (* j 27.0))))
   (if (<= t_2 -4e+165) t_1 (if (<= t_2 2e+244) (* -4.0 (* x i)) 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 = j * (k * -27.0);
	double t_2 = k * (j * 27.0);
	double tmp;
	if (t_2 <= -4e+165) {
		tmp = t_1;
	} else if (t_2 <= 2e+244) {
		tmp = -4.0 * (x * i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
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 = j * (k * (-27.0d0))
    t_2 = k * (j * 27.0d0)
    if (t_2 <= (-4d+165)) then
        tmp = t_1
    else if (t_2 <= 2d+244) then
        tmp = (-4.0d0) * (x * i)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k;
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 = j * (k * -27.0);
	double t_2 = k * (j * 27.0);
	double tmp;
	if (t_2 <= -4e+165) {
		tmp = t_1;
	} else if (t_2 <= 2e+244) {
		tmp = -4.0 * (x * i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a, b, c, i, j, k] = sort([x, y, z, t, a, b, c, i, j, k])
def code(x, y, z, t, a, b, c, i, j, k):
	t_1 = j * (k * -27.0)
	t_2 = k * (j * 27.0)
	tmp = 0
	if t_2 <= -4e+165:
		tmp = t_1
	elif t_2 <= 2e+244:
		tmp = -4.0 * (x * i)
	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(j * Float64(k * -27.0))
	t_2 = Float64(k * Float64(j * 27.0))
	tmp = 0.0
	if (t_2 <= -4e+165)
		tmp = t_1;
	elseif (t_2 <= 2e+244)
		tmp = Float64(-4.0 * Float64(x * i));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t, a, b, c, i, j, k = num2cell(sort([x, y, z, t, a, b, c, i, j, k])){:}
function tmp_2 = code(x, y, z, t, a, b, c, i, j, k)
	t_1 = j * (k * -27.0);
	t_2 = k * (j * 27.0);
	tmp = 0.0;
	if (t_2 <= -4e+165)
		tmp = t_1;
	elseif (t_2 <= 2e+244)
		tmp = -4.0 * (x * i);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(j * N[(k * -27.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(k * N[(j * 27.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+165], t$95$1, If[LessEqual[t$95$2, 2e+244], N[(-4.0 * N[(x * i), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 j #s(literal 27 binary64)) k) < -3.9999999999999996e165 or 2.00000000000000015e244 < (*.f64 (*.f64 j #s(literal 27 binary64)) k)

   1. Initial program 84.2%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

      \[\leadsto \color{blue}{-27 \cdot \left(j \cdot k\right)} \]
   4. 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 70.0

         \[\leadsto j \cdot \color{blue}{\left(k \cdot -27\right)} \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{j \cdot \left(k \cdot -27\right)} \]

   if -3.9999999999999996e165 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 2.00000000000000015e244

   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 inf

      \[\leadsto \color{blue}{-4 \cdot \left(i \cdot x\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{-4 \cdot \left(i \cdot x\right)} \]

      2. *-commutative N/A

         \[\leadsto -4 \cdot \color{blue}{\left(x \cdot i\right)} \]

      3. lower-*.f64 33.9

         \[\leadsto -4 \cdot \color{blue}{\left(x \cdot i\right)} \]

   5. Applied rewrites 33.9%

      \[\leadsto \color{blue}{-4 \cdot \left(x \cdot i\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \cdot \left(j \cdot 27\right) \leq -4 \cdot 10^{+165}:\\ \;\;\;\;j \cdot \left(k \cdot -27\right)\\ \mathbf{elif}\;k \cdot \left(j \cdot 27\right) \leq 2 \cdot 10^{+244}:\\ \;\;\;\;-4 \cdot \left(x \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(k \cdot -27\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 35.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} \mathbf{if}\;b \cdot c \leq -5 \cdot 10^{+62}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;b \cdot c \leq 4 \cdot 10^{+225}:\\ \;\;\;\;-4 \cdot \left(t \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot c\\ \end{array} \end{array} \]

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) -5e+62)
   (* b c)
   (if (<= (* b c) 4e+225) (* -4.0 (* t a)) (* 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) <= -5e+62) {
		tmp = b * c;
	} else if ((b * c) <= 4e+225) {
		tmp = -4.0 * (t * a);
	} 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) <= (-5d+62)) then
        tmp = b * c
    else if ((b * c) <= 4d+225) then
        tmp = (-4.0d0) * (t * a)
    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) <= -5e+62) {
		tmp = b * c;
	} else if ((b * c) <= 4e+225) {
		tmp = -4.0 * (t * a);
	} 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) <= -5e+62:
		tmp = b * c
	elif (b * c) <= 4e+225:
		tmp = -4.0 * (t * a)
	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) <= -5e+62)
		tmp = Float64(b * c);
	elseif (Float64(b * c) <= 4e+225)
		tmp = Float64(-4.0 * Float64(t * a));
	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) <= -5e+62)
		tmp = b * c;
	elseif ((b * c) <= 4e+225)
		tmp = -4.0 * (t * a);
	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], -5e+62], N[(b * c), $MachinePrecision], If[LessEqual[N[(b * c), $MachinePrecision], 4e+225], N[(-4.0 * N[(t * a), $MachinePrecision]), $MachinePrecision], N[(b * c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b c) < -5.00000000000000029e62 or 3.99999999999999971e225 < (*.f64 b c)

   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 b around inf

      \[\leadsto \color{blue}{b \cdot c} \]
   4. Step-by-step derivation

      1. lower-*.f64 54.8

         \[\leadsto \color{blue}{b \cdot c} \]

   5. Applied rewrites 54.8%

      \[\leadsto \color{blue}{b \cdot c} \]

   if -5.00000000000000029e62 < (*.f64 b c) < 3.99999999999999971e225

   1. Initial program 86.8%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-4 \cdot \left(a \cdot t\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{-4 \cdot \left(a \cdot t\right)} \]

      2. lower-*.f64 25.6

         \[\leadsto -4 \cdot \color{blue}{\left(a \cdot t\right)} \]

   5. Applied rewrites 25.6%

      \[\leadsto \color{blue}{-4 \cdot \left(a \cdot t\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot c \leq -5 \cdot 10^{+62}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;b \cdot c \leq 4 \cdot 10^{+225}:\\ \;\;\;\;-4 \cdot \left(t \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 30.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ \begin{array}{l} t_1 := -4 \cdot \left(x \cdot i\right)\\ \mathbf{if}\;i \leq -6.5 \cdot 10^{+108}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq -8.2 \cdot 10^{-268}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;i \leq 1050:\\ \;\;\;\;-4 \cdot \left(t \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* -4.0 (* x i))))
   (if (<= i -6.5e+108)
     t_1
     (if (<= i -8.2e-268) (* b c) (if (<= i 1050.0) (* -4.0 (* t a)) 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 = -4.0 * (x * i);
	double tmp;
	if (i <= -6.5e+108) {
		tmp = t_1;
	} else if (i <= -8.2e-268) {
		tmp = b * c;
	} else if (i <= 1050.0) {
		tmp = -4.0 * (t * a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
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) :: tmp
    t_1 = (-4.0d0) * (x * i)
    if (i <= (-6.5d+108)) then
        tmp = t_1
    else if (i <= (-8.2d-268)) then
        tmp = b * c
    else if (i <= 1050.0d0) then
        tmp = (-4.0d0) * (t * a)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b && b < c && c < i && i < j && j < k;
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 = -4.0 * (x * i);
	double tmp;
	if (i <= -6.5e+108) {
		tmp = t_1;
	} else if (i <= -8.2e-268) {
		tmp = b * c;
	} else if (i <= 1050.0) {
		tmp = -4.0 * (t * a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a, b, c, i, j, k] = sort([x, y, z, t, a, b, c, i, j, k])
def code(x, y, z, t, a, b, c, i, j, k):
	t_1 = -4.0 * (x * i)
	tmp = 0
	if i <= -6.5e+108:
		tmp = t_1
	elif i <= -8.2e-268:
		tmp = b * c
	elif i <= 1050.0:
		tmp = -4.0 * (t * a)
	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(-4.0 * Float64(x * i))
	tmp = 0.0
	if (i <= -6.5e+108)
		tmp = t_1;
	elseif (i <= -8.2e-268)
		tmp = Float64(b * c);
	elseif (i <= 1050.0)
		tmp = Float64(-4.0 * Float64(t * a));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t, a, b, c, i, j, k = num2cell(sort([x, y, z, t, a, b, c, i, j, k])){:}
function tmp_2 = code(x, y, z, t, a, b, c, i, j, k)
	t_1 = -4.0 * (x * i);
	tmp = 0.0;
	if (i <= -6.5e+108)
		tmp = t_1;
	elseif (i <= -8.2e-268)
		tmp = b * c;
	elseif (i <= 1050.0)
		tmp = -4.0 * (t * a);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, b, c, i, j, and k should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(-4.0 * N[(x * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -6.5e+108], t$95$1, If[LessEqual[i, -8.2e-268], N[(b * c), $MachinePrecision], If[LessEqual[i, 1050.0], N[(-4.0 * N[(t * a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if i < -6.4999999999999996e108 or 1050 < i

   1. Initial program 87.4%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{-4 \cdot \left(i \cdot x\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{-4 \cdot \left(i \cdot x\right)} \]

      2. *-commutative N/A

         \[\leadsto -4 \cdot \color{blue}{\left(x \cdot i\right)} \]

      3. lower-*.f64 52.8

         \[\leadsto -4 \cdot \color{blue}{\left(x \cdot i\right)} \]

   5. Applied rewrites 52.8%

      \[\leadsto \color{blue}{-4 \cdot \left(x \cdot i\right)} \]

   if -6.4999999999999996e108 < i < -8.1999999999999998e-268

   1. Initial program 82.5%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot c} \]
   4. Step-by-step derivation

      1. lower-*.f64 36.2

         \[\leadsto \color{blue}{b \cdot c} \]

   5. Applied rewrites 36.2%

      \[\leadsto \color{blue}{b \cdot c} \]

   if -8.1999999999999998e-268 < i < 1050

   1. Initial program 90.4%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-4 \cdot \left(a \cdot t\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{-4 \cdot \left(a \cdot t\right)} \]

      2. lower-*.f64 29.6

         \[\leadsto -4 \cdot \color{blue}{\left(a \cdot t\right)} \]

   5. Applied rewrites 29.6%

      \[\leadsto \color{blue}{-4 \cdot \left(a \cdot t\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -6.5 \cdot 10^{+108}:\\ \;\;\;\;-4 \cdot \left(x \cdot i\right)\\ \mathbf{elif}\;i \leq -8.2 \cdot 10^{-268}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;i \leq 1050:\\ \;\;\;\;-4 \cdot \left(t \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(x \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 24.1% accurate, 11.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i, j, k] = \mathsf{sort}([x, y, z, t, a, b, c, i, j, k])\\ \\ b \cdot c \end{array} \]

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 87.3%

   \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
2. Add Preprocessing

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{b \cdot c} \]
4. Step-by-step derivation

   1. lower-*.f64 21.9

      \[\leadsto \color{blue}{b \cdot c} \]

5. Applied rewrites 21.9%

   \[\leadsto \color{blue}{b \cdot c} \]
6. Add Preprocessing

Developer Target 1: 89.1% 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 2024219 
(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)))