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

Percentage Accurate: 96.0% → 97.8%

Time: 25.9s

Alternatives: 15

Speedup: 1.1×

• 27.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* (* a 27.0) b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((x * 2.0) - (((y * 9.0) * z) * t)) + ((a * 27.0) * b);
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((x * 2.0d0) - (((y * 9.0d0) * z) * t)) + ((a * 27.0d0) * b)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return ((x * 2.0) - (((y * 9.0) * z) * t)) + ((a * 27.0) * b);
}

Python

def code(x, y, z, t, a, b):
	return ((x * 2.0) - (((y * 9.0) * z) * t)) + ((a * 27.0) * b)

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(x * 2.0) - Float64(Float64(Float64(y * 9.0) * z) * t)) + Float64(Float64(a * 27.0) * b))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((x * 2.0) - (((y * 9.0) * z) * t)) + ((a * 27.0) * b);
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(x * 2.0), $MachinePrecision] - N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] + N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]

Initial Program: 96.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* (* a 27.0) b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((x * 2.0) - (((y * 9.0) * z) * t)) + ((a * 27.0) * b);
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((x * 2.0d0) - (((y * 9.0d0) * z) * t)) + ((a * 27.0d0) * b)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return ((x * 2.0) - (((y * 9.0) * z) * t)) + ((a * 27.0) * b);
}

Python

def code(x, y, z, t, a, b):
	return ((x * 2.0) - (((y * 9.0) * z) * t)) + ((a * 27.0) * b)

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(x * 2.0) - Float64(Float64(Float64(y * 9.0) * z) * t)) + Float64(Float64(a * 27.0) * b))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((x * 2.0) - (((y * 9.0) * z) * t)) + ((a * 27.0) * b);
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(x * 2.0), $MachinePrecision] - N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] + N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]

Alternative 1: 97.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 1.15 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 27, b, \mathsf{fma}\left(y, \left(z \cdot -9\right) \cdot t, x \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(a, 27 \cdot b, \mathsf{fma}\left(z, t \cdot \left(y \cdot -9\right), 0\right)\right)}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (if (<= z 1.15e+39)
   (fma (* a 27.0) b (fma y (* (* z -9.0) t) (* x 2.0)))
   (/ 1.0 (/ 1.0 (fma a (* 27.0 b) (fma z (* t (* y -9.0)) 0.0))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= 1.15e+39) {
		tmp = fma((a * 27.0), b, fma(y, ((z * -9.0) * t), (x * 2.0)));
	} else {
		tmp = 1.0 / (1.0 / fma(a, (27.0 * b), fma(z, (t * (y * -9.0)), 0.0)));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= 1.15e+39)
		tmp = fma(Float64(a * 27.0), b, fma(y, Float64(Float64(z * -9.0) * t), Float64(x * 2.0)));
	else
		tmp = Float64(1.0 / Float64(1.0 / fma(a, Float64(27.0 * b), fma(z, Float64(t * Float64(y * -9.0)), 0.0))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, 1.15e+39], N[(N[(a * 27.0), $MachinePrecision] * b + N[(y * N[(N[(z * -9.0), $MachinePrecision] * t), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(a * N[(27.0 * b), $MachinePrecision] + N[(z * N[(t * N[(y * -9.0), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.15000000000000006e39

   1. Initial program 94.8%

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

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

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

      7. associate-*l* N/A

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

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

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

      9. associate-*l* N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. *-lowering-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. *-lowering-*.f64 96.9

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

   4. Applied egg-rr 96.9%

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

   if 1.15000000000000006e39 < z

   1. Initial program 89.9%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. +-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{0 + -9 \cdot \left(y \cdot z\right)}, 27 \cdot \left(a \cdot b\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{-9 \cdot \left(y \cdot z\right) + 0}, 27 \cdot \left(a \cdot b\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{\left(y \cdot z\right) \cdot -9} + 0, 27 \cdot \left(a \cdot b\right)\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{fma}\left(y \cdot z, -9, 0\right)}, 27 \cdot \left(a \cdot b\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(\color{blue}{y \cdot z}, -9, 0\right), 27 \cdot \left(a \cdot b\right)\right) \]

      13. +-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(y \cdot z, -9, 0\right), \color{blue}{27 \cdot \left(a \cdot b\right) + 0}\right) \]

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(y \cdot z, -9, 0\right), \color{blue}{\mathsf{fma}\left(27, a \cdot b, 0\right)}\right) \]

      15. *-lowering-*.f64 73.6

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(y \cdot z, -9, 0\right), \mathsf{fma}\left(27, \color{blue}{a \cdot b}, 0\right)\right) \]

   5. Simplified 73.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(y \cdot z, -9, 0\right), \mathsf{fma}\left(27, a \cdot b, 0\right)\right)} \]
   6. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \color{blue}{\frac{\left(t \cdot \left(\left(y \cdot z\right) \cdot -9 + 0\right)\right) \cdot \left(t \cdot \left(\left(y \cdot z\right) \cdot -9 + 0\right)\right) - \left(27 \cdot \left(a \cdot b\right) + 0\right) \cdot \left(27 \cdot \left(a \cdot b\right) + 0\right)}{t \cdot \left(\left(y \cdot z\right) \cdot -9 + 0\right) - \left(27 \cdot \left(a \cdot b\right) + 0\right)}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot \left(\left(y \cdot z\right) \cdot -9 + 0\right) - \left(27 \cdot \left(a \cdot b\right) + 0\right)}{\left(t \cdot \left(\left(y \cdot z\right) \cdot -9 + 0\right)\right) \cdot \left(t \cdot \left(\left(y \cdot z\right) \cdot -9 + 0\right)\right) - \left(27 \cdot \left(a \cdot b\right) + 0\right) \cdot \left(27 \cdot \left(a \cdot b\right) + 0\right)}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot \left(\left(y \cdot z\right) \cdot -9 + 0\right) - \left(27 \cdot \left(a \cdot b\right) + 0\right)}{\left(t \cdot \left(\left(y \cdot z\right) \cdot -9 + 0\right)\right) \cdot \left(t \cdot \left(\left(y \cdot z\right) \cdot -9 + 0\right)\right) - \left(27 \cdot \left(a \cdot b\right) + 0\right) \cdot \left(27 \cdot \left(a \cdot b\right) + 0\right)}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{\left(t \cdot \left(\left(y \cdot z\right) \cdot -9 + 0\right)\right) \cdot \left(t \cdot \left(\left(y \cdot z\right) \cdot -9 + 0\right)\right) - \left(27 \cdot \left(a \cdot b\right) + 0\right) \cdot \left(27 \cdot \left(a \cdot b\right) + 0\right)}{t \cdot \left(\left(y \cdot z\right) \cdot -9 + 0\right) - \left(27 \cdot \left(a \cdot b\right) + 0\right)}}}} \]

      5. flip-+ N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{t \cdot \left(\left(y \cdot z\right) \cdot -9 + 0\right) + \left(27 \cdot \left(a \cdot b\right) + 0\right)}}} \]

   7. Applied egg-rr 83.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(a, 27 \cdot b, \mathsf{fma}\left(z, t \cdot \left(y \cdot -9\right), 0\right)\right)}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 87.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := t \cdot \left(z \cdot \left(y \cdot 9\right)\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+298}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(-9 \cdot t\right), y, \mathsf{fma}\left(x, 2, 0\right)\right)\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(z \cdot y, -9, 0\right), \mathsf{fma}\left(27, a \cdot b, 0\right)\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-39}:\\ \;\;\;\;\mathsf{fma}\left(27, a \cdot b, \mathsf{fma}\left(2, x, 0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot t, y \cdot -9, 27 \cdot \left(a \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* t (* z (* y 9.0)))))
   (if (<= t_1 -1e+298)
     (fma (* z (* -9.0 t)) y (fma x 2.0 0.0))
     (if (<= t_1 -5e+41)
       (fma t (fma (* z y) -9.0 0.0) (fma 27.0 (* a b) 0.0))
       (if (<= t_1 1e-39)
         (fma 27.0 (* a b) (fma 2.0 x 0.0))
         (fma (* z t) (* y -9.0) (* 27.0 (* a b))))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t * (z * (y * 9.0));
	double tmp;
	if (t_1 <= -1e+298) {
		tmp = fma((z * (-9.0 * t)), y, fma(x, 2.0, 0.0));
	} else if (t_1 <= -5e+41) {
		tmp = fma(t, fma((z * y), -9.0, 0.0), fma(27.0, (a * b), 0.0));
	} else if (t_1 <= 1e-39) {
		tmp = fma(27.0, (a * b), fma(2.0, x, 0.0));
	} else {
		tmp = fma((z * t), (y * -9.0), (27.0 * (a * b)));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(t * Float64(z * Float64(y * 9.0)))
	tmp = 0.0
	if (t_1 <= -1e+298)
		tmp = fma(Float64(z * Float64(-9.0 * t)), y, fma(x, 2.0, 0.0));
	elseif (t_1 <= -5e+41)
		tmp = fma(t, fma(Float64(z * y), -9.0, 0.0), fma(27.0, Float64(a * b), 0.0));
	elseif (t_1 <= 1e-39)
		tmp = fma(27.0, Float64(a * b), fma(2.0, x, 0.0));
	else
		tmp = fma(Float64(z * t), Float64(y * -9.0), Float64(27.0 * Float64(a * b)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t * N[(z * N[(y * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+298], N[(N[(z * N[(-9.0 * t), $MachinePrecision]), $MachinePrecision] * y + N[(x * 2.0 + 0.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e+41], N[(t * N[(N[(z * y), $MachinePrecision] * -9.0 + 0.0), $MachinePrecision] + N[(27.0 * N[(a * b), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-39], N[(27.0 * N[(a * b), $MachinePrecision] + N[(2.0 * x + 0.0), $MachinePrecision]), $MachinePrecision], N[(N[(z * t), $MachinePrecision] * N[(y * -9.0), $MachinePrecision] + N[(27.0 * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -9.9999999999999996e297

   1. Initial program 74.6%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{2 \cdot x - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]
   4. Step-by-step derivation

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. +-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{0 + -9 \cdot \left(y \cdot z\right)}, 2 \cdot x\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{-9 \cdot \left(y \cdot z\right) + 0}, 2 \cdot x\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{\left(y \cdot z\right) \cdot -9} + 0, 2 \cdot x\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{fma}\left(y \cdot z, -9, 0\right)}, 2 \cdot x\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(\color{blue}{y \cdot z}, -9, 0\right), 2 \cdot x\right) \]

      13. +-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(y \cdot z, -9, 0\right), \color{blue}{2 \cdot x + 0}\right) \]

      14. accelerator-lowering-fma.f64 74.6

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(y \cdot z, -9, 0\right), \color{blue}{\mathsf{fma}\left(2, x, 0\right)}\right) \]

   5. Simplified 74.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(y \cdot z, -9, 0\right), \mathsf{fma}\left(2, x, 0\right)\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto t \cdot \color{blue}{\left(\left(y \cdot z\right) \cdot -9\right)} + \left(2 \cdot x + 0\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y \cdot z\right) \cdot -9\right) \cdot t} + \left(2 \cdot x + 0\right) \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot \left(-9 \cdot t\right)} + \left(2 \cdot x + 0\right) \]

      4. +-rgt-identity N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. +-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(z \cdot \left(t \cdot -9\right), y, \color{blue}{2 \cdot x + 0}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z \cdot \left(t \cdot -9\right), y, \color{blue}{x \cdot 2} + 0\right) \]

      13. accelerator-lowering-fma.f64 91.3

          \[\leadsto \mathsf{fma}\left(z \cdot \left(t \cdot -9\right), y, \color{blue}{\mathsf{fma}\left(x, 2, 0\right)}\right) \]

   7. Applied egg-rr 91.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot \left(t \cdot -9\right), y, \mathsf{fma}\left(x, 2, 0\right)\right)} \]

   if -9.9999999999999996e297 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -5.00000000000000022e41

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. +-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{0 + -9 \cdot \left(y \cdot z\right)}, 27 \cdot \left(a \cdot b\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{-9 \cdot \left(y \cdot z\right) + 0}, 27 \cdot \left(a \cdot b\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{\left(y \cdot z\right) \cdot -9} + 0, 27 \cdot \left(a \cdot b\right)\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{fma}\left(y \cdot z, -9, 0\right)}, 27 \cdot \left(a \cdot b\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(\color{blue}{y \cdot z}, -9, 0\right), 27 \cdot \left(a \cdot b\right)\right) \]

      13. +-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(y \cdot z, -9, 0\right), \color{blue}{27 \cdot \left(a \cdot b\right) + 0}\right) \]

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(y \cdot z, -9, 0\right), \color{blue}{\mathsf{fma}\left(27, a \cdot b, 0\right)}\right) \]

      15. *-lowering-*.f64 92.5

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(y \cdot z, -9, 0\right), \mathsf{fma}\left(27, \color{blue}{a \cdot b}, 0\right)\right) \]

   5. Simplified 92.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(y \cdot z, -9, 0\right), \mathsf{fma}\left(27, a \cdot b, 0\right)\right)} \]

   if -5.00000000000000022e41 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 9.99999999999999929e-40

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(27, a \cdot b, \color{blue}{2 \cdot x + 0}\right) \]

      5. accelerator-lowering-fma.f64 92.5

         \[\leadsto \mathsf{fma}\left(27, a \cdot b, \color{blue}{\mathsf{fma}\left(2, x, 0\right)}\right) \]

   5. Simplified 92.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(27, a \cdot b, \mathsf{fma}\left(2, x, 0\right)\right)} \]

   if 9.99999999999999929e-40 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 88.7%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

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

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

      11. *-lowering-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. associate-*l* N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-lowering-*.f64 93.4

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

   4. Applied egg-rr 93.4%

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

   5. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 84.8

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

   7. Simplified 84.8%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot \left(y \cdot 9\right)\right) \leq -1 \cdot 10^{+298}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(-9 \cdot t\right), y, \mathsf{fma}\left(x, 2, 0\right)\right)\\ \mathbf{elif}\;t \cdot \left(z \cdot \left(y \cdot 9\right)\right) \leq -5 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(z \cdot y, -9, 0\right), \mathsf{fma}\left(27, a \cdot b, 0\right)\right)\\ \mathbf{elif}\;t \cdot \left(z \cdot \left(y \cdot 9\right)\right) \leq 10^{-39}:\\ \;\;\;\;\mathsf{fma}\left(27, a \cdot b, \mathsf{fma}\left(2, x, 0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot t, y \cdot -9, 27 \cdot \left(a \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 55.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := t \cdot \left(z \cdot \left(y \cdot 9\right)\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+41}:\\ \;\;\;\;-9 \cdot \left(y \cdot \left(z \cdot t\right)\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-146}:\\ \;\;\;\;x \cdot 2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-38}:\\ \;\;\;\;a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -9\right) \cdot \left(z \cdot t\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* t (* z (* y 9.0)))))
   (if (<= t_1 -5e+41)
     (* -9.0 (* y (* z t)))
     (if (<= t_1 5e-146)
       (* x 2.0)
       (if (<= t_1 5e-38) (* a (* 27.0 b)) (* (* y -9.0) (* z t)))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t * (z * (y * 9.0));
	double tmp;
	if (t_1 <= -5e+41) {
		tmp = -9.0 * (y * (z * t));
	} else if (t_1 <= 5e-146) {
		tmp = x * 2.0;
	} else if (t_1 <= 5e-38) {
		tmp = a * (27.0 * b);
	} else {
		tmp = (y * -9.0) * (z * t);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * (z * (y * 9.0d0))
    if (t_1 <= (-5d+41)) then
        tmp = (-9.0d0) * (y * (z * t))
    else if (t_1 <= 5d-146) then
        tmp = x * 2.0d0
    else if (t_1 <= 5d-38) then
        tmp = a * (27.0d0 * b)
    else
        tmp = (y * (-9.0d0)) * (z * t)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b;
public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t * (z * (y * 9.0));
	double tmp;
	if (t_1 <= -5e+41) {
		tmp = -9.0 * (y * (z * t));
	} else if (t_1 <= 5e-146) {
		tmp = x * 2.0;
	} else if (t_1 <= 5e-38) {
		tmp = a * (27.0 * b);
	} else {
		tmp = (y * -9.0) * (z * t);
	}
	return tmp;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	t_1 = t * (z * (y * 9.0))
	tmp = 0
	if t_1 <= -5e+41:
		tmp = -9.0 * (y * (z * t))
	elif t_1 <= 5e-146:
		tmp = x * 2.0
	elif t_1 <= 5e-38:
		tmp = a * (27.0 * b)
	else:
		tmp = (y * -9.0) * (z * t)
	return tmp

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(t * Float64(z * Float64(y * 9.0)))
	tmp = 0.0
	if (t_1 <= -5e+41)
		tmp = Float64(-9.0 * Float64(y * Float64(z * t)));
	elseif (t_1 <= 5e-146)
		tmp = Float64(x * 2.0);
	elseif (t_1 <= 5e-38)
		tmp = Float64(a * Float64(27.0 * b));
	else
		tmp = Float64(Float64(y * -9.0) * Float64(z * t));
	end
	return tmp
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp_2 = code(x, y, z, t, a, b)
	t_1 = t * (z * (y * 9.0));
	tmp = 0.0;
	if (t_1 <= -5e+41)
		tmp = -9.0 * (y * (z * t));
	elseif (t_1 <= 5e-146)
		tmp = x * 2.0;
	elseif (t_1 <= 5e-38)
		tmp = a * (27.0 * b);
	else
		tmp = (y * -9.0) * (z * t);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t * N[(z * N[(y * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+41], N[(-9.0 * N[(y * N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-146], N[(x * 2.0), $MachinePrecision], If[LessEqual[t$95$1, 5e-38], N[(a * N[(27.0 * b), $MachinePrecision]), $MachinePrecision], N[(N[(y * -9.0), $MachinePrecision] * N[(z * t), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -5.00000000000000022e41

   1. Initial program 85.5%

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

   3. Taylor expanded in y around inf

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

      1. +-lft-identity N/A

         \[\leadsto \color{blue}{0 + -9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{-9 \cdot \left(t \cdot \left(y \cdot z\right)\right) + 0} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot \left(y \cdot z\right)\right) \cdot -9} + 0 \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{t \cdot \left(\left(y \cdot z\right) \cdot -9\right)} + 0 \]

      5. *-commutative N/A

         \[\leadsto t \cdot \color{blue}{\left(-9 \cdot \left(y \cdot z\right)\right)} + 0 \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -9 \cdot \left(y \cdot z\right), 0\right)} \]

      7. +-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{0 + -9 \cdot \left(y \cdot z\right)}, 0\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{-9 \cdot \left(y \cdot z\right) + 0}, 0\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\left(y \cdot z\right) \cdot -9} + 0, 0\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{fma}\left(y \cdot z, -9, 0\right)}, 0\right) \]

      11. *-lowering-*.f64 71.1

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(\color{blue}{y \cdot z}, -9, 0\right), 0\right) \]

   5. Simplified 71.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(y \cdot z, -9, 0\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{t \cdot \left(\left(y \cdot z\right) \cdot -9 + 0\right)} \]

      2. +-rgt-identity N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 74.1

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

   7. Applied egg-rr 74.1%

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

   if -5.00000000000000022e41 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 4.99999999999999957e-146

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot x} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{2 \cdot x + 0} \]

      2. accelerator-lowering-fma.f64 52.3

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 0\right)} \]

   5. Simplified 52.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{2 \cdot x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot 2} \]

      3. *-lowering-*.f64 52.3

         \[\leadsto \color{blue}{x \cdot 2} \]

   7. Applied egg-rr 52.3%

      \[\leadsto \color{blue}{x \cdot 2} \]

   if 4.99999999999999957e-146 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 5.00000000000000033e-38

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

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

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

      7. associate-*l* N/A

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

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

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

      9. associate-*l* N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. *-lowering-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. *-lowering-*.f64 95.3

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

   4. Applied egg-rr 95.3%

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

   5. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{27 \cdot \left(a \cdot b\right)} \]

      2. *-lowering-*.f64 56.7

         \[\leadsto 27 \cdot \color{blue}{\left(a \cdot b\right)} \]

   7. Simplified 56.7%

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

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(27 \cdot a\right) \cdot b} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{b \cdot \left(27 \cdot a\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(b \cdot 27\right) \cdot a} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(b \cdot 27\right) \cdot a} \]

      5. *-lowering-*.f64 56.7

         \[\leadsto \color{blue}{\left(b \cdot 27\right)} \cdot a \]

   9. Applied egg-rr 56.7%

      \[\leadsto \color{blue}{\left(b \cdot 27\right) \cdot a} \]

   if 5.00000000000000033e-38 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 88.3%

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

   3. Taylor expanded in y around inf

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

      1. +-lft-identity N/A

         \[\leadsto \color{blue}{0 + -9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{-9 \cdot \left(t \cdot \left(y \cdot z\right)\right) + 0} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot \left(y \cdot z\right)\right) \cdot -9} + 0 \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{t \cdot \left(\left(y \cdot z\right) \cdot -9\right)} + 0 \]

      5. *-commutative N/A

         \[\leadsto t \cdot \color{blue}{\left(-9 \cdot \left(y \cdot z\right)\right)} + 0 \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -9 \cdot \left(y \cdot z\right), 0\right)} \]

      7. +-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{0 + -9 \cdot \left(y \cdot z\right)}, 0\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{-9 \cdot \left(y \cdot z\right) + 0}, 0\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\left(y \cdot z\right) \cdot -9} + 0, 0\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{fma}\left(y \cdot z, -9, 0\right)}, 0\right) \]

      11. *-lowering-*.f64 71.6

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(\color{blue}{y \cdot z}, -9, 0\right), 0\right) \]

   5. Simplified 71.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(y \cdot z, -9, 0\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{t \cdot \left(\left(y \cdot z\right) \cdot -9 + 0\right)} \]

      2. +-rgt-identity N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 69.1

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

   7. Applied egg-rr 69.1%

      \[\leadsto \color{blue}{\left(z \cdot t\right) \cdot \left(y \cdot -9\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot \left(y \cdot 9\right)\right) \leq -5 \cdot 10^{+41}:\\ \;\;\;\;-9 \cdot \left(y \cdot \left(z \cdot t\right)\right)\\ \mathbf{elif}\;t \cdot \left(z \cdot \left(y \cdot 9\right)\right) \leq 5 \cdot 10^{-146}:\\ \;\;\;\;x \cdot 2\\ \mathbf{elif}\;t \cdot \left(z \cdot \left(y \cdot 9\right)\right) \leq 5 \cdot 10^{-38}:\\ \;\;\;\;a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -9\right) \cdot \left(z \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := -9 \cdot \left(y \cdot \left(z \cdot t\right)\right)\\ t_2 := t \cdot \left(z \cdot \left(y \cdot 9\right)\right)\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-146}:\\ \;\;\;\;x \cdot 2\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-29}:\\ \;\;\;\;a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* -9.0 (* y (* z t)))) (t_2 (* t (* z (* y 9.0)))))
   (if (<= t_2 -5e+41)
     t_1
     (if (<= t_2 5e-146) (* x 2.0) (if (<= t_2 2e-29) (* a (* 27.0 b)) t_1)))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = -9.0 * (y * (z * t));
	double t_2 = t * (z * (y * 9.0));
	double tmp;
	if (t_2 <= -5e+41) {
		tmp = t_1;
	} else if (t_2 <= 5e-146) {
		tmp = x * 2.0;
	} else if (t_2 <= 2e-29) {
		tmp = a * (27.0 * b);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (-9.0d0) * (y * (z * t))
    t_2 = t * (z * (y * 9.0d0))
    if (t_2 <= (-5d+41)) then
        tmp = t_1
    else if (t_2 <= 5d-146) then
        tmp = x * 2.0d0
    else if (t_2 <= 2d-29) then
        tmp = a * (27.0d0 * b)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b;
public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = -9.0 * (y * (z * t));
	double t_2 = t * (z * (y * 9.0));
	double tmp;
	if (t_2 <= -5e+41) {
		tmp = t_1;
	} else if (t_2 <= 5e-146) {
		tmp = x * 2.0;
	} else if (t_2 <= 2e-29) {
		tmp = a * (27.0 * b);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	t_1 = -9.0 * (y * (z * t))
	t_2 = t * (z * (y * 9.0))
	tmp = 0
	if t_2 <= -5e+41:
		tmp = t_1
	elif t_2 <= 5e-146:
		tmp = x * 2.0
	elif t_2 <= 2e-29:
		tmp = a * (27.0 * b)
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(-9.0 * Float64(y * Float64(z * t)))
	t_2 = Float64(t * Float64(z * Float64(y * 9.0)))
	tmp = 0.0
	if (t_2 <= -5e+41)
		tmp = t_1;
	elseif (t_2 <= 5e-146)
		tmp = Float64(x * 2.0);
	elseif (t_2 <= 2e-29)
		tmp = Float64(a * Float64(27.0 * b));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp_2 = code(x, y, z, t, a, b)
	t_1 = -9.0 * (y * (z * t));
	t_2 = t * (z * (y * 9.0));
	tmp = 0.0;
	if (t_2 <= -5e+41)
		tmp = t_1;
	elseif (t_2 <= 5e-146)
		tmp = x * 2.0;
	elseif (t_2 <= 2e-29)
		tmp = a * (27.0 * b);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(-9.0 * N[(y * N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(z * N[(y * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+41], t$95$1, If[LessEqual[t$95$2, 5e-146], N[(x * 2.0), $MachinePrecision], If[LessEqual[t$95$2, 2e-29], N[(a * N[(27.0 * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -5.00000000000000022e41 or 1.99999999999999989e-29 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 86.7%

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

   3. Taylor expanded in y around inf

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

      1. +-lft-identity N/A

         \[\leadsto \color{blue}{0 + -9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{-9 \cdot \left(t \cdot \left(y \cdot z\right)\right) + 0} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot \left(y \cdot z\right)\right) \cdot -9} + 0 \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{t \cdot \left(\left(y \cdot z\right) \cdot -9\right)} + 0 \]

      5. *-commutative N/A

         \[\leadsto t \cdot \color{blue}{\left(-9 \cdot \left(y \cdot z\right)\right)} + 0 \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -9 \cdot \left(y \cdot z\right), 0\right)} \]

      7. +-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{0 + -9 \cdot \left(y \cdot z\right)}, 0\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{-9 \cdot \left(y \cdot z\right) + 0}, 0\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\left(y \cdot z\right) \cdot -9} + 0, 0\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{fma}\left(y \cdot z, -9, 0\right)}, 0\right) \]

      11. *-lowering-*.f64 71.7

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(\color{blue}{y \cdot z}, -9, 0\right), 0\right) \]

   5. Simplified 71.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(y \cdot z, -9, 0\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{t \cdot \left(\left(y \cdot z\right) \cdot -9 + 0\right)} \]

      2. +-rgt-identity N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 71.9

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

   7. Applied egg-rr 71.9%

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

   if -5.00000000000000022e41 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 4.99999999999999957e-146

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot x} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{2 \cdot x + 0} \]

      2. accelerator-lowering-fma.f64 52.3

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 0\right)} \]

   5. Simplified 52.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{2 \cdot x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot 2} \]

      3. *-lowering-*.f64 52.3

         \[\leadsto \color{blue}{x \cdot 2} \]

   7. Applied egg-rr 52.3%

      \[\leadsto \color{blue}{x \cdot 2} \]

   if 4.99999999999999957e-146 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 1.99999999999999989e-29

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

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

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

      7. associate-*l* N/A

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

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

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

      9. associate-*l* N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. *-lowering-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. *-lowering-*.f64 95.7

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

   4. Applied egg-rr 95.7%

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

   5. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{27 \cdot \left(a \cdot b\right)} \]

      2. *-lowering-*.f64 56.5

         \[\leadsto 27 \cdot \color{blue}{\left(a \cdot b\right)} \]

   7. Simplified 56.5%

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

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(27 \cdot a\right) \cdot b} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{b \cdot \left(27 \cdot a\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(b \cdot 27\right) \cdot a} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(b \cdot 27\right) \cdot a} \]

      5. *-lowering-*.f64 56.5

         \[\leadsto \color{blue}{\left(b \cdot 27\right)} \cdot a \]

   9. Applied egg-rr 56.5%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot \left(y \cdot 9\right)\right) \leq -5 \cdot 10^{+41}:\\ \;\;\;\;-9 \cdot \left(y \cdot \left(z \cdot t\right)\right)\\ \mathbf{elif}\;t \cdot \left(z \cdot \left(y \cdot 9\right)\right) \leq 5 \cdot 10^{-146}:\\ \;\;\;\;x \cdot 2\\ \mathbf{elif}\;t \cdot \left(z \cdot \left(y \cdot 9\right)\right) \leq 2 \cdot 10^{-29}:\\ \;\;\;\;a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;-9 \cdot \left(y \cdot \left(z \cdot t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := y \cdot \left(-9 \cdot \left(z \cdot t\right)\right)\\ t_2 := t \cdot \left(z \cdot \left(y \cdot 9\right)\right)\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-146}:\\ \;\;\;\;x \cdot 2\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-38}:\\ \;\;\;\;a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* y (* -9.0 (* z t)))) (t_2 (* t (* z (* y 9.0)))))
   (if (<= t_2 -5e+41)
     t_1
     (if (<= t_2 5e-146) (* x 2.0) (if (<= t_2 5e-38) (* a (* 27.0 b)) t_1)))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y * (-9.0 * (z * t));
	double t_2 = t * (z * (y * 9.0));
	double tmp;
	if (t_2 <= -5e+41) {
		tmp = t_1;
	} else if (t_2 <= 5e-146) {
		tmp = x * 2.0;
	} else if (t_2 <= 5e-38) {
		tmp = a * (27.0 * b);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = y * ((-9.0d0) * (z * t))
    t_2 = t * (z * (y * 9.0d0))
    if (t_2 <= (-5d+41)) then
        tmp = t_1
    else if (t_2 <= 5d-146) then
        tmp = x * 2.0d0
    else if (t_2 <= 5d-38) then
        tmp = a * (27.0d0 * b)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b;
public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y * (-9.0 * (z * t));
	double t_2 = t * (z * (y * 9.0));
	double tmp;
	if (t_2 <= -5e+41) {
		tmp = t_1;
	} else if (t_2 <= 5e-146) {
		tmp = x * 2.0;
	} else if (t_2 <= 5e-38) {
		tmp = a * (27.0 * b);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	t_1 = y * (-9.0 * (z * t))
	t_2 = t * (z * (y * 9.0))
	tmp = 0
	if t_2 <= -5e+41:
		tmp = t_1
	elif t_2 <= 5e-146:
		tmp = x * 2.0
	elif t_2 <= 5e-38:
		tmp = a * (27.0 * b)
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(y * Float64(-9.0 * Float64(z * t)))
	t_2 = Float64(t * Float64(z * Float64(y * 9.0)))
	tmp = 0.0
	if (t_2 <= -5e+41)
		tmp = t_1;
	elseif (t_2 <= 5e-146)
		tmp = Float64(x * 2.0);
	elseif (t_2 <= 5e-38)
		tmp = Float64(a * Float64(27.0 * b));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp_2 = code(x, y, z, t, a, b)
	t_1 = y * (-9.0 * (z * t));
	t_2 = t * (z * (y * 9.0));
	tmp = 0.0;
	if (t_2 <= -5e+41)
		tmp = t_1;
	elseif (t_2 <= 5e-146)
		tmp = x * 2.0;
	elseif (t_2 <= 5e-38)
		tmp = a * (27.0 * b);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y * N[(-9.0 * N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(z * N[(y * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+41], t$95$1, If[LessEqual[t$95$2, 5e-146], N[(x * 2.0), $MachinePrecision], If[LessEqual[t$95$2, 5e-38], N[(a * N[(27.0 * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -5.00000000000000022e41 or 5.00000000000000033e-38 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 86.9%

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

   3. Taylor expanded in y around inf

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

      1. +-lft-identity N/A

         \[\leadsto \color{blue}{0 + -9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{-9 \cdot \left(t \cdot \left(y \cdot z\right)\right) + 0} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot \left(y \cdot z\right)\right) \cdot -9} + 0 \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{t \cdot \left(\left(y \cdot z\right) \cdot -9\right)} + 0 \]

      5. *-commutative N/A

         \[\leadsto t \cdot \color{blue}{\left(-9 \cdot \left(y \cdot z\right)\right)} + 0 \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -9 \cdot \left(y \cdot z\right), 0\right)} \]

      7. +-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{0 + -9 \cdot \left(y \cdot z\right)}, 0\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{-9 \cdot \left(y \cdot z\right) + 0}, 0\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\left(y \cdot z\right) \cdot -9} + 0, 0\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{fma}\left(y \cdot z, -9, 0\right)}, 0\right) \]

      11. *-lowering-*.f64 71.4

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(\color{blue}{y \cdot z}, -9, 0\right), 0\right) \]

   5. Simplified 71.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(y \cdot z, -9, 0\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{t \cdot \left(\left(y \cdot z\right) \cdot -9 + 0\right)} \]

      2. +-rgt-identity N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 71.6

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

   7. Applied egg-rr 71.6%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 71.6

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

   9. Applied egg-rr 71.6%

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

   if -5.00000000000000022e41 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 4.99999999999999957e-146

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot x} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{2 \cdot x + 0} \]

      2. accelerator-lowering-fma.f64 52.3

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 0\right)} \]

   5. Simplified 52.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{2 \cdot x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot 2} \]

      3. *-lowering-*.f64 52.3

         \[\leadsto \color{blue}{x \cdot 2} \]

   7. Applied egg-rr 52.3%

      \[\leadsto \color{blue}{x \cdot 2} \]

   if 4.99999999999999957e-146 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 5.00000000000000033e-38

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

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

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

      7. associate-*l* N/A

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

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

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

      9. associate-*l* N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. *-lowering-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. *-lowering-*.f64 95.3

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

   4. Applied egg-rr 95.3%

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

   5. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{27 \cdot \left(a \cdot b\right)} \]

      2. *-lowering-*.f64 56.7

         \[\leadsto 27 \cdot \color{blue}{\left(a \cdot b\right)} \]

   7. Simplified 56.7%

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

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(27 \cdot a\right) \cdot b} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{b \cdot \left(27 \cdot a\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(b \cdot 27\right) \cdot a} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(b \cdot 27\right) \cdot a} \]

      5. *-lowering-*.f64 56.7

         \[\leadsto \color{blue}{\left(b \cdot 27\right)} \cdot a \]

   9. Applied egg-rr 56.7%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot \left(y \cdot 9\right)\right) \leq -5 \cdot 10^{+41}:\\ \;\;\;\;y \cdot \left(-9 \cdot \left(z \cdot t\right)\right)\\ \mathbf{elif}\;t \cdot \left(z \cdot \left(y \cdot 9\right)\right) \leq 5 \cdot 10^{-146}:\\ \;\;\;\;x \cdot 2\\ \mathbf{elif}\;t \cdot \left(z \cdot \left(y \cdot 9\right)\right) \leq 5 \cdot 10^{-38}:\\ \;\;\;\;a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-9 \cdot \left(z \cdot t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z \cdot t, y \cdot -9, 27 \cdot \left(a \cdot b\right)\right)\\ t_2 := t \cdot \left(z \cdot \left(y \cdot 9\right)\right)\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{-39}:\\ \;\;\;\;\mathsf{fma}\left(27, a \cdot b, \mathsf{fma}\left(2, x, 0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (* z t) (* y -9.0) (* 27.0 (* a b))))
        (t_2 (* t (* z (* y 9.0)))))
   (if (<= t_2 -5e+41)
     t_1
     (if (<= t_2 1e-39) (fma 27.0 (* a b) (fma 2.0 x 0.0)) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((z * t), (y * -9.0), (27.0 * (a * b)));
	double t_2 = t * (z * (y * 9.0));
	double tmp;
	if (t_2 <= -5e+41) {
		tmp = t_1;
	} else if (t_2 <= 1e-39) {
		tmp = fma(27.0, (a * b), fma(2.0, x, 0.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = fma(Float64(z * t), Float64(y * -9.0), Float64(27.0 * Float64(a * b)))
	t_2 = Float64(t * Float64(z * Float64(y * 9.0)))
	tmp = 0.0
	if (t_2 <= -5e+41)
		tmp = t_1;
	elseif (t_2 <= 1e-39)
		tmp = fma(27.0, Float64(a * b), fma(2.0, x, 0.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] * N[(y * -9.0), $MachinePrecision] + N[(27.0 * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(z * N[(y * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+41], t$95$1, If[LessEqual[t$95$2, 1e-39], N[(27.0 * N[(a * b), $MachinePrecision] + N[(2.0 * x + 0.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -5.00000000000000022e41 or 9.99999999999999929e-40 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 87.1%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

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

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

      11. *-lowering-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. associate-*l* N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-lowering-*.f64 93.5

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

   4. Applied egg-rr 93.5%

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

   5. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 86.7

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

   7. Simplified 86.7%

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

   if -5.00000000000000022e41 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 9.99999999999999929e-40

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(27, a \cdot b, \color{blue}{2 \cdot x + 0}\right) \]

      5. accelerator-lowering-fma.f64 92.5

         \[\leadsto \mathsf{fma}\left(27, a \cdot b, \color{blue}{\mathsf{fma}\left(2, x, 0\right)}\right) \]

   5. Simplified 92.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(27, a \cdot b, \mathsf{fma}\left(2, x, 0\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot \left(y \cdot 9\right)\right) \leq -5 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot t, y \cdot -9, 27 \cdot \left(a \cdot b\right)\right)\\ \mathbf{elif}\;t \cdot \left(z \cdot \left(y \cdot 9\right)\right) \leq 10^{-39}:\\ \;\;\;\;\mathsf{fma}\left(27, a \cdot b, \mathsf{fma}\left(2, x, 0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot t, y \cdot -9, 27 \cdot \left(a \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := t \cdot \left(z \cdot \left(y \cdot 9\right)\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot t, y \cdot -9, x \cdot 2\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-38}:\\ \;\;\;\;\mathsf{fma}\left(27, a \cdot b, \mathsf{fma}\left(2, x, 0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-9 \cdot \left(z \cdot t\right), y, x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* t (* z (* y 9.0)))))
   (if (<= t_1 -2e+80)
     (fma (* z t) (* y -9.0) (* x 2.0))
     (if (<= t_1 5e-38)
       (fma 27.0 (* a b) (fma 2.0 x 0.0))
       (fma (* -9.0 (* z t)) y (* x 2.0))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t * (z * (y * 9.0));
	double tmp;
	if (t_1 <= -2e+80) {
		tmp = fma((z * t), (y * -9.0), (x * 2.0));
	} else if (t_1 <= 5e-38) {
		tmp = fma(27.0, (a * b), fma(2.0, x, 0.0));
	} else {
		tmp = fma((-9.0 * (z * t)), y, (x * 2.0));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(t * Float64(z * Float64(y * 9.0)))
	tmp = 0.0
	if (t_1 <= -2e+80)
		tmp = fma(Float64(z * t), Float64(y * -9.0), Float64(x * 2.0));
	elseif (t_1 <= 5e-38)
		tmp = fma(27.0, Float64(a * b), fma(2.0, x, 0.0));
	else
		tmp = fma(Float64(-9.0 * Float64(z * t)), y, Float64(x * 2.0));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t * N[(z * N[(y * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+80], N[(N[(z * t), $MachinePrecision] * N[(y * -9.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-38], N[(27.0 * N[(a * b), $MachinePrecision] + N[(2.0 * x + 0.0), $MachinePrecision]), $MachinePrecision], N[(N[(-9.0 * N[(z * t), $MachinePrecision]), $MachinePrecision] * y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -2e80

   1. Initial program 83.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

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

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

      11. *-lowering-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. associate-*l* N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-lowering-*.f64 92.8

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

   4. Applied egg-rr 92.8%

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

   5. Taylor expanded in a around 0

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

      1. *-lowering-*.f64 82.0

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

   7. Simplified 82.0%

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

   if -2e80 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 5.00000000000000033e-38

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(27, a \cdot b, \color{blue}{2 \cdot x + 0}\right) \]

      5. accelerator-lowering-fma.f64 91.5

         \[\leadsto \mathsf{fma}\left(27, a \cdot b, \color{blue}{\mathsf{fma}\left(2, x, 0\right)}\right) \]

   5. Simplified 91.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(27, a \cdot b, \mathsf{fma}\left(2, x, 0\right)\right)} \]

   if 5.00000000000000033e-38 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 88.3%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

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

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

      11. *-lowering-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. associate-*l* N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-lowering-*.f64 93.2

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

   4. Applied egg-rr 93.2%

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

   5. Taylor expanded in a around 0

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

      1. *-lowering-*.f64 77.3

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

   7. Simplified 77.3%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 77.4

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

   9. Applied egg-rr 77.4%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot \left(y \cdot 9\right)\right) \leq -2 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot t, y \cdot -9, x \cdot 2\right)\\ \mathbf{elif}\;t \cdot \left(z \cdot \left(y \cdot 9\right)\right) \leq 5 \cdot 10^{-38}:\\ \;\;\;\;\mathsf{fma}\left(27, a \cdot b, \mathsf{fma}\left(2, x, 0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-9 \cdot \left(z \cdot t\right), y, x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-9 \cdot \left(z \cdot t\right), y, x \cdot 2\right)\\ t_2 := t \cdot \left(z \cdot \left(y \cdot 9\right)\right)\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-38}:\\ \;\;\;\;\mathsf{fma}\left(27, a \cdot b, \mathsf{fma}\left(2, x, 0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (* -9.0 (* z t)) y (* x 2.0))) (t_2 (* t (* z (* y 9.0)))))
   (if (<= t_2 -2e+80)
     t_1
     (if (<= t_2 5e-38) (fma 27.0 (* a b) (fma 2.0 x 0.0)) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((-9.0 * (z * t)), y, (x * 2.0));
	double t_2 = t * (z * (y * 9.0));
	double tmp;
	if (t_2 <= -2e+80) {
		tmp = t_1;
	} else if (t_2 <= 5e-38) {
		tmp = fma(27.0, (a * b), fma(2.0, x, 0.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = fma(Float64(-9.0 * Float64(z * t)), y, Float64(x * 2.0))
	t_2 = Float64(t * Float64(z * Float64(y * 9.0)))
	tmp = 0.0
	if (t_2 <= -2e+80)
		tmp = t_1;
	elseif (t_2 <= 5e-38)
		tmp = fma(27.0, Float64(a * b), fma(2.0, x, 0.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(-9.0 * N[(z * t), $MachinePrecision]), $MachinePrecision] * y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(z * N[(y * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+80], t$95$1, If[LessEqual[t$95$2, 5e-38], N[(27.0 * N[(a * b), $MachinePrecision] + N[(2.0 * x + 0.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -2e80 or 5.00000000000000033e-38 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 86.2%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

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

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

      11. *-lowering-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. associate-*l* N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-lowering-*.f64 93.0

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

   4. Applied egg-rr 93.0%

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

   5. Taylor expanded in a around 0

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

      1. *-lowering-*.f64 79.5

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

   7. Simplified 79.5%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 79.6

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

   9. Applied egg-rr 79.6%

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

   if -2e80 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 5.00000000000000033e-38

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(27, a \cdot b, \color{blue}{2 \cdot x + 0}\right) \]

      5. accelerator-lowering-fma.f64 91.5

         \[\leadsto \mathsf{fma}\left(27, a \cdot b, \color{blue}{\mathsf{fma}\left(2, x, 0\right)}\right) \]

   5. Simplified 91.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(27, a \cdot b, \mathsf{fma}\left(2, x, 0\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot \left(y \cdot 9\right)\right) \leq -2 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(-9 \cdot \left(z \cdot t\right), y, x \cdot 2\right)\\ \mathbf{elif}\;t \cdot \left(z \cdot \left(y \cdot 9\right)\right) \leq 5 \cdot 10^{-38}:\\ \;\;\;\;\mathsf{fma}\left(27, a \cdot b, \mathsf{fma}\left(2, x, 0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-9 \cdot \left(z \cdot t\right), y, x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := t \cdot \left(z \cdot \left(y \cdot 9\right)\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+80}:\\ \;\;\;\;-9 \cdot \left(y \cdot \left(z \cdot t\right)\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-38}:\\ \;\;\;\;\mathsf{fma}\left(27, a \cdot b, \mathsf{fma}\left(2, x, 0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -9\right) \cdot \left(z \cdot t\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* t (* z (* y 9.0)))))
   (if (<= t_1 -2e+80)
     (* -9.0 (* y (* z t)))
     (if (<= t_1 5e-38)
       (fma 27.0 (* a b) (fma 2.0 x 0.0))
       (* (* y -9.0) (* z t))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t * (z * (y * 9.0));
	double tmp;
	if (t_1 <= -2e+80) {
		tmp = -9.0 * (y * (z * t));
	} else if (t_1 <= 5e-38) {
		tmp = fma(27.0, (a * b), fma(2.0, x, 0.0));
	} else {
		tmp = (y * -9.0) * (z * t);
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(t * Float64(z * Float64(y * 9.0)))
	tmp = 0.0
	if (t_1 <= -2e+80)
		tmp = Float64(-9.0 * Float64(y * Float64(z * t)));
	elseif (t_1 <= 5e-38)
		tmp = fma(27.0, Float64(a * b), fma(2.0, x, 0.0));
	else
		tmp = Float64(Float64(y * -9.0) * Float64(z * t));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t * N[(z * N[(y * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+80], N[(-9.0 * N[(y * N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-38], N[(27.0 * N[(a * b), $MachinePrecision] + N[(2.0 * x + 0.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * -9.0), $MachinePrecision] * N[(z * t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -2e80

   1. Initial program 83.9%

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

   3. Taylor expanded in y around inf

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

      1. +-lft-identity N/A

         \[\leadsto \color{blue}{0 + -9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{-9 \cdot \left(t \cdot \left(y \cdot z\right)\right) + 0} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot \left(y \cdot z\right)\right) \cdot -9} + 0 \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{t \cdot \left(\left(y \cdot z\right) \cdot -9\right)} + 0 \]

      5. *-commutative N/A

         \[\leadsto t \cdot \color{blue}{\left(-9 \cdot \left(y \cdot z\right)\right)} + 0 \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -9 \cdot \left(y \cdot z\right), 0\right)} \]

      7. +-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{0 + -9 \cdot \left(y \cdot z\right)}, 0\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{-9 \cdot \left(y \cdot z\right) + 0}, 0\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\left(y \cdot z\right) \cdot -9} + 0, 0\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{fma}\left(y \cdot z, -9, 0\right)}, 0\right) \]

      11. *-lowering-*.f64 75.2

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(\color{blue}{y \cdot z}, -9, 0\right), 0\right) \]

   5. Simplified 75.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(y \cdot z, -9, 0\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{t \cdot \left(\left(y \cdot z\right) \cdot -9 + 0\right)} \]

      2. +-rgt-identity N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 78.5

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

   7. Applied egg-rr 78.5%

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

   if -2e80 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 5.00000000000000033e-38

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(27, a \cdot b, \color{blue}{2 \cdot x + 0}\right) \]

      5. accelerator-lowering-fma.f64 91.5

         \[\leadsto \mathsf{fma}\left(27, a \cdot b, \color{blue}{\mathsf{fma}\left(2, x, 0\right)}\right) \]

   5. Simplified 91.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(27, a \cdot b, \mathsf{fma}\left(2, x, 0\right)\right)} \]

   if 5.00000000000000033e-38 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 88.3%

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

   3. Taylor expanded in y around inf

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

      1. +-lft-identity N/A

         \[\leadsto \color{blue}{0 + -9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{-9 \cdot \left(t \cdot \left(y \cdot z\right)\right) + 0} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot \left(y \cdot z\right)\right) \cdot -9} + 0 \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{t \cdot \left(\left(y \cdot z\right) \cdot -9\right)} + 0 \]

      5. *-commutative N/A

         \[\leadsto t \cdot \color{blue}{\left(-9 \cdot \left(y \cdot z\right)\right)} + 0 \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -9 \cdot \left(y \cdot z\right), 0\right)} \]

      7. +-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{0 + -9 \cdot \left(y \cdot z\right)}, 0\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{-9 \cdot \left(y \cdot z\right) + 0}, 0\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\left(y \cdot z\right) \cdot -9} + 0, 0\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{fma}\left(y \cdot z, -9, 0\right)}, 0\right) \]

      11. *-lowering-*.f64 71.6

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(\color{blue}{y \cdot z}, -9, 0\right), 0\right) \]

   5. Simplified 71.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(y \cdot z, -9, 0\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{t \cdot \left(\left(y \cdot z\right) \cdot -9 + 0\right)} \]

      2. +-rgt-identity N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 69.1

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

   7. Applied egg-rr 69.1%

      \[\leadsto \color{blue}{\left(z \cdot t\right) \cdot \left(y \cdot -9\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot \left(y \cdot 9\right)\right) \leq -2 \cdot 10^{+80}:\\ \;\;\;\;-9 \cdot \left(y \cdot \left(z \cdot t\right)\right)\\ \mathbf{elif}\;t \cdot \left(z \cdot \left(y \cdot 9\right)\right) \leq 5 \cdot 10^{-38}:\\ \;\;\;\;\mathsf{fma}\left(27, a \cdot b, \mathsf{fma}\left(2, x, 0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -9\right) \cdot \left(z \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := t \cdot \left(z \cdot \left(y \cdot 9\right)\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+80}:\\ \;\;\;\;-9 \cdot \left(y \cdot \left(z \cdot t\right)\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-38}:\\ \;\;\;\;\mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -9\right) \cdot \left(z \cdot t\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* t (* z (* y 9.0)))))
   (if (<= t_1 -2e+80)
     (* -9.0 (* y (* z t)))
     (if (<= t_1 5e-38) (fma 2.0 x (* 27.0 (* a b))) (* (* y -9.0) (* z t))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t * (z * (y * 9.0));
	double tmp;
	if (t_1 <= -2e+80) {
		tmp = -9.0 * (y * (z * t));
	} else if (t_1 <= 5e-38) {
		tmp = fma(2.0, x, (27.0 * (a * b)));
	} else {
		tmp = (y * -9.0) * (z * t);
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(t * Float64(z * Float64(y * 9.0)))
	tmp = 0.0
	if (t_1 <= -2e+80)
		tmp = Float64(-9.0 * Float64(y * Float64(z * t)));
	elseif (t_1 <= 5e-38)
		tmp = fma(2.0, x, Float64(27.0 * Float64(a * b)));
	else
		tmp = Float64(Float64(y * -9.0) * Float64(z * t));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t * N[(z * N[(y * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+80], N[(-9.0 * N[(y * N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-38], N[(2.0 * x + N[(27.0 * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * -9.0), $MachinePrecision] * N[(z * t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < -2e80

   1. Initial program 83.9%

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

   3. Taylor expanded in y around inf

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

      1. +-lft-identity N/A

         \[\leadsto \color{blue}{0 + -9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{-9 \cdot \left(t \cdot \left(y \cdot z\right)\right) + 0} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot \left(y \cdot z\right)\right) \cdot -9} + 0 \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{t \cdot \left(\left(y \cdot z\right) \cdot -9\right)} + 0 \]

      5. *-commutative N/A

         \[\leadsto t \cdot \color{blue}{\left(-9 \cdot \left(y \cdot z\right)\right)} + 0 \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -9 \cdot \left(y \cdot z\right), 0\right)} \]

      7. +-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{0 + -9 \cdot \left(y \cdot z\right)}, 0\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{-9 \cdot \left(y \cdot z\right) + 0}, 0\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\left(y \cdot z\right) \cdot -9} + 0, 0\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{fma}\left(y \cdot z, -9, 0\right)}, 0\right) \]

      11. *-lowering-*.f64 75.2

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(\color{blue}{y \cdot z}, -9, 0\right), 0\right) \]

   5. Simplified 75.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(y \cdot z, -9, 0\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{t \cdot \left(\left(y \cdot z\right) \cdot -9 + 0\right)} \]

      2. +-rgt-identity N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 78.5

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

   7. Applied egg-rr 78.5%

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

   if -2e80 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t) < 5.00000000000000033e-38

   1. Initial program 99.8%

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

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

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

      7. associate-*l* N/A

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

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

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

      9. associate-*l* N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. *-lowering-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. *-lowering-*.f64 98.4

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

   4. Applied egg-rr 98.4%

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

   5. Taylor expanded in y around 0

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

      1. accelerator-lowering-fma.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-lowering-*.f64 91.5

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

   7. Simplified 91.5%

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

   if 5.00000000000000033e-38 < (*.f64 (*.f64 (*.f64 y #s(literal 9 binary64)) z) t)

   1. Initial program 88.3%

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

   3. Taylor expanded in y around inf

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

      1. +-lft-identity N/A

         \[\leadsto \color{blue}{0 + -9 \cdot \left(t \cdot \left(y \cdot z\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{-9 \cdot \left(t \cdot \left(y \cdot z\right)\right) + 0} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot \left(y \cdot z\right)\right) \cdot -9} + 0 \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{t \cdot \left(\left(y \cdot z\right) \cdot -9\right)} + 0 \]

      5. *-commutative N/A

         \[\leadsto t \cdot \color{blue}{\left(-9 \cdot \left(y \cdot z\right)\right)} + 0 \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -9 \cdot \left(y \cdot z\right), 0\right)} \]

      7. +-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{0 + -9 \cdot \left(y \cdot z\right)}, 0\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{-9 \cdot \left(y \cdot z\right) + 0}, 0\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\left(y \cdot z\right) \cdot -9} + 0, 0\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{fma}\left(y \cdot z, -9, 0\right)}, 0\right) \]

      11. *-lowering-*.f64 71.6

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(\color{blue}{y \cdot z}, -9, 0\right), 0\right) \]

   5. Simplified 71.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(y \cdot z, -9, 0\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{t \cdot \left(\left(y \cdot z\right) \cdot -9 + 0\right)} \]

      2. +-rgt-identity N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 69.1

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

   7. Applied egg-rr 69.1%

      \[\leadsto \color{blue}{\left(z \cdot t\right) \cdot \left(y \cdot -9\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot \left(y \cdot 9\right)\right) \leq -2 \cdot 10^{+80}:\\ \;\;\;\;-9 \cdot \left(y \cdot \left(z \cdot t\right)\right)\\ \mathbf{elif}\;t \cdot \left(z \cdot \left(y \cdot 9\right)\right) \leq 5 \cdot 10^{-38}:\\ \;\;\;\;\mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -9\right) \cdot \left(z \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 52.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(a \cdot 27\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+32}:\\ \;\;\;\;27 \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-58}:\\ \;\;\;\;x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* a 27.0) b)))
   (if (<= t_1 -4e+32) (* 27.0 (* a b)) (if (<= t_1 2e-58) (* x 2.0) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * 27.0) * b;
	double tmp;
	if (t_1 <= -4e+32) {
		tmp = 27.0 * (a * b);
	} else if (t_1 <= 2e-58) {
		tmp = x * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a * 27.0d0) * b
    if (t_1 <= (-4d+32)) then
        tmp = 27.0d0 * (a * b)
    else if (t_1 <= 2d-58) then
        tmp = x * 2.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b;
public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * 27.0) * b;
	double tmp;
	if (t_1 <= -4e+32) {
		tmp = 27.0 * (a * b);
	} else if (t_1 <= 2e-58) {
		tmp = x * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	t_1 = (a * 27.0) * b
	tmp = 0
	if t_1 <= -4e+32:
		tmp = 27.0 * (a * b)
	elif t_1 <= 2e-58:
		tmp = x * 2.0
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a * 27.0) * b)
	tmp = 0.0
	if (t_1 <= -4e+32)
		tmp = Float64(27.0 * Float64(a * b));
	elseif (t_1 <= 2e-58)
		tmp = Float64(x * 2.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a * 27.0) * b;
	tmp = 0.0;
	if (t_1 <= -4e+32)
		tmp = 27.0 * (a * b);
	elseif (t_1 <= 2e-58)
		tmp = x * 2.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+32], N[(27.0 * N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-58], N[(x * 2.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 a #s(literal 27 binary64)) b) < -4.00000000000000021e32

   1. Initial program 94.8%

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

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

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

      7. associate-*l* N/A

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

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

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

      9. associate-*l* N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. *-lowering-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. *-lowering-*.f64 99.7

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{27 \cdot \left(a \cdot b\right)} \]

      2. *-lowering-*.f64 69.2

         \[\leadsto 27 \cdot \color{blue}{\left(a \cdot b\right)} \]

   7. Simplified 69.2%

      \[\leadsto \color{blue}{27 \cdot \left(a \cdot b\right)} \]

   if -4.00000000000000021e32 < (*.f64 (*.f64 a #s(literal 27 binary64)) b) < 2.0000000000000001e-58

   1. Initial program 96.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot x} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{2 \cdot x + 0} \]

      2. accelerator-lowering-fma.f64 44.1

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 0\right)} \]

   5. Simplified 44.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{2 \cdot x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot 2} \]

      3. *-lowering-*.f64 44.1

         \[\leadsto \color{blue}{x \cdot 2} \]

   7. Applied egg-rr 44.1%

      \[\leadsto \color{blue}{x \cdot 2} \]

   if 2.0000000000000001e-58 < (*.f64 (*.f64 a #s(literal 27 binary64)) b)

   1. Initial program 87.5%

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

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

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

      7. associate-*l* N/A

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

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

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

      9. associate-*l* N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. *-lowering-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. *-lowering-*.f64 98.4

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

   4. Applied egg-rr 98.4%

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

   5. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{27 \cdot \left(a \cdot b\right)} \]

      2. *-lowering-*.f64 66.3

         \[\leadsto 27 \cdot \color{blue}{\left(a \cdot b\right)} \]

   7. Simplified 66.3%

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

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(27 \cdot a\right) \cdot b} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(27 \cdot a\right) \cdot b} \]

      3. *-lowering-*.f64 66.4

         \[\leadsto \color{blue}{\left(27 \cdot a\right)} \cdot b \]

   9. Applied egg-rr 66.4%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a \cdot 27\right) \cdot b \leq -4 \cdot 10^{+32}:\\ \;\;\;\;27 \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;\left(a \cdot 27\right) \cdot b \leq 2 \cdot 10^{-58}:\\ \;\;\;\;x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot 27\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \left(a \cdot 27\right) \cdot b\\ t_2 := 27 \cdot \left(a \cdot b\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+32}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-43}:\\ \;\;\;\;x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* a 27.0) b)) (t_2 (* 27.0 (* a b))))
   (if (<= t_1 -4e+32) t_2 (if (<= t_1 1e-43) (* x 2.0) t_2))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * 27.0) * b;
	double t_2 = 27.0 * (a * b);
	double tmp;
	if (t_1 <= -4e+32) {
		tmp = t_2;
	} else if (t_1 <= 1e-43) {
		tmp = x * 2.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (a * 27.0d0) * b
    t_2 = 27.0d0 * (a * b)
    if (t_1 <= (-4d+32)) then
        tmp = t_2
    else if (t_1 <= 1d-43) then
        tmp = x * 2.0d0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b;
public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * 27.0) * b;
	double t_2 = 27.0 * (a * b);
	double tmp;
	if (t_1 <= -4e+32) {
		tmp = t_2;
	} else if (t_1 <= 1e-43) {
		tmp = x * 2.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	t_1 = (a * 27.0) * b
	t_2 = 27.0 * (a * b)
	tmp = 0
	if t_1 <= -4e+32:
		tmp = t_2
	elif t_1 <= 1e-43:
		tmp = x * 2.0
	else:
		tmp = t_2
	return tmp

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a * 27.0) * b)
	t_2 = Float64(27.0 * Float64(a * b))
	tmp = 0.0
	if (t_1 <= -4e+32)
		tmp = t_2;
	elseif (t_1 <= 1e-43)
		tmp = Float64(x * 2.0);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a * 27.0) * b;
	t_2 = 27.0 * (a * b);
	tmp = 0.0;
	if (t_1 <= -4e+32)
		tmp = t_2;
	elseif (t_1 <= 1e-43)
		tmp = x * 2.0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$2 = N[(27.0 * N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+32], t$95$2, If[LessEqual[t$95$1, 1e-43], N[(x * 2.0), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 a #s(literal 27 binary64)) b) < -4.00000000000000021e32 or 1.00000000000000008e-43 < (*.f64 (*.f64 a #s(literal 27 binary64)) b)

   1. Initial program 89.9%

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

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

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

      7. associate-*l* N/A

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

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

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

      9. associate-*l* N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. *-lowering-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. *-lowering-*.f64 98.9

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

   4. Applied egg-rr 98.9%

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

   5. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{27 \cdot \left(a \cdot b\right)} \]

      2. *-lowering-*.f64 68.3

         \[\leadsto 27 \cdot \color{blue}{\left(a \cdot b\right)} \]

   7. Simplified 68.3%

      \[\leadsto \color{blue}{27 \cdot \left(a \cdot b\right)} \]

   if -4.00000000000000021e32 < (*.f64 (*.f64 a #s(literal 27 binary64)) b) < 1.00000000000000008e-43

   1. Initial program 96.5%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot x} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{2 \cdot x + 0} \]

      2. accelerator-lowering-fma.f64 43.9

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 0\right)} \]

   5. Simplified 43.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{2 \cdot x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot 2} \]

      3. *-lowering-*.f64 43.9

         \[\leadsto \color{blue}{x \cdot 2} \]

   7. Applied egg-rr 43.9%

      \[\leadsto \color{blue}{x \cdot 2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 97.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 9 \cdot 10^{+79}:\\ \;\;\;\;\mathsf{fma}\left(-9, y \cdot \left(z \cdot t\right), \mathsf{fma}\left(a, 27 \cdot b, x \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(z \cdot y, -9, 0\right), \mathsf{fma}\left(27, a \cdot b, 0\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (if (<= z 9e+79)
   (fma -9.0 (* y (* z t)) (fma a (* 27.0 b) (* x 2.0)))
   (fma t (fma (* z y) -9.0 0.0) (fma 27.0 (* a b) 0.0))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= 9e+79) {
		tmp = fma(-9.0, (y * (z * t)), fma(a, (27.0 * b), (x * 2.0)));
	} else {
		tmp = fma(t, fma((z * y), -9.0, 0.0), fma(27.0, (a * b), 0.0));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= 9e+79)
		tmp = fma(-9.0, Float64(y * Float64(z * t)), fma(a, Float64(27.0 * b), Float64(x * 2.0)));
	else
		tmp = fma(t, fma(Float64(z * y), -9.0, 0.0), fma(27.0, Float64(a * b), 0.0));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, 9e+79], N[(-9.0 * N[(y * N[(z * t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(27.0 * b), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(N[(z * y), $MachinePrecision] * -9.0 + 0.0), $MachinePrecision] + N[(27.0 * N[(a * b), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 8.99999999999999987e79

   1. Initial program 94.5%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

      8. +-commutative N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. metadata-eval N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-lowering-*.f64 96.4

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

   4. Applied egg-rr 96.4%

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

   if 8.99999999999999987e79 < z

   1. Initial program 90.4%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. +-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{0 + -9 \cdot \left(y \cdot z\right)}, 27 \cdot \left(a \cdot b\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{-9 \cdot \left(y \cdot z\right) + 0}, 27 \cdot \left(a \cdot b\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{\left(y \cdot z\right) \cdot -9} + 0, 27 \cdot \left(a \cdot b\right)\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{fma}\left(y \cdot z, -9, 0\right)}, 27 \cdot \left(a \cdot b\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(\color{blue}{y \cdot z}, -9, 0\right), 27 \cdot \left(a \cdot b\right)\right) \]

      13. +-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(y \cdot z, -9, 0\right), \color{blue}{27 \cdot \left(a \cdot b\right) + 0}\right) \]

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(y \cdot z, -9, 0\right), \color{blue}{\mathsf{fma}\left(27, a \cdot b, 0\right)}\right) \]

      15. *-lowering-*.f64 75.3

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(y \cdot z, -9, 0\right), \mathsf{fma}\left(27, \color{blue}{a \cdot b}, 0\right)\right) \]

   5. Simplified 75.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(y \cdot z, -9, 0\right), \mathsf{fma}\left(27, a \cdot b, 0\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 9 \cdot 10^{+79}:\\ \;\;\;\;\mathsf{fma}\left(-9, y \cdot \left(z \cdot t\right), \mathsf{fma}\left(a, 27 \cdot b, x \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(z \cdot y, -9, 0\right), \mathsf{fma}\left(27, a \cdot b, 0\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 96.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \mathsf{fma}\left(a \cdot 27, b, \mathsf{fma}\left(y, \left(z \cdot -9\right) \cdot t, x \cdot 2\right)\right) \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (fma (* a 27.0) b (fma y (* (* z -9.0) t) (* x 2.0))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	return fma((a * 27.0), b, fma(y, ((z * -9.0) * t), (x * 2.0)));
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	return fma(Float64(a * 27.0), b, fma(y, Float64(Float64(z * -9.0) * t), Float64(x * 2.0)))
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := N[(N[(a * 27.0), $MachinePrecision] * b + N[(y * N[(N[(z * -9.0), $MachinePrecision] * t), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.7%

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. *-lowering-*.f64 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

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

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

   7. associate-*l* N/A

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

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

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

   9. associate-*l* N/A

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

   10. accelerator-lowering-fma.f64 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. *-commutative N/A

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

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

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

   14. *-lowering-*.f64 N/A

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

   15. metadata-eval N/A

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

   16. *-lowering-*.f64 96.5

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

4. Applied egg-rr 96.5%

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

Alternative 15: 31.2% accurate, 6.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.7%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{2 \cdot x} \]
4. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{2 \cdot x + 0} \]

   2. accelerator-lowering-fma.f64 30.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 0\right)} \]

5. Simplified 30.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{2 \cdot x} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{x \cdot 2} \]

   3. *-lowering-*.f64 30.1

      \[\leadsto \color{blue}{x \cdot 2} \]

7. Applied egg-rr 30.1%

   \[\leadsto \color{blue}{x \cdot 2} \]
8. Add Preprocessing

Developer Target 1: 95.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y < 7.590524218811189 \cdot 10^{-161}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - 9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (< y 7.590524218811189e-161)
   (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* a (* 27.0 b)))
   (+ (- (* x 2.0) (* 9.0 (* y (* t z)))) (* (* a 27.0) b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y < 7.590524218811189e-161) {
		tmp = ((x * 2.0) - (((y * 9.0) * z) * t)) + (a * (27.0 * b));
	} else {
		tmp = ((x * 2.0) - (9.0 * (y * (t * z)))) + ((a * 27.0) * b);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y < 7.590524218811189d-161) then
        tmp = ((x * 2.0d0) - (((y * 9.0d0) * z) * t)) + (a * (27.0d0 * b))
    else
        tmp = ((x * 2.0d0) - (9.0d0 * (y * (t * z)))) + ((a * 27.0d0) * b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y < 7.590524218811189e-161) {
		tmp = ((x * 2.0) - (((y * 9.0) * z) * t)) + (a * (27.0 * b));
	} else {
		tmp = ((x * 2.0) - (9.0 * (y * (t * z)))) + ((a * 27.0) * b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y < 7.590524218811189e-161:
		tmp = ((x * 2.0) - (((y * 9.0) * z) * t)) + (a * (27.0 * b))
	else:
		tmp = ((x * 2.0) - (9.0 * (y * (t * z)))) + ((a * 27.0) * b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y < 7.590524218811189e-161)
		tmp = Float64(Float64(Float64(x * 2.0) - Float64(Float64(Float64(y * 9.0) * z) * t)) + Float64(a * Float64(27.0 * b)));
	else
		tmp = Float64(Float64(Float64(x * 2.0) - Float64(9.0 * Float64(y * Float64(t * z)))) + Float64(Float64(a * 27.0) * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y < 7.590524218811189e-161)
		tmp = ((x * 2.0) - (((y * 9.0) * z) * t)) + (a * (27.0 * b));
	else
		tmp = ((x * 2.0) - (9.0 * (y * (t * z)))) + ((a * 27.0) * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Less[y, 7.590524218811189e-161], N[(N[(N[(x * 2.0), $MachinePrecision] - N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(27.0 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * 2.0), $MachinePrecision] - N[(9.0 * N[(y * N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024195 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y 7590524218811189/100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (- (* x 2) (* (* (* y 9) z) t)) (* a (* 27 b))) (+ (- (* x 2) (* 9 (* y (* t z)))) (* (* a 27) b))))

  (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* (* a 27.0) b)))