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

Percentage Accurate: 91.1% → 96.0%

Time: 12.1s

Alternatives: 14

Speedup: 0.4×

• 28.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (x y z t a)
 :precision binary64
 (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))

C

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

Fortran

real(8) function code(x, y, z, t, a)
    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
    code = ((x * y) - ((z * 9.0d0) * t)) / (a * 2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t a)
 :precision binary64
 (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))

C

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

Fortran

real(8) function code(x, y, z, t, a)
    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
    code = ((x * y) - ((z * 9.0d0) * t)) / (a * 2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \frac{y}{a} \cdot \left(-4.5 \cdot \left(t \cdot \frac{z}{y}\right) + x \cdot 0.5\right)\\ t_2 := \frac{x \cdot y - t \cdot \left(z \cdot 9\right)}{a \cdot 2}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+303}:\\ \;\;\;\;\frac{\frac{x \cdot y}{a}}{2} - \frac{z \cdot \left(t \cdot 4.5\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ y a) (+ (* -4.5 (* t (/ z y))) (* x 0.5))))
        (t_2 (/ (- (* x y) (* t (* z 9.0))) (* a 2.0))))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 5e+303)
       (- (/ (/ (* x y) a) 2.0) (/ (* z (* t 4.5)) a))
       t_1))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (y / a) * ((-4.5 * (t * (z / y))) + (x * 0.5));
	double t_2 = ((x * y) - (t * (z * 9.0))) / (a * 2.0);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 5e+303) {
		tmp = (((x * y) / a) / 2.0) - ((z * (t * 4.5)) / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y / a) * ((-4.5 * (t * (z / y))) + (x * 0.5));
	double t_2 = ((x * y) - (t * (z * 9.0))) / (a * 2.0);
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= 5e+303) {
		tmp = (((x * y) / a) / 2.0) - ((z * (t * 4.5)) / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = (y / a) * ((-4.5 * (t * (z / y))) + (x * 0.5))
	t_2 = ((x * y) - (t * (z * 9.0))) / (a * 2.0)
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= 5e+303:
		tmp = (((x * y) / a) / 2.0) - ((z * (t * 4.5)) / a)
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(y / a) * Float64(Float64(-4.5 * Float64(t * Float64(z / y))) + Float64(x * 0.5)))
	t_2 = Float64(Float64(Float64(x * y) - Float64(t * Float64(z * 9.0))) / Float64(a * 2.0))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 5e+303)
		tmp = Float64(Float64(Float64(Float64(x * y) / a) / 2.0) - Float64(Float64(z * Float64(t * 4.5)) / a));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (y / a) * ((-4.5 * (t * (z / y))) + (x * 0.5));
	t_2 = ((x * y) - (t * (z * 9.0))) / (a * 2.0);
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= 5e+303)
		tmp = (((x * y) / a) / 2.0) - ((z * (t * 4.5)) / a);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / a), $MachinePrecision] * N[(N[(-4.5 * N[(t * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * y), $MachinePrecision] - N[(t * N[(z * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 5e+303], N[(N[(N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision] / 2.0), $MachinePrecision] - N[(N[(z * N[(t * 4.5), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) (*.f64 a #s(literal 2 binary64))) < -inf.0 or 4.9999999999999997e303 < (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) (*.f64 a #s(literal 2 binary64)))

   1. Initial program 77.9%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right), \color{blue}{\left(a \cdot 2\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y + \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(a \cdot 2\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(z \cdot \left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(t \cdot 9\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(t \cdot \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \left(a \cdot 2\right)\right) \]

      12. *-lowering-*.f64 77.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{2}\right)\right) \]

   3. Simplified 77.9%

      \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t \cdot -9\right)}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right) + \color{blue}{x \cdot \left(\frac{1}{2} \cdot \frac{y}{a}\right)} \]

      2. *-commutative N/A

         \[\leadsto x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right) + x \cdot \left(\frac{y}{a} \cdot \color{blue}{\frac{1}{2}}\right) \]

      3. associate-*r* N/A

         \[\leadsto x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right) + \left(x \cdot \frac{y}{a}\right) \cdot \color{blue}{\frac{1}{2}} \]

      4. associate-/l* N/A

         \[\leadsto x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right) + \frac{x \cdot y}{a} \cdot \frac{1}{2} \]

      5. *-commutative N/A

         \[\leadsto x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right) + \frac{1}{2} \cdot \color{blue}{\frac{x \cdot y}{a}} \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right) \cdot x + \color{blue}{\frac{1}{2}} \cdot \frac{x \cdot y}{a} \]

      7. associate-*l* N/A

         \[\leadsto \frac{-9}{2} \cdot \left(\frac{t \cdot z}{a \cdot x} \cdot x\right) + \color{blue}{\frac{1}{2}} \cdot \frac{x \cdot y}{a} \]

      8. fma-define N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2}, \color{blue}{\frac{t \cdot z}{a \cdot x} \cdot x}, \frac{1}{2} \cdot \frac{x \cdot y}{a}\right) \]

      9. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2}, \frac{\left(t \cdot z\right) \cdot x}{\color{blue}{a \cdot x}}, \frac{1}{2} \cdot \frac{x \cdot y}{a}\right) \]

      10. times-frac N/A

          \[\leadsto \mathsf{fma}\left(\frac{-9}{2}, \frac{t \cdot z}{a} \cdot \color{blue}{\frac{x}{x}}, \frac{1}{2} \cdot \frac{x \cdot y}{a}\right) \]

      11. *-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{-9}{2}, \frac{t \cdot z}{a} \cdot 1, \frac{1}{2} \cdot \frac{x \cdot y}{a}\right) \]

      12. *-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{-9}{2}, \frac{t \cdot z}{a} \cdot \frac{y}{\color{blue}{y}}, \frac{1}{2} \cdot \frac{x \cdot y}{a}\right) \]

      13. times-frac N/A

          \[\leadsto \mathsf{fma}\left(\frac{-9}{2}, \frac{\left(t \cdot z\right) \cdot y}{\color{blue}{a \cdot y}}, \frac{1}{2} \cdot \frac{x \cdot y}{a}\right) \]

      14. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{-9}{2}, \frac{t \cdot z}{a \cdot y} \cdot \color{blue}{y}, \frac{1}{2} \cdot \frac{x \cdot y}{a}\right) \]

      15. fma-define N/A

          \[\leadsto \frac{-9}{2} \cdot \left(\frac{t \cdot z}{a \cdot y} \cdot y\right) + \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{a}} \]

      16. associate-*l* N/A

          \[\leadsto \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot y}\right) \cdot y + \color{blue}{\frac{1}{2}} \cdot \frac{x \cdot y}{a} \]

   7. Simplified 90.6%

      \[\leadsto \color{blue}{\frac{y}{a} \cdot \left(-4.5 \cdot \left(t \cdot \frac{z}{y}\right) + 0.5 \cdot x\right)} \]

   if -inf.0 < (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) (*.f64 a #s(literal 2 binary64))) < 4.9999999999999997e303

   1. Initial program 98.8%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right), \color{blue}{\left(a \cdot 2\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y + \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(a \cdot 2\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(z \cdot \left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(t \cdot 9\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(t \cdot \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \left(a \cdot 2\right)\right) \]

      12. *-lowering-*.f64 98.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{2}\right)\right) \]

   3. Simplified 98.2%

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

      1. frac-2neg N/A

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

      2. distribute-neg-in N/A

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

      3. unsub-neg N/A

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

      4. sub-div N/A

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

      5. frac-2neg N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

         \[\leadsto \frac{x \cdot y}{a \cdot 2} - \frac{\left(z \cdot \left(\mathsf{neg}\left(9\right)\right)\right) \cdot t}{\mathsf{neg}\left(a \cdot 2\right)} \]

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

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

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

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

      11. frac-2neg N/A

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

      12. div-sub N/A

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

      13. div-inv N/A

          \[\leadsto \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right) \cdot \color{blue}{\frac{1}{a \cdot 2}} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{a \cdot 2} \cdot \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{a \cdot 2}\right), \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)}\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2 \cdot a}\right), \left(x \cdot \color{blue}{y} - \left(z \cdot 9\right) \cdot t\right)\right) \]

      17. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{a}\right), \left(\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2}\right), a\right), \left(\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t\right)\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \left(\color{blue}{x} \cdot y - \left(z \cdot 9\right) \cdot t\right)\right) \]

   6. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \left(x \cdot y + z \cdot \left(t \cdot -9\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \left(x \cdot y\right) \cdot \frac{\frac{1}{2}}{a} + \color{blue}{\left(z \cdot \left(t \cdot -9\right)\right) \cdot \frac{\frac{1}{2}}{a}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot \frac{1}{2}}{a} + \color{blue}{\left(z \cdot \left(t \cdot -9\right)\right)} \cdot \frac{\frac{1}{2}}{a} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left(y \cdot x\right) \cdot \frac{1}{2}}{a} + \left(z \cdot \left(t \cdot -9\right)\right) \cdot \frac{\frac{1}{2}}{a} \]

      4. associate-*r* N/A

         \[\leadsto \frac{y \cdot \left(x \cdot \frac{1}{2}\right)}{a} + \left(\color{blue}{z} \cdot \left(t \cdot -9\right)\right) \cdot \frac{\frac{1}{2}}{a} \]

      5. metadata-eval N/A

         \[\leadsto \frac{y \cdot \left(x \cdot \frac{1}{2}\right)}{a} + \left(z \cdot \left(t \cdot -9\right)\right) \cdot \frac{\frac{1}{2}}{a} \]

      6. div-inv N/A

         \[\leadsto \frac{y \cdot \frac{x}{2}}{a} + \left(z \cdot \left(t \cdot -9\right)\right) \cdot \frac{\frac{1}{2}}{a} \]

      7. associate-*l/ N/A

         \[\leadsto \frac{y}{a} \cdot \frac{x}{2} + \color{blue}{\left(z \cdot \left(t \cdot -9\right)\right)} \cdot \frac{\frac{1}{2}}{a} \]

      8. clear-num N/A

         \[\leadsto \frac{y}{a} \cdot \frac{x}{2} + \left(z \cdot \left(t \cdot -9\right)\right) \cdot \frac{1}{\color{blue}{\frac{a}{\frac{1}{2}}}} \]

      9. div-inv N/A

         \[\leadsto \frac{y}{a} \cdot \frac{x}{2} + \left(z \cdot \left(t \cdot -9\right)\right) \cdot \frac{1}{a \cdot \color{blue}{\frac{1}{\frac{1}{2}}}} \]

      10. metadata-eval N/A

          \[\leadsto \frac{y}{a} \cdot \frac{x}{2} + \left(z \cdot \left(t \cdot -9\right)\right) \cdot \frac{1}{a \cdot 2} \]

      11. un-div-inv N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

          \[\leadsto \frac{y}{a} \cdot \frac{x}{2} + \frac{t \cdot z}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      15. *-commutative N/A

          \[\leadsto \frac{y}{a} \cdot \frac{x}{2} + \frac{z \cdot t}{a} \cdot \frac{-9}{2} \]

      16. associate-/l* N/A

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

      17. metadata-eval N/A

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

      18. associate-*r* N/A

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

      19. metadata-eval N/A

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

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

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

      21. *-commutative N/A

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

   8. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{a}}{2} - \frac{z \cdot \left(t \cdot 4.5\right)}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y - t \cdot \left(z \cdot 9\right)}{a \cdot 2} \leq -\infty:\\ \;\;\;\;\frac{y}{a} \cdot \left(-4.5 \cdot \left(t \cdot \frac{z}{y}\right) + x \cdot 0.5\right)\\ \mathbf{elif}\;\frac{x \cdot y - t \cdot \left(z \cdot 9\right)}{a \cdot 2} \leq 5 \cdot 10^{+303}:\\ \;\;\;\;\frac{\frac{x \cdot y}{a}}{2} - \frac{z \cdot \left(t \cdot 4.5\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(-4.5 \cdot \left(t \cdot \frac{z}{y}\right) + x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := 4.5 \cdot \left(0 - \frac{z}{a}\right)\\ \mathbf{if}\;a \cdot 2 \leq 10^{+21}:\\ \;\;\;\;\frac{x \cdot y - \left(z \cdot t\right) \cdot 9}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, \frac{x}{2}, t \cdot t\_1\right) + \mathsf{fma}\left(t\_1, t, t \cdot \left(\frac{z}{a} \cdot 4.5\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* 4.5 (- 0.0 (/ z a)))))
   (if (<= (* a 2.0) 1e+21)
     (/ (- (* x y) (* (* z t) 9.0)) (* a 2.0))
     (+ (fma (/ y a) (/ x 2.0) (* t t_1)) (fma t_1 t (* t (* (/ z a) 4.5)))))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = 4.5 * (0.0 - (z / a));
	double tmp;
	if ((a * 2.0) <= 1e+21) {
		tmp = ((x * y) - ((z * t) * 9.0)) / (a * 2.0);
	} else {
		tmp = fma((y / a), (x / 2.0), (t * t_1)) + fma(t_1, t, (t * ((z / a) * 4.5)));
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(4.5 * Float64(0.0 - Float64(z / a)))
	tmp = 0.0
	if (Float64(a * 2.0) <= 1e+21)
		tmp = Float64(Float64(Float64(x * y) - Float64(Float64(z * t) * 9.0)) / Float64(a * 2.0));
	else
		tmp = Float64(fma(Float64(y / a), Float64(x / 2.0), Float64(t * t_1)) + fma(t_1, t, Float64(t * Float64(Float64(z / a) * 4.5))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(4.5 * N[(0.0 - N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * 2.0), $MachinePrecision], 1e+21], N[(N[(N[(x * y), $MachinePrecision] - N[(N[(z * t), $MachinePrecision] * 9.0), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / a), $MachinePrecision] * N[(x / 2.0), $MachinePrecision] + N[(t * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * t + N[(t * N[(N[(z / a), $MachinePrecision] * 4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a #s(literal 2 binary64)) < 1e21

   1. Initial program 92.5%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(t \cdot \left(z \cdot 9\right)\right)\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(t \cdot z\right) \cdot 9\right)\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(z \cdot t\right) \cdot 9\right)\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(z \cdot t\right), 9\right)\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      5. *-lowering-*.f64 92.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, t\right), 9\right)\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

   4. Applied egg-rr 92.5%

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

   if 1e21 < (*.f64 a #s(literal 2 binary64))

   1. Initial program 82.0%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right), \color{blue}{\left(a \cdot 2\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y + \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(a \cdot 2\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(z \cdot \left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(t \cdot 9\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(t \cdot \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \left(a \cdot 2\right)\right) \]

      12. *-lowering-*.f64 82.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{2}\right)\right) \]

   3. Simplified 82.0%

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

      1. frac-2neg N/A

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

      2. distribute-neg-in N/A

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

      3. unsub-neg N/A

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

      4. sub-div N/A

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

      5. frac-2neg N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

         \[\leadsto \frac{x \cdot y}{a \cdot 2} - \frac{\left(z \cdot \left(\mathsf{neg}\left(9\right)\right)\right) \cdot t}{\mathsf{neg}\left(a \cdot 2\right)} \]

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

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

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

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

      11. frac-2neg N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

          \[\leadsto \frac{x \cdot y}{a \cdot 2} - t \cdot \color{blue}{\frac{z \cdot 9}{a \cdot 2}} \]

      14. *-commutative N/A

          \[\leadsto \frac{y \cdot x}{a \cdot 2} - t \cdot \frac{z \cdot 9}{a \cdot 2} \]

      15. times-frac N/A

          \[\leadsto \frac{y}{a} \cdot \frac{x}{2} - \color{blue}{t} \cdot \frac{z \cdot 9}{a \cdot 2} \]

      16. prod-diff N/A

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

   6. Applied egg-rr 95.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a}, \frac{x}{2}, -\left(\frac{z}{a} \cdot 4.5\right) \cdot t\right) + \mathsf{fma}\left(-\frac{z}{a} \cdot 4.5, t, \left(\frac{z}{a} \cdot 4.5\right) \cdot t\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq 10^{+21}:\\ \;\;\;\;\frac{x \cdot y - \left(z \cdot t\right) \cdot 9}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, \frac{x}{2}, t \cdot \left(4.5 \cdot \left(0 - \frac{z}{a}\right)\right)\right) + \mathsf{fma}\left(4.5 \cdot \left(0 - \frac{z}{a}\right), t, t \cdot \left(\frac{z}{a} \cdot 4.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := 4.5 \cdot \frac{t}{a}\\ t_2 := 0 - t\_1\\ \mathbf{if}\;a \cdot 2 \leq 2 \cdot 10^{+35}:\\ \;\;\;\;\frac{x \cdot y - \left(z \cdot t\right) \cdot 9}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, \frac{x}{2}, z \cdot t\_2\right) + \mathsf{fma}\left(t\_2, z, z \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* 4.5 (/ t a))) (t_2 (- 0.0 t_1)))
   (if (<= (* a 2.0) 2e+35)
     (/ (- (* x y) (* (* z t) 9.0)) (* a 2.0))
     (+ (fma (/ y a) (/ x 2.0) (* z t_2)) (fma t_2 z (* z t_1))))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = 4.5 * (t / a);
	double t_2 = 0.0 - t_1;
	double tmp;
	if ((a * 2.0) <= 2e+35) {
		tmp = ((x * y) - ((z * t) * 9.0)) / (a * 2.0);
	} else {
		tmp = fma((y / a), (x / 2.0), (z * t_2)) + fma(t_2, z, (z * t_1));
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(4.5 * Float64(t / a))
	t_2 = Float64(0.0 - t_1)
	tmp = 0.0
	if (Float64(a * 2.0) <= 2e+35)
		tmp = Float64(Float64(Float64(x * y) - Float64(Float64(z * t) * 9.0)) / Float64(a * 2.0));
	else
		tmp = Float64(fma(Float64(y / a), Float64(x / 2.0), Float64(z * t_2)) + fma(t_2, z, Float64(z * t_1)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(4.5 * N[(t / a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.0 - t$95$1), $MachinePrecision]}, If[LessEqual[N[(a * 2.0), $MachinePrecision], 2e+35], N[(N[(N[(x * y), $MachinePrecision] - N[(N[(z * t), $MachinePrecision] * 9.0), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / a), $MachinePrecision] * N[(x / 2.0), $MachinePrecision] + N[(z * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * z + N[(z * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a #s(literal 2 binary64)) < 1.9999999999999999e35

   1. Initial program 92.6%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(t \cdot \left(z \cdot 9\right)\right)\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(t \cdot z\right) \cdot 9\right)\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(z \cdot t\right) \cdot 9\right)\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(z \cdot t\right), 9\right)\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      5. *-lowering-*.f64 92.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, t\right), 9\right)\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

   4. Applied egg-rr 92.6%

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

   if 1.9999999999999999e35 < (*.f64 a #s(literal 2 binary64))

   1. Initial program 80.5%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right), \color{blue}{\left(a \cdot 2\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y + \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(a \cdot 2\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(z \cdot \left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(t \cdot 9\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(t \cdot \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \left(a \cdot 2\right)\right) \]

      12. *-lowering-*.f64 80.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{2}\right)\right) \]

   3. Simplified 80.5%

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

      1. frac-2neg N/A

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

      2. distribute-neg-in N/A

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

      3. unsub-neg N/A

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

      4. sub-div N/A

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

      5. frac-2neg N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

         \[\leadsto \frac{x \cdot y}{a \cdot 2} - \frac{\left(z \cdot \left(\mathsf{neg}\left(9\right)\right)\right) \cdot t}{\mathsf{neg}\left(a \cdot 2\right)} \]

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

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

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

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

      11. frac-2neg N/A

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

      12. associate-*l* N/A

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

      13. associate-/l* N/A

          \[\leadsto \frac{x \cdot y}{a \cdot 2} - z \cdot \color{blue}{\frac{9 \cdot t}{a \cdot 2}} \]

      14. *-commutative N/A

          \[\leadsto \frac{y \cdot x}{a \cdot 2} - z \cdot \frac{9 \cdot t}{a \cdot 2} \]

      15. times-frac N/A

          \[\leadsto \frac{y}{a} \cdot \frac{x}{2} - \color{blue}{z} \cdot \frac{9 \cdot t}{a \cdot 2} \]

      16. prod-diff N/A

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

   6. Applied egg-rr 93.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a}, \frac{x}{2}, -\left(\frac{t}{a} \cdot 4.5\right) \cdot z\right) + \mathsf{fma}\left(-\frac{t}{a} \cdot 4.5, z, \left(\frac{t}{a} \cdot 4.5\right) \cdot z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq 2 \cdot 10^{+35}:\\ \;\;\;\;\frac{x \cdot y - \left(z \cdot t\right) \cdot 9}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, \frac{x}{2}, z \cdot \left(0 - 4.5 \cdot \frac{t}{a}\right)\right) + \mathsf{fma}\left(0 - 4.5 \cdot \frac{t}{a}, z, z \cdot \left(4.5 \cdot \frac{t}{a}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \frac{y}{a} \cdot \left(-4.5 \cdot \left(t \cdot \frac{z}{y}\right) + x \cdot 0.5\right)\\ t_2 := x \cdot y - t \cdot \left(z \cdot 9\right)\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+272}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+281}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(x \cdot y + z \cdot \left(t \cdot -9\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ y a) (+ (* -4.5 (* t (/ z y))) (* x 0.5))))
        (t_2 (- (* x y) (* t (* z 9.0)))))
   (if (<= t_2 -5e+272)
     t_1
     (if (<= t_2 1e+281) (* (/ 0.5 a) (+ (* x y) (* z (* t -9.0)))) t_1))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (y / a) * ((-4.5 * (t * (z / y))) + (x * 0.5));
	double t_2 = (x * y) - (t * (z * 9.0));
	double tmp;
	if (t_2 <= -5e+272) {
		tmp = t_1;
	} else if (t_2 <= 1e+281) {
		tmp = (0.5 / a) * ((x * y) + (z * (t * -9.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (y / a) * (((-4.5d0) * (t * (z / y))) + (x * 0.5d0))
    t_2 = (x * y) - (t * (z * 9.0d0))
    if (t_2 <= (-5d+272)) then
        tmp = t_1
    else if (t_2 <= 1d+281) then
        tmp = (0.5d0 / a) * ((x * y) + (z * (t * (-9.0d0))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y / a) * ((-4.5 * (t * (z / y))) + (x * 0.5));
	double t_2 = (x * y) - (t * (z * 9.0));
	double tmp;
	if (t_2 <= -5e+272) {
		tmp = t_1;
	} else if (t_2 <= 1e+281) {
		tmp = (0.5 / a) * ((x * y) + (z * (t * -9.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = (y / a) * ((-4.5 * (t * (z / y))) + (x * 0.5))
	t_2 = (x * y) - (t * (z * 9.0))
	tmp = 0
	if t_2 <= -5e+272:
		tmp = t_1
	elif t_2 <= 1e+281:
		tmp = (0.5 / a) * ((x * y) + (z * (t * -9.0)))
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(y / a) * Float64(Float64(-4.5 * Float64(t * Float64(z / y))) + Float64(x * 0.5)))
	t_2 = Float64(Float64(x * y) - Float64(t * Float64(z * 9.0)))
	tmp = 0.0
	if (t_2 <= -5e+272)
		tmp = t_1;
	elseif (t_2 <= 1e+281)
		tmp = Float64(Float64(0.5 / a) * Float64(Float64(x * y) + Float64(z * Float64(t * -9.0))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (y / a) * ((-4.5 * (t * (z / y))) + (x * 0.5));
	t_2 = (x * y) - (t * (z * 9.0));
	tmp = 0.0;
	if (t_2 <= -5e+272)
		tmp = t_1;
	elseif (t_2 <= 1e+281)
		tmp = (0.5 / a) * ((x * y) + (z * (t * -9.0)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / a), $MachinePrecision] * N[(N[(-4.5 * N[(t * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] - N[(t * N[(z * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+272], t$95$1, If[LessEqual[t$95$2, 1e+281], N[(N[(0.5 / a), $MachinePrecision] * N[(N[(x * y), $MachinePrecision] + N[(z * N[(t * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < -4.99999999999999973e272 or 1e281 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t))

   1. Initial program 68.3%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right), \color{blue}{\left(a \cdot 2\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y + \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(a \cdot 2\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(z \cdot \left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(t \cdot 9\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(t \cdot \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \left(a \cdot 2\right)\right) \]

      12. *-lowering-*.f64 68.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{2}\right)\right) \]

   3. Simplified 68.3%

      \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t \cdot -9\right)}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right) + \color{blue}{x \cdot \left(\frac{1}{2} \cdot \frac{y}{a}\right)} \]

      2. *-commutative N/A

         \[\leadsto x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right) + x \cdot \left(\frac{y}{a} \cdot \color{blue}{\frac{1}{2}}\right) \]

      3. associate-*r* N/A

         \[\leadsto x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right) + \left(x \cdot \frac{y}{a}\right) \cdot \color{blue}{\frac{1}{2}} \]

      4. associate-/l* N/A

         \[\leadsto x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right) + \frac{x \cdot y}{a} \cdot \frac{1}{2} \]

      5. *-commutative N/A

         \[\leadsto x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right) + \frac{1}{2} \cdot \color{blue}{\frac{x \cdot y}{a}} \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right) \cdot x + \color{blue}{\frac{1}{2}} \cdot \frac{x \cdot y}{a} \]

      7. associate-*l* N/A

         \[\leadsto \frac{-9}{2} \cdot \left(\frac{t \cdot z}{a \cdot x} \cdot x\right) + \color{blue}{\frac{1}{2}} \cdot \frac{x \cdot y}{a} \]

      8. fma-define N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2}, \color{blue}{\frac{t \cdot z}{a \cdot x} \cdot x}, \frac{1}{2} \cdot \frac{x \cdot y}{a}\right) \]

      9. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2}, \frac{\left(t \cdot z\right) \cdot x}{\color{blue}{a \cdot x}}, \frac{1}{2} \cdot \frac{x \cdot y}{a}\right) \]

      10. times-frac N/A

          \[\leadsto \mathsf{fma}\left(\frac{-9}{2}, \frac{t \cdot z}{a} \cdot \color{blue}{\frac{x}{x}}, \frac{1}{2} \cdot \frac{x \cdot y}{a}\right) \]

      11. *-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{-9}{2}, \frac{t \cdot z}{a} \cdot 1, \frac{1}{2} \cdot \frac{x \cdot y}{a}\right) \]

      12. *-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{-9}{2}, \frac{t \cdot z}{a} \cdot \frac{y}{\color{blue}{y}}, \frac{1}{2} \cdot \frac{x \cdot y}{a}\right) \]

      13. times-frac N/A

          \[\leadsto \mathsf{fma}\left(\frac{-9}{2}, \frac{\left(t \cdot z\right) \cdot y}{\color{blue}{a \cdot y}}, \frac{1}{2} \cdot \frac{x \cdot y}{a}\right) \]

      14. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{-9}{2}, \frac{t \cdot z}{a \cdot y} \cdot \color{blue}{y}, \frac{1}{2} \cdot \frac{x \cdot y}{a}\right) \]

      15. fma-define N/A

          \[\leadsto \frac{-9}{2} \cdot \left(\frac{t \cdot z}{a \cdot y} \cdot y\right) + \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{a}} \]

      16. associate-*l* N/A

          \[\leadsto \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot y}\right) \cdot y + \color{blue}{\frac{1}{2}} \cdot \frac{x \cdot y}{a} \]

   7. Simplified 87.9%

      \[\leadsto \color{blue}{\frac{y}{a} \cdot \left(-4.5 \cdot \left(t \cdot \frac{z}{y}\right) + 0.5 \cdot x\right)} \]

   if -4.99999999999999973e272 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < 1e281

   1. Initial program 99.0%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right), \color{blue}{\left(a \cdot 2\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y + \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(a \cdot 2\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(z \cdot \left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(t \cdot 9\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(t \cdot \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \left(a \cdot 2\right)\right) \]

      12. *-lowering-*.f64 98.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{2}\right)\right) \]

   3. Simplified 98.5%

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

      1. frac-2neg N/A

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

      2. distribute-neg-in N/A

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

      3. unsub-neg N/A

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

      4. sub-div N/A

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

      5. frac-2neg N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

         \[\leadsto \frac{x \cdot y}{a \cdot 2} - \frac{\left(z \cdot \left(\mathsf{neg}\left(9\right)\right)\right) \cdot t}{\mathsf{neg}\left(a \cdot 2\right)} \]

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

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

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

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

      11. frac-2neg N/A

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

      12. div-sub N/A

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

      13. div-inv N/A

          \[\leadsto \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right) \cdot \color{blue}{\frac{1}{a \cdot 2}} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{a \cdot 2} \cdot \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{a \cdot 2}\right), \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)}\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2 \cdot a}\right), \left(x \cdot \color{blue}{y} - \left(z \cdot 9\right) \cdot t\right)\right) \]

      17. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{a}\right), \left(\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2}\right), a\right), \left(\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t\right)\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \left(\color{blue}{x} \cdot y - \left(z \cdot 9\right) \cdot t\right)\right) \]

   6. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \left(x \cdot y + z \cdot \left(t \cdot -9\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - t \cdot \left(z \cdot 9\right) \leq -5 \cdot 10^{+272}:\\ \;\;\;\;\frac{y}{a} \cdot \left(-4.5 \cdot \left(t \cdot \frac{z}{y}\right) + x \cdot 0.5\right)\\ \mathbf{elif}\;x \cdot y - t \cdot \left(z \cdot 9\right) \leq 10^{+281}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(x \cdot y + z \cdot \left(t \cdot -9\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(-4.5 \cdot \left(t \cdot \frac{z}{y}\right) + x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := t \cdot \left(z \cdot 9\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+185}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(x \cdot y + z \cdot \left(t \cdot -9\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{z}{a} \cdot -4.5\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* t (* z 9.0))))
   (if (<= t_1 (- INFINITY))
     (* -4.5 (* z (/ t a)))
     (if (<= t_1 4e+185)
       (* (/ 0.5 a) (+ (* x y) (* z (* t -9.0))))
       (* t (* (/ z a) -4.5))))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = t * (z * 9.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -4.5 * (z * (t / a));
	} else if (t_1 <= 4e+185) {
		tmp = (0.5 / a) * ((x * y) + (z * (t * -9.0)));
	} else {
		tmp = t * ((z / a) * -4.5);
	}
	return tmp;
}

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t * (z * 9.0);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -4.5 * (z * (t / a));
	} else if (t_1 <= 4e+185) {
		tmp = (0.5 / a) * ((x * y) + (z * (t * -9.0)));
	} else {
		tmp = t * ((z / a) * -4.5);
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = t * (z * 9.0)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -4.5 * (z * (t / a))
	elif t_1 <= 4e+185:
		tmp = (0.5 / a) * ((x * y) + (z * (t * -9.0)))
	else:
		tmp = t * ((z / a) * -4.5)
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(t * Float64(z * 9.0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-4.5 * Float64(z * Float64(t / a)));
	elseif (t_1 <= 4e+185)
		tmp = Float64(Float64(0.5 / a) * Float64(Float64(x * y) + Float64(z * Float64(t * -9.0))));
	else
		tmp = Float64(t * Float64(Float64(z / a) * -4.5));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = t * (z * 9.0);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -4.5 * (z * (t / a));
	elseif (t_1 <= 4e+185)
		tmp = (0.5 / a) * ((x * y) + (z * (t * -9.0)));
	else
		tmp = t * ((z / a) * -4.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * N[(z * 9.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(-4.5 * N[(z * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+185], N[(N[(0.5 / a), $MachinePrecision] * N[(N[(x * y), $MachinePrecision] + N[(z * N[(t * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(N[(z / a), $MachinePrecision] * -4.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -inf.0

   1. Initial program 62.0%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right), \color{blue}{\left(a \cdot 2\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y + \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(a \cdot 2\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(z \cdot \left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(t \cdot 9\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(t \cdot \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \left(a \cdot 2\right)\right) \]

      12. *-lowering-*.f64 62.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{2}\right)\right) \]

   3. Simplified 62.0%

      \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t \cdot -9\right)}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \color{blue}{\left(\frac{t \cdot z}{a}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \mathsf{/.f64}\left(\left(t \cdot z\right), \color{blue}{a}\right)\right) \]

      3. *-lowering-*.f64 68.6%

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, z\right), a\right)\right) \]

   7. Simplified 68.6%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \left(\frac{z \cdot t}{a}\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \left(z \cdot \color{blue}{\frac{t}{a}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{t}{a}\right)}\right)\right) \]

      4. /-lowering-/.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(t, \color{blue}{a}\right)\right)\right) \]

   9. Applied egg-rr 99.9%

      \[\leadsto -4.5 \cdot \color{blue}{\left(z \cdot \frac{t}{a}\right)} \]

   if -inf.0 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 3.9999999999999999e185

   1. Initial program 94.8%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right), \color{blue}{\left(a \cdot 2\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y + \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(a \cdot 2\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(z \cdot \left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(t \cdot 9\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(t \cdot \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \left(a \cdot 2\right)\right) \]

      12. *-lowering-*.f64 94.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{2}\right)\right) \]

   3. Simplified 94.3%

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

      1. frac-2neg N/A

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

      2. distribute-neg-in N/A

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

      3. unsub-neg N/A

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

      4. sub-div N/A

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

      5. frac-2neg N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

         \[\leadsto \frac{x \cdot y}{a \cdot 2} - \frac{\left(z \cdot \left(\mathsf{neg}\left(9\right)\right)\right) \cdot t}{\mathsf{neg}\left(a \cdot 2\right)} \]

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

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

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

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

      11. frac-2neg N/A

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

      12. div-sub N/A

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

      13. div-inv N/A

          \[\leadsto \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right) \cdot \color{blue}{\frac{1}{a \cdot 2}} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{a \cdot 2} \cdot \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{a \cdot 2}\right), \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)}\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2 \cdot a}\right), \left(x \cdot \color{blue}{y} - \left(z \cdot 9\right) \cdot t\right)\right) \]

      17. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{a}\right), \left(\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2}\right), a\right), \left(\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t\right)\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \left(\color{blue}{x} \cdot y - \left(z \cdot 9\right) \cdot t\right)\right) \]

   6. Applied egg-rr 94.7%

      \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \left(x \cdot y + z \cdot \left(t \cdot -9\right)\right)} \]

   if 3.9999999999999999e185 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 72.1%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right), \color{blue}{\left(a \cdot 2\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y + \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(a \cdot 2\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(z \cdot \left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(t \cdot 9\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(t \cdot \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \left(a \cdot 2\right)\right) \]

      12. *-lowering-*.f64 72.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{2}\right)\right) \]

   3. Simplified 72.1%

      \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t \cdot -9\right)}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \color{blue}{\left(\frac{t \cdot z}{a}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \mathsf{/.f64}\left(\left(t \cdot z\right), \color{blue}{a}\right)\right) \]

      3. *-lowering-*.f64 72.1%

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, z\right), a\right)\right) \]

   7. Simplified 72.1%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      2. associate-/l* N/A

         \[\leadsto \left(t \cdot \frac{z}{a}\right) \cdot \frac{-9}{2} \]

      3. associate-*l* N/A

         \[\leadsto t \cdot \color{blue}{\left(\frac{z}{a} \cdot \frac{-9}{2}\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{z}{a} \cdot \frac{-9}{2}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(\frac{z}{a}\right), \color{blue}{\frac{-9}{2}}\right)\right) \]

      6. /-lowering-/.f64 96.3%

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, a\right), \frac{-9}{2}\right)\right) \]

   9. Applied egg-rr 96.3%

      \[\leadsto \color{blue}{t \cdot \left(\frac{z}{a} \cdot -4.5\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot 9\right) \leq -\infty:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \mathbf{elif}\;t \cdot \left(z \cdot 9\right) \leq 4 \cdot 10^{+185}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(x \cdot y + z \cdot \left(t \cdot -9\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{z}{a} \cdot -4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{\frac{y}{2}}{a}\\ \mathbf{elif}\;x \cdot y \leq 10^{-38}:\\ \;\;\;\;\frac{t \cdot \left(z \cdot -9\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{\frac{x}{2}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -2e-14)
   (* x (/ (/ y 2.0) a))
   (if (<= (* x y) 1e-38)
     (/ (* t (* z -9.0)) (* a 2.0))
     (/ y (/ a (/ x 2.0))))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e-14) {
		tmp = x * ((y / 2.0) / a);
	} else if ((x * y) <= 1e-38) {
		tmp = (t * (z * -9.0)) / (a * 2.0);
	} else {
		tmp = y / (a / (x / 2.0));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    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) :: tmp
    if ((x * y) <= (-2d-14)) then
        tmp = x * ((y / 2.0d0) / a)
    else if ((x * y) <= 1d-38) then
        tmp = (t * (z * (-9.0d0))) / (a * 2.0d0)
    else
        tmp = y / (a / (x / 2.0d0))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e-14) {
		tmp = x * ((y / 2.0) / a);
	} else if ((x * y) <= 1e-38) {
		tmp = (t * (z * -9.0)) / (a * 2.0);
	} else {
		tmp = y / (a / (x / 2.0));
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -2e-14:
		tmp = x * ((y / 2.0) / a)
	elif (x * y) <= 1e-38:
		tmp = (t * (z * -9.0)) / (a * 2.0)
	else:
		tmp = y / (a / (x / 2.0))
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -2e-14)
		tmp = Float64(x * Float64(Float64(y / 2.0) / a));
	elseif (Float64(x * y) <= 1e-38)
		tmp = Float64(Float64(t * Float64(z * -9.0)) / Float64(a * 2.0));
	else
		tmp = Float64(y / Float64(a / Float64(x / 2.0)));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -2e-14)
		tmp = x * ((y / 2.0) / a);
	elseif ((x * y) <= 1e-38)
		tmp = (t * (z * -9.0)) / (a * 2.0);
	else
		tmp = y / (a / (x / 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e-14], N[(x * N[(N[(y / 2.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e-38], N[(N[(t * N[(z * -9.0), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(y / N[(a / N[(x / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -2e-14

   1. Initial program 87.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right), \color{blue}{\left(a \cdot 2\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y + \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(a \cdot 2\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(z \cdot \left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(t \cdot 9\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(t \cdot \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \left(a \cdot 2\right)\right) \]

      12. *-lowering-*.f64 86.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{2}\right)\right) \]

   3. Simplified 86.2%

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

      1. frac-2neg N/A

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

      2. distribute-neg-in N/A

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

      3. unsub-neg N/A

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

      4. sub-div N/A

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

      5. frac-2neg N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

         \[\leadsto \frac{x \cdot y}{a \cdot 2} - \frac{\left(z \cdot \left(\mathsf{neg}\left(9\right)\right)\right) \cdot t}{\mathsf{neg}\left(a \cdot 2\right)} \]

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

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

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

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

      11. frac-2neg N/A

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

      12. div-sub N/A

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

      13. div-inv N/A

          \[\leadsto \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right) \cdot \color{blue}{\frac{1}{a \cdot 2}} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{a \cdot 2} \cdot \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{a \cdot 2}\right), \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)}\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2 \cdot a}\right), \left(x \cdot \color{blue}{y} - \left(z \cdot 9\right) \cdot t\right)\right) \]

      17. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{a}\right), \left(\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2}\right), a\right), \left(\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t\right)\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \left(\color{blue}{x} \cdot y - \left(z \cdot 9\right) \cdot t\right)\right) \]

   6. Applied egg-rr 87.5%

      \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \left(x \cdot y + z \cdot \left(t \cdot -9\right)\right)} \]

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \color{blue}{\left(x \cdot y\right)}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 71.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right) \]

   9. Simplified 71.8%

      \[\leadsto \frac{0.5}{a} \cdot \color{blue}{\left(x \cdot y\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{\frac{1}{2}}{a} \cdot x\right) \cdot \color{blue}{y} \]

       2. *-commutative N/A

          \[\leadsto y \cdot \color{blue}{\left(\frac{\frac{1}{2}}{a} \cdot x\right)} \]

       3. associate-*l/ N/A

          \[\leadsto y \cdot \frac{\frac{1}{2} \cdot x}{\color{blue}{a}} \]

       4. metadata-eval N/A

          \[\leadsto y \cdot \frac{\frac{1}{2} \cdot x}{a} \]

       5. associate-/r/ N/A

          \[\leadsto y \cdot \frac{\frac{1}{\frac{2}{x}}}{a} \]

       6. associate-/l* N/A

          \[\leadsto \frac{y \cdot \frac{1}{\frac{2}{x}}}{\color{blue}{a}} \]

       7. div-inv N/A

          \[\leadsto \frac{\frac{y}{\frac{2}{x}}}{a} \]

       8. associate-/r/ N/A

          \[\leadsto \frac{\frac{y}{2} \cdot x}{a} \]

       9. *-rgt-identity N/A

          \[\leadsto \frac{\frac{y}{2} \cdot x}{a \cdot \color{blue}{1}} \]

       10. times-frac N/A

           \[\leadsto \frac{\frac{y}{2}}{a} \cdot \color{blue}{\frac{x}{1}} \]

       11. /-rgt-identity N/A

           \[\leadsto \frac{\frac{y}{2}}{a} \cdot x \]

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

           \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{y}{2}}{a}\right), \color{blue}{x}\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{y}{2}\right), a\right), x\right) \]

       14. /-lowering-/.f64 71.7%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, 2\right), a\right), x\right) \]

   11. Applied egg-rr 71.7%

       \[\leadsto \color{blue}{\frac{\frac{y}{2}}{a} \cdot x} \]

   if -2e-14 < (*.f64 x y) < 9.9999999999999996e-39

   1. Initial program 94.7%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right), \color{blue}{\left(a \cdot 2\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y + \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(a \cdot 2\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(z \cdot \left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(t \cdot 9\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(t \cdot \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \left(a \cdot 2\right)\right) \]

      12. *-lowering-*.f64 94.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{2}\right)\right) \]

   3. Simplified 94.7%

      \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t \cdot -9\right)}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-9 \cdot \left(t \cdot z\right)\right)}, \mathsf{*.f64}\left(a, 2\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(t \cdot z\right) \cdot -9\right), \mathsf{*.f64}\left(\color{blue}{a}, 2\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(t \cdot \left(z \cdot -9\right)\right), \mathsf{*.f64}\left(\color{blue}{a}, 2\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(t \cdot \left(-9 \cdot z\right)\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(-9 \cdot z\right)\right), \mathsf{*.f64}\left(\color{blue}{a}, 2\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(z \cdot -9\right)\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      6. *-lowering-*.f64 75.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(z, -9\right)\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

   7. Simplified 75.0%

      \[\leadsto \frac{\color{blue}{t \cdot \left(z \cdot -9\right)}}{a \cdot 2} \]

   if 9.9999999999999996e-39 < (*.f64 x y)

   1. Initial program 86.2%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right), \color{blue}{\left(a \cdot 2\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y + \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(a \cdot 2\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(z \cdot \left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(t \cdot 9\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(t \cdot \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \left(a \cdot 2\right)\right) \]

      12. *-lowering-*.f64 86.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{2}\right)\right) \]

   3. Simplified 86.2%

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

      1. frac-2neg N/A

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

      2. distribute-neg-in N/A

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

      3. unsub-neg N/A

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

      4. sub-div N/A

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

      5. frac-2neg N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

         \[\leadsto \frac{x \cdot y}{a \cdot 2} - \frac{\left(z \cdot \left(\mathsf{neg}\left(9\right)\right)\right) \cdot t}{\mathsf{neg}\left(a \cdot 2\right)} \]

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

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

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

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

      11. frac-2neg N/A

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

      12. div-sub N/A

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

      13. div-inv N/A

          \[\leadsto \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right) \cdot \color{blue}{\frac{1}{a \cdot 2}} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{a \cdot 2} \cdot \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{a \cdot 2}\right), \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)}\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2 \cdot a}\right), \left(x \cdot \color{blue}{y} - \left(z \cdot 9\right) \cdot t\right)\right) \]

      17. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{a}\right), \left(\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2}\right), a\right), \left(\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t\right)\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \left(\color{blue}{x} \cdot y - \left(z \cdot 9\right) \cdot t\right)\right) \]

   6. Applied egg-rr 86.1%

      \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \left(x \cdot y + z \cdot \left(t \cdot -9\right)\right)} \]

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \color{blue}{\left(x \cdot y\right)}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 72.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right) \]

   9. Simplified 72.0%

      \[\leadsto \frac{0.5}{a} \cdot \color{blue}{\left(x \cdot y\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{\frac{1}{2}}{a} \cdot x\right) \cdot \color{blue}{y} \]

       2. *-commutative N/A

          \[\leadsto y \cdot \color{blue}{\left(\frac{\frac{1}{2}}{a} \cdot x\right)} \]

       3. associate-*l/ N/A

          \[\leadsto y \cdot \frac{\frac{1}{2} \cdot x}{\color{blue}{a}} \]

       4. metadata-eval N/A

          \[\leadsto y \cdot \frac{\frac{1}{2} \cdot x}{a} \]

       5. associate-/r/ N/A

          \[\leadsto y \cdot \frac{\frac{1}{\frac{2}{x}}}{a} \]

       6. associate-/l* N/A

          \[\leadsto \frac{y \cdot \frac{1}{\frac{2}{x}}}{\color{blue}{a}} \]

       7. div-inv N/A

          \[\leadsto \frac{\frac{y}{\frac{2}{x}}}{a} \]

       8. associate-/l/ N/A

          \[\leadsto \frac{y}{\color{blue}{a \cdot \frac{2}{x}}} \]

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

          \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(a \cdot \frac{2}{x}\right)}\right) \]

       10. remove-double-div N/A

           \[\leadsto \mathsf{/.f64}\left(y, \left(a \cdot \frac{1}{\color{blue}{\frac{1}{\frac{2}{x}}}}\right)\right) \]

       11. un-div-inv N/A

           \[\leadsto \mathsf{/.f64}\left(y, \left(\frac{a}{\color{blue}{\frac{1}{\frac{2}{x}}}}\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(a, \color{blue}{\left(\frac{1}{\frac{2}{x}}\right)}\right)\right) \]

       13. clear-num N/A

           \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(a, \left(\frac{x}{\color{blue}{2}}\right)\right)\right) \]

       14. /-lowering-/.f64 72.9%

           \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(a, \mathsf{/.f64}\left(x, \color{blue}{2}\right)\right)\right) \]

   11. Applied egg-rr 72.9%

       \[\leadsto \color{blue}{\frac{y}{\frac{a}{\frac{x}{2}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{\frac{y}{2}}{a}\\ \mathbf{elif}\;x \cdot y \leq 10^{-38}:\\ \;\;\;\;\frac{t \cdot \left(z \cdot -9\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{\frac{x}{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{\frac{y}{2}}{a}\\ \mathbf{elif}\;x \cdot y \leq 10^{-38}:\\ \;\;\;\;\frac{\left(z \cdot t\right) \cdot -4.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{\frac{x}{2}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -2e-14)
   (* x (/ (/ y 2.0) a))
   (if (<= (* x y) 1e-38) (/ (* (* z t) -4.5) a) (/ y (/ a (/ x 2.0))))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e-14) {
		tmp = x * ((y / 2.0) / a);
	} else if ((x * y) <= 1e-38) {
		tmp = ((z * t) * -4.5) / a;
	} else {
		tmp = y / (a / (x / 2.0));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    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) :: tmp
    if ((x * y) <= (-2d-14)) then
        tmp = x * ((y / 2.0d0) / a)
    else if ((x * y) <= 1d-38) then
        tmp = ((z * t) * (-4.5d0)) / a
    else
        tmp = y / (a / (x / 2.0d0))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e-14) {
		tmp = x * ((y / 2.0) / a);
	} else if ((x * y) <= 1e-38) {
		tmp = ((z * t) * -4.5) / a;
	} else {
		tmp = y / (a / (x / 2.0));
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -2e-14:
		tmp = x * ((y / 2.0) / a)
	elif (x * y) <= 1e-38:
		tmp = ((z * t) * -4.5) / a
	else:
		tmp = y / (a / (x / 2.0))
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -2e-14)
		tmp = Float64(x * Float64(Float64(y / 2.0) / a));
	elseif (Float64(x * y) <= 1e-38)
		tmp = Float64(Float64(Float64(z * t) * -4.5) / a);
	else
		tmp = Float64(y / Float64(a / Float64(x / 2.0)));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -2e-14)
		tmp = x * ((y / 2.0) / a);
	elseif ((x * y) <= 1e-38)
		tmp = ((z * t) * -4.5) / a;
	else
		tmp = y / (a / (x / 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e-14], N[(x * N[(N[(y / 2.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e-38], N[(N[(N[(z * t), $MachinePrecision] * -4.5), $MachinePrecision] / a), $MachinePrecision], N[(y / N[(a / N[(x / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -2e-14

   1. Initial program 87.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right), \color{blue}{\left(a \cdot 2\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y + \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(a \cdot 2\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(z \cdot \left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(t \cdot 9\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(t \cdot \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \left(a \cdot 2\right)\right) \]

      12. *-lowering-*.f64 86.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{2}\right)\right) \]

   3. Simplified 86.2%

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

      1. frac-2neg N/A

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

      2. distribute-neg-in N/A

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

      3. unsub-neg N/A

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

      4. sub-div N/A

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

      5. frac-2neg N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

         \[\leadsto \frac{x \cdot y}{a \cdot 2} - \frac{\left(z \cdot \left(\mathsf{neg}\left(9\right)\right)\right) \cdot t}{\mathsf{neg}\left(a \cdot 2\right)} \]

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

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

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

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

      11. frac-2neg N/A

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

      12. div-sub N/A

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

      13. div-inv N/A

          \[\leadsto \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right) \cdot \color{blue}{\frac{1}{a \cdot 2}} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{a \cdot 2} \cdot \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{a \cdot 2}\right), \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)}\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2 \cdot a}\right), \left(x \cdot \color{blue}{y} - \left(z \cdot 9\right) \cdot t\right)\right) \]

      17. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{a}\right), \left(\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2}\right), a\right), \left(\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t\right)\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \left(\color{blue}{x} \cdot y - \left(z \cdot 9\right) \cdot t\right)\right) \]

   6. Applied egg-rr 87.5%

      \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \left(x \cdot y + z \cdot \left(t \cdot -9\right)\right)} \]

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \color{blue}{\left(x \cdot y\right)}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 71.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right) \]

   9. Simplified 71.8%

      \[\leadsto \frac{0.5}{a} \cdot \color{blue}{\left(x \cdot y\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{\frac{1}{2}}{a} \cdot x\right) \cdot \color{blue}{y} \]

       2. *-commutative N/A

          \[\leadsto y \cdot \color{blue}{\left(\frac{\frac{1}{2}}{a} \cdot x\right)} \]

       3. associate-*l/ N/A

          \[\leadsto y \cdot \frac{\frac{1}{2} \cdot x}{\color{blue}{a}} \]

       4. metadata-eval N/A

          \[\leadsto y \cdot \frac{\frac{1}{2} \cdot x}{a} \]

       5. associate-/r/ N/A

          \[\leadsto y \cdot \frac{\frac{1}{\frac{2}{x}}}{a} \]

       6. associate-/l* N/A

          \[\leadsto \frac{y \cdot \frac{1}{\frac{2}{x}}}{\color{blue}{a}} \]

       7. div-inv N/A

          \[\leadsto \frac{\frac{y}{\frac{2}{x}}}{a} \]

       8. associate-/r/ N/A

          \[\leadsto \frac{\frac{y}{2} \cdot x}{a} \]

       9. *-rgt-identity N/A

          \[\leadsto \frac{\frac{y}{2} \cdot x}{a \cdot \color{blue}{1}} \]

       10. times-frac N/A

           \[\leadsto \frac{\frac{y}{2}}{a} \cdot \color{blue}{\frac{x}{1}} \]

       11. /-rgt-identity N/A

           \[\leadsto \frac{\frac{y}{2}}{a} \cdot x \]

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

           \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{y}{2}}{a}\right), \color{blue}{x}\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{y}{2}\right), a\right), x\right) \]

       14. /-lowering-/.f64 71.7%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, 2\right), a\right), x\right) \]

   11. Applied egg-rr 71.7%

       \[\leadsto \color{blue}{\frac{\frac{y}{2}}{a} \cdot x} \]

   if -2e-14 < (*.f64 x y) < 9.9999999999999996e-39

   1. Initial program 94.7%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right), \color{blue}{\left(a \cdot 2\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y + \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(a \cdot 2\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(z \cdot \left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(t \cdot 9\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(t \cdot \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \left(a \cdot 2\right)\right) \]

      12. *-lowering-*.f64 94.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{2}\right)\right) \]

   3. Simplified 94.7%

      \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t \cdot -9\right)}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \color{blue}{\left(\frac{t \cdot z}{a}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \mathsf{/.f64}\left(\left(t \cdot z\right), \color{blue}{a}\right)\right) \]

      3. *-lowering-*.f64 75.0%

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, z\right), a\right)\right) \]

   7. Simplified 75.0%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a}} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-9}{2} \cdot \left(t \cdot z\right)}{\color{blue}{a}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-9}{2} \cdot \left(t \cdot z\right)\right), \color{blue}{a}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-9}{2}, \left(t \cdot z\right)\right), a\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-9}{2}, \left(z \cdot t\right)\right), a\right) \]

      5. *-lowering-*.f64 75.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-9}{2}, \mathsf{*.f64}\left(z, t\right)\right), a\right) \]

   9. Applied egg-rr 75.0%

      \[\leadsto \color{blue}{\frac{-4.5 \cdot \left(z \cdot t\right)}{a}} \]

   if 9.9999999999999996e-39 < (*.f64 x y)

   1. Initial program 86.2%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right), \color{blue}{\left(a \cdot 2\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y + \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(a \cdot 2\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(z \cdot \left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(t \cdot 9\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(t \cdot \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \left(a \cdot 2\right)\right) \]

      12. *-lowering-*.f64 86.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{2}\right)\right) \]

   3. Simplified 86.2%

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

      1. frac-2neg N/A

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

      2. distribute-neg-in N/A

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

      3. unsub-neg N/A

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

      4. sub-div N/A

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

      5. frac-2neg N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

         \[\leadsto \frac{x \cdot y}{a \cdot 2} - \frac{\left(z \cdot \left(\mathsf{neg}\left(9\right)\right)\right) \cdot t}{\mathsf{neg}\left(a \cdot 2\right)} \]

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

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

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

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

      11. frac-2neg N/A

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

      12. div-sub N/A

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

      13. div-inv N/A

          \[\leadsto \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right) \cdot \color{blue}{\frac{1}{a \cdot 2}} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{a \cdot 2} \cdot \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{a \cdot 2}\right), \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)}\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2 \cdot a}\right), \left(x \cdot \color{blue}{y} - \left(z \cdot 9\right) \cdot t\right)\right) \]

      17. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{a}\right), \left(\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2}\right), a\right), \left(\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t\right)\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \left(\color{blue}{x} \cdot y - \left(z \cdot 9\right) \cdot t\right)\right) \]

   6. Applied egg-rr 86.1%

      \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \left(x \cdot y + z \cdot \left(t \cdot -9\right)\right)} \]

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \color{blue}{\left(x \cdot y\right)}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 72.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right) \]

   9. Simplified 72.0%

      \[\leadsto \frac{0.5}{a} \cdot \color{blue}{\left(x \cdot y\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{\frac{1}{2}}{a} \cdot x\right) \cdot \color{blue}{y} \]

       2. *-commutative N/A

          \[\leadsto y \cdot \color{blue}{\left(\frac{\frac{1}{2}}{a} \cdot x\right)} \]

       3. associate-*l/ N/A

          \[\leadsto y \cdot \frac{\frac{1}{2} \cdot x}{\color{blue}{a}} \]

       4. metadata-eval N/A

          \[\leadsto y \cdot \frac{\frac{1}{2} \cdot x}{a} \]

       5. associate-/r/ N/A

          \[\leadsto y \cdot \frac{\frac{1}{\frac{2}{x}}}{a} \]

       6. associate-/l* N/A

          \[\leadsto \frac{y \cdot \frac{1}{\frac{2}{x}}}{\color{blue}{a}} \]

       7. div-inv N/A

          \[\leadsto \frac{\frac{y}{\frac{2}{x}}}{a} \]

       8. associate-/l/ N/A

          \[\leadsto \frac{y}{\color{blue}{a \cdot \frac{2}{x}}} \]

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

          \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(a \cdot \frac{2}{x}\right)}\right) \]

       10. remove-double-div N/A

           \[\leadsto \mathsf{/.f64}\left(y, \left(a \cdot \frac{1}{\color{blue}{\frac{1}{\frac{2}{x}}}}\right)\right) \]

       11. un-div-inv N/A

           \[\leadsto \mathsf{/.f64}\left(y, \left(\frac{a}{\color{blue}{\frac{1}{\frac{2}{x}}}}\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(a, \color{blue}{\left(\frac{1}{\frac{2}{x}}\right)}\right)\right) \]

       13. clear-num N/A

           \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(a, \left(\frac{x}{\color{blue}{2}}\right)\right)\right) \]

       14. /-lowering-/.f64 72.9%

           \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(a, \mathsf{/.f64}\left(x, \color{blue}{2}\right)\right)\right) \]

   11. Applied egg-rr 72.9%

       \[\leadsto \color{blue}{\frac{y}{\frac{a}{\frac{x}{2}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{\frac{y}{2}}{a}\\ \mathbf{elif}\;x \cdot y \leq 10^{-38}:\\ \;\;\;\;\frac{\left(z \cdot t\right) \cdot -4.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{\frac{x}{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{\frac{y}{2}}{a}\\ \mathbf{elif}\;x \cdot y \leq 10^{-38}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{\frac{x}{2}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -2e-14)
   (* x (/ (/ y 2.0) a))
   (if (<= (* x y) 1e-38) (* -4.5 (/ (* z t) a)) (/ y (/ a (/ x 2.0))))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e-14) {
		tmp = x * ((y / 2.0) / a);
	} else if ((x * y) <= 1e-38) {
		tmp = -4.5 * ((z * t) / a);
	} else {
		tmp = y / (a / (x / 2.0));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    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) :: tmp
    if ((x * y) <= (-2d-14)) then
        tmp = x * ((y / 2.0d0) / a)
    else if ((x * y) <= 1d-38) then
        tmp = (-4.5d0) * ((z * t) / a)
    else
        tmp = y / (a / (x / 2.0d0))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e-14) {
		tmp = x * ((y / 2.0) / a);
	} else if ((x * y) <= 1e-38) {
		tmp = -4.5 * ((z * t) / a);
	} else {
		tmp = y / (a / (x / 2.0));
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -2e-14:
		tmp = x * ((y / 2.0) / a)
	elif (x * y) <= 1e-38:
		tmp = -4.5 * ((z * t) / a)
	else:
		tmp = y / (a / (x / 2.0))
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -2e-14)
		tmp = Float64(x * Float64(Float64(y / 2.0) / a));
	elseif (Float64(x * y) <= 1e-38)
		tmp = Float64(-4.5 * Float64(Float64(z * t) / a));
	else
		tmp = Float64(y / Float64(a / Float64(x / 2.0)));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -2e-14)
		tmp = x * ((y / 2.0) / a);
	elseif ((x * y) <= 1e-38)
		tmp = -4.5 * ((z * t) / a);
	else
		tmp = y / (a / (x / 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e-14], N[(x * N[(N[(y / 2.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e-38], N[(-4.5 * N[(N[(z * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(y / N[(a / N[(x / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -2e-14

   1. Initial program 87.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right), \color{blue}{\left(a \cdot 2\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y + \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(a \cdot 2\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(z \cdot \left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(t \cdot 9\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(t \cdot \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \left(a \cdot 2\right)\right) \]

      12. *-lowering-*.f64 86.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{2}\right)\right) \]

   3. Simplified 86.2%

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

      1. frac-2neg N/A

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

      2. distribute-neg-in N/A

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

      3. unsub-neg N/A

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

      4. sub-div N/A

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

      5. frac-2neg N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

         \[\leadsto \frac{x \cdot y}{a \cdot 2} - \frac{\left(z \cdot \left(\mathsf{neg}\left(9\right)\right)\right) \cdot t}{\mathsf{neg}\left(a \cdot 2\right)} \]

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

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

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

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

      11. frac-2neg N/A

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

      12. div-sub N/A

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

      13. div-inv N/A

          \[\leadsto \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right) \cdot \color{blue}{\frac{1}{a \cdot 2}} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{a \cdot 2} \cdot \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{a \cdot 2}\right), \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)}\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2 \cdot a}\right), \left(x \cdot \color{blue}{y} - \left(z \cdot 9\right) \cdot t\right)\right) \]

      17. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{a}\right), \left(\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2}\right), a\right), \left(\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t\right)\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \left(\color{blue}{x} \cdot y - \left(z \cdot 9\right) \cdot t\right)\right) \]

   6. Applied egg-rr 87.5%

      \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \left(x \cdot y + z \cdot \left(t \cdot -9\right)\right)} \]

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \color{blue}{\left(x \cdot y\right)}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 71.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right) \]

   9. Simplified 71.8%

      \[\leadsto \frac{0.5}{a} \cdot \color{blue}{\left(x \cdot y\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{\frac{1}{2}}{a} \cdot x\right) \cdot \color{blue}{y} \]

       2. *-commutative N/A

          \[\leadsto y \cdot \color{blue}{\left(\frac{\frac{1}{2}}{a} \cdot x\right)} \]

       3. associate-*l/ N/A

          \[\leadsto y \cdot \frac{\frac{1}{2} \cdot x}{\color{blue}{a}} \]

       4. metadata-eval N/A

          \[\leadsto y \cdot \frac{\frac{1}{2} \cdot x}{a} \]

       5. associate-/r/ N/A

          \[\leadsto y \cdot \frac{\frac{1}{\frac{2}{x}}}{a} \]

       6. associate-/l* N/A

          \[\leadsto \frac{y \cdot \frac{1}{\frac{2}{x}}}{\color{blue}{a}} \]

       7. div-inv N/A

          \[\leadsto \frac{\frac{y}{\frac{2}{x}}}{a} \]

       8. associate-/r/ N/A

          \[\leadsto \frac{\frac{y}{2} \cdot x}{a} \]

       9. *-rgt-identity N/A

          \[\leadsto \frac{\frac{y}{2} \cdot x}{a \cdot \color{blue}{1}} \]

       10. times-frac N/A

           \[\leadsto \frac{\frac{y}{2}}{a} \cdot \color{blue}{\frac{x}{1}} \]

       11. /-rgt-identity N/A

           \[\leadsto \frac{\frac{y}{2}}{a} \cdot x \]

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

           \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{y}{2}}{a}\right), \color{blue}{x}\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{y}{2}\right), a\right), x\right) \]

       14. /-lowering-/.f64 71.7%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, 2\right), a\right), x\right) \]

   11. Applied egg-rr 71.7%

       \[\leadsto \color{blue}{\frac{\frac{y}{2}}{a} \cdot x} \]

   if -2e-14 < (*.f64 x y) < 9.9999999999999996e-39

   1. Initial program 94.7%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right), \color{blue}{\left(a \cdot 2\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y + \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(a \cdot 2\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(z \cdot \left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(t \cdot 9\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(t \cdot \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \left(a \cdot 2\right)\right) \]

      12. *-lowering-*.f64 94.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{2}\right)\right) \]

   3. Simplified 94.7%

      \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t \cdot -9\right)}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \color{blue}{\left(\frac{t \cdot z}{a}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \mathsf{/.f64}\left(\left(t \cdot z\right), \color{blue}{a}\right)\right) \]

      3. *-lowering-*.f64 75.0%

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, z\right), a\right)\right) \]

   7. Simplified 75.0%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a}} \]

   if 9.9999999999999996e-39 < (*.f64 x y)

   1. Initial program 86.2%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right), \color{blue}{\left(a \cdot 2\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y + \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(a \cdot 2\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(z \cdot \left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(t \cdot 9\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(t \cdot \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \left(a \cdot 2\right)\right) \]

      12. *-lowering-*.f64 86.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{2}\right)\right) \]

   3. Simplified 86.2%

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

      1. frac-2neg N/A

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

      2. distribute-neg-in N/A

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

      3. unsub-neg N/A

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

      4. sub-div N/A

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

      5. frac-2neg N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

         \[\leadsto \frac{x \cdot y}{a \cdot 2} - \frac{\left(z \cdot \left(\mathsf{neg}\left(9\right)\right)\right) \cdot t}{\mathsf{neg}\left(a \cdot 2\right)} \]

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

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

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

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

      11. frac-2neg N/A

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

      12. div-sub N/A

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

      13. div-inv N/A

          \[\leadsto \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right) \cdot \color{blue}{\frac{1}{a \cdot 2}} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{a \cdot 2} \cdot \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{a \cdot 2}\right), \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)}\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2 \cdot a}\right), \left(x \cdot \color{blue}{y} - \left(z \cdot 9\right) \cdot t\right)\right) \]

      17. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{a}\right), \left(\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2}\right), a\right), \left(\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t\right)\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \left(\color{blue}{x} \cdot y - \left(z \cdot 9\right) \cdot t\right)\right) \]

   6. Applied egg-rr 86.1%

      \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \left(x \cdot y + z \cdot \left(t \cdot -9\right)\right)} \]

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \color{blue}{\left(x \cdot y\right)}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 72.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right) \]

   9. Simplified 72.0%

      \[\leadsto \frac{0.5}{a} \cdot \color{blue}{\left(x \cdot y\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{\frac{1}{2}}{a} \cdot x\right) \cdot \color{blue}{y} \]

       2. *-commutative N/A

          \[\leadsto y \cdot \color{blue}{\left(\frac{\frac{1}{2}}{a} \cdot x\right)} \]

       3. associate-*l/ N/A

          \[\leadsto y \cdot \frac{\frac{1}{2} \cdot x}{\color{blue}{a}} \]

       4. metadata-eval N/A

          \[\leadsto y \cdot \frac{\frac{1}{2} \cdot x}{a} \]

       5. associate-/r/ N/A

          \[\leadsto y \cdot \frac{\frac{1}{\frac{2}{x}}}{a} \]

       6. associate-/l* N/A

          \[\leadsto \frac{y \cdot \frac{1}{\frac{2}{x}}}{\color{blue}{a}} \]

       7. div-inv N/A

          \[\leadsto \frac{\frac{y}{\frac{2}{x}}}{a} \]

       8. associate-/l/ N/A

          \[\leadsto \frac{y}{\color{blue}{a \cdot \frac{2}{x}}} \]

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

          \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(a \cdot \frac{2}{x}\right)}\right) \]

       10. remove-double-div N/A

           \[\leadsto \mathsf{/.f64}\left(y, \left(a \cdot \frac{1}{\color{blue}{\frac{1}{\frac{2}{x}}}}\right)\right) \]

       11. un-div-inv N/A

           \[\leadsto \mathsf{/.f64}\left(y, \left(\frac{a}{\color{blue}{\frac{1}{\frac{2}{x}}}}\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(a, \color{blue}{\left(\frac{1}{\frac{2}{x}}\right)}\right)\right) \]

       13. clear-num N/A

           \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(a, \left(\frac{x}{\color{blue}{2}}\right)\right)\right) \]

       14. /-lowering-/.f64 72.9%

           \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(a, \mathsf{/.f64}\left(x, \color{blue}{2}\right)\right)\right) \]

   11. Applied egg-rr 72.9%

       \[\leadsto \color{blue}{\frac{y}{\frac{a}{\frac{x}{2}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{\frac{y}{2}}{a}\\ \mathbf{elif}\;x \cdot y \leq 10^{-38}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{\frac{x}{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{\frac{y}{2}}{a}\\ \mathbf{elif}\;x \cdot y \leq 10^{-38}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -2e-14)
   (* x (/ (/ y 2.0) a))
   (if (<= (* x y) 1e-38) (* -4.5 (/ (* z t) a)) (* 0.5 (* y (/ x a))))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e-14) {
		tmp = x * ((y / 2.0) / a);
	} else if ((x * y) <= 1e-38) {
		tmp = -4.5 * ((z * t) / a);
	} else {
		tmp = 0.5 * (y * (x / a));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    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) :: tmp
    if ((x * y) <= (-2d-14)) then
        tmp = x * ((y / 2.0d0) / a)
    else if ((x * y) <= 1d-38) then
        tmp = (-4.5d0) * ((z * t) / a)
    else
        tmp = 0.5d0 * (y * (x / a))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e-14) {
		tmp = x * ((y / 2.0) / a);
	} else if ((x * y) <= 1e-38) {
		tmp = -4.5 * ((z * t) / a);
	} else {
		tmp = 0.5 * (y * (x / a));
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -2e-14:
		tmp = x * ((y / 2.0) / a)
	elif (x * y) <= 1e-38:
		tmp = -4.5 * ((z * t) / a)
	else:
		tmp = 0.5 * (y * (x / a))
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -2e-14)
		tmp = Float64(x * Float64(Float64(y / 2.0) / a));
	elseif (Float64(x * y) <= 1e-38)
		tmp = Float64(-4.5 * Float64(Float64(z * t) / a));
	else
		tmp = Float64(0.5 * Float64(y * Float64(x / a)));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -2e-14)
		tmp = x * ((y / 2.0) / a);
	elseif ((x * y) <= 1e-38)
		tmp = -4.5 * ((z * t) / a);
	else
		tmp = 0.5 * (y * (x / a));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e-14], N[(x * N[(N[(y / 2.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e-38], N[(-4.5 * N[(N[(z * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -2e-14

   1. Initial program 87.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right), \color{blue}{\left(a \cdot 2\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y + \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(a \cdot 2\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(z \cdot \left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(t \cdot 9\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(t \cdot \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \left(a \cdot 2\right)\right) \]

      12. *-lowering-*.f64 86.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{2}\right)\right) \]

   3. Simplified 86.2%

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

      1. frac-2neg N/A

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

      2. distribute-neg-in N/A

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

      3. unsub-neg N/A

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

      4. sub-div N/A

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

      5. frac-2neg N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

         \[\leadsto \frac{x \cdot y}{a \cdot 2} - \frac{\left(z \cdot \left(\mathsf{neg}\left(9\right)\right)\right) \cdot t}{\mathsf{neg}\left(a \cdot 2\right)} \]

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

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

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

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

      11. frac-2neg N/A

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

      12. div-sub N/A

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

      13. div-inv N/A

          \[\leadsto \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right) \cdot \color{blue}{\frac{1}{a \cdot 2}} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{a \cdot 2} \cdot \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{a \cdot 2}\right), \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)}\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2 \cdot a}\right), \left(x \cdot \color{blue}{y} - \left(z \cdot 9\right) \cdot t\right)\right) \]

      17. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{a}\right), \left(\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2}\right), a\right), \left(\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t\right)\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \left(\color{blue}{x} \cdot y - \left(z \cdot 9\right) \cdot t\right)\right) \]

   6. Applied egg-rr 87.5%

      \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \left(x \cdot y + z \cdot \left(t \cdot -9\right)\right)} \]

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \color{blue}{\left(x \cdot y\right)}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 71.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right) \]

   9. Simplified 71.8%

      \[\leadsto \frac{0.5}{a} \cdot \color{blue}{\left(x \cdot y\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{\frac{1}{2}}{a} \cdot x\right) \cdot \color{blue}{y} \]

       2. *-commutative N/A

          \[\leadsto y \cdot \color{blue}{\left(\frac{\frac{1}{2}}{a} \cdot x\right)} \]

       3. associate-*l/ N/A

          \[\leadsto y \cdot \frac{\frac{1}{2} \cdot x}{\color{blue}{a}} \]

       4. metadata-eval N/A

          \[\leadsto y \cdot \frac{\frac{1}{2} \cdot x}{a} \]

       5. associate-/r/ N/A

          \[\leadsto y \cdot \frac{\frac{1}{\frac{2}{x}}}{a} \]

       6. associate-/l* N/A

          \[\leadsto \frac{y \cdot \frac{1}{\frac{2}{x}}}{\color{blue}{a}} \]

       7. div-inv N/A

          \[\leadsto \frac{\frac{y}{\frac{2}{x}}}{a} \]

       8. associate-/r/ N/A

          \[\leadsto \frac{\frac{y}{2} \cdot x}{a} \]

       9. *-rgt-identity N/A

          \[\leadsto \frac{\frac{y}{2} \cdot x}{a \cdot \color{blue}{1}} \]

       10. times-frac N/A

           \[\leadsto \frac{\frac{y}{2}}{a} \cdot \color{blue}{\frac{x}{1}} \]

       11. /-rgt-identity N/A

           \[\leadsto \frac{\frac{y}{2}}{a} \cdot x \]

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

           \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{y}{2}}{a}\right), \color{blue}{x}\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{y}{2}\right), a\right), x\right) \]

       14. /-lowering-/.f64 71.7%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, 2\right), a\right), x\right) \]

   11. Applied egg-rr 71.7%

       \[\leadsto \color{blue}{\frac{\frac{y}{2}}{a} \cdot x} \]

   if -2e-14 < (*.f64 x y) < 9.9999999999999996e-39

   1. Initial program 94.7%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right), \color{blue}{\left(a \cdot 2\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y + \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(a \cdot 2\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(z \cdot \left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(t \cdot 9\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(t \cdot \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \left(a \cdot 2\right)\right) \]

      12. *-lowering-*.f64 94.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{2}\right)\right) \]

   3. Simplified 94.7%

      \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t \cdot -9\right)}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \color{blue}{\left(\frac{t \cdot z}{a}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \mathsf{/.f64}\left(\left(t \cdot z\right), \color{blue}{a}\right)\right) \]

      3. *-lowering-*.f64 75.0%

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, z\right), a\right)\right) \]

   7. Simplified 75.0%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a}} \]

   if 9.9999999999999996e-39 < (*.f64 x y)

   1. Initial program 86.2%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right), \color{blue}{\left(a \cdot 2\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y + \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(a \cdot 2\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(z \cdot \left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(t \cdot 9\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(t \cdot \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \left(a \cdot 2\right)\right) \]

      12. *-lowering-*.f64 86.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{2}\right)\right) \]

   3. Simplified 86.2%

      \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t \cdot -9\right)}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{a}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{x \cdot y}{a}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{y \cdot x}{a}\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(y \cdot \color{blue}{\frac{x}{a}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{x}{a}\right)}\right)\right) \]

      5. /-lowering-/.f64 72.8%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 72.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(y \cdot \frac{x}{a}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{\frac{y}{2}}{a}\\ \mathbf{elif}\;x \cdot y \leq 10^{-38}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right)\\ \mathbf{elif}\;x \cdot y \leq 10^{-38}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -2e-14)
   (* x (* y (/ 0.5 a)))
   (if (<= (* x y) 1e-38) (* -4.5 (/ (* z t) a)) (* 0.5 (* y (/ x a))))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e-14) {
		tmp = x * (y * (0.5 / a));
	} else if ((x * y) <= 1e-38) {
		tmp = -4.5 * ((z * t) / a);
	} else {
		tmp = 0.5 * (y * (x / a));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    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) :: tmp
    if ((x * y) <= (-2d-14)) then
        tmp = x * (y * (0.5d0 / a))
    else if ((x * y) <= 1d-38) then
        tmp = (-4.5d0) * ((z * t) / a)
    else
        tmp = 0.5d0 * (y * (x / a))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e-14) {
		tmp = x * (y * (0.5 / a));
	} else if ((x * y) <= 1e-38) {
		tmp = -4.5 * ((z * t) / a);
	} else {
		tmp = 0.5 * (y * (x / a));
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -2e-14:
		tmp = x * (y * (0.5 / a))
	elif (x * y) <= 1e-38:
		tmp = -4.5 * ((z * t) / a)
	else:
		tmp = 0.5 * (y * (x / a))
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -2e-14)
		tmp = Float64(x * Float64(y * Float64(0.5 / a)));
	elseif (Float64(x * y) <= 1e-38)
		tmp = Float64(-4.5 * Float64(Float64(z * t) / a));
	else
		tmp = Float64(0.5 * Float64(y * Float64(x / a)));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -2e-14)
		tmp = x * (y * (0.5 / a));
	elseif ((x * y) <= 1e-38)
		tmp = -4.5 * ((z * t) / a);
	else
		tmp = 0.5 * (y * (x / a));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e-14], N[(x * N[(y * N[(0.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e-38], N[(-4.5 * N[(N[(z * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -2e-14

   1. Initial program 87.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right), \color{blue}{\left(a \cdot 2\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y + \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(a \cdot 2\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(z \cdot \left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(t \cdot 9\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(t \cdot \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \left(a \cdot 2\right)\right) \]

      12. *-lowering-*.f64 86.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{2}\right)\right) \]

   3. Simplified 86.2%

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

      1. frac-2neg N/A

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

      2. distribute-neg-in N/A

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

      3. unsub-neg N/A

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

      4. sub-div N/A

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

      5. frac-2neg N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

         \[\leadsto \frac{x \cdot y}{a \cdot 2} - \frac{\left(z \cdot \left(\mathsf{neg}\left(9\right)\right)\right) \cdot t}{\mathsf{neg}\left(a \cdot 2\right)} \]

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

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

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

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

      11. frac-2neg N/A

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

      12. div-sub N/A

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

      13. div-inv N/A

          \[\leadsto \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right) \cdot \color{blue}{\frac{1}{a \cdot 2}} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{a \cdot 2} \cdot \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{a \cdot 2}\right), \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)}\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2 \cdot a}\right), \left(x \cdot \color{blue}{y} - \left(z \cdot 9\right) \cdot t\right)\right) \]

      17. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{a}\right), \left(\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2}\right), a\right), \left(\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t\right)\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \left(\color{blue}{x} \cdot y - \left(z \cdot 9\right) \cdot t\right)\right) \]

   6. Applied egg-rr 87.5%

      \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \left(x \cdot y + z \cdot \left(t \cdot -9\right)\right)} \]

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \color{blue}{\left(x \cdot y\right)}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 71.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right) \]

   9. Simplified 71.8%

      \[\leadsto \frac{0.5}{a} \cdot \color{blue}{\left(x \cdot y\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{1}{2}}{a} \cdot \left(y \cdot \color{blue}{x}\right) \]

       2. associate-*r* N/A

          \[\leadsto \left(\frac{\frac{1}{2}}{a} \cdot y\right) \cdot \color{blue}{x} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{a} \cdot y\right), \color{blue}{x}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{a}\right), y\right), x\right) \]

       5. /-lowering-/.f64 71.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), y\right), x\right) \]

   11. Applied egg-rr 71.7%

       \[\leadsto \color{blue}{\left(\frac{0.5}{a} \cdot y\right) \cdot x} \]

   if -2e-14 < (*.f64 x y) < 9.9999999999999996e-39

   1. Initial program 94.7%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right), \color{blue}{\left(a \cdot 2\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y + \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(a \cdot 2\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(z \cdot \left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(t \cdot 9\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(t \cdot \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \left(a \cdot 2\right)\right) \]

      12. *-lowering-*.f64 94.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{2}\right)\right) \]

   3. Simplified 94.7%

      \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t \cdot -9\right)}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \color{blue}{\left(\frac{t \cdot z}{a}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \mathsf{/.f64}\left(\left(t \cdot z\right), \color{blue}{a}\right)\right) \]

      3. *-lowering-*.f64 75.0%

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, z\right), a\right)\right) \]

   7. Simplified 75.0%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a}} \]

   if 9.9999999999999996e-39 < (*.f64 x y)

   1. Initial program 86.2%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right), \color{blue}{\left(a \cdot 2\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y + \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(a \cdot 2\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(z \cdot \left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(t \cdot 9\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(t \cdot \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \left(a \cdot 2\right)\right) \]

      12. *-lowering-*.f64 86.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{2}\right)\right) \]

   3. Simplified 86.2%

      \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t \cdot -9\right)}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{a}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{x \cdot y}{a}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{y \cdot x}{a}\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(y \cdot \color{blue}{\frac{x}{a}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{x}{a}\right)}\right)\right) \]

      5. /-lowering-/.f64 72.8%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 72.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(y \cdot \frac{x}{a}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right)\\ \mathbf{elif}\;x \cdot y \leq 10^{-38}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := t \cdot \left(\frac{z}{a} \cdot -4.5\right)\\ \mathbf{if}\;t \leq -0.0002:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-12}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* t (* (/ z a) -4.5))))
   (if (<= t -0.0002) t_1 (if (<= t 3e-12) (* 0.5 (* y (/ x a))) t_1))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = t * ((z / a) * -4.5);
	double tmp;
	if (t <= -0.0002) {
		tmp = t_1;
	} else if (t <= 3e-12) {
		tmp = 0.5 * (y * (x / a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    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) :: t_1
    real(8) :: tmp
    t_1 = t * ((z / a) * (-4.5d0))
    if (t <= (-0.0002d0)) then
        tmp = t_1
    else if (t <= 3d-12) then
        tmp = 0.5d0 * (y * (x / a))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t * ((z / a) * -4.5);
	double tmp;
	if (t <= -0.0002) {
		tmp = t_1;
	} else if (t <= 3e-12) {
		tmp = 0.5 * (y * (x / a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = t * ((z / a) * -4.5)
	tmp = 0
	if t <= -0.0002:
		tmp = t_1
	elif t <= 3e-12:
		tmp = 0.5 * (y * (x / a))
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(t * Float64(Float64(z / a) * -4.5))
	tmp = 0.0
	if (t <= -0.0002)
		tmp = t_1;
	elseif (t <= 3e-12)
		tmp = Float64(0.5 * Float64(y * Float64(x / a)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = t * ((z / a) * -4.5);
	tmp = 0.0;
	if (t <= -0.0002)
		tmp = t_1;
	elseif (t <= 3e-12)
		tmp = 0.5 * (y * (x / a));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * N[(N[(z / a), $MachinePrecision] * -4.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.0002], t$95$1, If[LessEqual[t, 3e-12], N[(0.5 * N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.0000000000000001e-4 or 3.0000000000000001e-12 < t

   1. Initial program 83.7%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right), \color{blue}{\left(a \cdot 2\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y + \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(a \cdot 2\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(z \cdot \left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(t \cdot 9\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(t \cdot \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \left(a \cdot 2\right)\right) \]

      12. *-lowering-*.f64 82.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{2}\right)\right) \]

   3. Simplified 82.8%

      \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t \cdot -9\right)}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \color{blue}{\left(\frac{t \cdot z}{a}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \mathsf{/.f64}\left(\left(t \cdot z\right), \color{blue}{a}\right)\right) \]

      3. *-lowering-*.f64 56.4%

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, z\right), a\right)\right) \]

   7. Simplified 56.4%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      2. associate-/l* N/A

         \[\leadsto \left(t \cdot \frac{z}{a}\right) \cdot \frac{-9}{2} \]

      3. associate-*l* N/A

         \[\leadsto t \cdot \color{blue}{\left(\frac{z}{a} \cdot \frac{-9}{2}\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{z}{a} \cdot \frac{-9}{2}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(\frac{z}{a}\right), \color{blue}{\frac{-9}{2}}\right)\right) \]

      6. /-lowering-/.f64 63.4%

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, a\right), \frac{-9}{2}\right)\right) \]

   9. Applied egg-rr 63.4%

      \[\leadsto \color{blue}{t \cdot \left(\frac{z}{a} \cdot -4.5\right)} \]

   if -2.0000000000000001e-4 < t < 3.0000000000000001e-12

   1. Initial program 96.2%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right), \color{blue}{\left(a \cdot 2\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y + \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(a \cdot 2\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(z \cdot \left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(t \cdot 9\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(t \cdot \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \left(a \cdot 2\right)\right) \]

      12. *-lowering-*.f64 96.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{2}\right)\right) \]

   3. Simplified 96.2%

      \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t \cdot -9\right)}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{a}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{x \cdot y}{a}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{y \cdot x}{a}\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(y \cdot \color{blue}{\frac{x}{a}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{x}{a}\right)}\right)\right) \]

      5. /-lowering-/.f64 64.2%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 64.2%

      \[\leadsto \color{blue}{0.5 \cdot \left(y \cdot \frac{x}{a}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 67.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := -4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \mathbf{if}\;t \leq -2.8 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 10^{-11}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* -4.5 (* z (/ t a)))))
   (if (<= t -2.8e-5) t_1 (if (<= t 1e-11) (* 0.5 (* y (/ x a))) t_1))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = -4.5 * (z * (t / a));
	double tmp;
	if (t <= -2.8e-5) {
		tmp = t_1;
	} else if (t <= 1e-11) {
		tmp = 0.5 * (y * (x / a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    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) :: t_1
    real(8) :: tmp
    t_1 = (-4.5d0) * (z * (t / a))
    if (t <= (-2.8d-5)) then
        tmp = t_1
    else if (t <= 1d-11) then
        tmp = 0.5d0 * (y * (x / a))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = -4.5 * (z * (t / a));
	double tmp;
	if (t <= -2.8e-5) {
		tmp = t_1;
	} else if (t <= 1e-11) {
		tmp = 0.5 * (y * (x / a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = -4.5 * (z * (t / a))
	tmp = 0
	if t <= -2.8e-5:
		tmp = t_1
	elif t <= 1e-11:
		tmp = 0.5 * (y * (x / a))
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(-4.5 * Float64(z * Float64(t / a)))
	tmp = 0.0
	if (t <= -2.8e-5)
		tmp = t_1;
	elseif (t <= 1e-11)
		tmp = Float64(0.5 * Float64(y * Float64(x / a)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = -4.5 * (z * (t / a));
	tmp = 0.0;
	if (t <= -2.8e-5)
		tmp = t_1;
	elseif (t <= 1e-11)
		tmp = 0.5 * (y * (x / a));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(-4.5 * N[(z * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.8e-5], t$95$1, If[LessEqual[t, 1e-11], N[(0.5 * N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.79999999999999996e-5 or 9.99999999999999939e-12 < t

   1. Initial program 83.7%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right), \color{blue}{\left(a \cdot 2\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y + \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(a \cdot 2\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(z \cdot \left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(t \cdot 9\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(t \cdot \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \left(a \cdot 2\right)\right) \]

      12. *-lowering-*.f64 82.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{2}\right)\right) \]

   3. Simplified 82.8%

      \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t \cdot -9\right)}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \color{blue}{\left(\frac{t \cdot z}{a}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \mathsf{/.f64}\left(\left(t \cdot z\right), \color{blue}{a}\right)\right) \]

      3. *-lowering-*.f64 56.4%

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, z\right), a\right)\right) \]

   7. Simplified 56.4%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \left(\frac{z \cdot t}{a}\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \left(z \cdot \color{blue}{\frac{t}{a}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{t}{a}\right)}\right)\right) \]

      4. /-lowering-/.f64 65.1%

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(t, \color{blue}{a}\right)\right)\right) \]

   9. Applied egg-rr 65.1%

      \[\leadsto -4.5 \cdot \color{blue}{\left(z \cdot \frac{t}{a}\right)} \]

   if -2.79999999999999996e-5 < t < 9.99999999999999939e-12

   1. Initial program 96.2%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right), \color{blue}{\left(a \cdot 2\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y + \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(a \cdot 2\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(z \cdot \left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(t \cdot 9\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(t \cdot \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \left(a \cdot 2\right)\right) \]

      12. *-lowering-*.f64 96.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{2}\right)\right) \]

   3. Simplified 96.2%

      \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t \cdot -9\right)}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{a}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{x \cdot y}{a}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{y \cdot x}{a}\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(y \cdot \color{blue}{\frac{x}{a}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{x}{a}\right)}\right)\right) \]

      5. /-lowering-/.f64 64.2%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 64.2%

      \[\leadsto \color{blue}{0.5 \cdot \left(y \cdot \frac{x}{a}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 50.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 1.1 \cdot 10^{+35}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= a 1.1e+35) (* -4.5 (/ (* z t) a)) (* -4.5 (* z (/ t a)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= 1.1e+35) {
		tmp = -4.5 * ((z * t) / a);
	} else {
		tmp = -4.5 * (z * (t / a));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    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) :: tmp
    if (a <= 1.1d+35) then
        tmp = (-4.5d0) * ((z * t) / a)
    else
        tmp = (-4.5d0) * (z * (t / a))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= 1.1e+35) {
		tmp = -4.5 * ((z * t) / a);
	} else {
		tmp = -4.5 * (z * (t / a));
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if a <= 1.1e+35:
		tmp = -4.5 * ((z * t) / a)
	else:
		tmp = -4.5 * (z * (t / a))
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= 1.1e+35)
		tmp = Float64(-4.5 * Float64(Float64(z * t) / a));
	else
		tmp = Float64(-4.5 * Float64(z * Float64(t / a)));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= 1.1e+35)
		tmp = -4.5 * ((z * t) / a);
	else
		tmp = -4.5 * (z * (t / a));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[a, 1.1e+35], N[(-4.5 * N[(N[(z * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(-4.5 * N[(z * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.0999999999999999e35

   1. Initial program 92.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right), \color{blue}{\left(a \cdot 2\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y + \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(a \cdot 2\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(z \cdot \left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(t \cdot 9\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(t \cdot \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \left(a \cdot 2\right)\right) \]

      12. *-lowering-*.f64 92.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{2}\right)\right) \]

   3. Simplified 92.1%

      \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t \cdot -9\right)}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \color{blue}{\left(\frac{t \cdot z}{a}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \mathsf{/.f64}\left(\left(t \cdot z\right), \color{blue}{a}\right)\right) \]

      3. *-lowering-*.f64 47.1%

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, z\right), a\right)\right) \]

   7. Simplified 47.1%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a}} \]

   if 1.0999999999999999e35 < a

   1. Initial program 80.5%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right), \color{blue}{\left(a \cdot 2\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y + \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(a \cdot 2\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(z \cdot \left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(t \cdot 9\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(t \cdot \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \left(a \cdot 2\right)\right) \]

      12. *-lowering-*.f64 80.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{2}\right)\right) \]

   3. Simplified 80.5%

      \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t \cdot -9\right)}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \color{blue}{\left(\frac{t \cdot z}{a}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \mathsf{/.f64}\left(\left(t \cdot z\right), \color{blue}{a}\right)\right) \]

      3. *-lowering-*.f64 42.9%

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, z\right), a\right)\right) \]

   7. Simplified 42.9%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \left(\frac{z \cdot t}{a}\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \left(z \cdot \color{blue}{\frac{t}{a}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{t}{a}\right)}\right)\right) \]

      4. /-lowering-/.f64 47.0%

         \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(t, \color{blue}{a}\right)\right)\right) \]

   9. Applied egg-rr 47.0%

      \[\leadsto -4.5 \cdot \color{blue}{\left(z \cdot \frac{t}{a}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.1 \cdot 10^{+35}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 50.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ -4.5 \cdot \left(z \cdot \frac{t}{a}\right) \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a) :precision binary64 (* -4.5 (* z (/ t a))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	return -4.5 * (z * (t / a));
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    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
    code = (-4.5d0) * (z * (t / a))
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	return -4.5 * (z * (t / a));
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	return -4.5 * (z * (t / a))

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	return Float64(-4.5 * Float64(z * Float64(t / a)))
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp = code(x, y, z, t, a)
	tmp = -4.5 * (z * (t / a));
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := N[(-4.5 * N[(z * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.5%

   \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
2. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right), \color{blue}{\left(a \cdot 2\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y + \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(\color{blue}{a} \cdot 2\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right), \left(a \cdot 2\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\mathsf{neg}\left(z \cdot \left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(9 \cdot t\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(t \cdot 9\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(t \cdot \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, \left(\mathsf{neg}\left(9\right)\right)\right)\right)\right), \left(a \cdot 2\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \left(a \cdot 2\right)\right) \]

   12. *-lowering-*.f64 90.1%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(t, -9\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{2}\right)\right) \]

3. Simplified 90.1%

   \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t \cdot -9\right)}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \color{blue}{\left(\frac{t \cdot z}{a}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \mathsf{/.f64}\left(\left(t \cdot z\right), \color{blue}{a}\right)\right) \]

   3. *-lowering-*.f64 46.3%

      \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, z\right), a\right)\right) \]

7. Simplified 46.3%

   \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a}} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \left(\frac{z \cdot t}{a}\right)\right) \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \left(z \cdot \color{blue}{\frac{t}{a}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{t}{a}\right)}\right)\right) \]

   4. /-lowering-/.f64 49.6%

      \[\leadsto \mathsf{*.f64}\left(\frac{-9}{2}, \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(t, \color{blue}{a}\right)\right)\right) \]

9. Applied egg-rr 49.6%

   \[\leadsto -4.5 \cdot \color{blue}{\left(z \cdot \frac{t}{a}\right)} \]
10. Add Preprocessing

Developer Target 1: 93.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a < -2.090464557976709 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a < 2.144030707833976 \cdot 10^{+99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (< a -2.090464557976709e+86)
   (- (* 0.5 (/ (* y x) a)) (* 4.5 (/ t (/ a z))))
   (if (< a 2.144030707833976e+99)
     (/ (- (* x y) (* z (* 9.0 t))) (* a 2.0))
     (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a < -2.090464557976709e+86) {
		tmp = (0.5 * ((y * x) / a)) - (4.5 * (t / (a / z)));
	} else if (a < 2.144030707833976e+99) {
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	} else {
		tmp = ((y / a) * (x * 0.5)) - ((t / a) * (z * 4.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: tmp
    if (a < (-2.090464557976709d+86)) then
        tmp = (0.5d0 * ((y * x) / a)) - (4.5d0 * (t / (a / z)))
    else if (a < 2.144030707833976d+99) then
        tmp = ((x * y) - (z * (9.0d0 * t))) / (a * 2.0d0)
    else
        tmp = ((y / a) * (x * 0.5d0)) - ((t / a) * (z * 4.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a < -2.090464557976709e+86) {
		tmp = (0.5 * ((y * x) / a)) - (4.5 * (t / (a / z)));
	} else if (a < 2.144030707833976e+99) {
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	} else {
		tmp = ((y / a) * (x * 0.5)) - ((t / a) * (z * 4.5));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a < -2.090464557976709e+86:
		tmp = (0.5 * ((y * x) / a)) - (4.5 * (t / (a / z)))
	elif a < 2.144030707833976e+99:
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0)
	else:
		tmp = ((y / a) * (x * 0.5)) - ((t / a) * (z * 4.5))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a < -2.090464557976709e+86)
		tmp = Float64(Float64(0.5 * Float64(Float64(y * x) / a)) - Float64(4.5 * Float64(t / Float64(a / z))));
	elseif (a < 2.144030707833976e+99)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * Float64(9.0 * t))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(y / a) * Float64(x * 0.5)) - Float64(Float64(t / a) * Float64(z * 4.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a < -2.090464557976709e+86)
		tmp = (0.5 * ((y * x) / a)) - (4.5 * (t / (a / z)));
	elseif (a < 2.144030707833976e+99)
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	else
		tmp = ((y / a) * (x * 0.5)) - ((t / a) * (z * 4.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Less[a, -2.090464557976709e+86], N[(N[(0.5 * N[(N[(y * x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] - N[(4.5 * N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Less[a, 2.144030707833976e+99], N[(N[(N[(x * y), $MachinePrecision] - N[(z * N[(9.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / a), $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[(t / a), $MachinePrecision] * N[(z * 4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024150 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< a -209046455797670900000000000000000000000000000000000000000000000000000000000000000000000) (- (* 1/2 (/ (* y x) a)) (* 9/2 (/ t (/ a z)))) (if (< a 2144030707833976000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (- (* x y) (* z (* 9 t))) (* a 2)) (- (* (/ y a) (* x 1/2)) (* (/ t a) (* z 9/2))))))

  (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))