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

Percentage Accurate: 91.4% → 96.7%

Time: 13.0s

Alternatives: 11

Speedup: 0.6×

• 28.9% 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: 91.4% 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: 96.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := x \cdot y - \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+289}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a \cdot 2}, y, z \cdot \frac{t \cdot 4.5}{-a}\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot 0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(x, 0.5 \cdot \frac{y}{z \cdot a}, \frac{t \cdot -4.5}{a}\right)\\ \end{array} \end{array} \]

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
 (let* ((t_1 (- (* x y) (* (* z 9.0) t))))
   (if (<= t_1 -5e+289)
     (fma (/ x (* a 2.0)) y (* z (/ (* t 4.5) (- a))))
     (if (<= t_1 2e+303)
       (/ (* (fma z (* t -9.0) (* x y)) 0.5) a)
       (* z (fma x (* 0.5 (/ y (* z a))) (/ (* t -4.5) a)))))))

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 = (x * y) - ((z * 9.0) * t);
	double tmp;
	if (t_1 <= -5e+289) {
		tmp = fma((x / (a * 2.0)), y, (z * ((t * 4.5) / -a)));
	} else if (t_1 <= 2e+303) {
		tmp = (fma(z, (t * -9.0), (x * y)) * 0.5) / a;
	} else {
		tmp = z * fma(x, (0.5 * (y / (z * a))), ((t * -4.5) / a));
	}
	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(x * y) - Float64(Float64(z * 9.0) * t))
	tmp = 0.0
	if (t_1 <= -5e+289)
		tmp = fma(Float64(x / Float64(a * 2.0)), y, Float64(z * Float64(Float64(t * 4.5) / Float64(-a))));
	elseif (t_1 <= 2e+303)
		tmp = Float64(Float64(fma(z, Float64(t * -9.0), Float64(x * y)) * 0.5) / a);
	else
		tmp = Float64(z * fma(x, Float64(0.5 * Float64(y / Float64(z * a))), Float64(Float64(t * -4.5) / a)));
	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[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+289], N[(N[(x / N[(a * 2.0), $MachinePrecision]), $MachinePrecision] * y + N[(z * N[(N[(t * 4.5), $MachinePrecision] / (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+303], N[(N[(N[(z * N[(t * -9.0), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] / a), $MachinePrecision], N[(z * N[(x * N[(0.5 * N[(y / N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t * -4.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < -5.00000000000000031e289

   1. Initial program 59.7%

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

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

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

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

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

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

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

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

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

      10. associate-*l* N/A

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

      11. associate-/l* N/A

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

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

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

      13. *-commutative N/A

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

      14. associate-/r* N/A

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

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

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

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

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

      19. metadata-eval 99.9

          \[\leadsto \mathsf{fma}\left(\frac{x}{a \cdot 2}, y, -z \cdot \frac{t \cdot \color{blue}{4.5}}{a}\right) \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{a \cdot 2}, y, -z \cdot \frac{t \cdot 4.5}{a}\right)} \]

   if -5.00000000000000031e289 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < 2e303

   1. Initial program 97.3%

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

      1. associate-/l/ N/A

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

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

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

      3. div-inv N/A

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

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

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. associate-*l* N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

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

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

      13. metadata-eval N/A

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

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

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

      15. metadata-eval 98.8

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

   4. Applied egg-rr 98.8%

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

   if 2e303 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t))

   1. Initial program 61.0%

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

      1. associate-/l/ N/A

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

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

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

      3. div-inv N/A

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

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

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. associate-*l* N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

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

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

      13. metadata-eval N/A

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

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

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

      15. metadata-eval 61.0

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

   4. Applied egg-rr 61.0%

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

   5. Taylor expanded in z around inf

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      10. associate-*r/ N/A

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

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

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

      12. *-lowering-*.f64 89.9

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

   7. Simplified 89.9%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(x, \frac{y}{a \cdot z} \cdot 0.5, \frac{-4.5 \cdot t}{a}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -5 \cdot 10^{+289}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a \cdot 2}, y, z \cdot \frac{t \cdot 4.5}{-a}\right)\\ \mathbf{elif}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot 0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(x, 0.5 \cdot \frac{y}{z \cdot a}, \frac{t \cdot -4.5}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x}{a \cdot 2}, y, z \cdot \frac{t \cdot 4.5}{-a}\right)\\ t_2 := x \cdot y - \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+289}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+292}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot 0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (fma (/ x (* a 2.0)) y (* z (/ (* t 4.5) (- a)))))
        (t_2 (- (* x y) (* (* z 9.0) t))))
   (if (<= t_2 -5e+289)
     t_1
     (if (<= t_2 4e+292) (/ (* (fma z (* t -9.0) (* x y)) 0.5) 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 = fma((x / (a * 2.0)), y, (z * ((t * 4.5) / -a)));
	double t_2 = (x * y) - ((z * 9.0) * t);
	double tmp;
	if (t_2 <= -5e+289) {
		tmp = t_1;
	} else if (t_2 <= 4e+292) {
		tmp = (fma(z, (t * -9.0), (x * y)) * 0.5) / 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 = fma(Float64(x / Float64(a * 2.0)), y, Float64(z * Float64(Float64(t * 4.5) / Float64(-a))))
	t_2 = Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t))
	tmp = 0.0
	if (t_2 <= -5e+289)
		tmp = t_1;
	elseif (t_2 <= 4e+292)
		tmp = Float64(Float64(fma(z, Float64(t * -9.0), Float64(x * y)) * 0.5) / a);
	else
		tmp = 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[(N[(x / N[(a * 2.0), $MachinePrecision]), $MachinePrecision] * y + N[(z * N[(N[(t * 4.5), $MachinePrecision] / (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+289], t$95$1, If[LessEqual[t$95$2, 4e+292], N[(N[(N[(z * N[(t * -9.0), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] / a), $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)) < -5.00000000000000031e289 or 4.0000000000000001e292 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t))

   1. Initial program 61.7%

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

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

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

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

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

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

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

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

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

      10. associate-*l* N/A

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

      11. associate-/l* N/A

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

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

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

      13. *-commutative N/A

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

      14. associate-/r* N/A

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

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

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

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

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

      19. metadata-eval 93.1

          \[\leadsto \mathsf{fma}\left(\frac{x}{a \cdot 2}, y, -z \cdot \frac{t \cdot \color{blue}{4.5}}{a}\right) \]

   4. Applied egg-rr 93.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{a \cdot 2}, y, -z \cdot \frac{t \cdot 4.5}{a}\right)} \]

   if -5.00000000000000031e289 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < 4.0000000000000001e292

   1. Initial program 97.3%

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

      1. associate-/l/ N/A

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

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

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

      3. div-inv N/A

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

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

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. associate-*l* N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

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

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

      13. metadata-eval N/A

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

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

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

      15. metadata-eval 98.8

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

   4. Applied egg-rr 98.8%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -5 \cdot 10^{+289}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a \cdot 2}, y, z \cdot \frac{t \cdot 4.5}{-a}\right)\\ \mathbf{elif}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq 4 \cdot 10^{+292}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot 0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a \cdot 2}, y, z \cdot \frac{t \cdot 4.5}{-a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t \cdot \frac{z \cdot -4.5}{a}\\ \mathbf{elif}\;t\_1 \leq 10^{+264}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot 0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \end{array} \end{array} \]

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
 (let* ((t_1 (* (* z 9.0) t)))
   (if (<= t_1 (- INFINITY))
     (* t (/ (* z -4.5) a))
     (if (<= t_1 1e+264)
       (/ (* (fma z (* t -9.0) (* x y)) 0.5) a)
       (* -4.5 (* t (/ z a)))))))

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 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t * ((z * -4.5) / a);
	} else if (t_1 <= 1e+264) {
		tmp = (fma(z, (t * -9.0), (x * y)) * 0.5) / a;
	} else {
		tmp = -4.5 * (t * (z / a));
	}
	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(z * 9.0) * t)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(t * Float64(Float64(z * -4.5) / a));
	elseif (t_1 <= 1e+264)
		tmp = Float64(Float64(fma(z, Float64(t * -9.0), Float64(x * y)) * 0.5) / a);
	else
		tmp = Float64(-4.5 * Float64(t * Float64(z / a)));
	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[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(t * N[(N[(z * -4.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+264], N[(N[(N[(z * N[(t * -9.0), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] / a), $MachinePrecision], N[(-4.5 * N[(t * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 57.5%

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

   3. Taylor expanded in x around 0

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

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

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f64 94.9

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

   5. Simplified 94.9%

      \[\leadsto \color{blue}{-4.5 \cdot \left(t \cdot \frac{z}{a}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. times-frac N/A

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

      5. associate-/l* N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. div-inv N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 94.7

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

   7. Applied egg-rr 94.7%

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

      1. associate-*r/ N/A

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

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

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

      3. *-lowering-*.f64 94.9

         \[\leadsto t \cdot \frac{\color{blue}{z \cdot -4.5}}{a} \]

   9. Applied egg-rr 94.9%

      \[\leadsto t \cdot \color{blue}{\frac{z \cdot -4.5}{a}} \]

   if -inf.0 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 1.00000000000000004e264

   1. Initial program 94.1%

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

      1. associate-/l/ N/A

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

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

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

      3. div-inv N/A

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

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

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. associate-*l* N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

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

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

      13. metadata-eval N/A

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

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

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

      15. metadata-eval 95.4

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

   4. Applied egg-rr 95.4%

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

   if 1.00000000000000004e264 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 64.5%

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

   3. Taylor expanded in x around 0

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

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

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f64 100.0

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

   5. Simplified 100.0%

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

Alternative 4: 94.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+274}:\\ \;\;\;\;t \cdot \frac{z \cdot -4.5}{a}\\ \mathbf{elif}\;t\_1 \leq 10^{+264}:\\ \;\;\;\;\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \end{array} \end{array} \]

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
 (let* ((t_1 (* (* z 9.0) t)))
   (if (<= t_1 -5e+274)
     (* t (/ (* z -4.5) a))
     (if (<= t_1 1e+264)
       (* (fma z (* t -9.0) (* x y)) (/ 0.5 a))
       (* -4.5 (* t (/ z a)))))))

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 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -5e+274) {
		tmp = t * ((z * -4.5) / a);
	} else if (t_1 <= 1e+264) {
		tmp = fma(z, (t * -9.0), (x * y)) * (0.5 / a);
	} else {
		tmp = -4.5 * (t * (z / a));
	}
	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(z * 9.0) * t)
	tmp = 0.0
	if (t_1 <= -5e+274)
		tmp = Float64(t * Float64(Float64(z * -4.5) / a));
	elseif (t_1 <= 1e+264)
		tmp = Float64(fma(z, Float64(t * -9.0), Float64(x * y)) * Float64(0.5 / a));
	else
		tmp = Float64(-4.5 * Float64(t * Float64(z / a)));
	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[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+274], N[(t * N[(N[(z * -4.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+264], N[(N[(z * N[(t * -9.0), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(-4.5 * N[(t * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -4.9999999999999998e274

   1. Initial program 58.9%

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

   3. Taylor expanded in x around 0

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

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

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f64 95.5

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

   5. Simplified 95.5%

      \[\leadsto \color{blue}{-4.5 \cdot \left(t \cdot \frac{z}{a}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. times-frac N/A

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

      5. associate-/l* N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. div-inv N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 95.4

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

   7. Applied egg-rr 95.4%

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

      1. associate-*r/ N/A

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

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

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

      3. *-lowering-*.f64 95.5

         \[\leadsto t \cdot \frac{\color{blue}{z \cdot -4.5}}{a} \]

   9. Applied egg-rr 95.5%

      \[\leadsto t \cdot \color{blue}{\frac{z \cdot -4.5}{a}} \]

   if -4.9999999999999998e274 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 1.00000000000000004e264

   1. Initial program 94.5%

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

      1. div-inv N/A

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

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

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. metadata-eval N/A

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

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

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

      13. *-commutative N/A

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

      14. associate-/r* N/A

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

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

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

      16. metadata-eval 95.2

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

   4. Applied egg-rr 95.2%

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

   if 1.00000000000000004e264 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 64.5%

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

   3. Taylor expanded in x around 0

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

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

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f64 100.0

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

   5. Simplified 100.0%

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

Alternative 5: 93.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, z \cdot \left(t \cdot -9\right)\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a \cdot -0.2222222222222222}, t, y \cdot \frac{x}{a \cdot 2}\right)\\ \end{array} \end{array} \]

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 2.0) 5e-8)
   (/ (fma y x (* z (* t -9.0))) (* a 2.0))
   (fma (/ z (* a -0.2222222222222222)) t (* y (/ x (* a 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 ((a * 2.0) <= 5e-8) {
		tmp = fma(y, x, (z * (t * -9.0))) / (a * 2.0);
	} else {
		tmp = fma((z / (a * -0.2222222222222222)), t, (y * (x / (a * 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(a * 2.0) <= 5e-8)
		tmp = Float64(fma(y, x, Float64(z * Float64(t * -9.0))) / Float64(a * 2.0));
	else
		tmp = fma(Float64(z / Float64(a * -0.2222222222222222)), t, Float64(y * Float64(x / Float64(a * 2.0))));
	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_] := If[LessEqual[N[(a * 2.0), $MachinePrecision], 5e-8], N[(N[(y * x + N[(z * N[(t * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(z / N[(a * -0.2222222222222222), $MachinePrecision]), $MachinePrecision] * t + N[(y * N[(x / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a #s(literal 2 binary64)) < 4.9999999999999998e-8

   1. Initial program 92.1%

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. associate-*l* N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. metadata-eval 92.7

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

   4. Applied egg-rr 92.7%

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

   if 4.9999999999999998e-8 < (*.f64 a #s(literal 2 binary64))

   1. Initial program 79.1%

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

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

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

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

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

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

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

      9. associate-*l* N/A

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

      10. associate-/l* N/A

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

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

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

      12. *-commutative N/A

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

      13. associate-/r* N/A

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

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

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

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

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

      18. metadata-eval 93.4

          \[\leadsto \mathsf{fma}\left(\frac{y}{a \cdot 2}, x, -z \cdot \frac{t \cdot \color{blue}{4.5}}{a}\right) \]

   4. Applied egg-rr 93.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a \cdot 2}, x, -z \cdot \frac{t \cdot 4.5}{a}\right)} \]
   5. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. distribute-neg-frac N/A

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

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

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

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

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

      8. metadata-eval N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. associate-/r* N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

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

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

   6. Applied egg-rr 84.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{a \cdot -0.2222222222222222}, t, y \cdot \frac{x}{a \cdot 2}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 73.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{-29}:\\ \;\;\;\;x \cdot \frac{y}{a \cdot 2}\\ \mathbf{elif}\;x \cdot y \leq 10^{-20}:\\ \;\;\;\;t \cdot \frac{z \cdot -4.5}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a \cdot 2}\\ \end{array} \end{array} \]

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) -5e-29)
   (* x (/ y (* a 2.0)))
   (if (<= (* x y) 1e-20) (* t (/ (* z -4.5) a)) (* y (/ x (* a 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) <= -5e-29) {
		tmp = x * (y / (a * 2.0));
	} else if ((x * y) <= 1e-20) {
		tmp = t * ((z * -4.5) / a);
	} else {
		tmp = y * (x / (a * 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) <= (-5d-29)) then
        tmp = x * (y / (a * 2.0d0))
    else if ((x * y) <= 1d-20) then
        tmp = t * ((z * (-4.5d0)) / a)
    else
        tmp = y * (x / (a * 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) <= -5e-29) {
		tmp = x * (y / (a * 2.0));
	} else if ((x * y) <= 1e-20) {
		tmp = t * ((z * -4.5) / a);
	} else {
		tmp = y * (x / (a * 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) <= -5e-29:
		tmp = x * (y / (a * 2.0))
	elif (x * y) <= 1e-20:
		tmp = t * ((z * -4.5) / a)
	else:
		tmp = y * (x / (a * 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) <= -5e-29)
		tmp = Float64(x * Float64(y / Float64(a * 2.0)));
	elseif (Float64(x * y) <= 1e-20)
		tmp = Float64(t * Float64(Float64(z * -4.5) / a));
	else
		tmp = Float64(y * Float64(x / Float64(a * 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) <= -5e-29)
		tmp = x * (y / (a * 2.0));
	elseif ((x * y) <= 1e-20)
		tmp = t * ((z * -4.5) / a);
	else
		tmp = y * (x / (a * 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], -5e-29], N[(x * N[(y / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e-20], N[(t * N[(N[(z * -4.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -4.99999999999999986e-29

   1. Initial program 89.6%

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

      1. div-inv N/A

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

      2. flip3-- N/A

         \[\leadsto \color{blue}{\frac{{\left(x \cdot y\right)}^{3} - {\left(\left(z \cdot 9\right) \cdot t\right)}^{3}}{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(\left(z \cdot 9\right) \cdot t\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right) + \left(x \cdot y\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right)\right)}} \cdot \frac{1}{a \cdot 2} \]

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(\left(z \cdot 9\right) \cdot t\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right) + \left(x \cdot y\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right)\right)}{{\left(x \cdot y\right)}^{3} - {\left(\left(z \cdot 9\right) \cdot t\right)}^{3}}}} \cdot \frac{1}{a \cdot 2} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(\left(z \cdot 9\right) \cdot t\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right) + \left(x \cdot y\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right)\right)}{{\left(x \cdot y\right)}^{3} - {\left(\left(z \cdot 9\right) \cdot t\right)}^{3}}} \cdot \frac{1}{\color{blue}{2 \cdot a}} \]

      5. associate-/r* N/A

         \[\leadsto \frac{1}{\frac{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(\left(z \cdot 9\right) \cdot t\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right) + \left(x \cdot y\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right)\right)}{{\left(x \cdot y\right)}^{3} - {\left(\left(z \cdot 9\right) \cdot t\right)}^{3}}} \cdot \color{blue}{\frac{\frac{1}{2}}{a}} \]

      6. frac-times N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{1}{2}}{\frac{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(\left(z \cdot 9\right) \cdot t\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right) + \left(x \cdot y\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right)\right)}{{\left(x \cdot y\right)}^{3} - {\left(\left(z \cdot 9\right) \cdot t\right)}^{3}} \cdot a}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{1 \cdot \color{blue}{\frac{1}{2}}}{\frac{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(\left(z \cdot 9\right) \cdot t\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right) + \left(x \cdot y\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right)\right)}{{\left(x \cdot y\right)}^{3} - {\left(\left(z \cdot 9\right) \cdot t\right)}^{3}} \cdot a} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2}}}{\frac{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(\left(z \cdot 9\right) \cdot t\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right) + \left(x \cdot y\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right)\right)}{{\left(x \cdot y\right)}^{3} - {\left(\left(z \cdot 9\right) \cdot t\right)}^{3}} \cdot a} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2}}}{\frac{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(\left(z \cdot 9\right) \cdot t\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right) + \left(x \cdot y\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right)\right)}{{\left(x \cdot y\right)}^{3} - {\left(\left(z \cdot 9\right) \cdot t\right)}^{3}} \cdot a} \]

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

          \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\frac{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(\left(z \cdot 9\right) \cdot t\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right) + \left(x \cdot y\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right)\right)}{{\left(x \cdot y\right)}^{3} - {\left(\left(z \cdot 9\right) \cdot t\right)}^{3}} \cdot a}} \]

   4. Applied egg-rr 91.1%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{1}{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right)} \cdot a}} \]

   5. Taylor expanded in z around 0

      \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{a}{x \cdot y}}} \]
   6. Step-by-step derivation

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

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

      2. *-lowering-*.f64 76.0

         \[\leadsto \frac{0.5}{\frac{a}{\color{blue}{x \cdot y}}} \]

   7. Simplified 76.0%

      \[\leadsto \frac{0.5}{\color{blue}{\frac{a}{x \cdot y}}} \]
   8. Step-by-step derivation

      1. associate-/r/ N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. div-inv N/A

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

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

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

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

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

      11. *-lowering-*.f64 79.7

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

   9. Applied egg-rr 79.7%

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

   if -4.99999999999999986e-29 < (*.f64 x y) < 9.99999999999999945e-21

   1. Initial program 86.4%

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

   3. Taylor expanded in x around 0

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

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

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f64 80.8

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

   5. Simplified 80.8%

      \[\leadsto \color{blue}{-4.5 \cdot \left(t \cdot \frac{z}{a}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. times-frac N/A

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

      5. associate-/l* N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. div-inv N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 80.7

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

   7. Applied egg-rr 80.7%

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

      1. associate-*r/ N/A

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

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

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

      3. *-lowering-*.f64 80.7

         \[\leadsto t \cdot \frac{\color{blue}{z \cdot -4.5}}{a} \]

   9. Applied egg-rr 80.7%

      \[\leadsto t \cdot \color{blue}{\frac{z \cdot -4.5}{a}} \]

   if 9.99999999999999945e-21 < (*.f64 x y)

   1. Initial program 93.5%

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

      1. div-inv N/A

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

      2. flip3-- N/A

         \[\leadsto \color{blue}{\frac{{\left(x \cdot y\right)}^{3} - {\left(\left(z \cdot 9\right) \cdot t\right)}^{3}}{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(\left(z \cdot 9\right) \cdot t\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right) + \left(x \cdot y\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right)\right)}} \cdot \frac{1}{a \cdot 2} \]

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(\left(z \cdot 9\right) \cdot t\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right) + \left(x \cdot y\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right)\right)}{{\left(x \cdot y\right)}^{3} - {\left(\left(z \cdot 9\right) \cdot t\right)}^{3}}}} \cdot \frac{1}{a \cdot 2} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(\left(z \cdot 9\right) \cdot t\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right) + \left(x \cdot y\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right)\right)}{{\left(x \cdot y\right)}^{3} - {\left(\left(z \cdot 9\right) \cdot t\right)}^{3}}} \cdot \frac{1}{\color{blue}{2 \cdot a}} \]

      5. associate-/r* N/A

         \[\leadsto \frac{1}{\frac{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(\left(z \cdot 9\right) \cdot t\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right) + \left(x \cdot y\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right)\right)}{{\left(x \cdot y\right)}^{3} - {\left(\left(z \cdot 9\right) \cdot t\right)}^{3}}} \cdot \color{blue}{\frac{\frac{1}{2}}{a}} \]

      6. frac-times N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{1}{2}}{\frac{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(\left(z \cdot 9\right) \cdot t\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right) + \left(x \cdot y\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right)\right)}{{\left(x \cdot y\right)}^{3} - {\left(\left(z \cdot 9\right) \cdot t\right)}^{3}} \cdot a}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{1 \cdot \color{blue}{\frac{1}{2}}}{\frac{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(\left(z \cdot 9\right) \cdot t\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right) + \left(x \cdot y\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right)\right)}{{\left(x \cdot y\right)}^{3} - {\left(\left(z \cdot 9\right) \cdot t\right)}^{3}} \cdot a} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2}}}{\frac{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(\left(z \cdot 9\right) \cdot t\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right) + \left(x \cdot y\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right)\right)}{{\left(x \cdot y\right)}^{3} - {\left(\left(z \cdot 9\right) \cdot t\right)}^{3}} \cdot a} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2}}}{\frac{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(\left(z \cdot 9\right) \cdot t\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right) + \left(x \cdot y\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right)\right)}{{\left(x \cdot y\right)}^{3} - {\left(\left(z \cdot 9\right) \cdot t\right)}^{3}} \cdot a} \]

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

          \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\frac{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(\left(z \cdot 9\right) \cdot t\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right) + \left(x \cdot y\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right)\right)}{{\left(x \cdot y\right)}^{3} - {\left(\left(z \cdot 9\right) \cdot t\right)}^{3}} \cdot a}} \]

   4. Applied egg-rr 93.4%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{1}{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right)} \cdot a}} \]

   5. Taylor expanded in z around 0

      \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{a}{x \cdot y}}} \]
   6. Step-by-step derivation

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

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

      2. *-lowering-*.f64 75.1

         \[\leadsto \frac{0.5}{\frac{a}{\color{blue}{x \cdot y}}} \]

   7. Simplified 75.1%

      \[\leadsto \frac{0.5}{\color{blue}{\frac{a}{x \cdot y}}} \]
   8. Step-by-step derivation

      1. associate-/r/ N/A

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

      2. associate-*l* N/A

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

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

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-/r* N/A

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

      7. *-commutative N/A

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

      8. un-div-inv N/A

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

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

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

      10. *-lowering-*.f64 79.0

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

   9. Applied egg-rr 79.0%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{-29}:\\ \;\;\;\;x \cdot \frac{y}{a \cdot 2}\\ \mathbf{elif}\;x \cdot y \leq 10^{-20}:\\ \;\;\;\;t \cdot \frac{z \cdot -4.5}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-40}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{0.5}{a}\\ \mathbf{elif}\;x \cdot y \leq 10^{-20}:\\ \;\;\;\;t \cdot \frac{z \cdot -4.5}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a \cdot 2}\\ \end{array} \end{array} \]

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-40)
   (* (* x y) (/ 0.5 a))
   (if (<= (* x y) 1e-20) (* t (/ (* z -4.5) a)) (* y (/ x (* a 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-40) {
		tmp = (x * y) * (0.5 / a);
	} else if ((x * y) <= 1e-20) {
		tmp = t * ((z * -4.5) / a);
	} else {
		tmp = y * (x / (a * 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-40)) then
        tmp = (x * y) * (0.5d0 / a)
    else if ((x * y) <= 1d-20) then
        tmp = t * ((z * (-4.5d0)) / a)
    else
        tmp = y * (x / (a * 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-40) {
		tmp = (x * y) * (0.5 / a);
	} else if ((x * y) <= 1e-20) {
		tmp = t * ((z * -4.5) / a);
	} else {
		tmp = y * (x / (a * 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-40:
		tmp = (x * y) * (0.5 / a)
	elif (x * y) <= 1e-20:
		tmp = t * ((z * -4.5) / a)
	else:
		tmp = y * (x / (a * 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-40)
		tmp = Float64(Float64(x * y) * Float64(0.5 / a));
	elseif (Float64(x * y) <= 1e-20)
		tmp = Float64(t * Float64(Float64(z * -4.5) / a));
	else
		tmp = Float64(y * Float64(x / Float64(a * 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-40)
		tmp = (x * y) * (0.5 / a);
	elseif ((x * y) <= 1e-20)
		tmp = t * ((z * -4.5) / a);
	else
		tmp = y * (x / (a * 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-40], N[(N[(x * y), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e-20], N[(t * N[(N[(z * -4.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1.9999999999999999e-40

   1. Initial program 89.9%

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

      1. div-inv N/A

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

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

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. metadata-eval N/A

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

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

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

      13. *-commutative N/A

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

      14. associate-/r* N/A

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

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

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

      16. metadata-eval 91.4

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

   4. Applied egg-rr 91.4%

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

   5. Taylor expanded in z around 0

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

      1. *-lowering-*.f64 75.2

         \[\leadsto \color{blue}{\left(x \cdot y\right)} \cdot \frac{0.5}{a} \]

   7. Simplified 75.2%

      \[\leadsto \color{blue}{\left(x \cdot y\right)} \cdot \frac{0.5}{a} \]

   if -1.9999999999999999e-40 < (*.f64 x y) < 9.99999999999999945e-21

   1. Initial program 86.2%

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

   3. Taylor expanded in x around 0

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

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

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f64 81.3

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

   5. Simplified 81.3%

      \[\leadsto \color{blue}{-4.5 \cdot \left(t \cdot \frac{z}{a}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. times-frac N/A

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

      5. associate-/l* N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. div-inv N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 81.1

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

   7. Applied egg-rr 81.1%

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

      1. associate-*r/ N/A

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

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

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

      3. *-lowering-*.f64 81.2

         \[\leadsto t \cdot \frac{\color{blue}{z \cdot -4.5}}{a} \]

   9. Applied egg-rr 81.2%

      \[\leadsto t \cdot \color{blue}{\frac{z \cdot -4.5}{a}} \]

   if 9.99999999999999945e-21 < (*.f64 x y)

   1. Initial program 93.5%

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

      1. div-inv N/A

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

      2. flip3-- N/A

         \[\leadsto \color{blue}{\frac{{\left(x \cdot y\right)}^{3} - {\left(\left(z \cdot 9\right) \cdot t\right)}^{3}}{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(\left(z \cdot 9\right) \cdot t\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right) + \left(x \cdot y\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right)\right)}} \cdot \frac{1}{a \cdot 2} \]

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(\left(z \cdot 9\right) \cdot t\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right) + \left(x \cdot y\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right)\right)}{{\left(x \cdot y\right)}^{3} - {\left(\left(z \cdot 9\right) \cdot t\right)}^{3}}}} \cdot \frac{1}{a \cdot 2} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(\left(z \cdot 9\right) \cdot t\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right) + \left(x \cdot y\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right)\right)}{{\left(x \cdot y\right)}^{3} - {\left(\left(z \cdot 9\right) \cdot t\right)}^{3}}} \cdot \frac{1}{\color{blue}{2 \cdot a}} \]

      5. associate-/r* N/A

         \[\leadsto \frac{1}{\frac{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(\left(z \cdot 9\right) \cdot t\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right) + \left(x \cdot y\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right)\right)}{{\left(x \cdot y\right)}^{3} - {\left(\left(z \cdot 9\right) \cdot t\right)}^{3}}} \cdot \color{blue}{\frac{\frac{1}{2}}{a}} \]

      6. frac-times N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{1}{2}}{\frac{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(\left(z \cdot 9\right) \cdot t\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right) + \left(x \cdot y\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right)\right)}{{\left(x \cdot y\right)}^{3} - {\left(\left(z \cdot 9\right) \cdot t\right)}^{3}} \cdot a}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{1 \cdot \color{blue}{\frac{1}{2}}}{\frac{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(\left(z \cdot 9\right) \cdot t\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right) + \left(x \cdot y\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right)\right)}{{\left(x \cdot y\right)}^{3} - {\left(\left(z \cdot 9\right) \cdot t\right)}^{3}} \cdot a} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2}}}{\frac{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(\left(z \cdot 9\right) \cdot t\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right) + \left(x \cdot y\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right)\right)}{{\left(x \cdot y\right)}^{3} - {\left(\left(z \cdot 9\right) \cdot t\right)}^{3}} \cdot a} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2}}}{\frac{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(\left(z \cdot 9\right) \cdot t\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right) + \left(x \cdot y\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right)\right)}{{\left(x \cdot y\right)}^{3} - {\left(\left(z \cdot 9\right) \cdot t\right)}^{3}} \cdot a} \]

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

          \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\frac{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(\left(z \cdot 9\right) \cdot t\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right) + \left(x \cdot y\right) \cdot \left(\left(z \cdot 9\right) \cdot t\right)\right)}{{\left(x \cdot y\right)}^{3} - {\left(\left(z \cdot 9\right) \cdot t\right)}^{3}} \cdot a}} \]

   4. Applied egg-rr 93.4%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{1}{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right)} \cdot a}} \]

   5. Taylor expanded in z around 0

      \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{a}{x \cdot y}}} \]
   6. Step-by-step derivation

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

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

      2. *-lowering-*.f64 75.1

         \[\leadsto \frac{0.5}{\frac{a}{\color{blue}{x \cdot y}}} \]

   7. Simplified 75.1%

      \[\leadsto \frac{0.5}{\color{blue}{\frac{a}{x \cdot y}}} \]
   8. Step-by-step derivation

      1. associate-/r/ N/A

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

      2. associate-*l* N/A

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

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

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-/r* N/A

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

      7. *-commutative N/A

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

      8. un-div-inv N/A

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

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

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

      10. *-lowering-*.f64 79.0

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

   9. Applied egg-rr 79.0%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-40}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{0.5}{a}\\ \mathbf{elif}\;x \cdot y \leq 10^{-20}:\\ \;\;\;\;t \cdot \frac{z \cdot -4.5}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(x \cdot y\right) \cdot \frac{0.5}{a}\\ \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-40}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 10^{-20}:\\ \;\;\;\;t \cdot \frac{z \cdot -4.5}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* (* x y) (/ 0.5 a))))
   (if (<= (* x y) -2e-40)
     t_1
     (if (<= (* x y) 1e-20) (* t (/ (* z -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 = (x * y) * (0.5 / a);
	double tmp;
	if ((x * y) <= -2e-40) {
		tmp = t_1;
	} else if ((x * y) <= 1e-20) {
		tmp = t * ((z * -4.5) / 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 = (x * y) * (0.5d0 / a)
    if ((x * y) <= (-2d-40)) then
        tmp = t_1
    else if ((x * y) <= 1d-20) then
        tmp = t * ((z * (-4.5d0)) / 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 = (x * y) * (0.5 / a);
	double tmp;
	if ((x * y) <= -2e-40) {
		tmp = t_1;
	} else if ((x * y) <= 1e-20) {
		tmp = t * ((z * -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 = (x * y) * (0.5 / a)
	tmp = 0
	if (x * y) <= -2e-40:
		tmp = t_1
	elif (x * y) <= 1e-20:
		tmp = t * ((z * -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(x * y) * Float64(0.5 / a))
	tmp = 0.0
	if (Float64(x * y) <= -2e-40)
		tmp = t_1;
	elseif (Float64(x * y) <= 1e-20)
		tmp = Float64(t * Float64(Float64(z * -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 = (x * y) * (0.5 / a);
	tmp = 0.0;
	if ((x * y) <= -2e-40)
		tmp = t_1;
	elseif ((x * y) <= 1e-20)
		tmp = t * ((z * -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[(x * y), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -2e-40], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 1e-20], N[(t * N[(N[(z * -4.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.9999999999999999e-40 or 9.99999999999999945e-21 < (*.f64 x y)

   1. Initial program 91.9%

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

      1. div-inv N/A

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

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

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. metadata-eval N/A

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

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

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

      13. *-commutative N/A

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

      14. associate-/r* N/A

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

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

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

      16. metadata-eval 92.4

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

   4. Applied egg-rr 92.4%

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

   5. Taylor expanded in z around 0

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

      1. *-lowering-*.f64 75.1

         \[\leadsto \color{blue}{\left(x \cdot y\right)} \cdot \frac{0.5}{a} \]

   7. Simplified 75.1%

      \[\leadsto \color{blue}{\left(x \cdot y\right)} \cdot \frac{0.5}{a} \]

   if -1.9999999999999999e-40 < (*.f64 x y) < 9.99999999999999945e-21

   1. Initial program 86.2%

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

   3. Taylor expanded in x around 0

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

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

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f64 81.3

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

   5. Simplified 81.3%

      \[\leadsto \color{blue}{-4.5 \cdot \left(t \cdot \frac{z}{a}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. times-frac N/A

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

      5. associate-/l* N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. div-inv N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 81.1

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

   7. Applied egg-rr 81.1%

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

      1. associate-*r/ N/A

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

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

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

      3. *-lowering-*.f64 81.2

         \[\leadsto t \cdot \frac{\color{blue}{z \cdot -4.5}}{a} \]

   9. Applied egg-rr 81.2%

      \[\leadsto t \cdot \color{blue}{\frac{z \cdot -4.5}{a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 51.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ t \cdot \frac{z \cdot -4.5}{a} \end{array} \]

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 (* t (/ (* z -4.5) a)))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	return t * ((z * -4.5) / 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 = t * ((z * (-4.5d0)) / 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 t * ((z * -4.5) / a);
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	return t * ((z * -4.5) / a)

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	return Float64(t * Float64(Float64(z * -4.5) / a))
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp = code(x, y, z, t, a)
	tmp = t * ((z * -4.5) / 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[(t * N[(N[(z * -4.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.1%

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

3. Taylor expanded in x around 0

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

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

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

   2. associate-/l* N/A

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

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

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

   4. /-lowering-/.f64 52.6

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

5. Simplified 52.6%

   \[\leadsto \color{blue}{-4.5 \cdot \left(t \cdot \frac{z}{a}\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. times-frac N/A

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

   5. associate-/l* N/A

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

   6. associate-*l* N/A

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

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

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

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

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

   9. div-inv N/A

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

   10. *-commutative N/A

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

   11. associate-/r* N/A

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

   12. metadata-eval N/A

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

   13. associate-*r/ N/A

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

   14. metadata-eval N/A

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

   15. /-lowering-/.f64 52.6

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

7. Applied egg-rr 52.6%

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

   1. associate-*r/ N/A

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

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

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

   3. *-lowering-*.f64 52.6

      \[\leadsto t \cdot \frac{\color{blue}{z \cdot -4.5}}{a} \]

9. Applied egg-rr 52.6%

   \[\leadsto t \cdot \color{blue}{\frac{z \cdot -4.5}{a}} \]
10. Add Preprocessing

Alternative 10: 51.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ t \cdot \left(z \cdot \frac{-4.5}{a}\right) \end{array} \]

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 (* t (* z (/ -4.5 a))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	return t * (z * (-4.5 / 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 = t * (z * ((-4.5d0) / 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 t * (z * (-4.5 / a));
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	return t * (z * (-4.5 / a))

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	return Float64(t * Float64(z * Float64(-4.5 / a)))
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp = code(x, y, z, t, a)
	tmp = t * (z * (-4.5 / 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[(t * N[(z * N[(-4.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.1%

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

3. Taylor expanded in x around 0

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

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

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

   2. associate-/l* N/A

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

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

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

   4. /-lowering-/.f64 52.6

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

5. Simplified 52.6%

   \[\leadsto \color{blue}{-4.5 \cdot \left(t \cdot \frac{z}{a}\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. times-frac N/A

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

   5. associate-/l* N/A

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

   6. associate-*l* N/A

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

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

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

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

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

   9. div-inv N/A

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

   10. *-commutative N/A

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

   11. associate-/r* N/A

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

   12. metadata-eval N/A

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

   13. associate-*r/ N/A

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

   14. metadata-eval N/A

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

   15. /-lowering-/.f64 52.6

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

7. Applied egg-rr 52.6%

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

Alternative 11: 51.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ -4.5 \cdot \left(t \cdot \frac{z}{a}\right) \end{array} \]

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 (* t (/ z 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 * (t * (z / 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) * (t * (z / 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 * (t * (z / a));
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	return -4.5 * (t * (z / 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(t * Float64(z / 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 * (t * (z / 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[(t * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.1%

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

3. Taylor expanded in x around 0

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

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

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

   2. associate-/l* N/A

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

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

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

   4. /-lowering-/.f64 52.6

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

5. Simplified 52.6%

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

Developer Target 1: 93.4% accurate, 0.6× 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 2024205 
(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)))