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

Percentage Accurate: 91.1% → 97.6%

Time: 9.6s

Alternatives: 10

Speedup: 0.5×

• 28.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := y \cdot x - t \cdot \left(9 \cdot z\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+278}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a}, 0.5 \cdot y, \left(\frac{z}{a} \cdot 4.5\right) \cdot \left(-t\right)\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+299}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-9 \cdot t, z, y \cdot x\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, 4.5 \cdot \left(-t\right), \left(\frac{0.5}{a} \cdot x\right) \cdot y\right)\\ \end{array} \end{array} \]

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 (- (* y x) (* t (* 9.0 z)))))
   (if (<= t_1 -1e+278)
     (fma (/ x a) (* 0.5 y) (* (* (/ z a) 4.5) (- t)))
     (if (<= t_1 5e+299)
       (/ (fma (* -9.0 t) z (* y x)) (* 2.0 a))
       (fma (/ z a) (* 4.5 (- t)) (* (* (/ 0.5 a) x) y))))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * x) - (t * (9.0 * z));
	double tmp;
	if (t_1 <= -1e+278) {
		tmp = fma((x / a), (0.5 * y), (((z / a) * 4.5) * -t));
	} else if (t_1 <= 5e+299) {
		tmp = fma((-9.0 * t), z, (y * x)) / (2.0 * a);
	} else {
		tmp = fma((z / a), (4.5 * -t), (((0.5 / a) * x) * y));
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * x) - Float64(t * Float64(9.0 * z)))
	tmp = 0.0
	if (t_1 <= -1e+278)
		tmp = fma(Float64(x / a), Float64(0.5 * y), Float64(Float64(Float64(z / a) * 4.5) * Float64(-t)));
	elseif (t_1 <= 5e+299)
		tmp = Float64(fma(Float64(-9.0 * t), z, Float64(y * x)) / Float64(2.0 * a));
	else
		tmp = fma(Float64(z / a), Float64(4.5 * Float64(-t)), Float64(Float64(Float64(0.5 / a) * x) * y));
	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[(y * x), $MachinePrecision] - N[(t * N[(9.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+278], N[(N[(x / a), $MachinePrecision] * N[(0.5 * y), $MachinePrecision] + N[(N[(N[(z / a), $MachinePrecision] * 4.5), $MachinePrecision] * (-t)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+299], N[(N[(N[(-9.0 * t), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(z / a), $MachinePrecision] * N[(4.5 * (-t)), $MachinePrecision] + N[(N[(N[(0.5 / a), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < -9.99999999999999964e277

   1. Initial program 41.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. div-inv N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. times-frac N/A

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

      24. lower-*.f64 N/A

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

      25. metadata-eval N/A

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

      26. lower-/.f64 96.7

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

   4. Applied rewrites 96.7%

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

   if -9.99999999999999964e277 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < 5.0000000000000003e299

   1. Initial program 98.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 98.9

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 98.9

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

   4. Applied rewrites 98.9%

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

   if 5.0000000000000003e299 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t))

   1. Initial program 58.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. associate-/l* N/A

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

   4. Applied rewrites 91.0%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x - t \cdot \left(9 \cdot z\right) \leq -1 \cdot 10^{+278}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a}, 0.5 \cdot y, \left(\frac{z}{a} \cdot 4.5\right) \cdot \left(-t\right)\right)\\ \mathbf{elif}\;y \cdot x - t \cdot \left(9 \cdot z\right) \leq 5 \cdot 10^{+299}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-9 \cdot t, z, y \cdot x\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, 4.5 \cdot \left(-t\right), \left(\frac{0.5}{a} \cdot x\right) \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := y \cdot x - t \cdot \left(9 \cdot z\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+278}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a}, 0.5 \cdot y, \left(\frac{z}{a} \cdot 4.5\right) \cdot \left(-t\right)\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+299}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-9 \cdot t, z, y \cdot x\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{0.5}{a} \cdot x, \left(-4.5 \cdot \frac{z}{a}\right) \cdot t\right)\\ \end{array} \end{array} \]

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 (- (* y x) (* t (* 9.0 z)))))
   (if (<= t_1 -1e+278)
     (fma (/ x a) (* 0.5 y) (* (* (/ z a) 4.5) (- t)))
     (if (<= t_1 5e+299)
       (/ (fma (* -9.0 t) z (* y x)) (* 2.0 a))
       (fma y (* (/ 0.5 a) x) (* (* -4.5 (/ z a)) t))))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * x) - (t * (9.0 * z));
	double tmp;
	if (t_1 <= -1e+278) {
		tmp = fma((x / a), (0.5 * y), (((z / a) * 4.5) * -t));
	} else if (t_1 <= 5e+299) {
		tmp = fma((-9.0 * t), z, (y * x)) / (2.0 * a);
	} else {
		tmp = fma(y, ((0.5 / a) * x), ((-4.5 * (z / a)) * t));
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * x) - Float64(t * Float64(9.0 * z)))
	tmp = 0.0
	if (t_1 <= -1e+278)
		tmp = fma(Float64(x / a), Float64(0.5 * y), Float64(Float64(Float64(z / a) * 4.5) * Float64(-t)));
	elseif (t_1 <= 5e+299)
		tmp = Float64(fma(Float64(-9.0 * t), z, Float64(y * x)) / Float64(2.0 * a));
	else
		tmp = fma(y, Float64(Float64(0.5 / a) * x), Float64(Float64(-4.5 * Float64(z / a)) * t));
	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[(y * x), $MachinePrecision] - N[(t * N[(9.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+278], N[(N[(x / a), $MachinePrecision] * N[(0.5 * y), $MachinePrecision] + N[(N[(N[(z / a), $MachinePrecision] * 4.5), $MachinePrecision] * (-t)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+299], N[(N[(N[(-9.0 * t), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(0.5 / a), $MachinePrecision] * x), $MachinePrecision] + N[(N[(-4.5 * N[(z / a), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < -9.99999999999999964e277

   1. Initial program 41.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. div-inv N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. times-frac N/A

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

      24. lower-*.f64 N/A

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

      25. metadata-eval N/A

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

      26. lower-/.f64 96.7

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

   4. Applied rewrites 96.7%

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

   if -9.99999999999999964e277 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < 5.0000000000000003e299

   1. Initial program 98.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 98.9

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 98.9

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

   4. Applied rewrites 98.9%

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

   if 5.0000000000000003e299 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t))

   1. Initial program 58.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 61.4

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 61.4

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

   4. Applied rewrites 61.4%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 61.4

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 61.4

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

   6. Applied rewrites 61.4%

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

   8. Applied rewrites 85.2%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x - t \cdot \left(9 \cdot z\right) \leq -1 \cdot 10^{+278}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a}, 0.5 \cdot y, \left(\frac{z}{a} \cdot 4.5\right) \cdot \left(-t\right)\right)\\ \mathbf{elif}\;y \cdot x - t \cdot \left(9 \cdot z\right) \leq 5 \cdot 10^{+299}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-9 \cdot t, z, y \cdot x\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{0.5}{a} \cdot x, \left(-4.5 \cdot \frac{z}{a}\right) \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := y \cdot x - t \cdot \left(9 \cdot z\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+257}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{a} \cdot y, x, \left(\frac{z}{a} \cdot 4.5\right) \cdot \left(-t\right)\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+299}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-9 \cdot t, z, y \cdot x\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{0.5}{a} \cdot x, \left(-4.5 \cdot \frac{z}{a}\right) \cdot t\right)\\ \end{array} \end{array} \]

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 (- (* y x) (* t (* 9.0 z)))))
   (if (<= t_1 -5e+257)
     (fma (* (/ 0.5 a) y) x (* (* (/ z a) 4.5) (- t)))
     (if (<= t_1 5e+299)
       (/ (fma (* -9.0 t) z (* y x)) (* 2.0 a))
       (fma y (* (/ 0.5 a) x) (* (* -4.5 (/ z a)) t))))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * x) - (t * (9.0 * z));
	double tmp;
	if (t_1 <= -5e+257) {
		tmp = fma(((0.5 / a) * y), x, (((z / a) * 4.5) * -t));
	} else if (t_1 <= 5e+299) {
		tmp = fma((-9.0 * t), z, (y * x)) / (2.0 * a);
	} else {
		tmp = fma(y, ((0.5 / a) * x), ((-4.5 * (z / a)) * t));
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * x) - Float64(t * Float64(9.0 * z)))
	tmp = 0.0
	if (t_1 <= -5e+257)
		tmp = fma(Float64(Float64(0.5 / a) * y), x, Float64(Float64(Float64(z / a) * 4.5) * Float64(-t)));
	elseif (t_1 <= 5e+299)
		tmp = Float64(fma(Float64(-9.0 * t), z, Float64(y * x)) / Float64(2.0 * a));
	else
		tmp = fma(y, Float64(Float64(0.5 / a) * x), Float64(Float64(-4.5 * Float64(z / a)) * t));
	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[(y * x), $MachinePrecision] - N[(t * N[(9.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+257], N[(N[(N[(0.5 / a), $MachinePrecision] * y), $MachinePrecision] * x + N[(N[(N[(z / a), $MachinePrecision] * 4.5), $MachinePrecision] * (-t)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+299], N[(N[(N[(-9.0 * t), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(0.5 / a), $MachinePrecision] * x), $MachinePrecision] + N[(N[(-4.5 * N[(z / a), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < -5.00000000000000028e257

   1. Initial program 44.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. div-inv N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-/l* N/A

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

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

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

      20. lower-*.f64 N/A

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

   4. Applied rewrites 94.4%

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

   if -5.00000000000000028e257 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < 5.0000000000000003e299

   1. Initial program 98.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 98.8

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 98.8

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

   4. Applied rewrites 98.8%

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

   if 5.0000000000000003e299 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t))

   1. Initial program 58.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 61.4

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 61.4

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

   4. Applied rewrites 61.4%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 61.4

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 61.4

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

   6. Applied rewrites 61.4%

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

   8. Applied rewrites 85.2%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x - t \cdot \left(9 \cdot z\right) \leq -5 \cdot 10^{+257}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{a} \cdot y, x, \left(\frac{z}{a} \cdot 4.5\right) \cdot \left(-t\right)\right)\\ \mathbf{elif}\;y \cdot x - t \cdot \left(9 \cdot z\right) \leq 5 \cdot 10^{+299}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-9 \cdot t, z, y \cdot x\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{0.5}{a} \cdot x, \left(-4.5 \cdot \frac{z}{a}\right) \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{0.5}{a} \cdot x, \left(-4.5 \cdot \frac{z}{a}\right) \cdot t\right)\\ t_2 := y \cdot x - t \cdot \left(9 \cdot z\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+278}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+299}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-9 \cdot t, z, y \cdot x\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 y (* (/ 0.5 a) x) (* (* -4.5 (/ z a)) t)))
        (t_2 (- (* y x) (* t (* 9.0 z)))))
   (if (<= t_2 -1e+278)
     t_1
     (if (<= t_2 5e+299) (/ (fma (* -9.0 t) z (* y x)) (* 2.0 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(y, ((0.5 / a) * x), ((-4.5 * (z / a)) * t));
	double t_2 = (y * x) - (t * (9.0 * z));
	double tmp;
	if (t_2 <= -1e+278) {
		tmp = t_1;
	} else if (t_2 <= 5e+299) {
		tmp = fma((-9.0 * t), z, (y * x)) / (2.0 * 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(y, Float64(Float64(0.5 / a) * x), Float64(Float64(-4.5 * Float64(z / a)) * t))
	t_2 = Float64(Float64(y * x) - Float64(t * Float64(9.0 * z)))
	tmp = 0.0
	if (t_2 <= -1e+278)
		tmp = t_1;
	elseif (t_2 <= 5e+299)
		tmp = Float64(fma(Float64(-9.0 * t), z, Float64(y * x)) / Float64(2.0 * 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[(y * N[(N[(0.5 / a), $MachinePrecision] * x), $MachinePrecision] + N[(N[(-4.5 * N[(z / a), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * x), $MachinePrecision] - N[(t * N[(9.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+278], t$95$1, If[LessEqual[t$95$2, 5e+299], N[(N[(N[(-9.0 * t), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < -9.99999999999999964e277 or 5.0000000000000003e299 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t))

   1. Initial program 49.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 51.4

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 51.4

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

   4. Applied rewrites 51.4%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 50.0

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 50.0

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

   6. Applied rewrites 50.0%

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

   8. Applied rewrites 90.9%

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

   if -9.99999999999999964e277 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < 5.0000000000000003e299

   1. Initial program 98.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 98.9

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 98.9

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

   4. Applied rewrites 98.9%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x - t \cdot \left(9 \cdot z\right) \leq -1 \cdot 10^{+278}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{0.5}{a} \cdot x, \left(-4.5 \cdot \frac{z}{a}\right) \cdot t\right)\\ \mathbf{elif}\;y \cdot x - t \cdot \left(9 \cdot z\right) \leq 5 \cdot 10^{+299}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-9 \cdot t, z, y \cdot x\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{0.5}{a} \cdot x, \left(-4.5 \cdot \frac{z}{a}\right) \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := t \cdot \left(9 \cdot z\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+282}:\\ \;\;\;\;\left(\frac{t}{a} \cdot -4.5\right) \cdot z\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+282}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(-4.5 \cdot t\right) \cdot \frac{z}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* t (* 9.0 z))))
   (if (<= t_1 -5e+282)
     (* (* (/ t a) -4.5) z)
     (if (<= t_1 4e+282)
       (/ (fma (* -9.0 z) t (* y x)) (* 2.0 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 = t * (9.0 * z);
	double tmp;
	if (t_1 <= -5e+282) {
		tmp = ((t / a) * -4.5) * z;
	} else if (t_1 <= 4e+282) {
		tmp = fma((-9.0 * z), t, (y * x)) / (2.0 * 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(t * Float64(9.0 * z))
	tmp = 0.0
	if (t_1 <= -5e+282)
		tmp = Float64(Float64(Float64(t / a) * -4.5) * z);
	elseif (t_1 <= 4e+282)
		tmp = Float64(fma(Float64(-9.0 * z), t, Float64(y * x)) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(-4.5 * 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[(t * N[(9.0 * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+282], N[(N[(N[(t / a), $MachinePrecision] * -4.5), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$1, 4e+282], N[(N[(N[(-9.0 * z), $MachinePrecision] * t + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-4.5 * t), $MachinePrecision] * N[(z / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -4.99999999999999978e282

   1. Initial program 48.0%

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

   3. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 85.3

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

   5. Applied rewrites 85.3%

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

   7. Applied rewrites 85.6%

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

   9. Applied rewrites 85.3%

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

   if -4.99999999999999978e282 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 4.00000000000000013e282

   1. Initial program 93.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 93.9

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 93.9

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

   4. Applied rewrites 93.9%

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

   if 4.00000000000000013e282 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 27.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. frac-sub N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 26.8%

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

   5. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 93.7

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

   7. Applied rewrites 93.7%

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

   9. Applied rewrites 93.8%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(9 \cdot z\right) \leq -5 \cdot 10^{+282}:\\ \;\;\;\;\left(\frac{t}{a} \cdot -4.5\right) \cdot z\\ \mathbf{elif}\;t \cdot \left(9 \cdot z\right) \leq 4 \cdot 10^{+282}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(-4.5 \cdot t\right) \cdot \frac{z}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := t \cdot \left(9 \cdot z\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\frac{t}{a} \cdot -4.5\right) \cdot z\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+282}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, -9, y \cdot x\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(-4.5 \cdot t\right) \cdot \frac{z}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* t (* 9.0 z))))
   (if (<= t_1 (- INFINITY))
     (* (* (/ t a) -4.5) z)
     (if (<= t_1 4e+282)
       (* (fma (* t z) -9.0 (* y x)) (/ 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 = t * (9.0 * z);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((t / a) * -4.5) * z;
	} else if (t_1 <= 4e+282) {
		tmp = fma((t * z), -9.0, (y * x)) * (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(t * Float64(9.0 * z))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(t / a) * -4.5) * z);
	elseif (t_1 <= 4e+282)
		tmp = Float64(fma(Float64(t * z), -9.0, Float64(y * x)) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(-4.5 * 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[(t * N[(9.0 * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(t / a), $MachinePrecision] * -4.5), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$1, 4e+282], N[(N[(N[(t * z), $MachinePrecision] * -9.0 + N[(y * x), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(-4.5 * t), $MachinePrecision] * N[(z / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 39.0%

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

   3. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 82.9

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

   5. Applied rewrites 82.9%

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

   7. Applied rewrites 83.1%

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

   9. Applied rewrites 82.9%

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

   if -inf.0 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 4.00000000000000013e282

   1. Initial program 94.0%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. associate-/r* N/A

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

      21. lower-/.f64 N/A

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

      22. metadata-eval 93.8

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

   4. Applied rewrites 93.8%

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

   if 4.00000000000000013e282 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 27.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. frac-sub N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 26.8%

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

   5. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 93.7

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

   7. Applied rewrites 93.7%

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

   9. Applied rewrites 93.8%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(9 \cdot z\right) \leq -\infty:\\ \;\;\;\;\left(\frac{t}{a} \cdot -4.5\right) \cdot z\\ \mathbf{elif}\;t \cdot \left(9 \cdot z\right) \leq 4 \cdot 10^{+282}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, -9, y \cdot x\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(-4.5 \cdot t\right) \cdot \frac{z}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(\frac{y}{a} \cdot 0.5\right) \cdot x\\ \mathbf{if}\;y \cdot x \leq -7.5 \cdot 10^{-63}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \cdot x \leq 5 \cdot 10^{-44}:\\ \;\;\;\;\frac{\left(t \cdot z\right) \cdot -9}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* (/ y a) 0.5) x)))
   (if (<= (* y x) -7.5e-63)
     t_1
     (if (<= (* y x) 5e-44) (/ (* (* t z) -9.0) (* 2.0 a)) t_1))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = ((y / a) * 0.5) * x;
	double tmp;
	if ((y * x) <= -7.5e-63) {
		tmp = t_1;
	} else if ((y * x) <= 5e-44) {
		tmp = ((t * z) * -9.0) / (2.0 * 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 = ((y / a) * 0.5d0) * x
    if ((y * x) <= (-7.5d-63)) then
        tmp = t_1
    else if ((y * x) <= 5d-44) then
        tmp = ((t * z) * (-9.0d0)) / (2.0d0 * 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 = ((y / a) * 0.5) * x;
	double tmp;
	if ((y * x) <= -7.5e-63) {
		tmp = t_1;
	} else if ((y * x) <= 5e-44) {
		tmp = ((t * z) * -9.0) / (2.0 * a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = ((y / a) * 0.5) * x
	tmp = 0
	if (y * x) <= -7.5e-63:
		tmp = t_1
	elif (y * x) <= 5e-44:
		tmp = ((t * z) * -9.0) / (2.0 * 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(Float64(y / a) * 0.5) * x)
	tmp = 0.0
	if (Float64(y * x) <= -7.5e-63)
		tmp = t_1;
	elseif (Float64(y * x) <= 5e-44)
		tmp = Float64(Float64(Float64(t * z) * -9.0) / Float64(2.0 * a));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = ((y / a) * 0.5) * x;
	tmp = 0.0;
	if ((y * x) <= -7.5e-63)
		tmp = t_1;
	elseif ((y * x) <= 5e-44)
		tmp = ((t * z) * -9.0) / (2.0 * 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[(N[(y / a), $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[(y * x), $MachinePrecision], -7.5e-63], t$95$1, If[LessEqual[N[(y * x), $MachinePrecision], 5e-44], N[(N[(N[(t * z), $MachinePrecision] * -9.0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -7.5000000000000003e-63 or 5.00000000000000039e-44 < (*.f64 x y)

   1. Initial program 81.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 72.0

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

   5. Applied rewrites 72.0%

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

   if -7.5000000000000003e-63 < (*.f64 x y) < 5.00000000000000039e-44

   1. Initial program 92.9%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 84.8

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

   5. Applied rewrites 84.8%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -7.5 \cdot 10^{-63}:\\ \;\;\;\;\left(\frac{y}{a} \cdot 0.5\right) \cdot x\\ \mathbf{elif}\;y \cdot x \leq 5 \cdot 10^{-44}:\\ \;\;\;\;\frac{\left(t \cdot z\right) \cdot -9}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{a} \cdot 0.5\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(\frac{y}{a} \cdot 0.5\right) \cdot x\\ \mathbf{if}\;y \cdot x \leq -7.5 \cdot 10^{-63}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \cdot x \leq 5 \cdot 10^{-44}:\\ \;\;\;\;\frac{\left(-4.5 \cdot z\right) \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* (/ y a) 0.5) x)))
   (if (<= (* y x) -7.5e-63)
     t_1
     (if (<= (* y x) 5e-44) (/ (* (* -4.5 z) t) a) t_1))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = ((y / a) * 0.5) * x;
	double tmp;
	if ((y * x) <= -7.5e-63) {
		tmp = t_1;
	} else if ((y * x) <= 5e-44) {
		tmp = ((-4.5 * z) * t) / 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 = ((y / a) * 0.5d0) * x
    if ((y * x) <= (-7.5d-63)) then
        tmp = t_1
    else if ((y * x) <= 5d-44) then
        tmp = (((-4.5d0) * z) * t) / 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 = ((y / a) * 0.5) * x;
	double tmp;
	if ((y * x) <= -7.5e-63) {
		tmp = t_1;
	} else if ((y * x) <= 5e-44) {
		tmp = ((-4.5 * z) * t) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = ((y / a) * 0.5) * x
	tmp = 0
	if (y * x) <= -7.5e-63:
		tmp = t_1
	elif (y * x) <= 5e-44:
		tmp = ((-4.5 * z) * t) / 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(Float64(y / a) * 0.5) * x)
	tmp = 0.0
	if (Float64(y * x) <= -7.5e-63)
		tmp = t_1;
	elseif (Float64(y * x) <= 5e-44)
		tmp = Float64(Float64(Float64(-4.5 * z) * t) / a);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = ((y / a) * 0.5) * x;
	tmp = 0.0;
	if ((y * x) <= -7.5e-63)
		tmp = t_1;
	elseif ((y * x) <= 5e-44)
		tmp = ((-4.5 * z) * t) / 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[(N[(y / a), $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[(y * x), $MachinePrecision], -7.5e-63], t$95$1, If[LessEqual[N[(y * x), $MachinePrecision], 5e-44], N[(N[(N[(-4.5 * z), $MachinePrecision] * t), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -7.5000000000000003e-63 or 5.00000000000000039e-44 < (*.f64 x y)

   1. Initial program 81.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 72.0

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

   5. Applied rewrites 72.0%

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

   if -7.5000000000000003e-63 < (*.f64 x y) < 5.00000000000000039e-44

   1. Initial program 92.9%

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

   3. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 81.3

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

   5. Applied rewrites 81.3%

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

   7. Applied rewrites 84.9%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -7.5 \cdot 10^{-63}:\\ \;\;\;\;\left(\frac{y}{a} \cdot 0.5\right) \cdot x\\ \mathbf{elif}\;y \cdot x \leq 5 \cdot 10^{-44}:\\ \;\;\;\;\frac{\left(-4.5 \cdot z\right) \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{a} \cdot 0.5\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(\frac{y}{a} \cdot 0.5\right) \cdot x\\ \mathbf{if}\;y \cdot x \leq -7.5 \cdot 10^{-63}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \cdot x \leq 10^{-20}:\\ \;\;\;\;\left(\frac{-4.5}{a} \cdot t\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* (/ y a) 0.5) x)))
   (if (<= (* y x) -7.5e-63)
     t_1
     (if (<= (* y x) 1e-20) (* (* (/ -4.5 a) t) z) t_1))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = ((y / a) * 0.5) * x;
	double tmp;
	if ((y * x) <= -7.5e-63) {
		tmp = t_1;
	} else if ((y * x) <= 1e-20) {
		tmp = ((-4.5 / a) * t) * z;
	} 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 = ((y / a) * 0.5d0) * x
    if ((y * x) <= (-7.5d-63)) then
        tmp = t_1
    else if ((y * x) <= 1d-20) then
        tmp = (((-4.5d0) / a) * t) * z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = ((y / a) * 0.5) * x;
	double tmp;
	if ((y * x) <= -7.5e-63) {
		tmp = t_1;
	} else if ((y * x) <= 1e-20) {
		tmp = ((-4.5 / a) * t) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = ((y / a) * 0.5) * x
	tmp = 0
	if (y * x) <= -7.5e-63:
		tmp = t_1
	elif (y * x) <= 1e-20:
		tmp = ((-4.5 / a) * t) * z
	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(Float64(y / a) * 0.5) * x)
	tmp = 0.0
	if (Float64(y * x) <= -7.5e-63)
		tmp = t_1;
	elseif (Float64(y * x) <= 1e-20)
		tmp = Float64(Float64(Float64(-4.5 / a) * t) * z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = ((y / a) * 0.5) * x;
	tmp = 0.0;
	if ((y * x) <= -7.5e-63)
		tmp = t_1;
	elseif ((y * x) <= 1e-20)
		tmp = ((-4.5 / a) * t) * z;
	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[(N[(y / a), $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[(y * x), $MachinePrecision], -7.5e-63], t$95$1, If[LessEqual[N[(y * x), $MachinePrecision], 1e-20], N[(N[(N[(-4.5 / a), $MachinePrecision] * t), $MachinePrecision] * z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -7.5000000000000003e-63 or 9.99999999999999945e-21 < (*.f64 x y)

   1. Initial program 81.5%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 73.4

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

   5. Applied rewrites 73.4%

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

   if -7.5000000000000003e-63 < (*.f64 x y) < 9.99999999999999945e-21

   1. Initial program 91.6%

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

   3. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 79.9

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

   5. Applied rewrites 79.9%

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

   7. Applied rewrites 79.9%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -7.5 \cdot 10^{-63}:\\ \;\;\;\;\left(\frac{y}{a} \cdot 0.5\right) \cdot x\\ \mathbf{elif}\;y \cdot x \leq 10^{-20}:\\ \;\;\;\;\left(\frac{-4.5}{a} \cdot t\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{a} \cdot 0.5\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.2% accurate, 1.6× speedup

Math

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

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 a) t) z))

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 / a) * t) * z;
}

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

Python

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

Julia

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

Derivation

1. Initial program 85.9%

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

3. Taylor expanded in t around inf

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

   1. associate-*l/ N/A

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

   2. associate-*l* N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 49.7

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

5. Applied rewrites 49.7%

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

7. Applied rewrites 49.7%

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

8. Final simplification 49.7%

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

Developer Target 1: 93.6% 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 2024273 
(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)))