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

Percentage Accurate: 90.8% → 96.9%

Time: 8.4s

Alternatives: 10

Speedup: 1.1×

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

Specification

Math

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

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: 90.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := x \cdot y - \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+293}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \frac{0.5}{a}, x, \left(-t\right) \cdot \left(4.5 \cdot \frac{z}{a}\right)\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+249}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-9 \cdot t, z, y \cdot x\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, \left(-z\right) \cdot 4.5, \left(x \cdot \frac{0.5}{a}\right) \cdot y\right)\\ \end{array} \end{array} \]

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 (- (* x y) (* (* z 9.0) t))))
   (if (<= t_1 -5e+293)
     (fma (* y (/ 0.5 a)) x (* (- t) (* 4.5 (/ z a))))
     (if (<= t_1 1e+249)
       (/ (fma (* -9.0 t) z (* y x)) (* a 2.0))
       (fma (/ t a) (* (- z) 4.5) (* (* x (/ 0.5 a)) 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 = (x * y) - ((z * 9.0) * t);
	double tmp;
	if (t_1 <= -5e+293) {
		tmp = fma((y * (0.5 / a)), x, (-t * (4.5 * (z / a))));
	} else if (t_1 <= 1e+249) {
		tmp = fma((-9.0 * t), z, (y * x)) / (a * 2.0);
	} else {
		tmp = fma((t / a), (-z * 4.5), ((x * (0.5 / a)) * 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(x * y) - Float64(Float64(z * 9.0) * t))
	tmp = 0.0
	if (t_1 <= -5e+293)
		tmp = fma(Float64(y * Float64(0.5 / a)), x, Float64(Float64(-t) * Float64(4.5 * Float64(z / a))));
	elseif (t_1 <= 1e+249)
		tmp = Float64(fma(Float64(-9.0 * t), z, Float64(y * x)) / Float64(a * 2.0));
	else
		tmp = fma(Float64(t / a), Float64(Float64(-z) * 4.5), Float64(Float64(x * Float64(0.5 / a)) * 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[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+293], N[(N[(y * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] * x + N[((-t) * N[(4.5 * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+249], N[(N[(N[(-9.0 * t), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(t / a), $MachinePrecision] * N[((-z) * 4.5), $MachinePrecision] + N[(N[(x * N[(0.5 / a), $MachinePrecision]), $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)) < -5.00000000000000033e293

   1. Initial program 73.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 95.9%

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

   if -5.00000000000000033e293 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < 9.9999999999999992e248

   1. Initial program 98.5%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 98.5

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

      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 98.5

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

   4. Applied rewrites 98.5%

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

   if 9.9999999999999992e248 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t))

   1. Initial program 81.0%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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. 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. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

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

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. associate-/l* N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 97.5%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -5 \cdot 10^{+293}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \frac{0.5}{a}, x, \left(-t\right) \cdot \left(4.5 \cdot \frac{z}{a}\right)\right)\\ \mathbf{elif}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq 10^{+249}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-9 \cdot t, z, y \cdot x\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, \left(-z\right) \cdot 4.5, \left(x \cdot \frac{0.5}{a}\right) \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := x \cdot y - \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+293} \lor \neg \left(t\_1 \leq 10^{+164}\right):\\ \;\;\;\;\mathsf{fma}\left(y \cdot \frac{0.5}{a}, x, \left(-t\right) \cdot \left(4.5 \cdot \frac{z}{a}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-9 \cdot t, z, y \cdot x\right)}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* (* z 9.0) t))))
   (if (or (<= t_1 -5e+293) (not (<= t_1 1e+164)))
     (fma (* y (/ 0.5 a)) x (* (- t) (* 4.5 (/ z a))))
     (/ (fma (* -9.0 t) z (* y x)) (* a 2.0)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - ((z * 9.0) * t);
	double tmp;
	if ((t_1 <= -5e+293) || !(t_1 <= 1e+164)) {
		tmp = fma((y * (0.5 / a)), x, (-t * (4.5 * (z / a))));
	} else {
		tmp = fma((-9.0 * t), z, (y * x)) / (a * 2.0);
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t))
	tmp = 0.0
	if ((t_1 <= -5e+293) || !(t_1 <= 1e+164))
		tmp = fma(Float64(y * Float64(0.5 / a)), x, Float64(Float64(-t) * Float64(4.5 * Float64(z / a))));
	else
		tmp = Float64(fma(Float64(-9.0 * t), z, Float64(y * x)) / Float64(a * 2.0));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e+293], N[Not[LessEqual[t$95$1, 1e+164]], $MachinePrecision]], N[(N[(y * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] * x + N[((-t) * N[(4.5 * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-9.0 * t), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < -5.00000000000000033e293 or 1e164 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t))

   1. Initial program 82.2%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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. 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 96.3%

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

   if -5.00000000000000033e293 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < 1e164

   1. Initial program 98.5%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 98.4

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

      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 98.4

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

   4. Applied rewrites 98.4%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -5 \cdot 10^{+293} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \leq 10^{+164}\right):\\ \;\;\;\;\mathsf{fma}\left(y \cdot \frac{0.5}{a}, x, \left(-t\right) \cdot \left(4.5 \cdot \frac{z}{a}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-9 \cdot t, z, y \cdot x\right)}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\left(t \cdot z\right) \cdot -9}{a \cdot 2}\\ \mathbf{elif}\;t\_1 \leq 10^{+15}:\\ \;\;\;\;\frac{\left(0.5 \cdot y\right) \cdot x}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{t}{a} \cdot z\right) \cdot -4.5\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* z 9.0) t)))
   (if (<= t_1 -5e-7)
     (/ (* (* t z) -9.0) (* a 2.0))
     (if (<= t_1 1e+15) (/ (* (* 0.5 y) x) a) (* (* (/ t a) z) -4.5)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -5e-7) {
		tmp = ((t * z) * -9.0) / (a * 2.0);
	} else if (t_1 <= 1e+15) {
		tmp = ((0.5 * y) * x) / a;
	} else {
		tmp = ((t / a) * z) * -4.5;
	}
	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 = (z * 9.0d0) * t
    if (t_1 <= (-5d-7)) then
        tmp = ((t * z) * (-9.0d0)) / (a * 2.0d0)
    else if (t_1 <= 1d+15) then
        tmp = ((0.5d0 * y) * x) / a
    else
        tmp = ((t / a) * z) * (-4.5d0)
    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 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -5e-7) {
		tmp = ((t * z) * -9.0) / (a * 2.0);
	} else if (t_1 <= 1e+15) {
		tmp = ((0.5 * y) * x) / a;
	} else {
		tmp = ((t / a) * z) * -4.5;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = (z * 9.0) * t
	tmp = 0
	if t_1 <= -5e-7:
		tmp = ((t * z) * -9.0) / (a * 2.0)
	elif t_1 <= 1e+15:
		tmp = ((0.5 * y) * x) / a
	else:
		tmp = ((t / a) * z) * -4.5
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * 9.0) * t)
	tmp = 0.0
	if (t_1 <= -5e-7)
		tmp = Float64(Float64(Float64(t * z) * -9.0) / Float64(a * 2.0));
	elseif (t_1 <= 1e+15)
		tmp = Float64(Float64(Float64(0.5 * y) * x) / a);
	else
		tmp = Float64(Float64(Float64(t / a) * z) * -4.5);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (z * 9.0) * t;
	tmp = 0.0;
	if (t_1 <= -5e-7)
		tmp = ((t * z) * -9.0) / (a * 2.0);
	elseif (t_1 <= 1e+15)
		tmp = ((0.5 * y) * x) / a;
	else
		tmp = ((t / a) * z) * -4.5;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-7], N[(N[(N[(t * z), $MachinePrecision] * -9.0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+15], N[(N[(N[(0.5 * y), $MachinePrecision] * x), $MachinePrecision] / a), $MachinePrecision], N[(N[(N[(t / a), $MachinePrecision] * z), $MachinePrecision] * -4.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -4.99999999999999977e-7

   1. Initial program 90.3%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 79.2

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

   5. Applied rewrites 79.2%

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

   if -4.99999999999999977e-7 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 1e15

   1. Initial program 96.7%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 74.2

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

   5. Applied rewrites 74.2%

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

   7. Applied rewrites 80.2%

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

   if 1e15 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 89.1%

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

   3. Taylor expanded in x around 0

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

      1. 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 83.7

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

   5. Applied rewrites 83.7%

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

   7. Applied rewrites 80.5%

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

Alternative 4: 74.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-7}:\\ \;\;\;\;\frac{-4.5}{a} \cdot \left(t \cdot z\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+15}:\\ \;\;\;\;\frac{\left(0.5 \cdot y\right) \cdot x}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{t}{a} \cdot z\right) \cdot -4.5\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* z 9.0) t)))
   (if (<= t_1 -5e-7)
     (* (/ -4.5 a) (* t z))
     (if (<= t_1 1e+15) (/ (* (* 0.5 y) x) a) (* (* (/ t a) z) -4.5)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -5e-7) {
		tmp = (-4.5 / a) * (t * z);
	} else if (t_1 <= 1e+15) {
		tmp = ((0.5 * y) * x) / a;
	} else {
		tmp = ((t / a) * z) * -4.5;
	}
	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 = (z * 9.0d0) * t
    if (t_1 <= (-5d-7)) then
        tmp = ((-4.5d0) / a) * (t * z)
    else if (t_1 <= 1d+15) then
        tmp = ((0.5d0 * y) * x) / a
    else
        tmp = ((t / a) * z) * (-4.5d0)
    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 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -5e-7) {
		tmp = (-4.5 / a) * (t * z);
	} else if (t_1 <= 1e+15) {
		tmp = ((0.5 * y) * x) / a;
	} else {
		tmp = ((t / a) * z) * -4.5;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = (z * 9.0) * t
	tmp = 0
	if t_1 <= -5e-7:
		tmp = (-4.5 / a) * (t * z)
	elif t_1 <= 1e+15:
		tmp = ((0.5 * y) * x) / a
	else:
		tmp = ((t / a) * z) * -4.5
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * 9.0) * t)
	tmp = 0.0
	if (t_1 <= -5e-7)
		tmp = Float64(Float64(-4.5 / a) * Float64(t * z));
	elseif (t_1 <= 1e+15)
		tmp = Float64(Float64(Float64(0.5 * y) * x) / a);
	else
		tmp = Float64(Float64(Float64(t / a) * z) * -4.5);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (z * 9.0) * t;
	tmp = 0.0;
	if (t_1 <= -5e-7)
		tmp = (-4.5 / a) * (t * z);
	elseif (t_1 <= 1e+15)
		tmp = ((0.5 * y) * x) / a;
	else
		tmp = ((t / a) * z) * -4.5;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-7], N[(N[(-4.5 / a), $MachinePrecision] * N[(t * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+15], N[(N[(N[(0.5 * y), $MachinePrecision] * x), $MachinePrecision] / a), $MachinePrecision], N[(N[(N[(t / a), $MachinePrecision] * z), $MachinePrecision] * -4.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -4.99999999999999977e-7

   1. Initial program 90.3%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 91.9

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

      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 91.9

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

   4. Applied rewrites 91.9%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. *-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. div-inv N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. metadata-eval N/A

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

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

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

      15. *-commutative N/A

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

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

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

      17. *-commutative N/A

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

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

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

      19. div-inv N/A

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

   6. Applied rewrites 87.3%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   8. 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 76.3

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

   9. Applied rewrites 76.3%

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

   11. Applied rewrites 79.2%

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

   if -4.99999999999999977e-7 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 1e15

   1. Initial program 96.7%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 74.2

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

   5. Applied rewrites 74.2%

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

   7. Applied rewrites 80.2%

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

   if 1e15 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 89.1%

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

   3. Taylor expanded in x around 0

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

      1. 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 83.7

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

   5. Applied rewrites 83.7%

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

   7. Applied rewrites 80.5%

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

Alternative 5: 71.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-7}:\\ \;\;\;\;\frac{-4.5}{a} \cdot \left(t \cdot z\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+98}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{t}{a} \cdot z\right) \cdot -4.5\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* z 9.0) t)))
   (if (<= t_1 -5e-7)
     (* (/ -4.5 a) (* t z))
     (if (<= t_1 4e+98) (* x (* y (/ 0.5 a))) (* (* (/ t a) z) -4.5)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -5e-7) {
		tmp = (-4.5 / a) * (t * z);
	} else if (t_1 <= 4e+98) {
		tmp = x * (y * (0.5 / a));
	} else {
		tmp = ((t / a) * z) * -4.5;
	}
	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 = (z * 9.0d0) * t
    if (t_1 <= (-5d-7)) then
        tmp = ((-4.5d0) / a) * (t * z)
    else if (t_1 <= 4d+98) then
        tmp = x * (y * (0.5d0 / a))
    else
        tmp = ((t / a) * z) * (-4.5d0)
    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 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -5e-7) {
		tmp = (-4.5 / a) * (t * z);
	} else if (t_1 <= 4e+98) {
		tmp = x * (y * (0.5 / a));
	} else {
		tmp = ((t / a) * z) * -4.5;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = (z * 9.0) * t
	tmp = 0
	if t_1 <= -5e-7:
		tmp = (-4.5 / a) * (t * z)
	elif t_1 <= 4e+98:
		tmp = x * (y * (0.5 / a))
	else:
		tmp = ((t / a) * z) * -4.5
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * 9.0) * t)
	tmp = 0.0
	if (t_1 <= -5e-7)
		tmp = Float64(Float64(-4.5 / a) * Float64(t * z));
	elseif (t_1 <= 4e+98)
		tmp = Float64(x * Float64(y * Float64(0.5 / a)));
	else
		tmp = Float64(Float64(Float64(t / a) * z) * -4.5);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (z * 9.0) * t;
	tmp = 0.0;
	if (t_1 <= -5e-7)
		tmp = (-4.5 / a) * (t * z);
	elseif (t_1 <= 4e+98)
		tmp = x * (y * (0.5 / a));
	else
		tmp = ((t / a) * z) * -4.5;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-7], N[(N[(-4.5 / a), $MachinePrecision] * N[(t * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+98], N[(x * N[(y * N[(0.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t / a), $MachinePrecision] * z), $MachinePrecision] * -4.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -4.99999999999999977e-7

   1. Initial program 90.3%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 91.9

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

      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 91.9

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

   4. Applied rewrites 91.9%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. *-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. div-inv N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. metadata-eval N/A

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

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

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

      15. *-commutative N/A

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

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

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

      17. *-commutative N/A

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

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

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

      19. div-inv N/A

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

   6. Applied rewrites 87.3%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   8. 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 76.3

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

   9. Applied rewrites 76.3%

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

   11. Applied rewrites 79.2%

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

   if -4.99999999999999977e-7 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 3.99999999999999999e98

   1. Initial program 95.7%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 71.3

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

   5. Applied rewrites 71.3%

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

   7. Applied rewrites 73.4%

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

   if 3.99999999999999999e98 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 90.0%

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

   3. Taylor expanded in x around 0

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

      1. 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 93.6

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

   5. Applied rewrites 93.6%

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

   7. Applied rewrites 91.6%

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

Alternative 6: 71.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-7}:\\ \;\;\;\;\frac{-4.5}{a} \cdot \left(t \cdot z\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+98}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{t}{a} \cdot -4.5\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* z 9.0) t)))
   (if (<= t_1 -5e-7)
     (* (/ -4.5 a) (* t z))
     (if (<= t_1 4e+98) (* x (* y (/ 0.5 a))) (* (* (/ t a) -4.5) z)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -5e-7) {
		tmp = (-4.5 / a) * (t * z);
	} else if (t_1 <= 4e+98) {
		tmp = x * (y * (0.5 / a));
	} else {
		tmp = ((t / a) * -4.5) * z;
	}
	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 = (z * 9.0d0) * t
    if (t_1 <= (-5d-7)) then
        tmp = ((-4.5d0) / a) * (t * z)
    else if (t_1 <= 4d+98) then
        tmp = x * (y * (0.5d0 / a))
    else
        tmp = ((t / a) * (-4.5d0)) * z
    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 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -5e-7) {
		tmp = (-4.5 / a) * (t * z);
	} else if (t_1 <= 4e+98) {
		tmp = x * (y * (0.5 / a));
	} else {
		tmp = ((t / a) * -4.5) * z;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = (z * 9.0) * t
	tmp = 0
	if t_1 <= -5e-7:
		tmp = (-4.5 / a) * (t * z)
	elif t_1 <= 4e+98:
		tmp = x * (y * (0.5 / a))
	else:
		tmp = ((t / a) * -4.5) * z
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * 9.0) * t)
	tmp = 0.0
	if (t_1 <= -5e-7)
		tmp = Float64(Float64(-4.5 / a) * Float64(t * z));
	elseif (t_1 <= 4e+98)
		tmp = Float64(x * Float64(y * Float64(0.5 / a)));
	else
		tmp = Float64(Float64(Float64(t / a) * -4.5) * z);
	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 = (z * 9.0) * t;
	tmp = 0.0;
	if (t_1 <= -5e-7)
		tmp = (-4.5 / a) * (t * z);
	elseif (t_1 <= 4e+98)
		tmp = x * (y * (0.5 / a));
	else
		tmp = ((t / a) * -4.5) * z;
	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[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-7], N[(N[(-4.5 / a), $MachinePrecision] * N[(t * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+98], N[(x * N[(y * N[(0.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t / a), $MachinePrecision] * -4.5), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -4.99999999999999977e-7

   1. Initial program 90.3%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 91.9

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

      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 91.9

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

   4. Applied rewrites 91.9%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. *-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. div-inv N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. metadata-eval N/A

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

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

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

      15. *-commutative N/A

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

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

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

      17. *-commutative N/A

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

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

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

      19. div-inv N/A

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

   6. Applied rewrites 87.3%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   8. 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 76.3

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

   9. Applied rewrites 76.3%

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

   11. Applied rewrites 79.2%

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

   if -4.99999999999999977e-7 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 3.99999999999999999e98

   1. Initial program 95.7%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 71.3

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

   5. Applied rewrites 71.3%

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

   7. Applied rewrites 73.4%

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

   if 3.99999999999999999e98 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 90.0%

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

      1. lift--.f64 N/A

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

      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 89.8

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

      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 89.8

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

   4. Applied rewrites 89.8%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. *-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. div-inv N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. metadata-eval N/A

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

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

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

      15. *-commutative N/A

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

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

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

      17. *-commutative N/A

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

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

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

      19. div-inv N/A

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

   6. Applied rewrites 87.1%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   8. 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 91.6

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

   9. Applied rewrites 91.6%

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

Alternative 7: 92.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+246}:\\ \;\;\;\;\frac{x \cdot 0.5}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, -9, y \cdot x\right) \cdot \frac{0.5}{a}\\ \end{array} \end{array} \]

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
 (if (<= (* x y) -2e+246)
   (/ (* x 0.5) (/ a y))
   (* (fma (* t z) -9.0 (* y x)) (/ 0.5 a))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e+246) {
		tmp = (x * 0.5) / (a / y);
	} else {
		tmp = fma((t * z), -9.0, (y * x)) * (0.5 / a);
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -2e+246)
		tmp = Float64(Float64(x * 0.5) / Float64(a / y));
	else
		tmp = Float64(fma(Float64(t * z), -9.0, Float64(y * x)) * Float64(0.5 / a));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e+246], N[(N[(x * 0.5), $MachinePrecision] / N[(a / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t * z), $MachinePrecision] * -9.0 + N[(y * x), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2.00000000000000014e246

   1. Initial program 61.8%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 93.4

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

   5. Applied rewrites 93.4%

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

   7. Applied rewrites 93.5%

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

   9. Applied rewrites 93.6%

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

   if -2.00000000000000014e246 < (*.f64 x y)

   1. Initial program 95.3%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 95.3

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

   4. Applied rewrites 95.3%

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

Alternative 8: 51.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq 1.82 \cdot 10^{-73}:\\ \;\;\;\;\frac{-4.5}{a} \cdot \left(t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{t}{a} \cdot -4.5\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) 1.82e-73) (* (/ -4.5 a) (* t z)) (* (* (/ t a) -4.5) z)))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= 1.82e-73) {
		tmp = (-4.5 / a) * (t * z);
	} else {
		tmp = ((t / a) * -4.5) * z;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((x * y) <= 1.82d-73) then
        tmp = ((-4.5d0) / a) * (t * z)
    else
        tmp = ((t / a) * (-4.5d0)) * z
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= 1.82e-73) {
		tmp = (-4.5 / a) * (t * z);
	} else {
		tmp = ((t / a) * -4.5) * z;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= 1.82e-73:
		tmp = (-4.5 / a) * (t * z)
	else:
		tmp = ((t / a) * -4.5) * z
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= 1.82e-73)
		tmp = Float64(Float64(-4.5 / a) * Float64(t * z));
	else
		tmp = Float64(Float64(Float64(t / a) * -4.5) * z);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= 1.82e-73)
		tmp = (-4.5 / a) * (t * z);
	else
		tmp = ((t / a) * -4.5) * z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], 1.82e-73], N[(N[(-4.5 / a), $MachinePrecision] * N[(t * z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t / a), $MachinePrecision] * -4.5), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 1.82000000000000006e-73

   1. Initial program 94.7%

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

      1. lift--.f64 N/A

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

      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 95.2

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

      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 95.2

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

   4. Applied rewrites 95.2%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. *-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. div-inv N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. metadata-eval N/A

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

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

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

      15. *-commutative N/A

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

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

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

      17. *-commutative N/A

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

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

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

      19. div-inv N/A

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

   6. Applied rewrites 90.4%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   8. 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 59.1

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

   9. Applied rewrites 59.1%

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

   11. Applied rewrites 62.6%

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

   if 1.82000000000000006e-73 < (*.f64 x y)

   1. Initial program 90.5%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 90.5

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

      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 90.5

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

   4. Applied rewrites 90.5%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. *-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. div-inv N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. metadata-eval N/A

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

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

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

      15. *-commutative N/A

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

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

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

      17. *-commutative N/A

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

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

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

      19. div-inv N/A

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

   6. Applied rewrites 85.4%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   8. 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 37.8

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

   9. Applied rewrites 37.8%

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

Alternative 9: 91.2% accurate, 1.1× speedup

Math

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

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
 (/ (fma (* -9.0 t) z (* y x)) (* a 2.0)))

C

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

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	return Float64(fma(Float64(-9.0 * t), z, Float64(y * x)) / Float64(a * 2.0))
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[(-9.0 * t), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.4%

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

   1. lift--.f64 N/A

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

   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 93.7

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

   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 93.7

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

4. Applied rewrites 93.7%

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

Alternative 10: 52.1% accurate, 1.6× speedup

Math

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

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

C

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 93.4%

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

   1. lift--.f64 N/A

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

   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 93.7

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

   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 93.7

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

4. Applied rewrites 93.7%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. *-commutative N/A

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

   4. lift-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. distribute-rgt-in N/A

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

   7. div-inv N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. metadata-eval N/A

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

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

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

   15. *-commutative N/A

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

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

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

   17. *-commutative N/A

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

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

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

   19. div-inv N/A

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

6. Applied rewrites 88.8%

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

7. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
8. 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 52.3

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

9. Applied rewrites 52.3%

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

Developer Target 1: 93.0% 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 2024313 
(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)))